neutral hydrogen its connection with galaxies Simulating the cosmic distribution of and

neutral hydrogen its connection with galaxies Simulating the cosmic distribution of and
Simulating the cosmic distribution of
neutral hydrogen
and
its connection with galaxies
Alireza Rahmati
ISBN 978–9–46–191899–4
Cover by A. Rahmati
Front: The column density distribution of neutral hydrogen in a photoshoped
simulated galaxy. This design is motivated by the main findings of Chapter 4.
Back: Ancient constellations as published in The book of fixed stars by the famous
Iranian astronomer, Abd al-Rahman al-Sufi in 964 A.D. This highly influential
book has been reproduced numerous times during the last millennium. The
illustrations used for the design of the back-cover are taken from a version produced in 1430-1440 A.D. in Samarkand (Uzbekistan), which is accessible through
the following link:
http://gallica.bnf.fr/ark:/12148/btv1b60006156
Simulating the cosmic distribution of
neutral hydrogen
and
its connection with galaxies
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van de Rector Magnificus prof. mr. C. J. J. M. Stolker,
volgens besluit van het College voor Promoties
te verdedigen op dinsdag 15 oktober 2013
klokke 11:15 uur
door
Alireza Rahmati
geboren te Qom, Iran
in 1982
Promotiecommissie
Promotor:
Prof. dr. J. Schaye
Overige leden:
Prof. dr. M. Franx
Dr. A. H. Pawlik (Max-Planck Institute for Astrophysics)
Prof. dr. S. F. Portegies Zwart
Prof. dr. J. X. Prochaska (University of California, Santa Cruz)
Prof. dr. H. J. A. Röttgering
Prof. dr. P. P. van der Werf
Table of Contents
1
2
Introduction
1.1 Current standard model for galaxy formation and evolution
1.1.1 Cosmology: the backbone of galaxy formation . . . .
1.1.2 Main physical processes that drive galaxy evolution
1.2 Neutral hydrogen in galactic ecosystems . . . . . . . . . . .
1.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Cosmological hydrodynamical simulations . . . . . .
1.3.2 Radiative transfer . . . . . . . . . . . . . . . . . . . . .
1.3.3 Radiative transfer with TRAPHIC . . . . . . . . . . .
1.4 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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On the evolution of the Hi column density distribution in cosmological
simulations
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Simulation techniques . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Hydrodynamical simulations . . . . . . . . . . . . . . . . . .
2.2.2 Radiative transfer with TRAPHIC . . . . . . . . . . . . . . .
2.2.3 Ionizing background radiation . . . . . . . . . . . . . . . . .
2.2.4 Recombination radiation . . . . . . . . . . . . . . . . . . . .
2.2.5 The Hi column density distribution function . . . . . . . . .
2.2.6 Dust and molecular hydrogen . . . . . . . . . . . . . . . . .
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Comparison with observations . . . . . . . . . . . . . . . . .
2.3.2 The shape of the Hi CDDF . . . . . . . . . . . . . . . . . . .
2.3.3 Photoionization rate as a function of density . . . . . . . . .
2.3.4 The roles of diffuse recombination radiation and collisional
ionization at z = 3 . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A: Photoionization rate as a function of density . . . . . . . .
A1: Replacing the RT simulations with a fitting function . . . . . .
A2: The equilibrium hydrogen neutral fraction . . . . . . . . . . . .
Appendix B: The effects of box size, cosmological parameters and resolution on the Hi CDDF . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix C: RT convergence tests . . . . . . . . . . . . . . . . . . . . . .
C1: Angular resolution . . . . . . . . . . . . . . . . . . . . . . . . . .
C2: The number of ViP neighbors . . . . . . . . . . . . . . . . . . .
C3: Direct comparison with another RT method . . . . . . . . . . .
Appendix D: Approximated processes . . . . . . . . . . . . . . . . . . . .
D1: Multifrequency effects . . . . . . . . . . . . . . . . . . . . . . . .
1
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TABLE OF CONTENTS
D2: Helium treatment . . . . . . . . . . . . . . . . . . . . . . . . . .
3 The impact of local stellar radiation on the Hi column density distribution
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Photoionization rate in star-forming regions . . . . . . . . . . . . .
3.3 Simulation techniques . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Hydrodynamical simulations . . . . . . . . . . . . . . . . . .
3.3.2 Radiative transfer . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Ionizing background radiation and diffuse recombination
radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Stellar ionizing radiation . . . . . . . . . . . . . . . . . . . .
3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 The role of local stellar radiation in hydrogen ionization . .
3.4.2 Star-forming particles versus stellar particles . . . . . . . . .
3.4.3 Stellar ionizing radiation, its escape fraction and the
buildup of the UVB . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 The impact of local stellar radiation on the Hi column
density distribution . . . . . . . . . . . . . . . . . . . . . . .
3.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A: Hydrogen molecular fraction . . . . . . . . . . . . . . . . .
Appendix B: Resolution effects . . . . . . . . . . . . . . . . . . . . . . . .
B1: Limited spatial resolution at high densities . . . . . . . . . . . .
B2: The impact of a higher resolution on the RT . . . . . . . . . . .
Appendix C: Calculation of the escape fraction . . . . . . . . . . . . . . .
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4 Predictions for the relation between strong Hi absorbers and galaxies
at redshift 3
99
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2 Simulation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.1 Hydrodynamical simulations . . . . . . . . . . . . . . . . . . 102
4.2.2 Finding galaxies . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.3 Finding strong Hi absorbers . . . . . . . . . . . . . . . . . . 103
4.2.4 Connecting Hi absorbers to galaxies . . . . . . . . . . . . . . 107
4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3.1 Spatial distribution of Hi absorbers . . . . . . . . . . . . . . 110
4.3.2 The effect of a finite detection threshold . . . . . . . . . . . 111
4.3.3 Distribution of Hi absorbers relative to halos . . . . . . . . . 114
4.3.4 Resolution limit in simulations . . . . . . . . . . . . . . . . . 115
4.3.5 Correlations between absorbers and various properties of
their associated galaxies . . . . . . . . . . . . . . . . . . . . . 119
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TABLE OF CONTENTS
4.3.6
Are most strong Hi absorbers at z ∼ 3 around LymanBreak galaxies? . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A: Choosing the maximum allowed LOS Velocity difference .
Appendix B: Impact of feedback . . . . . . . . . . . . . . . . . . . . . . .
Appendix C: Impact of local stellar radiation . . . . . . . . . . . . . . . .
Appendix D: Resolution tests . . . . . . . . . . . . . . . . . . . . . . . . .
5
Genesis of the dusty Universe: modeling submillimetre source counts
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Model ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 CLF at z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 CLF evolution . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 SED Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 850 µm Observational Constraints . . . . . . . . . . . . . . . . . . .
5.3.1 Observed 850 µm source count . . . . . . . . . . . . . . . . .
5.3.2 Redshift distribution of bright 850 µm sources . . . . . . . .
5.4 Finding 850 µm best-fit Model . . . . . . . . . . . . . . . . . . . . . .
5.4.1 The source count curve: Amplitude vs. Shape . . . . . . . .
5.4.2 The luminosity evolution . . . . . . . . . . . . . . . . . . . .
5.5 Other necessary model ingredients . . . . . . . . . . . . . . . . . . .
5.5.1 The 850 µm best-fit model . . . . . . . . . . . . . . . . . . . .
5.6 Other wavelengths . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 Long submm wavelengths: 850 µm and 1100 µm . . . . . . .
5.6.2 SPIRE intermediate wavelengths: 500 µm, 350 µm and
250 µm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.3 Short wavelengths: 160 µm and 70 µm . . . . . . . . . . . . .
5.6.4 A best-fit model for all wavelengths . . . . . . . . . . . . . .
5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.1 The implied evolution scenario for dusty galaxies . . . . . .
5.7.2 Our best-fit model and previous models . . . . . . . . . . .
5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A: Some numerical details . . . . . . . . . . . . . . . . . . . . .
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Nederlandse samenvatting
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Publications
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Curriculum Vitae
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vii
Acknowledgements
193
1
Introduction
For more than a millennium, we have been aware of the existence of a celestial small cloud in the constellation of Andromeda (al-Sufi, 964), but it has been
less than a century since we observed that the small cloud, which is commonly
known to us as M31, is a spiral galaxy outside of our own Milky Way. In fact, almost everything we know about M31 and other galaxies has been learned during
the last century, thanks to the advent of large telescopes and new technologies.
During the last few decades, theoretical models have helped us to make sense
of the rapidly increasing wealth of data and have caused a revolution in our
understanding of the Universe. In particular, as a result of advancements in numerical techniques and computational recourses, cosmological simulations have
become an indispensable tool for probing the main processes involved in galaxy
formation and evolution.
This thesis is an attempt to add to our understanding of the Universe, by
using cosmological simulations for studying the distribution and evolution of
the neutral hydrogen around galaxies. In this work, I extensively use stateof-the-art cosmological hydrodynamical simulations of galaxy formation. To
set the stage, I begin this chapter with a very brief overview of our current
knowledge about how galaxies form and evolve. I continue by explaining the
importance of studying the neutral hydrogen for understanding galaxies. Then,
I discuss briefly how hydrodynamical and radiative transfer simulations work,
before ending this introductory chapter with the outline of this thesis.
Introduction
1.1 Current standard model for galaxy formation and
evolution
The theory of galaxy formation is where the properties of the largest scales in
the Universe meet the physics at tiny atomic scales. The huge dynamic range
of the relevant scales spanned by the different physical entities (e.g., length,
mass, time) that characterize galaxies, makes it a daunting task to model these
complex systems. Despite the monstrous size of this problem, in principle, the
formation and evolution of galaxies should be understandable in terms of the
known physical laws. Using these physical laws to explain the observed trends
among galaxies has been the main challenge of the theory of galaxy formation
and evolution. As a result of the hard work of many great minds, we now have
an understanding of how galaxies form and evolve, which can explain, with a
good degree of accuracy, what we see in the Universe. However, as in other
natural sciences, any progress in understanding galaxies reveals new puzzles to
solve. As a result, there is (and there will be) a huge number of phenomena that
we do not fully understand. In the following, I briefly review what I call the
standard model of galaxy formation and evolution, which consists of ideas that
glue together large sets of observed properties, and are commonly accepted by
most experts who work in this field.
1.1.1 Cosmology: the backbone of galaxy formation
Cosmology serves as the backbone of galaxy formation by describing the
physical laws that govern the formation and evolution of structures on large
scales. In this context, the ΛCDM paradigm has enjoyed great success by
explaining a large number of observables, such as the temperature fluctuations in the radiation we receive from the early Universe (i.e., the Cosmic
Microwave Background), the accelerated expansion of the Universe and the
growth of structures. The constraints on the parameters of the ΛCDM concordance cosmological model have been improving rapidly in recent years,
thanks to precise observational experiments like the Cosmic Background Explorer (COBE: Mather et al., 1990), the Wilkinson Microwave Anisotropy Probe
(WMAP: Bennett et al., 2003; Spergel et al., 2003; Komatsu et al., 2011) and recently, the Planck satellite (Planck Collaboration et al., 2013). Based on the above
mentioned measurements, and other experiments, we know that the vast majority of the energy density of the Universe is in the form of Dark Energy (i.e.,
Λ), which causes the accelerating expansion of the Universe. The rest consists
mainly of cold dark matter (i.e., CDM) with tiny fractions of baryonic matter and
radiation. The exact values of the above mentioned fractions define the dynamics of the Universe on large scales and the growth of small scale density fluctuations in the nearly homogeneous initial distribution of (Dark and baryonic)
matter after the Big Bang.
2
Galaxy formation
As the Universe expands, the separation between small scale density fluctuations increases. At the same time, the deviation between the density of fluctuations and the mean density of the Universe also increases. In other words,
over-dense regions attract more matter at the expense of draining under-dense
regions. When the density fluctuation (i.e., ∆ρ) is much smaller than the mean
density of the Universe (i.e., ∆ρ/ρ ≪ 1), its physical size increases with the expansion of the Universe as the magnitude of ∆ρ increases (i.e., the linear regime).
There is, however, a turn-around when ∆ρ/ρ ∼ 1, after which the physical size
of the fluctuation starts to decrease (i.e., the non-linear regime). The result of the
latter stage is the formation of self-gravitating bound structures. Since the majority of matter is in the form of collisionless dark matter, collapsing structures
(i.e., dark mater haloes) are regularized through violent relaxation. On large
scales and low over-densities, the baryonic matter is bound to the dark matter,
which is the dominant gravitational actor. In the non-linear regime, however,
baryons reveal their collisional nature and become shock heated to very high
temperatures through the efficient conversion of gravitational energy into the
internal energy of the baryonic gas, as it collapses into the inner parts of the
gravitational potential wells. At this stage, the fate of baryons in the collapsed
structures is significantly affected by the laws of small-scale atomic physics and
related complex and collective processes like radiative cooling and star formation. At the same time, the large-scale evolution of structures continues and
brings more matter into the collapsed structures and merges them.
1.1.2 Main physical processes that drive galaxy evolution
Despite the great success of the ΛCDM paradigm in explaining the invisible
side of galaxy formation by accurately predicting the formation and growth of
dark matter haloes (e.g., Zehavi et al., 2011; Heymans et al., 2013), the complex
physical processes that control the evolution of gas and stars are far from understood. To first order, the baryonic content of collapsing dark matter haloes,
which is initially shock heated to high temperatures, loses its energy through radiative cooling. The conservation of angular momentum dictates the cooling gas
to form a rotating disk as it looses energy and falls deeper into the potential well
of the dark matter halo. The density of baryonic gas increases until the radiative
cooling becomes inefficient and the gaseous disk approaches a quasi-equilibrium
state. The small scale instabilities in the gaseous disk, however, continue to grow
into dense molecular clouds which evolve and collapse to form stars.
The advent of stars makes the lives of galaxies much more complicated. Stars
are energetic sources of feedback and significantly affect the fate of their parent
galaxies. They heat up the gas and ionize it with their radiation, and they produce heavy elements as they evolve. These processes change the evolution of
the gas around stars by changing its cooling/heating. Stars also transport large
quantities of kinetic energy and heavy elements into their surroundings by injecting winds into the interstellar medium (ISM). Massive stars, which evolve
3
Introduction
faster than stars with lower masses, end their lives dramatically in energetic explosions (i.e., Supernovae; SNe) that inject huge amounts of energy into the ISM
of their host galaxies, affecting subsequent star formation and launching largescale galactic winds. In addition to energetic SNe explosions, radiation pressure
from very luminous young stars removes gas from star-forming regions, and
possibly makes a significant contribution to the launching of galactic winds.
The presence of very massive black holes, the end result of the death of very
massive stars, causes more complications in our understanding of how galaxies
evolve. Super-massive black holes, which are found at the centers of many (if
not all) galaxies we see in the Universe, are very efficient in converting mass
into energy. The huge amount of energy they inject into their host galaxies in
different ways, changes their fate dramatically.
There is much solid observational evidence and there are strong theoretical
arguments for the presence and importance of the above mentioned mechanisms in the formation and evolution of galaxies. Understanding how they work
individually and together to control the lives of galaxies at different epochs has
been an active area of research during the last few decades, an area of research
which is still vibrantly active due to its complexity.
1.2 Neutral hydrogen in galactic ecosystems
As mentioned above, galaxies are influenced on the one hand by the force of
gravity, which forms haloes, brings fresh material into the already existing haloes and keeps baryonic and dark matter structures together, and on the other
hand, by the feedback mechanisms that fight against the force of gravity and
try to unbind structures. The interaction between these two fronts creates a
complex ecosystem in and around galaxies, and imprints the history of galaxies in the distribution of baryons around them (i.e., the circumgalactic medium;
CGM). In this context, understanding the distribution of neutral hydrogen (HI)
is of particular importance. The main reason for this is that Hi is the main fuel
for the formation of molecular clouds, the birth places of stars, which makes
studying the distribution of Hi and its evolution crucial for our understanding
of various aspects of star formation.
We know that shortly after the Big Bang, the initially hot plasma cools down
as the Universe expands and electrons and protons recombine. Hydrogen, which
is the most abundant element in the Universe, becomes highly ionized again
by z ∼ 6 due to the formation of the first stars and galaxies (i.e., the reionization). After reionization, the mean-free-path of ionizing photons (i.e., the
distance photons can travel before being significantly absorbed) increases with
time, as the average star formation activity of the Universe increases and the
Universe becomes less dense. The relatively uniform distribution of sources of
ionizing radiation on large scales creates a relatively uniform ultraviolet background radiation (i.e., the UVB) which is dominated by stellar photons at z & 3
4
Neutral hydrogen in galactic ecosystems
and quasars at lower redshifts (e.g., Becker & Bolton, 2013). As a result, after
reionization most hydrogen atoms are kept ionized by the UVB radiation they
receive from all the ionizing sources they can see in the Universe.
The complexity of processes that set the ionization state of hydrogen depends
on the Hi column density. At low Hi column densities (i.e., NHI . 1017 cm−2 ,
corresponding to the so-called Lyman-α forest), hydrogen is highly ionized by
the UVB radiation and largely transparent to the ionizing radiation. For these
systems, the Hi column densities can therefore be accurately computed in the
optically thin limit. At higher Hi column densities (i.e., NHI & 1017 cm−2 , corresponding to the so-called Lyman Limit and Damped Lyman-α systems), the
gas becomes optically thick and self-shielded. As a result, the ionization state of
hydrogen in these systems is more sensitive to various radiative transfer effects
such as self-shielding, shadowing and the fluctuations of the UVB radiation on
small scales.
Studying the distribution of Hi is proven to be challenging. In the local Universe, the Hi content of galaxies can be probed by observing 21-cm emission, but
at higher redshifts this will not be possible until the advent of significantly more
powerful telescopes, such as the Square Kilometer Array1. At z . 6, i.e., after
reionization, the neutral gas can be probed through absorption signatures that
are imprinted by the intervening Hi systems on the spectra of bright background
sources, such as quasars. The analysis of these absorption features provides an
alternative probe of the distribution of matter at high redshifts, compared to
studying the Universe through emission. The large distances that separate most
absorbers from their background QSOs make it unlikely that there is a physical
connection between them. This opens up a window to study an unbiased sample
of matter that resides between us and the background QSOs.
Constraining the statistical properties of the Hi distribution has been the focus of many observational studies during the last few decades (e.g., Tytler, 1987;
Kim et al., 2002; Péroux et al., 2005; O’Meara et al., 2007; Noterdaeme et al.,
2009; Prochaska et al., 2009; O’Meara et al., 2013). Thanks to a significant increase in the number of observed quasars and improved observational techniques, more recent studies have extended these observations to both lower and
higher Hi column densities and to higher redshifts. Therefore, it is important to
study the properties of the Hi distribution in cosmological simulations to better
understand the observed trends and to put them in the context of the standard
theory of galaxy formation and evolution, which is the focus of this thesis. Once
simulations agree with observations, one can use them to predict what future
observations reveal. These predictions can be used to validate the underlying
models in the simulations and to examine the importance of different processes
that are relevant to the formation and evolution of galaxies, or in case of disagreement, to point us to necessary improvements.
1 http://www.skatelescope.org/
5
Introduction
1.3 Simulations
Hydrodynamical simulations attempt to model complex baryonic interactions
by combining various physically motivated and empirical ingredients. Modern
state-of-the-art cosmological simulations of galaxy formation try to put together
most of what we know about the astrophysical and cosmological processes that
are shaping the Universe on different scales and they try to reproduce different
observables.
In this context, the vital importance of radiation and radiative transfer (RT)
processes for galaxy evolution in general, and for producing the Hi distribution
in particular, is evident. However, RT is often ignored or poorly approximated
in the simulations. The main reason for this is the high dimensionality of the
underlying calculation which makes RT an enormous computational challenge
for cosmological simulations. Thanks to increasingly more powerful computers,
and in the light of new algorithms, the accurate treatment of radiative effects is
now becoming possible.
Since in this thesis we extensively use hydrodynamical simulations and RT,
we will briefly discuss how they work.
1.3.1 Cosmological hydrodynamical simulations
Cosmological simulations calculate the evolution of the Universe by starting
from an approximately uniform density of matter with small fluctuations on
different scales. These fluctuations are set by the observed statistical properties
of the fluctuations in the early stages of the Universe (i.e., the CMB). To be able
to trace the evolution of the Universe numerically, different techniques are adopted to discretize the continuous distribution of matter into a finite number of
resolution elements (e.g., particles). In addition, to keep the numerical calculations tractable, one needs to adopt a finite volume which is assumed to be a
representative sample of the whole Universe. Periodic boundary conditions can
then be used which assume that the distribution of matter on scales beyond the
extent of the simulation box is statistically similar to that inside it.
The gravitational interactions between all particles in the simulation are followed directly while the large-scale cosmological expansion of the Universe is
accounted for by a change of coordinate system. Different techniques are used
to accelerate the computation of the gravitational field without losing accuracy
significantly. For instance, calculating the gravitational interaction between two
groups of particles that are far away from each other (compared to the typical
distances between the particles in each group) is possible even if we neglect
the small scale distribution of the group members and replace the whole group
with a single gravitationally interacting element (see for an example the TreePM
algorithm explained in Bagla, 2002).
Calculating the hydrodynamical forces, which are short-range forces compared to gravity, is simpler due to their local nature. In other words, only neigh6
Simulations
boring resolution elements interact hydrodynamically. One of the most successful methods to simulate the evolution of the gas fluid is the smoothed particle
hydrodynamic (SPH) technique which was introduced by Gingold & Monaghan
(1977) and Lucy (1977). In the SPH prescription, the fluid is discretized into individual particles that are smoothed. This means that the relevant properties of
the fluid that are represented by each particle are distributed smoothly in space
around the particle. Then, the value of each relevant quantity (in most cases
only density), for any given position, is calculated by adding the contribution of
all SPH particles that are close enough to have significant contributions. Then,
the evolution of the fluid is calculated based on the known physical laws that
affect the fluid, like the laws of mass, momentum and energy conservation.
By combining the gravitational and hydrodynamical forces, it is possible to
start from small initial fluctuations and trace the evolution of dark matter and
baryonic gas as the Universe evolves. As the formation of structure proceeds,
the baryonic content of collapsing structures, which is initially shock heated,
cools down due to radiative cooling. The radiative cooling is calculated in the
simulations by including several important atomic processes. These processes
are mainly sensitive to the density, temperature, the abundances of different
elements (i.e., metallicity) and the properties of the radiation field. Because of
the high-dimensionality of the equations that control the cooling, their net effect
is included in the simulations using fitting functions and tables. Due to cooling,
the density of baryons increases which further complicates the simulation of the
evolution of baryons.
The limited spatial resolution of cosmological simulations, which is usually
of the order of a kilo-parsec, is not enough to resolve the complex and multiphase structure of the ISM. Because of this, an effective equation of state is often
adopted to model the collective hydrodynamical properties of the ISM gas. This
effective equation of state can also be used to prevent artificial fragmentation of
dense gas on scales close to the resolution limit (Schaye & Dalla Vecchia, 2008).
As the gas density increases, the dense ISM should be converted into stars. Since
it is not possible to simulate the complex process of star formation at kpc resolution, simplified algorithms are used to convert gas into stars. These star
formation prescriptions are often tuned to match the observed relation between
the gas (column) density and star formation rate in real galaxies, on kpc scales
(i.e., the Kenicutt-Schmidt relation; Kennicutt, 1998).
The resolution elements (e.g., SPH particles) that satisfy the adopted star
formation criteria are converted into stellar particles with typical masses of
105 − 106 M⊙ . In other words, each stellar particle in cosmological simulations represent groups of stars which are comparable to observed stellar clusters.
The subsequent evolution of this population of stars, which are assumed to be
formed at the same time, is followed by assuming an initial mass function (IMF)
and using stellar population synthesis models that trace the stellar evolution.
As the simulation continues, the stars that are assumed to be embedded in each
stellar particle, age and produce metals. The fraction of the stars that are massive
7
Introduction
enough to explode as SNe, end their lives in dramatic explosions and heat up
the gas which also launches high velocity flows that create large-scale galactic
winds. All these processes, together with other complicated phenomena, are
poorly understood and their detailed simulation requires much higher resolution than what is affordable in current cosmological simulations. Therefore, they
are included in the simulations through approximate rules, similar to the formation of stars explained above, which are commonly known as sub-grid models.
The sub-grid models in cosmological simulations try to incorporate our physical
knowledge or observed empirical relations into simulations. Due to the large
uncertainties, they are often tuned to produce the desired observables, such as
the average star formation activity of the Universe or the stellar mass function
at different epochs. Because of the complexity of how different subgrid models work individually and in combination, implementing them reliably in the
simulations and understanding how they regulate galaxies is an active area of
research.
By combining all the above mentioned ingredients (i.e., gravity, hydrodynamics and subgrid models), it is possible to start from initial conditions that are set
by cosmological observations, and to simulate the formation and evolution of
galaxies and the cosmic distribution of gas.
1.3.2 Radiative transfer
In addition to hydrodynamical and gravitational interactions, radiation plays an
important role in shaping the cosmos as we see it. The interaction between light
and matter has important consequences for the evolution of structures across a
range of different scales. While the radiation pressure close to stars, and perhaps
on galactic scales, changes the dynamics of the gas, the UVB radiation, which
is the superposition of radiation from large numbers of sources distributed on
large scales, affects the cooling/heating of the baryons significantly. In addition,
it is impossible to interpret the observations without understanding the relation between the measured intensity of radiation and different phenomena that
produce/affect photons.
Although it is essential to include the impact of radiation accurately in cosmological simulations, it is tremendously challenging to follow the production
of photons and their propagation as they interact with matter (i.e., radiative
transfer). First of all, the details of the RT depend on frequency, which can
change as photons travel through space. In addition, photons can travel to large
distances which makes them important over a wide range of scales. Because
radiation can propagate far from the locations where photons are generated, radiation is more similar to gravity than hydrodynamical interactions. However,
unlike gravity, for calculating the radiation field it is important to know what
is between the sources of radiation and the point at which the radiation field is
calculated. The time-dependency of the radiation field adds further complexity
to RT calculations.
8
Simulations
Figure 1.1: Different steps in the photon transport with TRAPHIC : 1- Emission (top panels): Photons are emitted isotropically from source particles to their SPH neighbors and
then travel down-stream based on their propagation directions. 2- Transport (middle panels): photon packets are distributed among the neighboring particles that are within the
transmission cone. If there is no neighboring particle inside the emission/transmission
cones, virtual particles are created to transport photons along their propagation directions (not shown). 3- Merging (bottom panels): if two or more of the photon packets
that a particle receives have close enough propagation directions, they are merged into a
single photon packet with appropriately averaged direction.
9
Introduction
1.3.3 Radiative transfer with
TRAPHIC
Because of the reasons mentioned above, RT is sensitive to a large number of
variables which makes it computationally expensive. Various techniques have
been developed to reduce the computational cost of the RT, mainly by adopting
different simplifying assumptions. Among different methods, TRAPHIC (TRAnsport of PHotons In Cones; Pawlik & Schaye, 2008, 2011) is a unique RT method
with several important advantages, as is explained below, for radiation transport in cosmological simulations that use the SPH prescription. Since we use
SPH simulations in this work, TRAPHIC is a natural choice to compute the RT. In
the following we briefly review how photon transfer is done with TRAPHIC (see
also Figure 1.1).
TRAPHIC is an explicitly photon-conserving RT method designed to transport radiation directly on the irregular distribution of SPH particles. This means
that unlike many RT methods that use coarse grids to simplify the calculations,
TRAPHIC exploits the full dynamic range that is available based on the underlying SPH simulation. Another challenge that RT methods face is the large number of sources in cosmological simulations. The computational cost of most
RT methods is proportional to the number of radiation sources, which poses a
big challenge for cosmological RT simulations. TRAPHIC solves this problem
by tracing photon packets inside a discrete number of cones which renders the
computational cost of the RT independent of the number of radiation sources.
The two above mentioned advantages make TRAPHIC particularly well-suited
for RT calculation in cosmological density fields with a large dynamic range in
densities and large numbers of sources.
The photon transport in TRAPHIC proceeds in two steps: the isotropic
emission of photon packets by source particles and their subsequent directed propagation on the irregular distribution of SPH particles. After sources
emit photon packets isotropically to their neighbors, the photon packets travel
along their propagation directions to neighboring SPH particles which are inside their transmission cones. Transmission cones are regular cones with solid
angle 4π/NTC and are centered on the propagation direction. The parameter
NTC sets the angular resolution of the RT. The transmission cones are defined
locally at the transmitting particle, and hence the angular resolution of the RT
with TRAPHIC is independent of the distance from the source.
It can happen that transmission cones do not contain any neighboring SPH
particles. In this case, additional particles (virtual particles, ViPs) are placed
inside the transmission cones to accomplish the photon transport. The ViPs,
which enable the particle-to-particle transport of photons along any direction
independently of the spatially inhomogeneous distribution of the particles, do
not affect the SPH simulation and are deleted after the photon packets have been
transferred.
An important feature of the RT with TRAPHIC is the merging of photon packets, which guarantees the independence of the RT computational cost from the
10
This thesis
number of sources. Photon packets received by each SPH particle are binned
based on their propagation directions in NRC reception cones. Then, photon
packets with identical frequencies that fall in the same reception cone are merged
into a single photon packet with a new direction set by the luminosity-weighted
sum of the directions of the original photon packets.
Photon packets are transported along their propagation direction until they
reach the distance they are allowed to travel within the RT time step by the finite
speed of light. At the end of each time step the ionization states of the particles
are updated based on the number of absorbed ionizing photons.
By following the aforementioned steps, TRAPHIC calculates the radiation
field in cosmological simulations and the ionization state of different species
accurately (see Pawlik & Schaye, 2008, 2011). In this work, we mainly use
TRAPHIC to calculate the ionization state of hydrogen by taking into account
different sources of radiation.
1.4 This thesis
Observational studies are moving rapidly beyond studying only the stellar components of galaxies, towards probing the important but complex processes that
affect the distribution of gas in and around galaxies. Hydrodynamical cosmological simulations of galaxy formation are also improving rapidly by including
better numerical techniques, more physically motivated and better-understood
subgrid models and by using higher resolution. It is important, if not essential,
to compare observations and simulations in order to improve our understanding
of the observational results and to test/improve the simulations.
The neutral hydrogen distribution and its evolution is closely related to various aspects of star formation. This makes understanding and modeling the Hi
distribution critically important for studying galaxy evolution. The main focus
of this thesis is therefore the study of the cosmic distribution of neutral hydrogen using hydrodynamical cosmological simulations. To do this, we combine
hydrodynamical cosmological simulations based on the OverWhelmingly Large
Simulations (OWLS; Schaye et al., 2010) with accurate radiative transfer, and account for different photoionizing processes.
In Chapter 2, we include the radiative transfer effects of the metagalactic
UVB radiation and diffuse recombination radiation for hydrogen ionization at
redshifts z = 5 − 0. We focus on studying Hi column densities NHI > 1016 cm−2 ,
where RT effects can be important. By modeling more than 12 billion years of
evolution of the Hi distribution, we show that the predicted Hi column density
distribution is in excellent agreement with observations and evolves only weakly
from z = 5 to z = 0. We find that the UVB is the dominant source of Hi ionization at z & 1, but that collisional ionization becomes more important at lower
redshift, which affects the self-shielding significantly. Based on our simulations,
we present fitting functions that can be used to accurately calculate the neutral
11
Introduction
hydrogen fractions without RT. Given the difficulty of RT simulations, these fitting functions are particularly useful for the next generation of high resolution
cosmological simulations.
Stars typically form at very high column densities, where the gas is selfshielded against the external ionizing radiation. This makes local stellar radiation an important source of ionization at these column densities. However,
simulating the effect of ionizing stellar radiation for the large numbers of sources
that are typical of cosmological simulations, is extremely challenging. As a result, different studies have found different (and often inconclusive) results regarding the importance of local stellar radiation on the Hi distribution. In Chapter
3, we tackle this problem by combining cosmological simulations with RT using
TRAPHIC , which is designed to handle large numbers of sources efficiently. We
simulate the ionizing radiation from stars together with the UVB and recombination radiation. We show that the local stellar radiation can significantly change
the Hi column density distribution at column densities relevant for the so-called
Damped Lyα (DLA) and Lyman-Limit (LL) systems. We also show that the
main source of disagreement between previous works is insufficient resolution,
a problem we solve by using star-forming particles as ionizing sources. We also
show that the absence of a fully resolved ISM in cosmological simulations is a
bottle-neck for modelling the properties of strong DLAs (i.e., NHI ≥ 1021 cm−2 ).
Strong Hi absorbers, such as DLAs, are likely to be representative of the cold
gas in, or close to, the ISM in high-redshift galaxies. Because of this, they provide
a unique opportunity to define an absorption-selected galaxy sample and to
study the ISM, particularly at the early stages of galaxy formation. However,
because observational studies are limited by the small number of known strong
Hi absorbers and are missing low-mass galaxies in surveys for their counterparts in emission, it is very difficult to probe the relation between Hi absorbers
and their host galaxies observationally and we have to resort to cosmological
simulations to help us understand the link between the two. In Chapter 4, we
use the hydrodynamical cosmological simulations that we have shown to match
the Hi observations very well (see Chapter 2), to study the link between strong
Hi systems and galaxies at z = 3. We show that most strong Hi absorbers are
associated with low-mass galaxies too faint to be detected in current observations. We demonstrate, however, that our predictions are in good agreement
with the existing observations. We show that there is a strong anti-correlation
between the column density of strong Hi absorbers and the impact parameters that connect them to their closest galaxies. We also investigate correlations
between the column density of strong Hi absorbers and different properties of
their associated galaxies.
Similar to the neutral hydrogen, which provides fuel for star formation, the
dust content of galaxies has a strong connection with their star formation activity. This makes studying the distribution and evolution of the dust also very
important for understanding the evolution of galaxies. The low angular resolution of observations at long wavelengths makes identification and spectroscopy
12
REFERENCES
of individual distant infrared galaxies a daunting task. Significant information
about the evolution and statistical properties of these objects is encoded in the
surface density of sources as a function of brightness (i.e., the source count).
In Chapter 5, we present a model for the evolution of dusty galaxies, constrained
by the 850µm source counts and redshift distribution. We use a simple formalism for the evolution of the luminosity function and the color distribution of
infrared galaxies. Using a novel algorithm for calculating the source counts, we
analyze how individual free parameters in the model are constrained by observational data. The model is shown to successfully reproduce the observed
source count and redshift distributions at wavelengths 70µm . λ . 1100µm, and
to be in excellent agreement with the most recent Herschel and SCUBA 2 results.
References
al-Sufi, Abd al-Rahman, Book of Fixed Stars, 964, Isfahan, Iran
Bagla, J. S. 2002, Journal of Astrophysics and Astronomy, 23, 185
Becker, G. D., & Bolton, J. S. 2013, arXiv:1307.2259
Bennett, C. L., Halpern, M., Hinshaw, G., et al. 2003, ApJS, 148, 1
Gingold, R. A., & Monaghan, J. J. 1977, MNRAS, 181, 375
Heymans, C., Grocutt, E., Heavens, A., et al. 2013, MNRAS, 432, 2433
Kennicutt, R. C., Jr. 1998, ApJ, 498, 541
Kim, T.-S., Carswell, R. F., Cristiani, S., D’Odorico, S., & Giallongo, E. 2002,
MNRAS, 335, 555
Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18
Lucy, L. B. 1977, AJ, 82, 1013
Mather, J. C., Cheng, E. S., Eplee, R. E., Jr., et al. 1990, ApJL, 354, L37
Noterdaeme, P., Petitjean, P., Ledoux, C., & Srianand, R. 2009, A&A, 505, 1087
O’Meara, J. M., Prochaska, J. X., Burles, S., et al. 2007, ApJ, 656, 666
O’Meara, J. M., Prochaska, J. X., Worseck, G., Chen, H.-W., & Madau, P. 2013,
ApJ, 765, 137
Pawlik, A. H., & Schaye, J. 2008, MNRAS, 389, 651
Pawlik, A. H., & Schaye, J. 2011, MNRAS, 412, 1943
Péroux, C., Dessauges-Zavadsky, M., D’Odorico, S., Sun Kim, T., & McMahon,
R. G. 2005, MNRAS, 363, 479
Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2013, arXiv:1303.5076
Prochaska, J. X., Worseck, G., & O’Meara, J. M. 2009, ApJL, 705, L113
Schaye, J., & Dalla Vecchia, C. 2008, MNRAS, 383, 1210
Schaye, J., Dalla Vecchia, C., Booth, C. M., et al. 2010, MNRAS, 402, 1536
Spergel, D. N., Verde, L., Peiris, H. V., et al. 2003, ApJS, 148, 175
Tytler, D. 1987, ApJ, 321, 49
Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2011, ApJ, 736, 59
13
2
On the evolution of the Hi
column density distribution in
cosmological simulations
We use a set of cosmological simulations combined with radiative transfer
calculations to investigate the distribution of neutral hydrogen in the postreionization Universe. We assess the contributions from the metagalactic ionizing background, collisional ionization and diffuse recombination radiation to
the total ionization rate at redshifts z = 0 − 5. We find that the densities above
which hydrogen self-shielding becomes important are consistent with analytic
calculations and previous work. However, because of diffuse recombination radiation, whose intensity peaks at the same density, the transition between highly
ionized and self-shielded regions is smoother than what is usually assumed. We
provide fitting functions to the simulated photoionization rate as a function of
density and show that post-processing simulations with the fitted rates yields
results that are in excellent agreement with the original radiative transfer calculations. The predicted neutral hydrogen column density distributions agree
very well with the observations. In particular, the simulations reproduce the
remarkable lack of evolution in the column density distribution of Lyman limit
and weak damped Lyα systems below z = 3. The evolution of the low column
density end is affected by the increasing importance of collisional ionization with
decreasing redshift. On the other hand, the simulations predict the abundance
of strong damped Lyα systems to broadly track the cosmic star formation rate
density.
Alireza Rahmati, Andreas H. Pawlik, Milan Raičevic̀, Joop Schaye
Monthly Notices of the Royal Astronomical Society
Volume 430, Issue 3, pp. 2427-2445 (2013)
On the evolution of the Hi CDDF
2.1 Introduction
A substantial fraction of the interstellar medium (ISM) in galaxies consists of
atomic hydrogen. This makes studying the distribution of neutral hydrogen
(Hi) and its evolution crucial for our understanding of various aspects of star
formation. In the local universe, the Hi content of galaxies is measured through
21-cm observations, but at higher redshifts this will not be possible until the
advent of significantly more powerful telescopes such as the Square Kilometer
Array1. However, at z . 6, i.e., after reionization, the neutral gas can already
be probed through the absorption signatures imprinted by the intervening Hi
systems on the spectra of bright background sources, such as quasars (QSOs).
The early observational constraints on the Hi column density distribution
function (Hi CDDF hereafter), from quasar absorption spectroscopy at z . 3,
were well described by a single power-law in the range NHI ∼ 1013 − 1021 cm−2
(Tytler, 1987). Thanks to a significant increase in the number of observed
quasars and improved observational techniques, more recent studies have extended these observations to both lower and higher Hi column densities and
to higher redshifts (e.g., Kim et al., 2002; Péroux et al., 2005; O’Meara et al.,
2007; Noterdaeme et al., 2009; Prochaska et al., 2009; Prochaska & Wolfe, 2009;
O’Meara et al., 2013; Noterdaeme et al., 2012). These studies have revealed a
much more complex shape which has been described using several different
power-law functions (e.g., Prochaska et al., 2010; O’Meara et al., 2013).
The shape of the Hi CDDF is determined by both the distribution and ionization state of hydrogen. Consequently, determining the distribution function
of Hi column densities requires not only accurate modeling of the cosmological
distribution of gas, but also radiative transfer (RT) of ionizing photons. As a
starting point, the Hi CDDF can be modeled by assuming a certain gas profile and exposing it to an ambient ionizing radiation field (e.g., Petitjean et al.,
1992; Zheng & Miralda-Escudé, 2002). Although this approach captures the effect of self-shielding, it cannot be used to calculate the detailed shape and normalization of the Hi CDDF which results from the cumulative effect of large
numbers of objects with different profiles, total gas contents, temperatures and
sizes. Moreover, the interaction between galaxies and the circum-galactic medium through accretion and various feedback mechanisms, and its impact on
the overall gas distribution are not easily captured by simplified models. Therefore, it is important to complement these models with cosmological simulations
that model the evolution of the large-scale structure of the Universe and the
formation of galaxies.
The complexity of the RT calculation depends on the Hi column density.
At low Hi column densities (i.e., NHI . 1017 cm−2 , corresponding to the socalled Lyman-α forest), hydrogen is highly ionized by the metagalactic ultraviolet background radiation (hereafter UVB) and largely transparent to the ion1 http://www.skatelescope.org/
16
Introduction
izing radiation. For these systems, the Hi column densities can therefore be
accurately computed in the optically thin limit. At higher Hi column densities
(i.e., NHI & 1017 cm−2 , corresponding to the so-called Lyman Limit and Damped
Lyman-α systems), the gas becomes optically thick and self-shielded. As a result,
the accurate computation of the Hi column densities in these systems requires
precise RT simulations. On the other hand, at the highest Hi column densities
where the gas is fully self-shielded and the recombination rate is high, nonlocal RT effects are not very important and the gas remains largely neutral. At
these column densities, the hydrogen ionization rate may, however, be strongly
affected by the local sources of ionization (Miralda-Escudé, 2005; Schaye, 2006;
Rahmati et al., 2013). In addition, other processes like H2 formation (Schaye,
2001b; Krumholz et al., 2009; Altay et al., 2011) or mechanical feedback from
young stars and / or AGNs (Erkal et al., 2012), can also affect the highest Hi
column densities.
Despite the importance of RT effects, most of the previous theoretical
works on the Hi column density distribution did not attempt to model RT effects in detail (e.g., Katz et al., 1996; Gardner et al., 1997; Haehnelt et al., 1998;
Gardner et al., 2001; Cen et al., 2003; Nagamine et al., 2004, 2007). Only very recent works incorporated RT, primarily to account for the attenuation of the UVB
(Razoumov et al., 2006; Pontzen et al., 2008; Fumagalli et al., 2011; Altay et al.,
2011; McQuinn et al., 2011) and found a sharp transition between optically thin
and self-shielded gas that is expected from the exponential nature of extinction.
The aforementioned studies focused mainly on redshifts z = 2 − 3, for which
observational constraints are strongest, without investigating the evolution of the
Hi distribution. They found that the Hi CDDF in current cosmological simulations is in reasonable agreement with observations in a large range of Hi column
densities. Only at the highest Hi column densities (i.e., NHI & 1021 cm−2 ) the
agreement is poor. However, it is worth noting that the interpretation of these
Hi systems is complicated due to the complex physics of the ISM and ionization
by local sources. Moreover, the observational uncertainties are also larger for
these rare high NHI systems.
In this chapter, we investigate the cosmological Hi distribution and its evolution during the last & 12 billion years (i.e., z . 5). For this purpose, we use a set
of cosmological simulations which include star formation, feedback and metalline cooling in the presence of the UVB. These simulations are based on the
Overwhelmingly Large Simulations (OWLS) presented in Schaye et al. (2010).
To obtain the Hi CDDF, we post-processed the simulations with RT, accounting
for both ionizing UVB radiation and ionizing recombination radiation (RR). In
contrast to previous works, we account for the impact of recombination radiation explicitly, by propagating RR photons. Using these simulations we study
the evolution of the Hi CDDF in the range of redshifts z = 0 − 5 for column
densities NHI & 1016 cm−2 . We discuss how the individual contributions from
the UVB, RR and collisional ionization to the total ionization rate shape the Hi
CDDF and assess their relative importance at different redshifts.
17
On the evolution of the Hi CDDF
The structure of this chapter is as follows. In §2.2 we describe the details of
the hydrodynamical simulations and of the RT, including the treatment of the
UVB and recombination radiation. In §2.3 we present the simulated Hi CDDF
and its evolution and compare it with observations. In the same section we also
discuss the contributions of different ionizing processes to the total ionization
rate and provide fitting functions for the total photoionization rate as a function
of density which reproduce the RT results. Finally, we conclude in §2.4.
2.2 Simulation techniques
2.2.1 Hydrodynamical simulations
We use density fields from a set of cosmological simulations performed using a
modified version of the smoothed particle hydrodynamics code GADGET-3 (last
described in Springel, 2005). The subgrid physics is identical to that used in
the reference simulation of the OWLS project (Schaye et al., 2010). Star formation is pressure dependent and reproduces the observed Kennicutt-Schmidt
law (Schaye & Dalla Vecchia, 2008). Chemical evolution is followed using the
model of Wiersma et al. (2009a), which traces the abundance evolution of eleven elements by following stellar evolution assuming a Chabrier (2003) initial
mass function. Moreover, a radiative heating and cooling implementation based
on Wiersma et al. (2009b) calculates cooling rates element-by-element (i.e., using the above mentioned 11 elements) in the presence of the uniform cosmic
microwave background and the UVB model given by Haardt & Madau (2001).
About 40 per cent of the available kinetic energy in type II SNe is injected in
winds with initial velocity of 600 kms−1 and a mass loading parameter η = 2
(Dalla Vecchia & Schaye, 2008). Our tests show that varying the implementation
of the kinetic feedback only changes the Hi CDDF in the highest column densities (NHI & 1021 cm−2 ). However, the differences caused by these variations are
smaller than the evolution in the Hi CDDF and observational uncertainties (see
Altay et al. in prep.).
We adopt fiducial cosmological parameters consistent with the most recent
WMAP 7-year results: Ωm = 0.272, Ωb = 0.0455, ΩΛ = 0.728, σ8 = 0.81, ns =
0.967 and h = 0.704 (Komatsu et al., 2011). We also use cosmological simulations from the OWLS project which are performed with a cosmology consistent
with WMAP 3-year values with Ωm = 0.238, Ωb = 0.0418, ΩΛ = 0.762, σ8 =
0.74, ns = 0.951 and h = 0.73. We use those simulations to avoid expensive resimulation with a WMAP 7-year cosmology. Instead, we correct for the difference
in the cosmological parameters as explained in Appendix B.
Our simulations have box sizes in the range L = 6.25 − 100 comoving h −1 Mpc
and baryonic particle masses in the range 1.7 × 105 h−1 M⊙ − 8.7 × 107 h−1 M⊙ .
The suite of simulations allows us to study the dependence of our results on the
box size and mass resolution. Characteristic parameters of the simulations are
18
Simulation techniques
summarized in Table 2.1.
2.2.2 Radiative transfer with TRAPHIC
The RT is performed using TRAPHIC (Pawlik & Schaye, 2008, 2011). TRAPHIC is
an explicitly photon-conserving RT method designed to transport radiation directly on the irregular distribution of SPH particles using its full dynamic range.
Moreover, by tracing photon packets inside a discrete number of cones, the
computational cost of the RT becomes independent of the number of radiation
sources. TRAPHIC is therefore particularly well-suited for RT calculation in cosmological density fields with a large dynamical range in densities and large
numbers of sources. In the following we briefly describe how TRAPHIC works.
More details, as well as various RT tests, can be found in Pawlik & Schaye (2008,
2011).
The photon transport in TRAPHIC proceeds in two steps: the isotropic emission of photon packets with a characteristic frequency ν by source particles
and their subsequent directed propagation on the irregular distribution of SPH
particles. The spatial resolution of the RT is set by the number of neighbors
for which we generally use the same number of SPH neighbors used for the
underlying hydrodynamical simulations, i.e., Nngb = 48.
After source particles emit photon packets isotropically to their neighbors,
the photon packets travel along their propagation directions to other neighboring SPH particles which are inside their transmission cones. Transmission cones
are regular cones with opening solid angle 4π/NTC and are centered on the
propagation direction. The parameter NTC sets the angular resolution of the RT,
and we adopt NTC = 64. We demonstrate convergence of our results with the
angular resolution in Appendix C. Note that the transmission cones are defined
locally at the transmitting particle, and hence the angular resolution of the RT is
independent of the distance from the source.
It can happen that transmission cones do not contain any neighboring SPH
particles. In this case, additional particles (virtual particles, ViPs) are placed
inside the transmission cones to accomplish the photon transport. The ViPs,
which enable the particle-to-particle transport of photons along any direction
independent of the spatially inhomogeneous distribution of the particles, do not
affect the SPH simulation and are deleted after the photon packets have been
transferred.
An important feature of the RT with TRAPHIC is the merging of photon packets which guarantees the independence of the computational cost from the number of sources. Different photon packets which are received by each SPH particle
are binned based on their propagation directions in NRC reception cones. Then,
photon packets with identical frequencies that fall in the same reception cone
are merged into a single photon packet with a new direction set by the weighted
sum of the directions of the original photon packets. Consequently, each SPH
particle holds at most NRC × Nν photon packets, where Nν is the number of
19
On the evolution of the Hi CDDF
Simulation
L006N256
L006N128
L012N256
L025N512
L006N128-W3
L025N512-W3
L025N128-W3
L050N256-W3
L050N512-W3
L100N512-W3
L
(h−1 Mpc)
6.25
6.25
12.50
25.00
6.25
25.00
25.00
50.00
50.00
100.00
N
2563
1283
2563
5123
1283
5123
1283
2563
5123
5123
mb
( h −1 M ⊙ )
1.7 × 105
1.4 × 106
1.4 × 106
1.4 × 106
1.4 × 106
1.4 × 106
8.7 × 107
8.7 × 107
1.1 × 107
8.7 × 107
mdm
( h −1 M ⊙ )
7.9 × 105
6.3 × 106
6.3 × 106
6.3 × 106
6.3 × 106
6.3 × 106
4.1 × 108
4.1 × 108
5.1 × 107
4.1 × 108
ǫcom
(h−1 kpc)
0.98
1.95
1.95
1.95
1.95
1.95
7.81
7.81
3.91
7.81
ǫprop
(h−1 kpc)
0.25
0.50
0.50
0.50
0.50
0.50
2.00
2.00
1.00
2.00
zend
Cosmology
RT
2
0
2
2
2
2
0
0
0
0
WMAP7
WMAP7
WMAP7
WMAP7
WMAP3
WMAP3
WMAP3
WMAP3
WMAP3
WMAP3
✓
✓
✓
✗
✓
✗
✓
✓
✓
✗
20
Table 2.1: List of cosmological simulations used in this work. All the simulations use model ingredients identical to the reference simulation
of Schaye et al. (2010). From left to right the columns show: simulation identifier; comoving box size; number of dark matter particles (there
are equally many baryonic particles); initial baryonic particle mass; dark matter particle mass; comoving (Plummer-equivalent) gravitational
softening; maximum physical softening; final redshift; cosmology. The last column shows whether the simulation was post-processed with
RT. In simulations without RT, the Hi distribution is obtained by using a fit to the photoionization rates as a function of density measured
from simulations with RT.
Simulation techniques
frequency bins. We set NRC = 8 for which our tests yield converged results.
Photon packets are transported along their propagation direction until they
reach the distance they are allowed to travel within the RT time step by the finite
speed of light, i.e., c∆t. Photon packets that cross the simulation box boundaries
are assumed to be lost from
domain. We use a time step
thecomputational
4
128
∆t = 1 Myr 6.25 Lhbox
−1 Mpc
1+ z
NSPH , where NSPH is the number of SPH
particles in each dimension. We verified that our results are insensitive to the
exact value of the RT time step: values that are smaller or larger by a factor
of two produce essentially identical results. This is mostly because we evolve
the ionization balance on smaller subcycling steps, and because we iterate for
the equilibrium solution, as we discuss below. At the end of each time step the
ionization states of the particles are updated based on the number of absorbed
ionizing photons.
The number of ionizing photons that are absorbed during the propagation
of a photon packet from one particle to its neighbor is given by δNabs,ν =
δNin,ν [1 − exp(−τ (ν))] where δNin,ν and τ (ν) are, respectively, the initial number of ionizing photons in the photon packet with frequency ν and the total optical depth of all the absorbing species. In this work we mainly consider hydrogen ionization, but in general the total optical depth is the sum τ (ν) = ∑α τα (ν)
of the optical depth of each absorbing species (i.e., α ∈ {HI, HeI, HeII}). Assuming that neighboring SPH particles have similar densities, we approximate the
optical depth of each species using τα (ν) = σα (ν)nα dabs , where nα is the number
density of species, dabs is the absorption distance between the SPH particle and
its neighbor and σα (ν) is the absorption cross section (Verner et al., 1996). Note
that ViPs are deleted after each transmission, and hence the photons they absorb
need to be distributed among their SPH neighbors. However, in order to decrease the amount of smoothing associated with this redistribution of photons,
ViPs are assigned only 5 (instead of 48) SPH neighbors. We demonstrate convergence of our results with the number of ViP neighbors in Appendix C.
At the end of each RT time step, every SPH particle has a total number of
ionizing photons that have been absorbed by each species, ∆Nabs,α (ν). This
number is used in order to calculate the photoionization rate of every species for
that SPH particle. For instance, the hydrogen photoionization rate is given by:
ΓHI =
∑ν ∆Nabs,HI (ν)
,
ηHI NH ∆t
(2.1)
where NH is the total number of hydrogen atoms inside the SPH particle and
ηHI ≡ nHI /nH is the hydrogen neutral fraction.
Once the photoionization rate is known, the evolution of the ionization states
is calculated. For instance, the equation which governs the ionization state of
hydrogen is
dηHI
= αHII ne (1 − ηHI ) − ηHI (ΓHI + Γe,H ne ),
(2.2)
dt
21
On the evolution of the Hi CDDF
where ne is the free electron number density, Γe,H is the collisional ionization
rate and αHII is the HII recombination rate. The differential equations which
govern the ionization balance (e.g., equation 2.2) are solved using a subcycling
time step, δt = min( f τeq , ∆t) where τeq ≡ τion τrec /(τion + τrec ), and f is a dimensionless factor which controls the integration accuracy (we set it to 10−3),
τrec ≡ 1/ ∑i ne αi and τion ≡ 1/ ∑i (Γi + ne Γe,i ). The subcycling scheme allows
the RT time step to be chosen independently of the photoionization and recombination time scales without compromising the accuracy of the ionization state
calculations2 .
We employ separate frequency bins to transport UVB and RR photons. Because the propagation directions of photons in different frequency bins are
merged separately, this allows us to track the individual radiation components,
i.e., UVB and RR, and to compute their contributions to the total photoionization rate. The implementation of the UVB and the recombination radiation is
described in § 2.2.3 and § 2.2.4 below.
At the start of the RT, the hydrogen is assumed to be neutral. In addition, we use a common simplification (e.g., Faucher-Giguère et al., 2009;
McQuinn & Switzer, 2010; Altay et al., 2011) by assuming a hydrogen mass fraction of unity, i.e., we ignore helium (only for the RT). To calculate recombination
and collisional ionizations rates, we set, in post-processing, the temperatures of
star-forming gas particles with densities nH > 0.1 cm−3 to TISM = 104 K, which
is typical of the observed warm-neutral phase of the ISM. This is needed because in our hydrodynamical simulations the star-forming gas particles follow
a polytropic equation of state which defines their effective temperatures. These
temperatures are only a measure of the imposed pressure and do not represent
physical temperatures (see Schaye & Dalla Vecchia, 2008). To speed up convergence, the hydrogen at low densities (i.e., nH < 10−3 cm−3 ) or high temperatures
(i.e., T > 105 K) is assumed to be in ionization equilibrium with the UVB and
the collisional ionization rate (see Appendix A2). Typically, the neutral fraction
of the box and the resulting Hi CDDF do not evolve after 2-3 light-crossing times
(the light-crossing time for the extended box with Lbox = 6.25 comoving h −1 Mpc
is ≈ 7.5 Myr at z = 3).
2.2.3 Ionizing background radiation
Although our hydrodynamical simulations are performed using periodic boundary conditions, we use absorbing boundary conditions for the RT. This is necessary because our box size is much smaller than the mean free path of ionizing
photons. We simulate the ionizing background radiation as plane-parallel radi2 Other considerations prevent the use of arbitrarily large RT time steps. The RT assumes that
species fractions and hence opacities do not evolve within a RT time step. This approximation
becomes increasingly inaccurate with increasing RT time steps. Note that in this work, we iterate for
ionization equilibrium which help to render our results robust against changes in the RT time step,
as our convergence studies confirm.
22
Simulation techniques
Table 2.2: Hydrogen photoionization rate, absorption cross-section, equivalent gray approximation frequency and the self-shielding density threshold (i.e., based on equation 2.13) for three UVB models: Haardt & Madau (2001) (HM01; used in this work),
Haardt & Madau (2012) (HM12) and Faucher-Giguère et al. (2009) (FG09) at different redshifts. For the calculation of the photoionization rate and absorption cross-sections, only
photons with energies below 54.4 eV are taken into account, effectively assuming that
more energetic photons are absorbed by He.
Redshift
UVB
ΓUVB (s−1 )
z=0
HM01
HM12
FG09
HM01
HM12
FG09
HM01
HM12
FG09
HM01
HM12
FG09
HM01
HM12
FG09
HM01
HM12
FG09
8.34 × 10−14
2.27 × 10−14
3.99 × 10−14
7.39 × 10−13
3.42 × 10−13
3.03 × 10−13
1.50 × 10−12
8.98 × 10−13
6.00 × 10−13
1.16 × 10−12
8.74 × 10−13
5.53 × 10−13
7.92 × 10−13
6.14 × 10−13
4.31 × 10−13
5.43 × 10−13
4.57 × 10−13
3.52 × 10−13
z=1
z=2
z=3
z=4
z=5
σ̄νHI ( cm2 )
3.27 × 10−18
2.68 × 10−18
2.59 × 10−18
2.76 × 10−18
2.62 × 10−18
2.37 × 10−18
2.55 × 10−18
2.61 × 10−18
2.27 × 10−18
2.49 × 10−18
2.61 × 10−18
2.15 × 10−18
2.45 × 10−18
2.60 × 10−18
2.02 × 10−18
2.45 × 10−18
2.58 × 10−18
1.94 × 10−18
Eeq (eV)
16.9
18.1
18.3
17.9
18.2
18.8
18.3
18.2
19.1
18.5
18.2
19.5
18.6
18.3
19.9
18.6
18.3
20.1
nH,SSh ( cm−3 )
1.1 × 10−3
5.1 × 10−4
7.7 × 10−4
5.1 × 10−3
3.3 × 10−3
3.1 × 10−3
8.7 × 10−3
6.1 × 10−3
5.1 × 10−3
7.4 × 10−3
6.0 × 10−3
5.0 × 10−3
5.8 × 10−3
4.7 × 10−3
4.4 × 10−3
4.5 × 10−3
3.9 × 10−3
4.0 × 10−3
23
On the evolution of the Hi CDDF
ation entering the simulation box from its sides. At the beginning of each RT
step, we generate a large number of photon packets, Nbg , on the nodes of a regular grid at each side of the simulation box and set their propagation directions
perpendicular to the sides. The number of photon packets is chosen to obtain
converged results. Furthermore, to avoid numerical artifacts close to the edges
of the box, we use the periodicity of our simulations to extend the simulation
box by the typical size of the region where we generate the background radiation
(i.e., 2% of the box size from each side). These extended regions are excluded
from the analysis, thereby removing the artifacts without losing any information
contained in the original simulation box.
The photon content of each packet is normalized such that in the absence of
any absorption (i.e., assuming the optically thin limit), the total photon density
of the box corresponds to the desired uniform hydrogen photoionization rate. If
we assume that all the photons with frequencies higher than νHeII are absorbed
by helium, then the hydrogen photoionization rate can be written as:
ΓUVB =
Z ν
HeII
Jν
σHI, ˚ dν
hν
νHI
Z ν
HeII Jν
4π σ̄νHI
≡
dν,
h
ν
νHI
4π
(2.3)
where Jν is the radiation intensity (in units erg cm−2 s−1 sr−1 Hz−1 ), νHI and
νHeII are respectively the frequency at the Lyman-limit and the frequency at the
HeII ionization edge, and σHI, ˚ is the neutral hydrogen absorption cross-section
for ionizing photons. In the last equation we have defined the gray absorption
cross-section,
R νHeII
Jν /ν σHI, ˚ dν
ν
.
(2.4)
σ̄νHI ≡ HIR νHeII
Jν /ν dν
ν
HI
The radiation intensity is related to the photon energy density, uν ,
Jν =
uν c
nν hν c
=
,
4π
4π
(2.5)
where nν is the number density of photons inside the box. Combining Equations
2.3-2.5 yields
ΓUVB = nνHI c σ̄νHI ,
(2.6)
where nνHI is the number density of ionizing photons inside the box. The total
number of ionizing photons in the box is therefore given by
nνHI L3box = nγ 6 Nbg
Lbox
,
c ∆t
(2.7)
where nγ is the number of ionizing photons carried by each photon packet. Now
we can calculate the photon content of each packet that must be injected into the
24
Simulation techniques
box during each step in order to achieve the desired Hi photoionization rate:
nγ =
ΓUVB L2box ∆t
,
6 σ̄νHI Nbg
(2.8)
We use the redshift-dependent UVB spectrum of Haardt & Madau (2001) to
calculate ΓUVB and σ̄νHI . The Haardt & Madau (2001) UVB model successfully
reproduces the relative strengths of the observed metal absorption lines in the
intergalactic medium (Aguirre et al., 2008) and has been used to calculate heating/cooling in our cosmological simulations 3 . We note however that using
more recent models for the UVB is not expected to change our main results.
On can show that varying the UVB photoionization rate by a factor of 3, only
changes the HI CDDF by less than 0.2 dex for LLSs (e.g., Altay et al., 2011). As
shown in Table 2.2, the differences between photoionization rates in different
UVB models are smaller that a factor of 3, particularly at z > 1, where the photoionization by the UVB is not subdominant (see §2.3.5). The variations in the
adopted UVB model is even less important for systems with higher HI column
densities (i.e., DLAs) which remain highly neutral for reasonable UVB models
(e.g., Haardt & Madau, 2012; Faucher-Giguère et al., 2009).
To reduce the computational cost, we treat the multi-frequency problem in
the gray approximation. In other words, we transport the UVB radiation using
a single frequency bin, inside which photons are absorbed using the gray crosssection σ̄νHI defined in equation 2.4. Note that the gray approximation ignores
the spectral hardening of the radiation field that would occur in multifrequency
simulations. In Appendix D we show the result of repeating our simulations using multiple frequency bins, and also explicitly accounting for the absorption of
photons by helium. These results clearly show the expected spectral hardening.
The impact of spectral hardening on the hydrogen neutral fractions and the Hi
CDDF is small. However, we note that spectral hardening can change the temperature of the gas in self-shielded regions and that this effect is not captured in
our simulations.
Hydrogen photoionization rates and average absorption cross-sections for
UVB radiation at different redshifts are listed in Table 2.2 for our fiducial UVB
model based on Haardt & Madau (2001) together with Haardt & Madau (2012)
and Faucher-Giguère et al. (2009). The photoionization rate peaks at z ≈ 2 − 3
in those models and the equivalent effective photon energy4 of the background
radiation changes only weakly with redshift, compared to the total photoionization rate.
3 Note
that during the hydrodynamical simulations, photoheating from the UVB is applied to
all gas particles. This ignores the self-shielding of hydrogen atoms against the UVB that occurs at
densities nH & 10−3 − 10−2 cm3 . This inconsistency, which could affect both collisional ionization
rates and the small-scale structure of the absorbers, has been found to have no significant impact on
the simulated Hi CDDF (Pontzen et al., 2008; McQuinn & Switzer, 2010; Altay et al., 2011).
4 We defined the equivalent effective photon energy, E , which corresponds to the absorption
eq
cross section, σ̄νHI , as: Eeq ≡ 13.6 eV (σ̄νHI /σ0 ) −1/3 where σ0 = 6.3 × 10−18 cm2 .
25
On the evolution of the Hi CDDF
2.2.4 Recombination radiation
Photons produced by the recombination of positive ions and electrons can also
ionize the gas. If the recombining gas is optically thin, recombination radiation
can escape and its ionizing effects can be ignored (i.e., the so-called Case A).
However, for regions in which the gas is optically thick, the proper approximation is to assume the ionizing recombination radiation is absorbed on the spot.
In this case, the effective recombination rate can be approximated by excluding the transitions that produce ionizing photons (e.g., Osterbrock & Ferland,
2006). This scenario is usually called Case B. A possible way to take into account the effect of recombination radiation is to use Case A recombination at
low densities and Case B recombination at high densities (e.g., Altay et al., 2011;
McQuinn et al., 2011), but this will be inaccurate in the transition regime.
In this work we explicitly treat the ionizing photons emitted by recombining
hydrogen atoms and follow their propagation through the simulation box. This
is facilitated by the fact that the computational cost of RT with TRAPHIC is
independent of the number of sources. This is particularly important noting
that every SPH particle is potentially a source. The photon production rates of
SPH particles depend on their recombination rates and the radiation is emitted
isotropically once at the beginning of every RT time step (see Raicevic et al. in
prep. for full details).
We do not take into account the redshifting of the recombination photons
by peculiar velocities of the emitters, or the Hubble flow. Instead, we assume
that all recombination photons are monochromatic with energy 13.6 eV. In reality, recombination photons cannot travel to large cosmological distances without
being redshifted to frequencies below the Lyman edge. Therefore, neglecting the
cosmological redshifting of RR will result in overestimation of its photoionization rate on large scales. However, because of the small size of our simulation
box, the total photoionization rate that is produced by RR on these scales remains negligible compared to the UVB photoionization rate. Consequently, the
neglect of RR redshifting is not expected to affect our results.
2.2.5 The Hi column density distribution function
In order to compare the simulation results with observations, we compute the
CDDF of neutral hydrogen, f ( NHI , z), a quantity that is somewhat straightforward to measure in QSO absorption line studies and is defined as the number
of absorbers per unit column density, per unit absorption length, dX:
f ( NHI , z) ≡
26
1
d2 n H ( z )
d2 n
.
≡
dNHI dX
dNHI dz H0 (1 + z)2
(2.9)
Simulation techniques
We project the Hi content of the simulation box along each axis onto a grid with
50002 or 100002 pixels (for the 1283 − 2563 and 5123 simulations, respectively)5.
This is done using the actual kernels of SPH particles and for each of the three
axes. The projection may merge distinct systems along the line of sight. However, for the small box sizes and high column densities with NHI > 1017 cm−2 ,
which are the focus of this work, the chance of overlap between multiple absorbing systems in projection is negligible. Based on our numerical experiments, we
expect that this projection effect starts to appear only at NHI < 1016 cm−2 if one
uses a single slice for the projection of the entire L50N512-W3 simulation box at
z = 3. To make sure our results are insensitive to this effect, we use, depending
on redshift, 25 or 50 slices for projecting the L50N512-W3 simulation.
To produce a converged f ( NHI , z) from simulations, one needs to use cosmological boxes that are large enough to capture the relevant range of overdensities. This is particularly demanding at very high Hi column densities: for instance, van de Voort et al. (2012) showed that most of the gas with
NHI > 1021 cm−2 resides in galaxies with halo masses & 1011 M⊙ which are relatively rare. As we show in Appendix B, the box size required to produce a
converged Hi CDDF up to NHI ∼ 1022 cm−2 is L & 50 comoving h−1 Mpc. Simulating RT in such a large volume is expensive. However, as we show in §2.3.4,
at a given redshift the photoionization rates are fit very well by a function of
the hydrogen number density. This relation is conserved with respect to both
box size6 and resolution and can therefore be applied to our highest resolution
simulation (i.e., L50N512-W3), allowing us to keep the numerical cost tractable.
Finally, since repeating the high-resolution simulations is expensive, we apply a
redshift-independent correction which accounts for the difference between the
WMAP year 3 parameters used for L50N512-W3 simulation and the WMAP year
7 values. This is done by multiplying all the Hi CDDFs produced based on the
WMAP year 3 cosmology by the ratio between the Hi CDDFs for L25N512 and
L25N512-W3 at z = 3.
2.2.6 Dust and molecular hydrogen
Dust and star formation are highly correlated and infrared observations indicate
that the prevalence of dusty galaxies follows the average star formation history
of the Universe (e.g., Rahmati & van der Werf, 2011). Nevertheless, dust extinction is a physical processes that is not treated in our simulations. Assuming a
constant dust-to-gas ratio, the typical dust absorption cross-section per atom is
orders of magnitudes lower than the typical hydrogen absorption cross-section
5 Using 50002 cells, the corresponding cell size is similar to the minimum smoothing length, and
∼ 100 times smaller than the mean smoothing length, of SPH particles at z = 3 in the L06N128
simulation.
6 One should note that the box size can indirectly change the resulting photoionization rate profile. For instance, self-shielding can be affected by collisional ionization, which become stronger at
lower redshifts and whose importance depends on the abundance of massive objects, which is more
sensitive to the box size.
27
On the evolution of the Hi CDDF
Figure 2.1: CDDF of neutral gas at different redshifts in the presence of the UVB and diffuse recombination radiation for L50N512-W3. A column density dependent amplitude
correction has been applied to make the results consistent with WMAP year 7 cosmological parameters. The observational data points represent a compilation of various quasar
absorption line observations at high redshifts (i.e., z = [1.7, 5.5]) taken from Péroux et al.
(2005) with z = [1.8, 3.5], O’Meara et al. (2007) with z = [1.7, 4.5], Noterdaeme et al.
(2009) with z = [2.2, 5.5] and Prochaska & Wolfe (2009) with z = [2.2, 5.5]. The colored
data points in the top-left corner of the left panel are taken from Kim et al. (2002) with
z = [2.9, 3.5] and z = [1.7, 2.4] for the yellow crosses and orange diamonds, respectively.
The orange filled circles show the best-fit based on the low-redshift 21-cm observations
of Zwaan et al. (2005). The high column density end of the Hi distribution is magnified
in the right panel and for clarity only the simulated Hi CDDF of redshifts z = 1, 3 & 5 are
shown. The top-section of each panel shows the ratio between the Hi CDDFs at different
redshifts and the Hi CDDF at z = 3. The simulation results are in reasonably good agreement with the observations and, like the observations, show only a remarkably weak
evolution for Lyman Limit and weak damped Lyα systems below z = 3.
28
Simulation techniques
Figure 2.2: The high column density end of the Hi distribution shown in Figure 2.1 is
magnified here. For clarity, only the simulated Hi CDDF of redshifts z = 1, 3 & 5 are
shown. The top-section shows the ratio between the Hi CDDFs at different redshifts and
the Hi CDDF at z = 3. The simulation results are in reasonably good agreement with
the observations and, like the observations, show only a weak evolution for Lyman Limit
and weak damped Lyα systems below z = 3.
29
On the evolution of the Hi CDDF
for ionizing photons (Weingartner & Draine, 2001). In other words, the absorption of ionizing photons by dust particles is not significant compared to the absorption by the neutral hydrogen. Consequently, as also found in cosmological
simulations with ionizing radiation (Gnedin et al., 2008), dust absorption does
not noticeably alter the overall distribution of ionizing photons and hydrogen
neutral fractions.
The observed cut-off in the abundance of very high NHI systems may be
related to the conversion of atomic hydrogen into H2 (e.g., Schaye, 2001b;
Krumholz et al., 2009; Prochaska & Wolfe, 2009; Altay et al., 2011). Following Altay et al. (2011) and Duffy et al. (2012), we adopt an observationally
driven scaling relation between gas pressure and hydrogen molecular fraction
(Blitz & Rosolowsky, 2006) in post-processing, which reduces the amount of observable Hi at high densities. This scaling relation is based on observations of
low-redshift galaxies and may not cover the low metallicities relevant for higher
redshifts. This could be an issue, since the Hi-H2 relation is known to be sensitive to the dust content and hence to the metallicity (e.g., Schaye, 2001b, 2004;
Krumholz et al., 2009a).
2.3 Results
In this section we report our findings based on various RT simulations which include UVB ionizing radiation and diffuse recombination radiation from ionized
gas. As we demonstrate in §2.3.3, the dependence of the photoionization rate
on density obtained from our RT simulations shows a generic trend for different
resolutions and box sizes. Therefore, we can use the results of RT calculations
obtained from smaller boxes (e.g., L06N128 or L06N256) which are computationally cheaper, to calculate the neutral hydrogen distribution in larger boxes. The
last column of Table 2.1 indicates for which simulations this was done.
In the following, we will first present the predicted Hi CDDF and compare it
with observations. Next we discuss other aspects of our RT results and the effects
of ionization by the UVB, recombination radiation and collisional ionization on
the resulting Hi distributions at different redshifts.
2.3.1 Comparison with observations
In Figure 2.1 & 2.2 we compare the simulation results with a compilation of
observed Hi CDDFs, after converting both to the WMAP year 7 cosmology. The
data points with error bars show results from high-redshift (z = 1.7 − 5.5) QSO
absorption line studies and the orange filled circles show the fitting function
reported by Zwaan et al. (2005) based on 21-cm observations of nearby galaxies.
The latter observations only probe column densities NHI & 1019 cm−2 .
We note that the OWLS simulations have already been shown to agree with
observations by Altay et al. (2011), but only for z = 3 and based on a different
30
Results
RT method (see Appendix C3 for a comparison). Overall, our RT results are
also in good agreement with the observations. At high column densities (i.e.,
NHI > 1017 cm−2 ) the observations probing 0 < z < 5.5 are consistent with
each other. This implies weak or no evolution with redshift. The simulation
is consistent with this remarkable observational result, predicting only weak
evolution (. 50%) for 1017 cm−2 < NHI < 1021 cm−2 (i.e., Lyman limit systems,
LLSs, and weak Damped Ly-α systems, DLAs) especially at z . 3.
The simulation predicts some variation with redshift for strong DLAs (NHI &
1021 cm−2 ). The abundance of strong DLAs in the simulations follows a similar
redshift-dependent trend as the average star formation density in our simulations which peaks at z ≈ 2 − 3 (Schaye et al., 2010). This result is consistent
with the DLA evolution found by Cen (2012) in two zoomed simulations of a
cluster and a void. One should, however, note that at very high column densities (e.g., NHI & 1021.5 cm−2 ) both observations and simulations are limited by
small number statistics and the simulation results are more sensitive to the adopted feedback scheme (Altay et al. in prep.). Moreover, as we will show in
Rahmati et al. (2013), including local stellar ionizing radiation can decrease the
Hi CDDF by up to ≈ 1 dex for NHI & 1021 cm−2 , especially at redshifts z ≈ 2 − 3
for which the average star formation activity of the Universe is near its peak.
At low column densities (i.e., NHI . 1017 cm−2 ) the simulation results agree
very well with the observations. This is apparent from the agreement between
the simulated f ( NHI , z) at z = 3 and z = 4, and the observed values for redshifts
2.9 < z < 3.5 (Kim et al., 2002) which are shown by the yellow crosses in Figure
2.1. The simulated f ( NHI , z) at lower and higher redshifts deviate from those
at z ≈ 3 showing the abundance of those systems decreases with decreasing
redshift and remains nearly constant at z . 2. This is consistent with the Lyα forest observations at lower redshifts (Kim et al., 2002; Janknecht et al., 2006;
Lehner et al., 2007; Prochaska & Wolfe, 2009; Ribaudo et al., 2011), as illustrated
with the orange diamonds which correspond to z ≈ 2 observations, in the topleft corner of Figure 2.1.
The evolution of the Hi CDDF with redshift results from a combination of
the expanding Universe and the growing intensity of the UVB radiation down
to redshifts z ≈ 2 − 3. At low redshifts (i.e., z ≈ 0) the intensity of the UVB
radiation has dropped by more than one order of magnitude leading to higher
hydrogen neutral fractions and higher Hi column densities. However, as we
show in §2.3.5, at lower redshifts an increasing fraction of low-density gas is
shock-heated to temperatures sufficiently high to become collisionally ionized
and this compensates for the weaker UVB radiation at low redshifts.
The simulated Hi CDDFs at all redshifts are consistent with each other and
the observations. However, as illustrated in Figure 2.2, there is a ≈ 0.2 dex difference between the simulation results and the observations of LLS and DLAs at
all redshifts. We found that the normalization of the Hi CDDF in those regimes
is sensitive to the adopted cosmological parameters (see Appendix B). Notably,
the cosmology consistent with the WMAP 7 year results that is shown here, pro31
On the evolution of the Hi CDDF
Figure 2.3: nHI -weighted total hydrogen number density as a function of NHI . The brown
solid curve shows the RT results and the purple dotted curve shows the optically thin
limit. Blue dot-dashed, dot-dot-dot-dashed and long dashed curves assume models with
self-shielding density thresholds of nH,SSh = 10−1 , 10−2 and 10−3 cm−3 , respectively.
All of the above mentioned curves show the median of the nHI -weighted total hydrogen
number density at a given NHI . The gray thin lines show the expected Jeans scaling
relations for optically thin gas (equation 2.10; diagonal solid line) and for neutral gas
(equation 2.11; steeper dotted line). A second solid line with the same slope expected
from equation (2.11) but a different normalization is illustrated by the second solid line
which is identical to the dotted line but shifted by 0.5 dex to higher NHI . The pink and
blue shaded areas in the right panel indicate the 70% and 99% scatter respectively, while
the solid curves shows the median for the RT result. All the other curves are also medians.
This shows that the nH − NHI relationship can be explained by the Jeans scaling and that
the flattening in the CDDF is due to self-shielding.
32
Results
Figure 2.4: HI CDDF in the presence of the UVB and diffuse recombination radiation
for simulation L06N256. Simulations shown with different curves are identical to those
in Figure 2.3. In addition, the effect of H2 formation is shown by the green solid curve
which deviates from the brown solid curve at NHI & 3 × 1021 cm−2 . Finally the red
dashed curve, which is indistinguishable from the brown solid curve, shows the result of
assuming the median of the photoionization rate profile of the RT results to calculate the
neutral fractions (see §2.3.3 and Appendix A1). The top-section in the right panel shows
the ratio between different Hi CDDFs and the one resulting from the RT simulations.
33
On the evolution of the Hi CDDF
duces a better match to the observations than a cosmology based on the WMAP
3 year results with smaller values for Ωb and σ8 . This suggests that a higher
value of σ8 may explain the small discrepancy between the simulation results
and the observations.
2.3.2 The shape of the Hi CDDF
The shape of the Hi CDDF is determined by the distribution of hydrogen and
by the different ionizing processes that set the hydrogen neutral fractions of the
absorbers. One can assume that over-dense hydrogen resides in self-gravitating
systems that are in local hydrostatic equilibrium. Then, the typical scales of
the systems can be calculated as a function of the gas density based on a Jeans
scaling argument (Schaye, 2001). Assuming that absorbers have universal baΩ
ryon fractions (i.e., f g = Ωmb ) and typical temperatures of T4 ≡ ( T/104 K ) ∼ 1
(i.e., collisional ionization is unimportant), one can calculate the total hydrogen
column density (Schaye, 2001):
NH ∼ 1.6 × 10
21
cm
−2
1/2
nH
T41/2
fg
0.17
1/2
.
(2.10)
Assuming that the gas is highly ionized and in ionization equilibrium with
the ambient ionizing radiation field with the photoionization rate, Γ−12 =
Γ/10−12 s−1 , one gets (Schaye, 2001):
3/2
nH
NHI ∼ 2.3 × 1013 cm−2
10−5 cm−3
f g 1/2
1
× T4−0.26 Γ−
.
−12
0.17
(2.11)
At high densities where the gas is nearly neutral, equation (2.10) provides
a relation between NHI and nH . Equation (2.11) on the other hand, gives the
relation for optically thin, highly ionized gas. The latter is derived assuming
that the UVB photoionization is the dominant source of ionization, which is a
good assumption at high redshifts and explains the relation between density
and column density in Lyα forest simulations (e.g., Davé et al., 2010; Altay et al.,
2011; McQuinn et al., 2011; Tepper-García et al., 2012). However, as we will show
in the following sections, photoionization domination breaks down at lower redshifts where collisional ionization plays a significant role.
The column density at which hydrogen starts to be self-shielded against the
UVB radiation follows from setting τHI = 1:
NHI,SSh ∼ 4 × 1017 cm−2
σ̄νHI
2.49 × 10−18 cm2
−1
(2.12)
which can be used together with equation (2.11) to find the typical densities at
34
Results
which the self-shielding begins (e.g., Furlanetto et al., 2005):
−3
cm
Γ2/3
−12
nH,SSh ∼ 6.73 × 10
×
T40.17
−3
fg
0.17
σ̄νHI
2.49 × 10−18 cm2
−1/3
.
−2/3
(2.13)
These relations are compared with the nHI -weighted total hydrogen number
density as a function of NHI in the L06N256 simulation at z = 3 in Figure 2.3.
The solid curve shows the median and the red (blue) shaded area represents
the central 70% (90%) percentile. The diagonal gray solid line which converges
with the simulation results at low column densities, shows equation (2.11) and
the steeper gray dotted line which converges with the simulation results at high
column densities is based on equation (2.10). The agreement between the expected slopes of the nH − NHI relation and the simulations at low and high
column densities confirms our initial assumption that hydrogen resides in selfgravitating systems which are close to local hydrostatic equilibrium7 .
As expected from equation (2.13), at low densities the gas is optically thin and
follows the Jeans scaling relation of the highly ionized gas. At nH & 0.01 cm−3
however, the relation between density and column density starts to deviate from
equation (2.11) and approaches that of a nearly neutral gas. Consequently, for
densities above the self-shielding threshold the Hi column density increases rapidly over a narrow range of densities, leading to a flattening in the nH − NHI
relation and in the resulting Hi CDDF at NHI & 1018 cm−2 (see Figure 2.4).
The results from the RT simulation deviate from the magenta dotted lines,
which are obtained assuming optically thin gas, at NHI & 4 × 1017 cm−2 . As
the dotted line in Figure 2.4 shows, in the absence of self-shielding, the slope
of f ( NHI , z) ∝ NHI β is constant all the way up to DLAs at β Lyα ≈ −1.6.
However, because of self-shielding, the Hi CDDF flattens to β LLS ≈ −1.1 at
1018 cm−2 . NHI . 1020 cm−2 in the RT simulation (solid curve). These predicted slopes are in excellent agreement with the latest observational constraints
of β Lyα & −1.6 for 1015 cm−2 < NHI < 1017 cm−2 to β LLS ≈ −1 in the LLS regime
(O’Meara et al., 2013). We also note that β Lyα & −1.6 is predicted to be almost
the same for all redshifts, which agrees well with observations (Janknecht et al.,
2006; Lehner et al., 2007; Ribaudo et al., 2011).
At densities nH & 0.1 cm−3 the gas is nearly neutral and the Jeans scaling in
equation (2.10) controls the nH − NHI relation. Consequently, the rate at which
NHI responds to changes in nH slows down, causing a steepening in the resulting
f ( NHI , z) in the DLA range (i.e., NHI & 1021 cm−2 ). However, as the thick solid
7 One should note that the above mentioned Jeans argument provides an order of magnitude
calculation due to its simplifying assumptions (e.g., uniform density, universal baryon fraction, etc.).
Although we may expect the predicted scaling relations to be correct, the very close agreement of
the normalization with the simulations at low densities is coincidental. As the steeper gray dotted
line which is based on equation (2.10) shows, the simulated NHI for a given nH is ≈ 0.5 dex higher
than implied by the Jeans scaling for the nearly neutral case (i.e., steep, gray solid line).
35
On the evolution of the Hi CDDF
curve in Figure 2.4 illustrates, the slope of f ( NHI , z) remains constant for NHI =
1021 − 1022 cm−2 . This is in contrast with observed trends indicating a sharp cutoff at NHI & 3 × 1021 cm−2 (Prochaska et al., 2010; O’Meara et al., 2013; but see
Noterdaeme et al., 2012). At those column densities a large fraction of hydrogen
is expected to form H2 molecules and be absent from Hi observations (Schaye,
2001b; Krumholz et al., 2009; Altay et al., 2011). As the thin solid line in Figure
2.4 shows, accounting for H2 using the empirical relation between H2 fraction
and pressure, based on z = 0 observations (Blitz & Rosolowsky, 2006), does
reproduce a sharp cut-off. If the observed relation does not cut off sharply
(Noterdaeme et al., 2012), then this may imply that H2 fractions are lower at
z = 3 than at z = 0. We also note that the ionizing effect of local sources
(Rahmati et al., 2013), increasing the efficiency of stellar feedback, e.g., by using a
top-heavy IMF, and AGN feedback can also affect these high Hi column densities
(Altay et al. in prep.).
To first order, one can mimic the effect of RT by assuming gas with nH <
nH,SSh to be optically thin (i.e., Case A recombination) and gas with nH > nH,SSh
to be fully neutral. Simulations with three different self-shielding density
thresholds are shown in Figure 2.3 & 2.4. The dot-dashed, dot-dot-dot-dashed
and long dashed curves correspond to nH,SSh = 10−1, 10−2 and 10−3 cm−3 , respectively. Although all of these simulations predict the flattening of f ( NHI , z),
they produce a transition between optically thin and neutral gas that is too steep.
In contrast, the RT results show a transition between highly ionized and highly
neutral gas that is more gradual, as observed.
2.3.3 Photoionization rate as a function of density
Figure 2.5 illustrates the RT results for neutral fractions and photoionization
rates as a function of density in the presence of UVB radiation and diffuse recombination radiation for the L06N128, L06N256 and L12N256 simulations at z
= 3. For comparison, the results for the optically thin limit are shown by the
green dotted curves. The sharp transition between highly ionized and neutral
gas and its deviation from the optically thin case are evident in the left panel.
This transition can also be seen in the photoionization rate (right panel) which
drops at nH & 0.01 cm−3 , consistent with equation (2.13) and previous studies (Tajiri & Umemura, 1998; Razoumov et al., 2006; Faucher-Giguère et al., 2010;
Nagamine et al., 2010; Fumagalli et al., 2011; Altay et al., 2011).
The medians and the scatter around them are insensitive to the resolution of
the underlying simulation and to the box size. This suggests that one can use
the photoionization rate profile obtained from the RT simulations for calculating
the hydrogen neutral fractions in other simulations for which no RT has been
performed.
Moreover, as we show in §2.3.5, the total photoionization rate as a function
of the hydrogen number density has the same shape at different redshifts. This
shape can be characterized by three features: i) a knee at densities around the
36
37
Results
Figure 2.5: The hydrogen neutral fraction (left) and the photoionization rate (right) as a function of hydrogen number density do not change
by varying the simulation box size or mass resolution. This is shown for different simulations at z = 3 in the presence of the UVB and
recombination radiation. Purple solid, blue dashed and red dot-dashed lines show, respectively, the results for L12N256, L06N128 and
L06N256. The green dotted line indicates the results for the L06N128 simulation if the gas is assumed to be optically thin to the UVB
radiation (i.e., no RT calculation is performed). The deviation between the optically thin hydrogen neutral fractions and RT results at
nH & 10−2 cm−3 shows the impact of self-shielding. The lines show the medians and the shaded areas indicate the 15% − 85% percentiles.
At the top of each panel we show Hi column densities corresponding to each density.
On the evolution of the Hi CDDF
self-shielding density threshold, ii) a relatively steep fall-off at densities higher
than the self-shielding threshold and iii) a flattening in the fall-off after the photoionization rate has dropped by ∼ 2 dex from its maximum value which is
caused by the RR photoionization. These features are captured by the following
fitting formula:
ΓPhot
= 0.98
ΓUVB
"
1+
nH
nH,SSh
1.64 # −2.28
+0.02 1 +
nH
nH,SSh
−0.84
,
(2.14)
where ΓUVB is the background photoionization rate and ΓPhot is the total photoionization rate. Moreover, the self-shielding density threshold, nH,SSh , is given
by equation (2.13) and is thus a function of ΓUVB and σ̄νHI which vary with
redshift. As explained in more detail in Appendix A1, the numerical parameters representing the shape of the profile are chosen to provide a redshift
independent best fit to our RT results. In addition, the parametrization is based
on the main RT related quantities, namely the intensity of UVB radiation and
its spectral shape. It can therefore be used for UVB models similar to the
Haardt & Madau (2001) model we used in this work (e.g., Faucher-Giguère et al.,
2009; Haardt & Madau, 2012). For a given UVB model, one only needs to know
ΓUVB and σ̄νHI in order to determine the corresponding nH,SSh from (2.13) (see
also Table 2.2). Then, after using equation (2.14) to calculate the photoionization
rate as a function of density, the equilibrium hydrogen neutral fraction for different densities, temperatures and redshifts can be readily calculated as explained
in Appendix A1.
We note that the parameters used in equation (2.14) are only accurate for
photoionization dominated cases. As we show in §2.3.5, at z ∼ 0 the collisional
ionization rate is greater than the total photoionization rate around the selfshielding density threshold. Consequently, equation (2.13) does not provide an
accurate estimate of the self-shielding density threshold at low redshifts. In Appendix A1 we therefore report the parameters that best reproduce our RT results
at z = 0. Our tests show that simulations that use equation (2.14) reproduce the
f ( NHI , z) accurately to within 10% for z & 1 where photoionization is dominant
(see Appendix A1).
Although using the relation between the median photoionization rate and
the gas density is a computationally efficient way of calculating equilibrium
neutral fractions in big simulations, it comes at the expense of the information
encoded in the scatter around the median photoionization rate at a given density.
However, our experiments show that the error in f ( NHI , z) that results from
neglecting the scatter in the photoionization rate profile is negligible for NHI &
1018 cm−3 and less than . 0.1 dex at lower column densities (see Appendix A1).
38
Results
Figure 2.6: Ionization rates due to different sources of ionization as a function of hydrogen number density. Blue solid, green dashed and red dotted curves show, respectively,
the UVB photoionization rate, the recombination radiation photoionization rate and the
collisional ionization rate. The curves show the medians and the shaded areas around the
medians indicate the 15% − 85% percentiles. Hi column densities corresponding to each
density are shown along the top x-axis. While the UVB is the dominant source of ionization below the self-shielding (i.e., nH . 10−2 cm−3 ), recombination radiation dominates
the ionization at higher densities.
2.3.4 The roles of diffuse recombination radiation and collisional ionization at z = 3
To study the interplay between different ionizing processes and their effects on
the distribution of Hi, we compare their ionization rates at different densities.
We start the analysis by presenting the results at z = 3 and extend it to other
redshifts in §2.3.5.
The total photoionization rate profiles shown in the right panel of Figure 2.5
are almost flat at low densities and decrease with increasing density, starting
at densities nH ∼ 10−4 cm−3 . Just below nH = 10−2 cm−3 self-shielding causes
a sharp drop, but the fall-off becomes shallower for nH > 10−2 cm−3 and the
photoionization rate starts to increase at nH > 10 cm−3 . As shown in Figure 2.6,
the shallower fall-off in the total photoionization rate with increasing density
is caused by RR. The increase in the photoionization rate with density at the
highest densities on the other hand, is an artifact of the imposed temperature
for ISM particles (i.e., T = 104 K) which produces a rising collisional ionization rate with increasing density. As the comparison between the UVB and RR
photoionization profiles shows (see Figure 2.6), RR only starts to dominate the
39
40
On the evolution of the Hi CDDF
Figure 2.7: The hydrogen neutral fractions (left) and the UVB photoionization rate profiles (right) as a function of density in RT simulations
with different models for recombination radiation for the L06N256 simulation at z = 3. The red dashed curve shows the reference simulation
where recombination radiation is modeled self-consistently. The blue solid and green dot-dashed curves show simulations in which
recombination radiation is substituted by the use of Case A and Case B recombination rates, respectively. The curves show the medians
and the shaded areas around the medians indicate the 15% − 85% percentiles. Hi column densities corresponding to each density are shown
along the top x-axes. The effect of recombination radiation on the hydrogen neutral fractions is similar to the use of Case A recombination
at low densities (i.e., nH . 10−3 cm−3 ) and to the use of Case B recombination at higher densities (i.e., nH & 10−1 cm−3 ). However,
recombination radiation can penetrate into the self-shielded regions, an effect that is not captured by the use of Case B recombination.
Results
total photoionization rate at nH > 10−2 cm−3 , where the UVB photoionization
rate has dropped by more than one order of magnitude and the gas is no longer
highly ionized. RR reduces the total Hi content of high-density gas by ≈ 20%.
Although ionization rates remain non-negligible at higher densities, they cannot
keep the hydrogen highly ionized. For instance at nH ∼ 1 cm−3 , a photoionization rate of Γ ∼ 10−14 s−1 can only ionize the gas by . 20%.
The shape of the photoionization rate profile produced by diffuse RR can be
understood by noting that the production rate of RR increases with the density of ionized gas. At number densities nH < 10−2 cm−3 , where the gas is
highly ionized, the photoionization rate due to recombination photons is proportional to the density (i.e., ΓRR ∝ nH ). At higher densities on the other hand,
the gas becomes neutral. As a result, the density of ionized gas decreases with
increasing density and the production rate of recombination photons decreases.
Therefore, there is a peak in the photoionization rate due to RR around the
self-shielding density. At very low densities, the superposition of recombination
photons which have escaped from higher densities becomes dominant and the
net photoionization rate of recombination photons flattens. Note that our simulations may underestimate this asymptotic rate because our simulation volumes
are small compared to the mean free path for ionizing radiation (which is ∼ 100
Mpc at z ∼ 3). On the other hand, the neglect of cosmological redshifting for RR
will result in overestimation of its photoionization rate on large scales. Recombination photons also leak from lower densities to self-shielded regions, smoothing the transition between highly ionized and highly neutral gas. At high densities, in the absence of the UVB ionizing photons, RR and collisional ionization
can boost each other by providing more free electrons and ions.
In Figure 2.7 we compare hydrogen neutral fraction and photoionization rate
profiles for different assumptions about RR. The hydrogen neutral fraction profile based on a precise RT calculation of RR is close to the Case A result at low
densities (nH . 10−3 cm−3 ) but converges to the Case B result at high densities
(nH & 10−1 cm−3 ). This suggests that the neutral fraction profile, though not the
ionization rate, can be modeled by switching from Case A to Case B recombination at nH ∼ nH,SSh (e.g., Altay et al., 2011; McQuinn et al., 2011).
2.3.5 Evolution
The general trends in the profile of the photoionization rates with density and
their influences on the distribution of Hi are not very sensitive to redshift.
However, as shown in table 2.2, the intensity and hardness of the UVB radiation change with redshift which, in turn, changes the self-shielding density.
Moreover, as the Universe expands, the average density of absorbers decreases
and their distributions evolve. The larger structures that form at lower redshifts
drastically change the temperature structure of the gas at low and intermediate densities where collisional ionization becomes the dominant process. In the
top-left panel of Figure 2.8, the evolution of the hydrogen neutral fraction is il41
On the evolution of the Hi CDDF
Figure 2.8: Evolution of the hydrogen neutral fraction profile (top-left) and various ionization rates as a function of density. Top-right, bottom-left and bottom-right panels show,
respectively, the total photoionization rates, collisional ionization rates and RR photoionization rates. All the Hi fractions and collisional ionization rates which are sensitive to
both collisional ionization and photoionization are taken from the L50N512-W3 simulation. Photoionization rates at z ≥ 2 are based on the L06N128 simulation. At lower
redshifts (i.e., z = 0 and 1), where the box size become important because of the collisional ionization and its effect on changing the self-shielding, we used a representative
sub-volume of the L50N512-W3 simulation to calculate the photoionization rate profile.
with density While the overall shape of the UVB photoionization rate profile is similar at
different redshifts, the collisional ionization becomes increasingly stronger at lower redshifts and strongly reduces the hydrogen neutral fractions at densities nH . 10−3 cm−3 .
42
Results
lustrated for the L50N512-W3 simulation. As discussed in §2.3.3 and Appendix
A1, since the photoionization rate profiles are converged with box size and resolution, we apply the profiles derived from a RT simulation of a smaller box, or a
subset of the big box at lower redshifts8 , to calculate the neutral fractions in this
big box. Figure 2.8 shows that the neutral fraction profiles are similar in shape at
high redshifts but that at z ≤ 1 the profiles are largely different, particularly at
low hydrogen number densities, due to the evolving collisional ionization rates.
The evolution of the collisional ionization rate profiles is shown in the
bottom-left panel of Figure 2.8. At z ≥ 2 and for nH < 10−2 cm−3 , the collisional ionization rate is not high enough to compete with the UVB photoionization rate. At lower redshifts and for number densities nH . 3 × 10−3 cm−3 , on
the other hand, collisional ionization dominates9 . Indeed, the median collisional
ionization rates are more than 100 times higher than the UVB photoionization
rate at densities around the expected self-shielding thresholds. Collisional ionization therefore helps the UVB ionizing photons to penetrate to higher densities
without being significantly absorbed. As a result, self-shielding starts at densities higher than expected from equation (2.13). The signature of collisional
ionization on the hydrogen neutral fraction is more dramatic at low densities
and partly compensates for the lower UVB intensity at z = 0. This results in a
flattening of f ( NHI , z) at column densities NHI . 1016 cm−2 as shown in Figure
2.1.
As mentioned above, at low redshifts (e.g., z = 0) the collisional ionization
rate peaks at densities higher than the expected self-shielding threshold against
the UVB. As a result, the total photoionization rate falls off rapidly together with
the drop in the collisional ionization rate. Therefore, the drop in the hydrogen
ionized fraction, and hence the resulting free electron density, is much sharper
at lower redshifts. This causes a steeper high-density fall-off in the collisional
ionization rate as shown in the bottom-left panel of Figure 2.8.
The differences between the total photoionization rates at different redshifts
shown in the top-right panel of Figure 2.8, are caused by the evolution of
the UVB intensity and its hardness, which affects the self-shielding density
thresholds (see equation 2.13). On the other hand, as we showed in the previous section, the peak of the photoionization rate produced by RR tracks the
self-shielding density. As a result the peak of the RR photoionization rate also
changes with redshift as illustrated in the bottom-right panel of Figure 2.8).
The filled circles in Figure 2.9 indicate the UVB photoionization rate versus
the number density at which the RR photoionization rate peaks. The selfshielding density expected from the Jeans scaling argument (equation 2.13) is
also shown (green dotted line). The peaks in the RR photoionization rate in RT
8 Strong collisional ionization at low redshifts can change the self-shielding. Therefore, for our
RT simulation at low redshifts (i.e., z . 1) we used a representative sub-volume of the L50N512-W3
simulation sufficiently large for the collisional ionization rates to be converged.
9
We note that at redshift z . 1 box sizes LBox & 25 comoving h−1 Mpc are required to fully capture
the large-scale accretion shocks and to produce converged collisional ionization rates.
43
On the evolution of the Hi CDDF
Figure 2.9: The photoionization from recombination radiation peaks at the expected
self-shielding density and smooths the transition between highly ionized gas and selfshielded gas. This is illustrated by showing different characteristic densities for various
redshifts. Filled circles show the density at which recombination radiation peaks, while
open circles and squares show the Hi number density corresponding to hydrogen neutral
fractions of 10−2 and 0.5, respectively. The self-shielding density threshold for a given
photoionization rate expected from the Jeans scaling argument (equation 2.13) is also indicated by the green dotted line. The Jeans scaling argument works well, except at z = 0
when collisional ionization is important.
simulations follow this expected scaling for z ≥ 1. However, the z = 0 result deviates from this trend since collisional ionization affects the self-shielding
density threshold, a factor that is not captured by equation (2.13).
As a result of the RR photoionization rate peaking around the self-shielding
density threshold, the transition between highly ionized and nearly neutral gas
becomes more extended at all redshifts. To illustrate the smoothness of this
transition, the densities at which the median hydrogen neutral fractions are 10−2
and 0.5 are shown in Figure 2.9 with open circles and squares, respectively.
The densities at which the hydrogen neutral fraction is 10−2 are slightly higher
than the densities at which the RR photoionization rate peaks (filled circles).
The evolution agrees with the trend expected from the Jeans scaling argument
and the self-shielding density. The exception is again z = 0, where the large
collisional ionization rate at number densities nH ∼ 10−3 − 10−2 cm−3 shifts the
transition to neutral fraction of 0.5 to densities that are ≈ 1 dex higher. However,
the relation between the photoionization rate and the density still follows the
slope expected from equation (2.13).
It is interesting to note that the UVB spectral shape at z = 4 is slightly harder
than at z = 1 while the UVB intensities at these two redshifts are similar. This
44
Conclusions
results in a deeper penetration of ionizing photons at z = 4. Consequently, the
densities corresponding to the indicated neutral fractions (i.e., 10−2 and 0.5) at
z = 1 are lower than their counterparts at z = 4.
2.4 Conclusions
We combined a set of cosmological hydrodynamical simulations with an accurate RT simulation of the UVB radiation to compute the Hi column density
distribution function and its evolution. We ignored the effect of local sources of
ionizing radiation, but we did include a self-consistent treatment of recombination radiation.
Our RT results for the distribution of photoionization rates at different densities are converged with respect to the simulation box size and resolution. Therefore, the resulting photoionization rate can be expressed as a function of the
hydrogen density and the UVB. We provided a fit for the median total photoionization rate as a function of density that can be used with any desired UVB
model to take into account the effect of Hi self-shielding in cosmological simulations without the need to perform RT.
The CDDF, f ( NHI , z), predicted by our RT simulations is in excellent agreement with observational constraints at all redshifts (z = 0 − 5) and reproduces
the slopes of the observed f ( NHI , z) function for a wide range of Hi column
densities. At low Hi column densities, the CDDF is a steep function which
decreases with increasing NHI before it flattens at NHI & 1018 cm−2 due to selfshielding. At NHI & 1021 cm−3 on the other hand, f ( NHI , z) is determined mainly
by the intrinsic distribution of total hydrogen and the H2 fraction.
We showed that the NHI − nH relationship can be explained by a simple Jeans
scaling. This argument assumes Hi absorbers to be self-gravitating systems close
to local hydrostatic equilibrium (Schaye, 2001) and to be either neutral or in
photoionization equilibrium in the presence of an ionizing radiation field. However, at z = 0 the analytic treatment underestimates the self-shielding density
threshold due to its neglect of collisional ionization.
The high Hi column density end of the predicted f ( NHI , z) evolves only
weakly from z = 5 to z = 0, consistent with observations. In the Lyman limit
range of the distribution function, the slope of f ( NHI , z) remains the same at all
redshifts. However, at z > 3 the number of absorbers increases with redshift as
the Universe becomes denser while the UVB intensity remains similar. At lower
redshifts, on the other hand, the combination of a decreasing UVB intensity and
the expansion of the Universe results in a non-evolving f ( NHI , z). In contrast,
the number of absorbers with lower Hi column densities (i.e., the Lyα forest)
decreases significantly from z ∼ 3. We showed that this results in part from
the stronger collisional ionization at redshifts z . 1, which compensates for the
lower intensity of the UVB. The increasing importance of collisional ionization
is due to the rise in the fraction of hot gas due to shock-heating associated with
45
On the evolution of the Hi CDDF
the formation of structure.
The inclusion of diffuse recombination radiation smooths the transition
between optically thin and thick gas. Consequently, the transition to highly
neutral gas is not as sharp as what has been assumed in some previous works
(e.g., Nagamine et al., 2010; Yajima et al., 2011; Goerdt et al., 2012). For instance,
the difference in the gas density at which hydrogen is highly ionized (i.e.,
nHI /nH . 0.01) and the density at which gas is highly neutral (i.e., nHI /nH & 0.5)
is more than one order of magnitude (see Figure 2.9). As a result, assuming a
sharp self-shielding density threshold at the density for which the optical depth
of ionizing photons is ∼ 1, overestimates the resulting neutral hydrogen mass
by a factor of a few.
Our simulations adopted some commonly used approximations (e.g., neglecting helium RT effects, using a gray approximation in order to mimic the
UVB spectra, neglecting absorption by dust and local sources of ionizing radiation). Our tests show that most of those approximations have negligible effects
on our results. But there are some assumptions which require further investigation. For instance, the presence of young stars in high-density regions could
change the Hi CDDF, especially at high Hi column densities through feedback
and emission of ionizing photons. Indeed, we will show in Rahmati et al. (2013)
that for very high column densities the ionizing radiation from young stars can
reduce the f ( NHI , z) by 0.5-1 dex.
Acknowledgments
We thank the anonymous referee for a helpful report. We thank Garbriel Altay
for providing us with his simulation results and a compilation of the observed
Hi CDDF. We also would like to thank Kristian Finlator, J. Xavier Prochaska,
Tom Theuns and all the members of the OWLS team for valuable discussions
and Marcel Haas, Joakim Rosdahl, Maryam Shirazi and Freeke van de Voort for
helpful comments on an earlier version of the paper this chapter is based on. The
simulations presented here were run on the Cosmology Machine at the Institute
for Computational Cosmology in Durham (which is part of the DiRAC Facility
jointly funded by STFC, the Large Facilities Capital Fund of BIS, and Durham
University) as part of the Virgo Consortium research programme and on Stella,
the LOFAR BlueGene/L system in Groningen. This work was sponsored by
the National Computing Facilities Foundation (NCF) for the use of supercomputer facilities, with financial support from the Netherlands Organization for
Scientific Research (NWO), also through a VIDI grant and an NWO open competition grant. We also benefited from funding from NOVA, from the European
Research Council under the European Union’s Seventh Framework Programme
(FP7/2007-2013) / ERC Grant agreement 278594-GasAroundGalaxies and from
the Marie Curie Training Network CosmoComp (PITN-GA-2009-238356). AHP
receives funding from the European Union’s Seventh Framework Programme
46
References
(FP7/2007-2013) under grant agreement number 301096.
References
Aguirre, A., Dow-Hygelund, C., Schaye, J., & Theuns, T. 2008, ApJ, 689, 851
Altay, G., Theuns, T., Schaye, J., Crighton, N. H. M., & Dalla Vecchia, C. 2011,
ApJL, 737, L37
Blitz, L., & Rosolowsky, E. 2006, ApJ, 650, 933
Cen, R., Ostriker, J. P., Prochaska, J. X., & Wolfe, A. M. 2003, ApJ, 598, 741
Cen, R. 2012, ApJ, 748, 121
Chabrier, G. 2003, PASP, 115, 763
Dalla Vecchia, C., & Schaye, J. 2008, MNRAS, 387, 1431
Davé, R., Oppenheimer, B. D., Katz, N., Kollmeier, J. A., & Weinberg, D. H.
2010, MNRAS, 408, 2051
Duffy, A. R., Kay, S. T., Battye, R. A., et al. 2012, MNRAS, 420, 2799
Erkal, D., Gnedin, N. Y., & Kravtsov, A. V. 2012, arXiv:1201.3653
Faucher-Giguère, C.-A., Lidz, A., Zaldarriaga, M., & Hernquist, L. 2009, ApJ,
703, 1416
Faucher-Giguère, C.-A., Kereš, D., Dijkstra, M., Hernquist, L., & Zaldarriaga,
M. 2010, ApJ, 725, 633
Friedrich, M. M., Mellema, G., Iliev, I. T., & Shapiro, P. R. 2012, MNRAS, 2385
Fumagalli, M., Prochaska, J. X., Kasen, D., et al. 2011, MNRAS, 418, 1796
Furlanetto, S. R., Schaye, J., Springel, V., & Hernquist, L. 2005, ApJ, 622, 7
Gardner, J. P., Katz, N., Hernquist, L., & Weinberg, D. H. 1997, ApJ, 484, 31
Gardner, J. P., Katz, N., Hernquist, L., & Weinberg, D. H. 2001, ApJ, 559, 131
Gnedin, N. Y., Kravtsov, A. V., & Chen, H.-W. 2008, ApJ, 672, 765
Goerdt, T., Dekel, A., Sternberg, A., Gnat, O., & Ceverino, D. 2012,
arXiv:1205.2021
Haardt F., Madau P., 2001, in Clusters of Galaxies and the High Redshift Universe Observed in X-rays, Neumann D. M., Tran J. T. V., eds.
Haardt, F., & Madau, P. 2012, ApJ, 746, 125
Haehnelt, M. G., Steinmetz, M., & Rauch, M. 1998, ApJ, 495, 647
Hui, L., & Gnedin, N. Y. 1997, MNRAS, 292, 27
Janknecht, E., Reimers, D., Lopez, S., & Tytler, D. 2006, A&A, 458, 427
Katz, N., Weinberg, D. H., Hernquist, L., & Miralda-Escude, J. 1996, ApJL, 457,
L57
Kim, T.-S., Carswell, R. F., Cristiani, S., D’Odorico, S., & Giallongo, E. 2002,
MNRAS, 335, 555
Komatsu, E., et al. 2011, ApJS, 192, 18
Krumholz, M. R., McKee, C. F., & Tumlinson, J. 2009a, ApJ, 693, 216
Krumholz, M. R., Ellison, S. L., Prochaska, J. X., & Tumlinson, J. 2009b, ApJL,
701, L12
Lehner, N., Savage, B. D., Richter, P., et al. 2007, ApJ, 658, 680
47
REFERENCES
McQuinn, M., & Switzer, E. R. 2010, MNRAS, 408, 1945
McQuinn, M., Oh, S. P., & Faucher-Giguère, C.-A. 2011, ApJ, 743, 82
Miralda-Escudé, J. 2005, ApJL, 620, L91
Nagamine, K., Springel, V., & Hernquist, L. 2004, MNRAS, 348, 421
Nagamine, K., Wolfe, A. M., Hernquist, L., & Springel, V. 2007, ApJ, 660, 945
Nagamine, K., Choi, J.-H., & Yajima, H. 2010, ApJL, 725, L219
Noterdaeme, P., Petitjean, P., Ledoux, C., & Srianand, R. 2009, A&A, 505, 1087
Noterdaeme, P., Petitjean, P., Carithers, W. C., et al. 2012, arXiv:1210.1213
O’Meara, J. M., Prochaska, J. X., Burles, S., et al. 2007, ApJ, 656, 666
O’Meara, J. M., Prochaska, J. X., Worseck, G., Chen, H.-W., & Madau, P. 2012,
arXiv:1204.3093
Osterbrock, D. E., & Ferland, G. J. 2006, Astrophysics of gaseous nebulae and
active galactic nuclei, 2nd. ed. by D.E. Osterbrock and G.J. Ferland. Sausalito,
CA: University Science Books, 2006,
Pawlik, A. H., & Schaye, J. 2008, MNRAS, 389, 651
Pawlik, A. H., & Schaye, J. 2011, MNRAS, 412, 1943
Péroux, C., Dessauges-Zavadsky, M., D’Odorico, S., Sun Kim, T., & McMahon,
R. G. 2005, MNRAS, 363, 479
Petitjean, P., Bergeron, J., & Puget, J. L. 1992, A&A, 265, 375
Pontzen, A., Governato, F., Pettini, M., et al. 2008, MNRAS, 390, 1349
Prochaska, J. X., & Wolfe, A. M. 2009, ApJ, 696, 1543
Prochaska, J. X., Worseck, G., & O’Meara, J. M. 2009, ApJL, 705, L113
Prochaska, J. X., O’Meara, J. M., & Worseck, G. 2010, ApJ, 718, 392
Rahmati, A., & van der Werf, P. P. 2011, MNRAS, 418, 176
Rahmati, A., Schaye, J., Pawlik, A. H., & Raičević, M. 2013, MNRAS, 431, 2261
Razoumov, A. O., Norman, M. L., Prochaska, J. X., & Wolfe, A. M. 2006, ApJ,
645, 55
Ribaudo, J., Lehner, N., & Howk, J. C. 2011, ApJ, 736, 42
Robbins, D. 1978, Amer. Math. Monthly 85, 278
Schaye, J. 2001a, ApJ, 559, 507
Schaye, J. 2001b, ApJL, 562, 95
Schaye, J. 2004, ApJ, 609, 667
Schaye, J. 2006, ApJ, 643, 59
Schaye, J., & Dalla Vecchia, C. 2008, MNRAS, 383, 1210
Schaye, J., Dalla Vecchia, C., Booth, C. M., et al. 2010, MNRAS, 402, 1536
Springel, V. 2005, MNRAS, 364, 1105
Tepper-García, T., Richter, P., Schaye, J., Booth, C. M.; Dalla Vecchia, C.; Theuns,
T. 2012, MNRAS, 425, 1640
Tytler, D. 1987, ApJ, 321, 49
Tajiri, Y., & Umemura, M. 1998, ApJ, 502, 59
Theuns, T., Leonard, A., Efstathiou, G., Pearce, F. R., & Thomas, P. A. 1998,
MNRAS, 301, 478
van de Voort, F., Schaye, J., Altay, G., & Theuns, T. 2012, MNRAS, 421, 2809
Verner, D. A., Ferland, G. J., Korista, K. T., & Yakovlev, D. G. 1996, ApJ, 465, 487
48
Appendix A: Photoionization rate as a function of density
Weingartner, J. C., & Draine, B. T. 2001, ApJ, 548, 296
Wiersma, R. P. C., Schaye, J., & Smith, B. D. 2009a, MNRAS, 393, 99
Wiersma, R. P. C., Schaye, J., Theuns, T., Dalla Vecchia, C., & Tornatore, L.
2009b, MNRAS, 399, 574
Yajima, H., Choi, J.-H., & Nagamine, K. 2011, arXiv:1112.5691
Zheng, Z., & Miralda-Escudé, J. 2002, ApJL, 568, L71
Zwaan, M. A., van der Hulst, J. M., Briggs, F. H., Verheijen, M. A. W., & RyanWeber, E. V. 2005, MNRAS, 364, 1467
Appendix A: Photoionization rate as a function of
density
A1: Replacing the RT simulations with a fitting function
In §2.3.3 we demonstrated that the median of the simulated relation between the
total photoionization rate, ΓPhot , and density is converged with respect to resolution and box size. We used this result and provided fits to the median of this
relation. We have exploited these fits to compute the neutral hydrogen fraction
in cosmological simulations under the assumption of ionization equilibrium (see
Appendix A2), without performing the computationally demanding RT. In this
section, we discuss the accuracy of these fits.
The left panel of Figure 2.10 shows that using the median photoionization
rates produces an Hi CDDF in very good agreement with the Hi CDDF obtained
from the corresponding RT simulation (orange solid curve) at NHI & 1018 cm−2 .
However, there is a small systematic difference at lower column densities. One
may think that this small difference is caused by the loss of information contained in the scatter in the photoionization rates at fixed density. We tested this
hypothesis by including a log-normal random scatter around the median photoionization rate consistent with the scatter exhibited by the RT result. However,
after accounting for the random scatter, the f ( NHI , z) is slightly overproduced
compared to the full RT result at nearly all Hi column densities.
We exploit the insensitivity of the shape of the ΓPhot -density relation to the
redshift, and propose the following fit to the photoionization rate, ΓPhot ,
"
β # α1
n α2
nH
ΓPhot
,
(2.15)
= (1 − f ) 1 +
+ f 1+ H
ΓUVB
n0
n0
where ΓUVB is the photoionization rate due to the ionizing background, and n0 ,
α1 , α2 , and β are parameters of the fit. The best-fit values of these parameters are
listed in Table 2.3 and the photoionization rate-density relations they produce are
compared with the RT simulations at redshifts z = 0 and z = 4 in Figure 2.11.
At all redshifts, the best-fit value of n0 is almost identical to the self-shielding
density threshold, nH,SSh , defined in equation (2.13), and the characteristic slopes
49
On the evolution of the Hi CDDF
Figure 2.10: Left: The ratio between the Hi CDDF calculated using the RT based ΓPhot − nH
relationship and the actual RT results for the L06N128 simulation in the presence of
the UVB and diffuse recombination radiation, at z = 3. The orange solid line shows
the result of using the median photoionization rate-density profile predicted by the RT
simulation. The blue dashed curve shows the result of including the scatter around
the median in the calculations. Right: Hi CDDFs calculated using the ΓPhot − nH fitting
function (i.e., equation 2.14) are compared to the Hi CDDFs for which the actual ΓPhot density relation from the RT simulations are used. Blue and green curves are for z = 0
and z = 2 respectively and the red curve is for z = 4. The difference between the RT
result and the result of using the fitting function at z = 0 is due to the importance of
collisional ionization at z = 0. To capture this effect and to reproduce the RT results at
z = 0, we advise using the best-fit parameters shown in Table 2.4. All the CDDFs are for
the L50N512-W3 simulation and in the presence of the UVB and diffuse recombination
radiation.
of the photoionization rate-density relation are similar. This suggest that one
can find a single set of best-fit values to reproduce the RT results at z & 1.
The corresponding best-fit parameter values are (see also equation 2.14) α1 =
−2.28 ± 0.31, α2 = −0.84 ± 0.11, n0 = (1.003 ± 0.005) × nH,SSh , β = 1.64 ± 0.19
and f = 0.02 ± 0.0089.
In the right panel of Figure 2.10, the ratio between the Hi CDDF calculated
using the fitting function presented in equation (2.14) and the RT based Hi CDDF
(i.e., calculated using the median of the photoionization rate-density relation
in the RT simulations) is shown for the L50N512-W3 and at z = 0, 2 and 4.
This illustrates that the fitting function reproduces the RT results accurately,
except at z = 0. As explained in §2.3, this is expected since at low redshifts
collisional ionization affects the self-shielding and the resulting photoionization
rate-density profile. However, a separate fit can be obtained using converged RT
results at z = 0. The parameters that define such a fit are shown in Table 2.4.
A2: The equilibrium hydrogen neutral fraction
In this section we explain how to derive the neutral fraction in ionization equilibrium. Equating the total number of ionizations per unit time per unit volume
50
Appendix A: Photoionization rate as a function of density
Table 2.3: The best-fit parameters for equation (2.15) at different redshifts based on RT
results in the L06N128 simulation.
Redshift
log [n0 ] ( cm−3 )
α1
α2
β
z = 1-5
z=0
z=1
z=2
z=3
z=4
z=5
log [nH,SSh ]
-2.94
-2.29
-2.06
-2.13
-2.23
-2.35
-2.28
-3.98
-2.94
-2.22
-1.99
-2.05
-2.63
-0.84
-1.09
-0.90
-1.09
-0.88
-0.75
-0.57
1.64
1.29
1.21
1.75
1.72
1.93
1.77
1− f
0.98
0.99
0.97
0.97
0.96
0.98
0.99
Figure 2.11: Comparisons between the total photoionization rates as a function of density
in the L06N128 simulation. Photoionization rates based on the RT simulations and best-fit
functions at z = 4 and z = 0 are shown in the left and right panels, respectively. In each
panel, the RT result is shown with the orange solid curve. The best fit to the RT result
at a given redshift (equation 2.15 and Table 2.3) is shown with the blue dashed curve
and the best fit to the RT results at z = 1 − 5 (equation 2.14) is shown with the purple
dotted curve. As shown in the right panel, because of the impact of collisional ionization
on self-shielding, the low redshift photoionization curve (the blue dashed curve) deviates
from the best fit to the results at higher redshifts (the purple dotted curve). To resolve
this issue and to capture the impact of collisional ionization, we advise using the best-fit
parameters shown in Table 2.4 for z = 0.
Table 2.4: The best-fit parameters for equation (2.15) at z = 0 based on RT results for the
L50N512 simulation. To capture the impact of collisional ionization on the self-shielding,
one needs to use large cosmological simulations. The simulation with a box size of
50 h−1 Mpc results in converged collisional ionizations.
Redshift
log [n0 ] ( cm−3 )
α1
α2
β
z=0
-2.56
-1.86
-0.51
2.83
1− f
0.99
51
On the evolution of the Hi CDDF
with the total number of recombinations per unit time per unit volume, we obtain
nHI ΓTOT = αA ne nHII ,
(2.16)
where nHI , ne and nHII are the number densities of neutral hydrogen atoms, free
electrons and protons, respectively. ΓTOT is the total ionization rate per neutral
hydrogen atom and αA is the Case A recombination rate10 for which we use the
fitting function given by Hui & Gnedin (1997):
αA = 1.269 × 10−13 λ1.503
0.47
1 + (λ/0.522)
3 −1
1.923 cm s ,
(2.17)
where λ = 315614/T.
Defining the hydrogen neutral fraction as the ratio between the number densities of neutral hydrogen and total hydrogen, η = nHI /nH , and ignoring helium
(which is an excellent approximation, see Appendix D2), we can rewrite equation (2.17) as:
η ΓTOT = αA (1 − η )2 nH .
(2.18)
Furthermore, we can assume that the total ionization rate, ΓTOT , consists of two
components: the total photoionization rate, ΓPhot , and the collisional ionization
rate, ΓCol :
ΓTOT = ΓPhot + ΓCol ,
(2.19)
where ΓCol = ΛT (1 − η ) nH . The photoionization rate can be expressed as
a function of density using equation (2.14). For ΛT , which depends only on
temperature, we use a relation given in Theuns et al. (1998):
ΛT = 1.17 × 10−10
T 1/2 exp(−157809/T )
√
cm3 s−1 .
1 + T/105
(2.20)
We can now rearrange equation (2.18) as a quadratic equation:
A η 2 − B η + C = 0,
with A = αA + ΛT , B = 2αA +
ΓPhot
nH
η=
(2.21)
+ ΛT and C = αA which gives:
B−
√
B2 − 4AC
.
2A
(2.22)
Using the last equation one can calculate the equilibrium hydrogen neutral
fraction for a given nH and temperature.
10 The use of Case B is more appropriate for n > n
H,SSh . However, we assume the photoionization
H
due to RR is included in ΓTOT , e.g., by using the best-fit function that is presented in equation 2.14.
Therefore, Case A recombination should be adopted even at high densities.
52
Appendix B: Box size, cosmology and resolution
Appendix B: The effects of box size, cosmological
parameters and resolution on the Hi CDDF
The size of the simulation box may limit the abundance and the density of the
densest systems captured by the simulation. In other words, very massive structures, which may be associated with the highest Hi column densities, cannot be
formed in a small cosmological box. Indeed, as shown in the top panels of Figure
2.12, one needs to use cosmological boxes larger than & 25 comoving h−1 Mpc in
order to achieve convergence in the Hi distribution (see also Altay et al., 2011).
On the other hand, the bottom-right panel of Figure 2.12 shows that changing
the resolution of the cosmological simulations also affects f ( NHI , z), although
the effect is small.
The adopted cosmological parameters also affect the gas distribution and
hence the Hi CDDF. For instance, one expects that the number of absorbers at a
given density varies with the density parameter Ωb , and the root mean square
amplitude of density fluctuations σ8 . The bottom-left panel of Figure 2.12 shows
the ratio of column densities in simulations assuming WMAP 7-year and 3-year
parameters. The ratio is only weakly dependent on the box size of the simulation
and its resolution. This motivates us to use this ratio to convert the Hi CDDF
between the two cosmologies for all box sizes and resolutions (at any given
redshift). While this is an approximate way of correcting for the difference in
the cosmological parameters, it does not affect the main conclusions presented
in this work (e.g., the lack of evolution of f ( NHI , z)).
Appendix C: RT convergence tests
C1: Angular resolution
The left-hand panel of Figure 2.13 shows the dependence of photoionization
rates on the adopted angular resolution, i.e., the opening angle of the transmission cones 4π/NTC . The photoionization rates are converged for NTC = 64 (our
fiducial value) or higher.
C2: The number of ViP neighbors
The right panel of Figure 2.13 shows the dependence of the photoionization rates
on the number of SPH neighbors of ViPs. As discussed in §2.2.2, ViPs distribute
the ionizing photons they absorb among their NGBViP nearest SPH neighbors.
The larger the number of neighbors, the larger the volume over which photons
are distributed, and the more extended is the transition between highly ionized
and self-shielded gas. The photoionization rates converge for . 5 ViP neighbors
(our fiducial value is 5).
53
On the evolution of the Hi CDDF
Figure 2.12: The relative changes in the Hi CDDF using different resolutions, box sizes
and cosmologies in the presence of the UVB and diffuse recombination radiation. The
top-left panel shows the effect of box size on f ( NHI , z) for a fixed resolution at z = 3,
where the orange solid (blue dashed) curve shows the difference between using a box
size of L = 25 (50) comoving h−1 Mpc and a box size of L = 100 comoving h−1 Mpc. The
top-right panel shows the same effect but for smaller box sizes: the orange solid (blue
dashed) curve shows the difference between using a box size of L = 6 (12) comoving
h−1 Mpc and a box size of L = 25 comoving h−1 Mpc. The bottom-left shows the effect of
using a cosmology consistent with WMAP 3-year results instead of using a cosmology
based on the WMAP 7-year constraints. The orange solid and blue dashed curves show
this effect for simulations with box sizes of L = 6 and 25 comoving h−1 Mpc, respectively.
The bottom-right panel shows the effect of resolution.
54
Appendix C: RT convergence tests
Figure 2.13: The UVB photoionization rate is converged for our adopted angular resolution, i.e., NTC = 64, as shown in the left panel and our adopted number of ViP neighbors,
i.e., NGBViP = 5, as shown in the right panel. Photoionization rate profiles are shown for
the L06N128 simulation in the presence of the UVB radiation where the Case A recombination is adopted. The curves show the medians and the shaded areas around them
indicate the 15% − 85% percentiles.
C3: Direct comparison with another RT method
Altay et al. (2011) used cosmological simulations from the reference model of
the OWLS project (Schaye et al., 2010), i.e., a simulation run with the same hydro code as we used in this work, to investigate the effect of the UVB on the
Hi CDDF at z = 3. However, they employed a ray-tracing method very different from the RT method we use here. Furthermore, they did not explicitly
treat the transfer of recombination radiation. In Figure 2.14, we compare one of
our UVB photoionization rate profiles11 with the photoionization rate found by
Altay et al. (2011) in a similar simulation. The overall agreement is very good,
but the comparison also reveals important differences.
Altay et al. (2011) calculate the average optical depth around every SPH
particle within a distance of 100 proper kpc, assuming the UVB is unattenuated
at larger distances. Then, they use this optical depth to calculate the attenuation
of the UVB photoionization rate for every particle. This procedure may underestimate the small but non-negligible absorption of UVB ionizing photons on
large scales. Indeed, by tracing the self-consistent propagation of photons inside
the simulation box, we have found that the UVB photoionization rate decreases
gradually with increasing density up to the density of self-shielding. However,
we note that the small differences between our UVB photoionization rates and
those calculated by Altay et al. (2011) at densities below the self-shielding, become slightly smaller by increasing the angular resolution in our RT calculations
11
Note that in our simulations the UVB photoionization rate is converged with the box size and
the resolution as shown in §2.3.3.
55
On the evolution of the Hi CDDF
Figure 2.14: Left: Median UVB photoionization rate as a function of density at z = 3 using
different RT methods. The red dashed curve shows the results based on the method
that has been used in Altay et al. (2011) and the blue solid curve shows the result of
this work. Right: The Hi CDDF of the L25N512 simulation at z = 3 using different
RT methods and without RR. The red dashed curve shows the ratio between the Hi
CDDF given in Altay et al. (2011) and our results. This comparison shows that despite the
overall agreement between our results and Altay et al. (2011), there are some important
differences.
(see the left panel of Figure 2.13).
Appendix D: Approximated processes
D1: Multifrequency effects
As discussed in §2.2.3, in our RT simulations we have treated the multifrequency
nature of the UVB radiation in the gray approximation (see equation 2.4). This
approach does not capture the spectral hardening which is a consequence of
variation of the absorption cross-sections with frequencies. We tested the impact of spectral hardening on the Hi fractions by repeating the L06N128 simulation at z = 3 with the UVB using 3 frequency bins. We used energy intervals
[13.6 − 16.6], [16.6 − 24.6] and [24.6 − 54.4] eV and assumed that photons with
higher frequencies are absorbed by He. The result is illustrated in the top section
of the left panel in Figure 2.15, by plotting the ratio between the resulting hydrogen neutral fraction, η, and the same quantity in the original simulation that
uses the gray approximation. This comparison shows that the simulation that
uses multifrequency predicts hydrogen neutral fractions < 10% lower at low
densities (i.e., nH . 10−4 cm−3 ). This does not change the resulting f ( NHI , z)
noticeably at the column densities of interest here.
The spectral hardening captured in the simulation with 3 frequency bins is
illustrated in the right panel of Figure 2.15. This figure shows the fractional
contribution of different frequencies to the total UVB photoionization rate as a
function of density. The red solid curve shows the contribution of the bin with
56
Appendix D: Approximated processes
Figure 2.15: Spectral hardening and multifrequency treatment do not change the Hi distribution significantly. Left: The ratio between hydrogen neutral fractions, η, obtained
by using 3 frequency bins and by using the gray approximation is shown in the top-left
apnel. The ratio between hydrogen neutral fractions resulting from a simulation with 4
frequency bins and explicit He treatment and the same quantity using the gray approximation and without explicit He treatment is shown in the bottom-left panel. The vertical
lines with different lengths indicate the 15% − 85% percentiles. Right: The fractional
contribution of different frequency bins to the total UVB photoionization rate for the simulations with 3 frequency bins. All the RT calculations are performed using the L06N128
simulation at z = 3 in the presence of the UVB and assuming Case A recombination.
the lowest frequency and drops at the self-shielding density threshold. On the
other hand, the fractional contribution of the hardest frequency bin increases at
higher densities, as shown with the blue dashed curve. Despite the differences
in the fractional contributions to the total UVB photoionization rate, the absolute photoionization rates drop rapidly at densities higher that the self-shielding
threshold for all frequency bins.
D2: Helium treatment
A simplifying assumption frequently used in RT simulations which aim to calculate the distribution of neutral hydrogen is to ignore helium in the ionization processes (e.g., Faucher-Giguère et al., 2009; McQuinn & Switzer, 2010; Altay et al.,
2011). We adopted the same assumption in our RT calculations which implies
that we implicitly assumed the ionization state of neutral helium and its interaction with free electrons to be similar to the trends followed by neutral hydrogen. This has been shown to be a good assumption (Osterbrock & Ferland,
2006; McQuinn & Switzer, 2010; Friedrich et al., 2012). Nevertheless, we tested
the validity of our approximate helium treatment by repeating the L06N128 simulation at z = 3 with the UVB using 4 frequency bins and an explicit He treatment. The first three frequency bins are identical to the bins used in the previous section (i.e., [13.6 − 16.6], [16.6 − 24.6] and [24.6 − 54.4] eV) and the last
bin is chosen to cover higher frequencies which are capable of HeII ionization.
57
On the evolution of the Hi CDDF
We adopted a helium mass fraction of 25% and a Case A recombination rate.
The ratio between the resulting hydrogen neutral fraction and the same quantity
when a single frequency is used and helium is not treated explicitly is illustrated
in the bottom-left panel of Figure 2.15. The hydrogen neutral fractions are very
close in the two simulations. However, the simulation with multifrequency and
explicit He treatment results in hydrogen neutral fractions that are < 10% higher
at low densities (i.e., nH . 10−4 cm−3 ). This difference is barely noticeable in the
comparison between the two Hi CDDFs (not shown).
58
3
The impact of local stellar
radiation on the Hi column
density distribution
It is often assumed that local sources of ionizing radiation have little impact on
the distribution of neutral hydrogen in the post-reionization Universe. While
this is a good assumption for the intergalactic medium, analytic arguments suggest that local sources may typically be more important than the meta-galactic
background radiation for high column density absorbers (NHI > 1017 cm−2 ). We
post-process cosmological, hydrodynamical simulations with accurate radiation
transport to investigate the impact of local stellar sources on the column density
distribution function of neutral hydrogen. We demonstrate that the limited numerical resolution and the simplified treatment of the interstellar medium (ISM)
that are typical of the current generation of cosmological simulations provide
significant challenges, but that many of the problems can be overcome by taking two steps. First, using ISM particles rather than stellar particles as sources
results in a much better sampling of the source distribution, effectively mimicking higher-resolution simulations. Second, by rescaling the source luminosities
so that the amount of radiation escaping into the intergalactic medium agrees
with that required to produce the observed background radiation, many of the
results become insensitive to errors in the predicted fraction of the radiation that
escapes the immediate vicinity of the sources. By adopting this strategy and by
drastically varying the assumptions about the structure of the unresolved ISM,
we conclude that we can robustly estimate the effect of local sources for column
densities NHI ≪ 1021 cm−2 . However, neither the escape fraction of ionizing
radiation nor the effect of local sources on the abundance of NHI & 1021 cm−2
systems can be predicted with confidence. We find that local stellar radiation
is unimportant for NHI ≪ 1017 cm−2 , but that it can affect Lyman Limit and
Damped Lyα systems. For 1018 < NHI < 1021 cm−2 the impact of local sources
increases with redshift. At z = 5 the abundance of absorbers is substantially
reduced for NHI ≫ 1017 cm−2 , but at z = 0 the effect only becomes significant
for NHI & 1021 cm−2 .
The impact of local stellar radiation on the Hi CDDF
Alireza Rahmati, Joop Schaye, Andreas H. Pawlik, Milan Raičevic̀
Monthly Notices of the Royal Astronomical Society
Volume 431, Issue 3, pp. 2261-2277 (2013)
60
Introduction
3.1 Introduction
After the reionization of the Universe at z & 6, hydrogen residing in the intergalactic medium (IGM) is kept highly ionized primarily by the meta-galactic
UV background (UVB). The UVB is the integrated radiation that has been able
to escape from sources into the IGM. Because the mean free path of ionizing
photons is large compared to the scale at which the sources of ionizing radiation
cluster, the UVB is expected to be close to uniform in the IGM. However, close
to galaxies, the radiation field is dominated by local sources and hence more
inhomogeneous.
Observations of neutral hydrogen (Hi) in the Lyα forest mostly probe the
low-density IGM which is typically far from star-forming regions. The statistical
properties of the Lyα forest are therefore insensitive to the small-scale fluctuations in the UVB (e.g., Zuo, 1992; Croft, 2004). On the other hand, the neutral
hydrogen in Damped Lyα (DLA; i.e., NHI > 1020.3 cm−2 ) and Lyman Limit systems (LL; i.e., 1017.2 < NHI ≤ 1020.3 cm−2 ), which are thought to originate inside or close to galaxies, might be substantially affected by radiation from local
sources that are stronger than the ambient UVB (Gnedin, 2010). As a result, the
abundances of the high Hi column densities may also change significantly by
locally produced radiation.
Indeed, Schaye (2006) and Miralda-Escudé (2005) have used analytic arguments to show that the impact of local radiation may be substantial for LL
and (sub-)DLA systems, but should generally be very small at lower column
densities. However, relatively little has been done to go beyond idealized analytic arguments and to simulate the effect of local radiation by taking into
account the inhomogeneous distribution of sources and gas in and around
galaxies. One of the main reasons for this is the computational expense
of radiative transfer (RT) calculations in simulations with large numbers of
sources. In addition, high resolution is required to capture the distribution
of gas on small scales accurately. Because of these difficulties, most simulations of the cosmological Hi distribution have ignored the impact of local
radiation and focused only on the effect of the UVB (e.g., Katz et al., 1996;
Gardner et al., 1997; Haehnelt et al., 1998; Cen et al., 2003; Nagamine et al., 2004;
Razoumov et al., 2006; Pontzen et al., 2008; Altay et al., 2011; McQuinn et al.,
2011; van de Voort et al., 2012; Rahmati et al., 2013; Bird et al., 2013). A few studies have taken into account local stellar radiation but their results are inconclusive: while Nagamine et al. (2010) and Yajima et al. (2012) found that local stellar
radiation has a negligible impact on the distribution of HI, Fumagalli et al. (2011)
found that the Hi column density distribution above the Lyman limit is reduced
by ∼ 0.5 dex due to local stellar radiation.
In this chapter, we investigate the impact of local stellar radiation on the Hi
distribution by combining cosmological hydrodynamical simulations with accurate RT. We find that the inclusion of local sources can dramatically change
the predicted abundance of strong DLA systems and, depending on redshift,
61
The impact of local stellar radiation on the Hi CDDF
also those of LL and weak DLA systems. On the other hand, lower Hi column
densities are hardly affected. We also show that resolution effects have a major
impact. For instance, the resolution accessible to current cosmological simulations is insufficient to resolve the interstellar medium (ISM) on the scales relevant
for the propagation of ionizing photons. The limited resolution also affects the
source distribution which can change the resulting Hi distribution, especially in
low-mass galaxies. On top of that, assumptions about the structure of the unresolved multiphase ISM significantly affect the escape of stellar radiation into the
IGM. Therefore, any attempt to use cosmological simulations to investigate the
impact of local stellar radiation on the Hi distribution may suffer from serious
numerical artifacts.
Some of these difficulties can be circumvented by tuning the luminosities of
the sources such that the escaped radiation can account for the observed UVB.
Then the interaction between the radiation that reaches the IGM and the intervening gas can be studied on scales that are properly resolved in the simulations.
We adopt this procedure to generate the observed UVB for various ISM models
but find that our fiducial simulation reproduces the observed UVB without any
tuning.
Among the known sources of radiation, quasars and massive stars are the
most efficient producers of hydrogen ionizing photons and are therefore thought
to be the main contributors to the UVB. Star-forming galaxies however are
thought to be the dominant producers of the UVB at z & 3 (e.g., Haehnelt et al.,
2001; Bolton et al., 2005; Faucher-Giguère et al., 2008a). Therefore, we only account for local radiation that is produced by star formation in our simulations.
Moreover, we assume that the ionizing emissivity of baryons strictly follows the
star formation rate. We use star-forming gas particles rather than young stellar
particles as ionizing sources. Since the gas consumption time scale in the ISM
is much larger than the lifetime of massive stars (∼ 109 yr vs. ∼ 107 yr), there
are many more star-forming gas particles than there are stellar particles young
enough to efficiently emit ionizing radiation. Therefore, using star-forming gas
particles as ionizing sources allows us to sample the source distribution better
and hence to reduce the impact of the limited resolution of cosmological simulations.
The structure of the chapter is as follows. In §3.2 we use the observed relation between the star formation rate and gas surface densities to provide an
analytic estimate for the photoionization rate that is produced by young stars
in a uniform and optically thick ISM that is in good agreement with our simulation results. In §3.3 we discuss the details of our numerical simulations and
RT calculations. The simulation results are presented in §3.4. We show how the
local stellar radiation can generate the observed UVB. We also discuss the effect
of unresolved ISM on the escape fraction of ionizing radiation and we investigate the impact of local stellar radiation on the Hi column density distribution.
Finally, we conclude in §3.5.
62
Photoionization rate in star-forming regions
3.2 Photoionization rate in star-forming regions
A strong correlation between gas surface density and star formation rate has
been observed in low-redshift galaxies (e.g., Kennicutt, 1998; Bigiel et al., 2008).
In principle, such a relation can be combined with simplified assumptions to
derive a typical photoionization rate that is expected from stellar radiation. In
this section, we use this approach to estimate the average ionization rate that
is produced by star formation as a function of gas (surface) density on galactic
scales.
Assuming a Chabrier (2003) initial mass function (IMF), the observed
Kennicutt-Schmidt law provides a relation between gas surface density, Σgas ,
and star formation rate surface density, Σ̇⋆ , on kilo-parsec scales (Kennicutt,
1998):
1.4
Σgas
−4
−1
−2
.
(3.1)
Σ̇⋆ ≈ 1.5 × 10 M⊙ yr kpc
1M⊙ pc−2
Equation (3.1) can be used to derive a relation between ionizing emissivity (per
unit area), Σ̇γ , and the gas surface density. As we will discuss in §3.3.4, stellar population synthesis models indicate that the typical number of ionizing
photons produced per unit time by a constant star formation rate is
SFR
53 −1
Q̇γ ∼ 2 × 10 s
.
(3.2)
1 M⊙ yr−1
Furthermore, as both models and observations suggest, the escape fraction of ionizing photons from galaxies is ≪ 1 (e.g., Shapley et al., 2006;
Gnedin et al., 2008; Vanzella et al., 2010; Yajima et al., 2012; Paardekooper et al.,
2011; Kim et al., 2012). This allows us to assume that most of the ionizing
photons that are produced by star-forming gas are absorbed on scales . kpc.
Therefore, the hydrogen photoionization rate, on galactic scales, can be computed:
Q̇γ Σ̇⋆
Σ̇γ
=
,
(3.3)
Γ⋆ =
NH
NH
where NH is the hydrogen column density which can be obtained from the gas
surface density:
Σgas
X
19
−2
,
(3.4)
NH ≈ 9.4 × 10 cm
0.75
1M⊙ pc−2
where X is the hydrogen mass fraction. After assuming X = 0.75 and substituting equations (3.1), (3.2) and (3.4) in equation (3.3), one gets
0.4
NH
−14 −1
.
(3.5)
Γ⋆ ∼ 8.5 × 10
s
1021 cm−2
If the scale height of the disk is similar to the local Jeans scale, the hydrogen
column density can be computed as a function of the hydrogen number density
63
The impact of local stellar radiation on the Hi CDDF
(Schaye, 2001, 2004; Schaye & Dalla Vecchia, 2008)
NH ∼ 2.8 × 1021 cm−2
nH 1/2 1/2
T4
1 cm−3
fg
f th
1/2
,
(3.6)
where T4 ≡ T/104 K, f g is the total mass fraction in gas and f th is the fraction
of the pressure that is thermal (i.e., Pth = f th PTOT). In addition, for deriving
equation (3.6) we have adopted an adiabatic index, γ = 5/3, mean particle mass,
µ = 1.23 mH . After eliminating NH between equation (3.5) and equation (3.6),
the photoionization rate can be written as a function of the hydrogen number
density
0.2 f g 0.2
n
−13 −1
H
T40.2 .
(3.7)
Γ⋆ ∼ 1.3 × 10
s
f th
1 cm−3
Based on equation (3.7), the photoionization rate produced by star formation
is only weakly sensitive to the temperature, total gas fraction, f g , and the fraction
of the pressure that is thermal, f th . The photoionization rate in equation (3.7) is
also very weakly dependent on the gas density. As we will discuss in §3.4.1, our
simulations show the same trend and are in excellent agreement with this analytic estimate. However, one should note that due to the simplified assumptions
we adopted to derive equation (3.7), it provides only an order-of-magnitude estimate for the typical photoionization rate that is produced by young stars in
the ISM, on kilo-parsec scales. This relation does not capture the inhomogeneity
of the ISM that may result in large fluctuations in the radiation field on smaller
scales. We note that the relation we found here between the gas density and stellar photoionization rate is a direct consequence of the underlying star formation
law and is independent of the total amount of star formation in a galaxy.
3.3 Simulation techniques
In this section we describe different parts of our simulations. We start with
discussing the details of the hydrodynamical simulations that are post-processed
with RT calculations. Then we explain our RT that includes the radiation from
local stellar radiation, the UVB and recombination radiation.
3.3.1 Hydrodynamical simulations
We use a set of cosmological simulations that are performed using a modified and extended version of the smoothed particle hydrodynamics (SPH) code
GADGET-3 (last described in Springel, 2005). The physical processes that are
included in the simulations are identical to what has been used in the reference simulation of the Overwhelmingly Large Simulations (OWLS) described in
Schaye et al. (2010). Briefly, we use a subgrid pressure-dependent star formation
prescription of Schaye & Dalla Vecchia (2008) which reproduces the observed
64
Simulation techniques
Kennicutt-Schmidt law. We use the chemodynamics model of Wiersma et al.
(2009b) which follows the abundances of eleven elements assuming a Chabrier
(2003) IMF. These abundances are used for calculating radiative cooling/heating
rates, element-by-element and in the presence of the uniform cosmic microwave
background and the Haardt & Madau (2001) UVB model (Wiersma et al., 2009a).
We note that the Haardt & Madau (2001) UVB model has been shown to be consistent with observations of Hi (Altay et al., 2011; Rahmati et al., 2013) and metal
absorption lines (Aguirre et al., 2008). We model galactic winds driven by star
formation using a kinetic feedback recipe that assumes 40% of the kinetic energy generated by Type II SNe is injected as outflows with initial velocity of
600 kms−1 and with a mass loading parameter η = 2 (Dalla Vecchia & Schaye,
2008).
We adopt cosmological parameters consistent with the most recent WMAP
year 7 results: {Ωm = 0.272, Ωb = 0.0455, ΩΛ = 0.728, σ8 = 0.81, ns =
0.967, h = 0.704} (Komatsu et al., 2011). Our reference simulation has a periodic
box of L = 6.25 comoving h−1 Mpc and contains 1283 dark matter particles with
mass 6.3 × 106 h−1 M⊙ and an equal number of baryons with initial mass 1.4 ×
106 h−1 M⊙ . The Plummer equivalent gravitational softening length is set to
ǫcom = 1.95 h−1 kpc and is limited to a minimum physical scale of ǫprop =
0.5 h−1 kpc. We also use a simulation with a box size identical to our reference
simulation but with 8 (2) times better mass (spatial) resolution to assess the effect
of resolution on our findings (see Appendix B2).
In our hydrodynamical simulations, ISM gas particles (which all have densities nH > 0.1 cm−3 ) follow a polytropic equation of state that defines their temperatures. These temperatures are not physical and only measure the imposed
pressure (Schaye & Dalla Vecchia, 2008). Therefore, when calculating recombination and collisional ionization rates, we set the temperature of ISM particles
to TISM = 104 K which is the typical temperature of the warm-neutral phase
of the ISM. Furthermore, we simplify our RT calculations by assuming that helium and hydrogen absorb the same amount of ionizing radiation per unit mass
and by ignoring dust absorption and the possibility that some hydrogen may be
molecular (see Rahmati et al., 2013 for more discussion).
3.3.2 Radiative transfer
For the RT calculations we use TRAPHIC (see Pawlik & Schaye, 2008, 2011;
Rahmati et al., 2013). TRAPHIC is an explicitly photon-conserving RT method
designed to exploit the full spatial resolution of SPH simulations by transporting radiation directly on the irregular distribution of SPH particles.
The RT calculation in TRAPHIC starts with source particles emitting photon
packets to their neighbors. This is done using a set of NEC tessellating emission
cones, each subtending a solid angle of 4π/NEC . The propagation directions of
the photon packets are initially parallel to the central axes of the emission cones.
In order to improve the angular sampling of the RT, the orientations of these
65
The impact of local stellar radiation on the Hi CDDF
emission cones are randomly rotated between emission time steps. We adopt
NEC = 8 for computational efficiency. However, we note that our results are
insensitive to the precise value of NEC , thanks to the random rotations.
After emission, photon packets travel along their propagation direction from
one SPH particle to its neighbors. Only neighbors that are inside transmission
cones can receive photons. Transmission cones are defined as regular cones
with opening solid angle 4π/NTC , and are centered on the propagation direction. The parameter NTC sets the angular resolution of the RT and we adopt
NTC = 64 which produces converged results (see Appendix C1 of Rahmati et al.,
2013). To guarantee the independence of the RT from the distribution of SPH
particles, additional virtual particles (ViP) are introduced. This is done wherever
a transmission cone contains no neighboring SPH particle. ViPs do not affect the
underlying SPH simulation and are deleted after the photon packets are transferred.
Furthermore, arriving photon packets are merged into a discrete number of
reception cones, NRC = 8. This makes the computational cost of RT calculations
with TRAPHIC independent of the number of radiation sources. This feature is
particularly important for the purpose of the present work as it enables the RT
calculations in cosmological density fields with large numbers of sources.
Reception cones are also used for emission from gas particles (e.g., ionizing
radiation from star-forming gas particles, recombination radiation). This reduces
computational expenses while yielding accurate results. Photon packets are isotropically emitted into reception cones which are already in place at each SPH
particle. This obviates the need for constructing any additional emission cone
tessellations. Using this recipe, the emission of radiation by star-forming gas
particles is identical to that of the emission of diffuse radiation by recombining
gas particles, and is described in more detail in Raicevic et al. (in prep.).
Photon packets emitted by the three main radiation sources we consider in
our study (i.e., UVB, RR and stellar radiation) are channeled in separate frequency bins and are not merged with each other. This enables us to compute
the contribution of each component to the total photoionization rate. The total
amount of the absorbed radiation, summed over all frequency bins, determines
the ionization state of the absorbing SPH particles. The time-dependent differential equations that control the evolution of the ionization states of different
species (e.g., H, He) are solved using a sub-cycling scheme that allows us to
choose the RT time step independent of the photoionization and recombination
times.
In our RT calculations we use a set of numerical parameters identical to
that used in Rahmati et al. (2013). These parameters produce converged results. In addition to the parameters mentioned above, we use 48 neighbors
for
4
SPH particles, 5 neighbors for ViPs and an RT time step of ∆t = 1 Myr 1+
z .
Ionizing photons are propagated at the speed of light, inside the simulation
box with absorbing boundaries, until the equilibrium solution for the hydro-
66
Simulation techniques
Figure 3.1: The number of hydrogen ionizing photons produced for a constant star formation rate of 1 M⊙ yr−1 as a function of time since the onset of star formation. These
results are calculated using STARBURST99. Red, green, blue and purple curves indicate metallicities of Z/Z⊙ = 1.6, 6.6 × 10−1 3.3 × 10−1 , & 3.3 × 10−2 , respectively. Solid
and dashed curves show results for the Kroupa and Salpeter IMF, respectively. All the
curves with different metallicities and IMFs converge to similar equilibrium values at
Q̇ γ ∼ 2 × 1053 photons per second, after about ∼ 5 − 10 Myr.
gen neutral fractions is reached. To facilitate this process, the RT calculation
is starting from an initial neutral state, except for the gas at low densities (i.e.,
ngas < 1 × 10−3 cm−3 ) and high temperatures (i.e., T > 105 K) which is assumed
to be in equilibrium with the UVB photoionization rate and its collisional ionization rate. Typically, the average neutral fraction in the simulation box does not
evolve after 2-3 light-crossing times (the light-crossing time for the Lbox = 6.25
comoving h−1 Mpc is ≈ 7.2 Myr at z = 3).
We continue this section by briefly describing the implementation of the UVB,
diffuse recombination radiation and stellar radiation.
3.3.3 Ionizing background radiation and diffuse recombination
radiation
In principle, local sources of ionization inside the simulation box should be able
to generate the UVB. However, the box size of our simulations is smaller than the
mean free path of ionizing photons which makes the simulated volume too small
to generate the observed UVB intensity (see §3.4.3). In addition, a considerable
fraction of the UVB at z . 3 is produced by quasars which are not included in
67
The impact of local stellar radiation on the Hi CDDF
our simulations. Therefore, we impose an additional UVB in our simulation box.
The implementation of the UVB is identical to that of Rahmati et al. (2013).
The ionizing background radiation is simulated as plain-parallel radiation entering the simulation box from its sides and the injection rate of the UVB ionizing
photons is normalized to the desired photoionization rate in the absence of any
absorption (i.e., the so-called optically thin limit).
We set the effective photoionization rate and spectral shape of the UVB radiation at different redshifts based on the UVB model of Haardt & Madau (2001)
for quasars and galaxies. The same UVB model has been used to calculate heating/cooling in our hydrodynamical simulations and has been shown to be consistent with observations of Hi (Altay et al., 2011; Rahmati et al., 2013) and metal
absorption lines (Aguirre et al., 2008) at z ∼ 3. We adopt the gray approximation instead of an explicit multifrequency treatment for the UVB radiation. Our
tests show that a multifrequency treatment of the UVB radiation does not significantly change the resulting hydrogen neutral fractions (see Appendix D1 of
Rahmati et al., 2013)
In addition, hydrogen recombination radiation (RR) is simulated by making all SPH particles isotropic radiation sources with emissivities based on their
recombination rates. We do not account for the redshifting of the recombination photons and assume that they are monochromatic with energy 13.6 eV (see
Raicevic et al. in prep.).
3.3.4 Stellar ionizing radiation
The ionizing photon production rate of star-forming galaxies is dominated by
young and massive stars which have relatively short life times. In cosmological simulations with limited mass resolutions, the spatial distribution of newly
formed stellar particles (e.g., with ages less than a few tens of Myr) may sample
the locally imposed star formation rates relatively poorly. As we show in §3.4.2,
the distribution of star formation is better sampled by star-forming SPH particles
and this is particularly important for low and intermediate halo masses. For this
reason, we use star-forming particles as sources of stellar ionizing radiation.
For a constant star formation rate, the production rate of ionizing photons
reaches an equilibrium value within ∼ 5 − 10 Myr (i.e., the typical life-time of
massive stars). This equilibrium photon production rate per unit star formation
rate can be calculated using stellar population synthesis models. We used STARBURST99 (Leitherer et al., 1999) to calculate this emissivity for the Kroupa (2001)
and Salpeter (1955) IMF and a metallicity consistent with the typical metallicities of star-forming particles in our simulations (the median metallicity of starforming particles in our simulations are within the range 10−1 . Z/Z⊙ . 1 at
redshifts 0 - 5). We found that for a constant star formation rate of 1 M⊙ yr−1
and after ∼ 10 Myr, the photon production rate of ionizing photons converges
to Q̇γ ∼ 2 × 1053 s−1 , which is the value we used in equation (3.2). As shown in
68
Simulation techniques
69
Figure 3.2: The Hi column density distribution of a Milky Way-like galaxy in the reference simulation at z = 3 (top) and z = 0 (bottom). The
left panels show the total hydrogen column density distribution. The middle panels show the Hi column densities in the presence of collisional
ionization, photoionization from the UVB and recombination radiation. In the right panel, the Hi distribution is shown after photoionization
from local stellar radiation is added to the other sources of ionization. The images are 100 h−1 kpc (proper) on a side. Comparison between
the right and middle panels shows that local stellar radiation substantially changes the Hi column density distribution at NHI ∼ 1021 cm−2 .
The impact of local stellar radiation on the Hi CDDF
Figure 3.1, the ionizing photon production rate varies only by . ±0.2 dex if the
metallicity changes by ∼ ±1 dex. Also, reasonable variations in the IMF (e.g.,
the Chabrier IMF) do not significantly change our adopted value for the photon
production rate. For the spectral shape of the stellar radiation, we adopt a blackbody spectrum with temperature Tbb = 5 × 104 K which is consistent with the
spectrum of massive young stars.
3.4 Results and discussion
In this section, we report our findings based on RT simulations that include the
UVB, recombination radiation and local stellar radiation. The RT calculations
are performed by post-processing a hydrodynamical simulation with 1283 SPH
particles in a 6.25 comoving h−1 Mpc box and at redshifts z = 0 − 5. In §3.4.1,
we compare different photoionizing components and assess the impact of local
stellar radiation on the Hi distribution. In §3.4.2 we illustrate the importance
of using star-forming SPH particles instead of using young stellar particles as
ionizing sources. In §3.4.3 we show that in our simulations, the predicted intensity of stellar radiation in the IGM is consistent with the observed intensity
of the UVB and calculate the implied average escape fraction. In §3.4.4 we show
that assumptions about the unresolved properties of the ISM are very important
and we discuss the impact of local stellar radiation on the Hi column density
distribution.
3.4.1 The role of local stellar radiation in hydrogen ionization
As we showed in Rahmati et al. (2013), the UVB radiation and collisional ionization are the dominant sources of ionization at low densities where the gas
is shock heated and optically thin. However, self-shielding prevents the UVB
radiation from penetrating high-density gas. In addition, much of the ISM has
temperatures that are too low for collisional ionization to be efficient. Therefore, because UVB photoionization and collisional ionization are both inefficient
in those regions, the ionizing radiation from young stars becomes the primary
source of ionization.
Figure 3.2 shows the distribution of neutral hydrogen in and around a galaxy
in our reference simulation at redshifts z = 3 (top row) and 0 (bottom row) and
illustrates that the distribution of Hi in galaxies may change significantly as a
result of local stellar radiation. The total mass of the halo hosting this galaxy
at z = 3 and 0 is M200 = 2 × 1011M⊙ and 1.1 × 1012M⊙ , respectively. Comparing the panels in the left and middle columns, we see that while collisional
ionization and photoionization by the UVB ionize the low-density gas around
the galaxy, they do not change the high Hi column densities in the inner regions. Comparing the yellow regions in the right and middle panels of Figure
3.2 demonstrates that local stellar radiation significantly changes the distribu70
Results and discussion
tion of the Hi at NHI ∼ 1021 cm−2 . In the left panel of Figure 3.3 the contributions of the photoionization rates from different radiation components are
plotted against the total hydrogen number density at z = 3. The purple solid
line shows the total photoionization rate. The blue dashed, green dotted and red
dot-dashed lines show respectively the contribution of the UVB, recombination
radiation (RR) and local stellar radiation (LSR). The resulting hydrogen neutral
fraction as a function of gas density is shown by the purple solid curve in the
right panel of Figure 3.3 which illustrates the significant impact of LSR at high
densities. We note that there is a sharp feature in the hydrogen neutral fraction
at 10−1 < nH < 10−2 cm−3 in the presence of local stellar radiation. This feature
corresponds to a sharp drop-off in the photoionization rate from local stellar
radiation at the same densities (see the red dot-dashed curve in the left panel of
Figure 3.3). The left panel of Figure 3.4 shows photoionization rate profiles in
spherical shells around haloes with 10.5 < log10 M200 < 11 M⊙ at z = 3. The
right panel of Figure 3.4 shows the resulting neutral hydrogen fraction profile
(purple solid curve) which is compared with the simulation that does not include local stellar radiation (green dashed curve). We note that the trends in
the photoionization rate and hydrogen neutral fraction profiles do not strongly
depend on the chosen mass bin as long as log10 M200 & 9.5 M⊙ .
As Figures 3.3 and 3.4 show, at low densities (i.e., at large distances from the
centers of the halos) the gas is highly ionized by a combination of the UVB and
collisional ionization and this optically thin gas does not absorb a significant
fraction of the ambient ionizing radiation. Consequently, at nH . 10−4 cm−3 ,
which corresponds to typical distances R & 1 comoving Mpc from the centers
of the haloes, the UVB photoionization rate does not change with density. By
increasing the density, or equivalently by decreasing the distance to the center
of the haloes, the optical depth increases and eventually the gas becomes selfshielded against the UVB. This causes a sharp drop in the UVB photoionization
rate at the self-shielding density (see the blue dashed curves in the left panels
of Figures 3.3 and 3.4). As shown in Rahmati et al. (2013), the UVB photoionization rate shows a similar behavior in the absence of local stellar radiation.
However, local stellar radiation increases the photoionization rate at high and
intermediate densities. This decreases the hydrogen neutral fractions around
the self-shielding densities, allowing the UVB radiation to penetrate to higher
densities. Consequently, the local stellar radiation increases the effective selfshielding threshold against the UVB radiation slightly (by ∼ 0.1 dex) compared
to the simulation without the radiation from local stars (not shown).
As shown by the red dot-dashed curve in the left panel of Figure 3.3, at
densities 0.1 . nH . 1 cm−3 the median photoionization rate due to local stellar
radiation increases with decreasing density. The main reason for this is the superposition of radiation from multiple sources (i.e., star-forming SPH particles)
as the mean free path of ionizing photons increases with decreasing density (see
Figure 3.11). This is also seen in the left panel of Figure 3.4 as increasing stellar photoionization rate with increasing the distance from the center of halos
71
72
The impact of local stellar radiation on the Hi CDDF
Figure 3.3: Left: Photoionization rate as a function of density due to different radiation components in the reference simulation at z = 3.
The purple solid curve shows the total photoionization rate. The blue dashed, green dotted and red dot-dashed curves show respectively
the photoionization rates due to the UVB, diffuse recombination radiation (RR) and local stellar radiation (LSR). Right: The hydrogen
neutral fraction as a function of density for the same simulation is shown with the purple solid line. For comparison, the results for a
simulation without local stellar radiation (green dashed curve) and a simulation with the UVB radiation in the optically thin limit (i.e., no
absorption; brown dotted curve) are also shown. The curves show the medians and the shaded areas around them indicate the 15% − 85%
percentiles. Hi column densities corresponding to each density in the presence of all ionization sources are shown along the top x-axis.
The photoionization due to local stellar radiation exceeds the UVB photoionization rate at high densities and compensates the effect of
self-shielding. This produces a hydrogen neutral fraction profile that is very similar to what is expected from the UVB in the optically thin
limit.
Results and discussion
73
Figure 3.4: Left: Median photoionization rate profiles (comoving) due to different radiation components for haloes with 10.5 < log10 M200 <
11 M⊙ at z = 3. Blue solid, green dotted and red dot-dashed curves show the photoionization rates of the UVB, recombination radiation
(RR) and local stellar radiation (LSR). Right: Median hydrogen neutral fraction profiles for the same halos with local stellar radiation
(purple solid curve), without local stellar radiation (green dashed curve) and with only the UVB radiation in the optically thin limit (dotted
brown). In both panels, the vertical dotted line indicates the median R200 radius of the halos in the chosen mass bin. The shaded areas
around the medians indicate the 15% − 85% percentiles. The top axis in each panel shows the median density at a given comoving distance
from the center of the haloes. The photoionization due to local stellar radiation exceeds the UVB photoionization rate close to galaxies and
compensates the effect of self-shielding. This produces a hydrogen neutral fraction profile that is very similar to what is expected from the
UVB in the optically thin limit.
The impact of local stellar radiation on the Hi CDDF
for R < 1 comoving kpc. On the other hand, at densities lower than the star
formation threshold (i.e., nH < 0.1 cm−3 ), the gas is typically at larger distances
from the star-forming regions. Therefore, the photoionization rate of local stellar
radiation drops rapidly with decreasing density. The star formation rate density
averaged on larger and larger scales becomes increasingly more uniform. This
causes the photoionization from galaxies that are emitting ionizing radiation
to produce a more uniform photoionization rate at the lowest densities. Note
that at the highest densities, the photoionization rate from local stellar radiation
agrees well with the analytic estimate presented in §3.2.
In the absence of local stellar radiation, recombination radiation photoionization rate peaks at around the self-shielding density (Rahmati et al., 2013). The
reason for this is that the production rate of ionizing photons by recombination
radiation is proportional to the density of the ionized gas. At densities below
the self-shielding threshold, hydrogen is highly ionized and recombination rate
is proportional to the total hydrogen density. At much higher densities, the
gas is nearly neutral and little recombination radiation is generated. As the
left panel in Figure 3.3 shows, at high densities the situation changes dramatically if we include local stellar radiation. Since the gas at high densities (e.g.,
nH ≫ 10−2 cm−3 ) is optically thick, recombination radiation ionizing photons
produced at these densities are absorbed locally. In equilibrium, recombination
radiation photoionization rate at densities above the self-shielding is therefore
a constant fraction of the total ionization rate1. Below the self-shielding density, the recombination radiation photoionization rate decreases with decreasing
density and asymptotes to a background rate. However, in reality recombination
photons cannot travel to large cosmological distances without being redshifted
to frequencies below the Lyman edge. Therefore, our neglect of the cosmological
redshifting of recombination radiation leads us to overestimate the photoionization rate due to recombination radiation on large scales. On the other hand,
because of the small size of our simulation box, the total photoionization rate
that is produced by recombination radiation remains negligible compared to the
UVB photoionization rate and neglecting the redshifting of recombination radiation is not expected to affect our results.
The purple solid curve in the right panel of Figure 3.3 shows the hydrogen
neutral fractions in the presence of the UVB, recombination radiation and local
stellar radiation. For comparison, the hydrogen neutral fractions in the absence
of local stellar radiation, and for the optically thin gas that is photoionized only
by the UVB, are also shown (with the green dashed and brown dotted curves
respectively). Hydrogen at densities nH & 10−1 cm−3 is self-shielded and mostly
neutral if the UVB and recombination radiation are the only sources of photoion1 This can be explained noting that in equilibrium, recombination and ionization rates are equal.
Depending on temperature, ≈ 40% of recombination photons are ionizing photons which are absorbed on the spot in optically thick gas (e.g., Osterbrock & Ferland, 2006). Therefore, the photoionization rate produced by recombination radiation is also ≈ 40% of the total ionization rate in dense
and optically thick gas with T ∼ 104 K.
74
Results and discussion
Figure 3.5: Star-formation activity of haloes in the reference simulation (6.25 Mpc , 1283 )
at z = 3. In the left panel the blue crosses show the total instantaneous star formation
rate for a given halo computed using the gas particles and the red circles show the star
formation rate calculated by dividing the total stellar mass formed during the last 20 Myr
by 20 Myr. Haloes with zero star formation rates are shown in the bottom of the plot.
For massive haloes the two measures agree but as a result of limited mass resolution and
the stochastic nature of the star formation algorithm, they start to differ substantially for
haloes with M200 . 1010 M⊙ . A large fraction of low-mass haloes does not contain any
young stellar particles and the median star formation rate calculated using young stellar
particles drops to zero. The right panel shows the number of ionizing sources for different
halos in the same simulation. While the blue crosses show the number of star-forming
gas particles in each halo, red circles indicate the number of young stellar particles (i.e.,
younger than 20 Myr). In both panels the horizontal dotted lines correspond to a single
stellar particle and the blue dashed and red dotted curves show respectively the medians
for star-forming gas and young stellar particles.
ization. However, local stellar radiation significantly ionizes the gas at intermediate and high densities. As mentioned in §3.2, the typical photoionization rate
that is produced by star-forming gas is ΓSF ∼ 10−13 s−1 , which is comparable
to the photoionization rate of the unattenuated UVB at z ∼ 0. Consequently,
the hydrogen neutral fractions in the presence of local stellar radiation are much
closer to the optically thin case. It is also worth noting that the scatter around
the median hydrogen neutral fractions is largest for intermediate densities (i.e.,
10−3 . nH . 1 cm−3 . This is closely related to the large scatter in the photoionization rate produced by local stellar radiation (see the left panel in Figure 3.3).
At these densities, the large scatter in the distances to the nearest sources and
RT effects like shadowing produce a large range of photoionization rates. For
the gas at very high and very low densities on the other hand, the scatter becomes smaller because the relative distribution of sources with respect to the
absorbing gas becomes more uniform. The trends discussed so far are qualitatively similar at redshifts other than z = 3: Although the star formation rates
75
The impact of local stellar radiation on the Hi CDDF
evolve at high densities, the photoionization rate due to local stars is set by the
underlying star formation law (see §3.2) which does not change with time. The
peak of the stellar photoionization rate at nH ∼ 10−1 cm−3 , produced by the
superposition of multiple sources, exists at all redshifts but it moves to slightly
higher densities at lower redshifts. At densities immediately below the adopted
star formation threshold (i.e., nH . 0.1 cm−3 ), the photoionization rate produced
by local sources drops rapidly with decreasing density. The distribution of star
formation in the simulation box is almost uniform on large scales. Therefore, at
the lowest densities, local stellar radiation produces a photoionization rate which
is not changing strongly with density. However, at low redshifts (e.g., z . 1), the
star formation density decreases significantly and becomes highly non-uniform
on the scales probed by our small simulation box. Consequently, the resulting
photoionization rate due to young stars does not converge to a constant value at
low densities at these redshifts. For the same reason, at low densities the scatter
around the median stellar photoionization rate increases with decreasing redshift (not shown). As we will discuss in §3.4.3, if we correct for the small size of
our simulation box, the asymptotic photoionization rate due to stellar radiation
that has reached the IGM, is consistent with the intensity of the observed UVB
at the same redshift.
3.4.2 Star-forming particles versus stellar particles
In cosmological simulations, the integrated instantaneous star formation rate
of the simulation box closely matches the total amount of mass converted into
stellar particles. However, due to limited resolution, the spatial distribution of
young stellar particles (e.g., those with ages ∼ 10 Myr) may not sample the
spatial distribution of star formation in individual haloes very well.
This issue is illustrated in the left panel of Figure 3.5, where the blue crosses
show the total instantaneous star formation rates inside galaxies in our reference
simulation at z = 3. These star formation rates are calculated by the summation
of the star formation rates of all SPH particles in a given galaxy. The star formation rates can also be estimated by measuring the average rate by which stellar
particles are formed in a given galaxy over some small time interval. The star
formation rate calculated using this method (i.e., measuring the rate by which
stellar particles are formed during the last 20 Myr), is indicted in the left panel of
Figure 3.5 by the red circles. For massive galaxies, with high star formation rates,
the formation rate of stellar particles agrees reasonably well with the star formation rates computed from the gas distribution. However, most haloes with . 103
SPH particles (i.e., M200 . 1010 M⊙ ) do not contain any young stellar particles
despite having non-zero instantaneous star formation rates. Consequently, for
the few low-mass galaxies that by chance contain one or more young stellar
particles, the implied star formation rates are much higher than the instantaneous rate that corresponds to the gas distribution. Moreover, as the right panel
of Figure 3.5 shows, the number of star-forming particles in a given simulated
76
77
Results and discussion
Figure 3.6: Median photoionization rate profiles (comoving) for local stellar radiation emitted by star-forming SPH particles (SF-SPH; blue
solid curves) and young (< 20 Myr) stellar particles (Y Stars; red dotted curves) at z = 3. The left panel shows the photoionization rate
profiles for a halo with M200 = 2 × 1011 M⊙ while the right panel shows the same profiles for haloes with 10 < log10 M200 < 10.5 M⊙ . In
both panels the vertical dotted line indicates the median R200 radius of the halos in the illustrated mass bin. The shaded areas around the
medians indicate the 15% − 85% percentiles. The top axis in each panel shows the median density at a given comoving distance from the
centers of the haloes. While the photoionization rate profiles produced by star-forming gas particles and young stellar particles are similar
for massive haloes (left), they are substantially different for haloes with M200 . 1010 M⊙ (right).
The impact of local stellar radiation on the Hi CDDF
galaxy is ∼ 102 times larger than that of young stellar particles. This ratio could
be understood by noting that the observed star formation law implies a gas consumption time scale in the ISM that is ∼ 102 times longer than the life times
of massive stars. Hence, if star formation is implemented by the stochastic conversion of gas particles into stellar particles, as is the case here, the number
of star-forming (i.e., ISM) particles is expected to be ∼ 102 times larger than
the number of young stellar particles. This ratio will be somewhat smaller in
starbursts or if gas particles are allowed to spawn multiple stellar particles, but
under realistic conditions, star formation will be sampled substantially better by
gas particles than by young stellar particles.
As a result of the above mentioned sampling effects, using stellar particles
as ionizing sources (as was done in all previous work, e.g., Nagamine et al.,
2010; Fumagalli et al., 2011; Yajima et al., 2012) would underestimate the impact
of local stellar radiation on the Hi distribution for a large fraction of simulated
galaxies. We therefore use star-forming SPH particles instead of young stellar
particles as local sources of radiation.
Figure 3.6 illustrates the difference between the photoionization rate profiles
produced by star-forming SPH particles (blue solid curves) and young stellar
particles (red dotted curves) at z = 3. The left panel of Figure 3.6 shows this
for a halo with M200 = 2 × 1011 M⊙ (see the top panels of Figure 3.2), while the
right panel illustrates the results for haloes with 10 < log10 M200 < 10.5 M⊙ . For
this comparison we imposed the same total number of emitted photons in the
simulation box in both cases. This was done by setting the photon production
rates of individual gas particles proportional to their star formation rates (see
equation 3.2) and setting the photon production rate of individual young stellar
particles proportional to their masses.
For the massive halo, simulating local stellar radiation using SPH particles
results in photoionization rate profiles similar to those produced by using stellar
particles. This similarity is mainly due to the agreement between the total star
formation rate and the rate by which gas is converted into young stellar particles
(see the left panel of Figure 3.5). However, this galaxy has ∼ 50 times fewer
young stellar particles than star-forming SPH particles (see the right panel of
Figure 3.5) and as the shaded areas in the left panel of Figure 3.6 show, the
scatter in the photoionization rate profile is much larger when stellar particles
are used.
For haloes with slightly lower masses, shown in the right panel of Figure
3.6, the rates by which gas is converted into young stellar particles are similar
to the total star formation rates (see the left panel of Figure 3.5). Despite this
agreement, the median photoionization rates for these two cases differ dramatically, being much higher when star-forming gas particles are used as sources.
We conclude that using stellar particles as sources results in the underestimation of the ionization impact of local stellar radiation on most of the galaxies in
the simulation. The intensity of the UVB produced by the simulation that uses
young stellar particles as sources does not agree with the observed UVB intens78
Results and discussion
Figure 3.7: Left: The simulated UVB photoionization rate produced by stellar radiation in
our reference simulation is shown with filled circles for different redshifts. The observed
mean free paths of ionizing photons have been used to correct for the small size of the
simulation box (see text). The error bars represent the 1 − σ errors in these mean free
paths. The dashed curve shows the Haardt & Madau (2001) UVB photoionization rates
that have been used in our simulations as the UVB. The observational measurement
of the UVB from the Lyα effective opacity by Bolton & Haehnelt (2007) is shown using
orange diamonds. Right: the average escape fraction of stellar ionizing radiation into the
IGM calculated based on equation 3.13. Our simulation reproduces the observed UVB
photoionization rates between z = 2 and 5. The implied average escape fractions are
10−2 < f esc < 10−1 between z = 2 and 5 and they decrease with decreasing redshift
below z = 4
ity, while using star-forming particles resolves this issue, after correcting for the
box size (see §3.4.3).
Although in this section we have shown that using star-forming particles
as sources of ionizing radiation helps to resolve the above mentioned sampling
issues in post-processing RT simulations, we note that doing the same may not
be a good solution for simulations with sufficient resolution to model the cold,
interstellar gas phase. Such simulations can capture the effects of the relative
motions of stars and gas, e.g., the effect of stars moving out of, or destroying,
their parent molecular clouds.
3.4.3 Stellar ionizing radiation, its escape fraction and the
buildup of the UVB
Observations show that the luminosity function of bright quasars drops sharply
at redshifts z & 2 (e.g., Hopkins et al., 2007) and models for the cosmic ionizing background indicate that star-forming galaxies dominate the production
of hydrogen ionizing photons at z & 3 (e.g., Haehnelt et al., 2001; Bolton et al.,
2005; Faucher-Giguère et al., 2008a). In the simulations, it should therefore be
possible to build up the background radiation from the radiation produced by
79
The impact of local stellar radiation on the Hi CDDF
Table 3.1: The comoving mean free path of hydrogen Lyman-limit photons, λmfp , at different redshifts. From left to right, columns respectively show redshift, λmfp in comoving
Mpc (cMpc) and the references from which the mean free path values are taken, i.e., B07:
Bolton & Haehnelt (2007), FG08: Faucher-Giguère et al. (2008a), P09: Prochaska et al.
(2009) and SC10: Songaila & Cowie (2010).
Redshift
z
z
z
z
=2
=3
=4
=5
λmfp
(cMpc)
909 ± 252
337 ± 170
170 ± 15.5
83.4 ± 21.6
Reference
FG08, SC10
FG08, SC10
FG08, P09
B07, SC10
star-forming galaxies at z & 3. In this section, we quantify the contribution of the
ionizing photons that are produced in stars to the build-up of the UVB radiation
in our simulations. In addition, we calculate the implied average escape fraction
that is required to generate the observed UVB.
3.4.3.1 Generating the UVB
Our simulation box is smaller than the typical mean free path of ionizing
photons, λmfp (see Table 3.1). Therefore, the IGM gas can receive stellar ionizing radiation from a region that is larger than the simulation box. We note
that using periodic boundaries to propagate the stellar ionizing photons is not a
good solution to resolve this issue. In addition to increasing the computational
expense, using periodic boundaries for RT would require us to account for the
cosmological redshifting of ionizing photons. Moreover, because of the small
size of our box, stellar photons traveling along paths that are nearly parallel
to a side of the box may never intersect an optically thick absorber, if periodic
boundaries are being used for RT. However, we can correct for the small size of
the simulation box by requiring consistency between the simulated UV intensities that local sources produce in the IGM and existing observations/models of
the UVB. In order to do that, we assume an isotropic and homogeneous universe
that is in photoionization equilibrium, and λmfp /(1 + z) ≪ c/H (z), so that we
can ignore evolution and redshifting during the travel time of the photons. Then,
the volume-weighted mean ionizing flux, F⋆ , from stars is given by:
F⋆ =
Z ∞
0
u⋆ e
−λ
r
mfp
dr = ū⋆ λmfp ,
(3.8)
where ū⋆ is the photon production rate per unit comoving volume. Moreover,
one can express the mean ionizing flux in equation (3.8) in terms of the volumeweighted mean ionizing flux produced by source that are inside the simulation
80
Results and discussion
box, F⋆in
F⋆in =
Z αL
box
0
u⋆ e
−λ
r
mfp
dr,
(3.9)
where α . 1 is a geometrical factor. In the absence of any absorption and for uniform and isotropic distributions of gas and sources, α is set by the average distance between two random points inside a cube. For a cube with unit length this
would yield α = 0.66 (Robbins, 1978). Based on the average distance between
the low-density gas (e.g., SPH particles with nH . 10−5 cm−3 ) and sources (i.e.,
star-forming particles) in our simulations, we find that the average value of α is
0.66 which varies mildly with redshift from α0 = 0.54 at z = 0 to α5 = 0.79 at
z = 5.
As Table 3.1 shows, our simulation box is much smaller than the mean free
path of ionizing photons. Therefore, we can use Lbox ≪ λmfp in equation (3.9)
and get:
F⋆in ≈ ū⋆ αLbox ,
(3.10)
Using equation (3.8) and (3.10), the total stellar UVB radiation flux can thus be
written as:
λmfp in
F .
(3.11)
F⋆ ≈
α Lbox ⋆
Therefore, the volume-weighted photoionization rate due to radiation produced
by stars in the simulation, Γin
⋆ , which is close to the median photoionization rate
in low-density gas, can be used to calculate the implied UVB photoionization
rate after correcting for the small box size of the simulations:
ΓUVB,⋆ ≈
λmfp in
Γ .
α Lbox ⋆
(3.12)
As mentioned earlier, our simulations show that the photoionization rate
from local stellar radiation approaches a density independent rate at low densities. We take this photoionization rate as Γin
⋆ . In addition, we use a compilation of
available Lyman-limit mean free path measurements at different redshifts from
the literature (i.e., from Bolton & Haehnelt, 2007; Faucher-Giguère et al., 2008a;
Prochaska et al., 2009; Songaila & Cowie, 2010; see Table 3.1 and O’Meara et al.,
2013 and Fumagalli et al., 2013 for new measurements). After converting the
Lyman-limit mean free paths into the typical mean free path of ionizing photons
with our assumed stellar spectrum2 , we use equation (3.12) to derive the implied
UVB photoionization rate.
Figure 3.7 shows the predicted contribution of stellar radiation to the UVB
(filled circles). The error bars reflect the quoted error in the mean free path
measurements. For comparison, the observational measurement of the UVB
from the Lyα effective opacity by Bolton & Haehnelt (2007) and the modeled
2 Because the effective hydrogen ionization cross section of stellar ionizing photons that we use in
this work, σ̄⋆ = 2.9 × 10−18 cm2 , their typical mean free paths are ∼ 2 times longer than the mean
free path of Lyman-limit photons which have hydrogen ionizing cross sections σ0 = 6.8 × 10−18 cm2 .
81
The impact of local stellar radiation on the Hi CDDF
Haardt & Madau (2001) UVB photoionization rates are also shown by orange
diamonds and green dashed curve, respectively. Both the observational measurements and modeled UVB intensities are in good agreement with our simulation results for z > 2. However, their UVB intensity is slightly higher than ours
at z = 2. The reason for this could be the absence of radiation from quasars in
our simulations.
3.4.3.2 Average escape fractions
The average star formation rate density in our simulations is in good agreement
with the observed cosmic star formation rate (Schaye et al., 2010). Therefore,
the good agreement between the UVB intensities in our simulation and the ones
inferred from the observations suggests that at z & 3 the average escape fractions
of stellar ionizing photons in our simulations are also reasonable. However, as
we will discuss in Appendix B2, the structure of the ISM is unresolved in our
simulations. Therefore, the fact that the produced escape fractions are reasonable
may be coincidental.
Since in our simulations both the intensity of stellar radiation in the IGM and
the photon production rate are known, we can measure the mean star formation
rate-weighted escape fraction of ionizing photons from galaxies into the IGM
(see Appendix C):
f esc
∼ 10−2
×
×
Γin
⋆
10−14 s−1
σ̄⋆
2.9 × 10−18 cm2
−1
−1
ρ̇⋆
0.15 M⊙ yr−1 cMpc−3
−1 α −1 L
1 + z −2
box
0.7
10 cMpc
4
.
(3.13)
The implied escape fractions are shown in the right panel of Figure 3.7. The
simulated escape fractions are 10−2 < f esc < 10−1 between z = 2 and 5 and they
decrease with decreasing redshift below z = 4. This result is consistent with
previous observational and theoretical studies (e.g., Shapley et al., 2006; Schaye,
2006; Gnedin et al., 2008; Kuhlen & Faucher-Giguère, 2012).
3.4.4 The impact of local stellar radiation on the Hi column
density distribution
The observed distribution of neutral hydrogen is often quantified by measuring the distribution of Hi absorbers with different strengths in the spectra of
background quasars (e.g., Kim et al., 2002; Péroux et al., 2005; O’Meara et al.,
2007; Noterdaeme et al., 2009; Prochaska et al., 2009; Prochaska & Wolfe, 2009;
O’Meara et al., 2013; Noterdaeme et al., 2012). The Hi column density distribution function (CDDF) is defined as the number of systems at a given column
82
Results and discussion
Figure 3.8: Left: The Hi column density distribution function (CDDF) at z = 3 with different ionizing sources. The blue solid curve shows our reference simulation which includes
the UVB, local stellar radiation (LSR) and recombination radiation and the green dashed
curve indicates the simulation without local stellar radiation (i.e., with the UVB and recombination radiation). While in the reference simulation star-forming SPH particles are
used as ionizing sources, the Hi CDDF that is shown with the red dotted curve indicates
a simulation in which young stellar particles are used as sources. Using star-forming SPH
particles as sources lowers the Hi CDDF by ≈ 0.5 dex for NHI & 1021 cm−2 . However,
using young stellar particles, which results in sampling issues (see §3.4.2), has a weak
impact on the Hi CDDF. For calculating the Hi CDDF, the neutral gas is assumed to be
fully atomic (i.e., no H2 ). Right: The Hi CDDF at z = 3 in the presence of local stellar
radiation (i.e., star-forming SPH particles as sources) for different assumptions about the
ISM. The blue solid curve shows our reference simulation in which all H atoms contribute to the absorption but molecular hydrogen does not contribute to the Hi CDDF.
The orchid dot-dashed curve shows a porous ISM model (see the text) where molecular
hydrogen does not absorb ionizing radiation during the RT calculation (it is assumed to
have a very small covering fraction). In order to reproduce the observed UVB intensity
in this model, the local stellar radiation has been reduced by a factor of 3 compared to
the fiducial model. The red long-dashed curve is identical to the orchid dot-dashed curve
(porous ISM) but molecular hydrogen is assumed to dissociate into atomic hydrogen before calculating the Hi CDDF. At higher Hi column densities (NHI & 1021 cm−2 ) the Hi
CDDF is highly sensitive to the assumptions about the unresolved ISM.
83
The impact of local stellar radiation on the Hi CDDF
density, per unit column density, per unit absorption length, dX:
f ( NHI , z) ≡
d2 n
1
d2 n H ( z )
≡
.
dNHI dX
dNHI dz H0 (1 + z)2
(3.14)
In order to study the effect of local stellar radiation on the Hi CDDF, we project
our simulation boxes on a two-dimensional grid and use this to calculate the
column densities (see Rahmati et al., 2013 for more details).
3.4.4.1 Local stellar radiation and the Hi CDDF at z = 3
The simulated Hi CDDF at z = 3 is shown in the left panel of Figure 3.8 for
different ionizing sources. The blue solid curve shows the Hi CDDF in our reference simulation which includes local stellar radiation, the UVB and recombination radiation. For comparison, the Hi CDDF without local stellar radiation (i.e.,
only including the UVB and recombination radiation) is shown with the green
dashed curve. As the ratio between these two Hi CDDFs in the top section of
the left panel in Figure 3.8 illustrates, the effect of local stellar radiation increases
with Hi column density and reaches a ∼ 0.5 dex reduction at NHI & 1021 cm−2 .
For Hi column densities lower than 1017 cm−2 on the other hand, the Hi CDDF
is insensitive to local stellar radiation. These trends are consistent with previous
analytic arguments (Miralda-Escudé, 2005; Schaye, 2006) and numerical simulations performed by Fumagalli et al. (2011).
As we discussed in §3.4.2, using star-forming SPH particles as ionizing
sources (i.e., our reference model) results in a better sampling than using young
stellar particles as sources. We showed that if one uses young stellar particles as
ionizing sources, the impact of local stellar radiation will be under-estimated in
a large fraction of low-mass haloes in our simulations (see Figure 3.6 and 3.5).
To illustrate this difference, the Hi CDDF for the simulation in which young
stellar particles are used, is shown with the red dotted curve in the left panel of
Figure 3.8. Indeed, the impact of local sources on the Hi CDDF is much weaker
if we use young stellar particles as ionizing sources. This may partly explain
why Yajima et al. (2012) found that local sources did not affect the Hi CDDF
significantly.
3.4.4.2 The impact of the unresolved ISM
In our simulations, the ISM is modeled by enforcing a polytropic equation of
state on SPH particles with densities nH > 10−1 cm−3 . This means that our
simulations do not include a cold (T ≪ 104 K) interstellar gas phase. This
simple ISM modeling introduces uncertainties in the hydrogen neutral fraction
calculations of the dense regions in our simulations. In addition, for gas with
nH ≫ 10−2 cm−3 the mean free path of ionizing photons is unresolved (see
Appendix B1).
84
Results and discussion
To estimate the impact of the structure of the ISM on the Hi CDDF and the effect of local sources, we introduce a porous ISM model in which we assume that
molecular hydrogen is confined to clouds with such small covering fractions that
we can ignore them when performing the RT. Following Rahmati et al. (2013),
we use the observationally inferred pressure law of Blitz & Rosolowsky (2006) to
compute the molecular fraction (see Appendix A), but now we not only subtract
the H2 fraction when projecting the Hi distribution to compute the Hi CDDF,
we also subtract it before doing the RT calculation (note that in the fiducial case
we assumed H2 fractions to be zero during the RT calculation). The conversion
of diffuse atomic hydrogen into compact molecular clouds which do not absorb
ionizing photons should facilitate the propagation of photons from star-forming
regions into the IGM. Indeed, we find that replacing our reference uniform ISM
model with the porous ISM model increases the resulting UVB photoionization
rate by a factor of ∼ 3. Therefore, we decreased the photon production rate of
local stellar radiation by a factor of 3, such that the model with a porous ISM
yields the same UVB intensity that is generated by our reference simulation and
is in agreement with the observed UVB. This factor of 3 reduction could be interpreted as reflecting absorptions in molecular clouds that are hosting young
stars3 . One should note that despite the enforced agreement between the generated UVBs, the stellar photoionization rates at high and intermediate densities
are different in the two simulations.
The right panel of Figure 3.8 shows that for NHI < 1021 cm−2 the Hi CDDF in
the simulation with a porous ISM (orchid dot-dashed curve) is almost identical
to the Hi CDDF in our reference simulation (blue solid curve). This suggests
that for these column densities the predicted impact of local stellar radiation
on the Hi CDDF is robust to uncertainties regarding the small scale structure
of the ISM. Instead, its impact is controlled by the amount of radiation that is
propagating through the properly resolved intermediate densities towards the
IGM, which is constrained by the observed UVB intensity. Note, however, that
the impact of local sources on the high Hi column density part of the distribution
(i.e., NHI & 1021 cm−2 ) is in fact highly dependent on the assumptions that are
made about the physics of the ISM.
It is also possible that the intense stellar ionizing radiation within the ISM
will effectively increase the Hi column densities by dissociating hydrogen molecules. However, we implicitly neglected this effect in our simulation with a
porous ISM by assuming that molecular clouds are not affected by local stellar
radiation. To put an upper limit on the impact of this effect, one can assume
that all molecular clouds are completely dissociated by the absorption of stellar
radiation. The result of this exercise is shown by the red long-dashed curve in
Figure 3.8. This shows that accounting for H2 dissociation could reduce (or even
reverse) the impact of local stellar radiation only at the very high column density
end of the Hi CDDF (i.e., NHI & 1021 cm−2 ).
3
Note that this is not equivalent to f esc = 1/3. The radiation that leaves star-forming regions is
still subject to significant absorption before it reaches the IGM.
85
The impact of local stellar radiation on the Hi CDDF
Figure 3.9: The ratio between the Hi CDDF with and without local stellar radiation at
different redshifts. For calculating the Hi CDDF, the neutral gas is assumed to be fully
atomic (i.e., no H2 ). The impact of local stellar radiation decreases the Hi CDDF by up to
1 dex. The impact of local stellar radiation on LLSs increases with redshift.
We stress that none of our ISM models are realistic. However, by considering
very different models, we can nevertheless get an idea of the possible impact
of our simplified treatment of the ISM and we conclude that our results are
relatively robust for NHI ≪ 1021 cm−2 .
3.4.4.3 Evolution
The evolution of the impact of local stellar radiation on the Hi CDDF is illustrated in Figure 3.9 for our fiducial model. Each curve in this figure shows the
ratio between the Hi CDDF with and without local stellar radiation at a given
redshift. To avoid the uncertainties about the conversion of atomic gas into H2 ,
the Hi CDDFs are computed assuming that the neutral gas is fully atomic (i.e.,
no H2 ). For all redshifts local stellar radiation has only a very small impact on
the Hi CDDF for NHI ≪ 1017 cm−2 but significantly reduces the abundance of
systems with NHI & 1021 cm−2 . The impact of local stellar radiation increases
with redshift for 1018 < NHI < 1021 cm−2 . While local sources significantly reduce the Hi CDDF for NHI ≫ 1017 cm−2 at z = 5, their effect only becomes
significant for NHI & 1021 cm−2 at z = 0. This might be attributed to decrease in
the proper sizes of galaxies with redshift.
In Rahmati et al. (2013) we used simulations that include only the UVB and
recombination radiation to show that for 1018 . NHI . 1020 cm−2 , the Hi CDDF
86
Discussion and conclusions
does not evolve at z ≤ 3 and increases with increasing redshift for z > 3. Figure
3.10 shows that if we also include local stellar radiation, the weak evolution of
the Hi CDDF in the Lyman Limit range extends to even higher redshifts (i.e.,
z . 4).
The predicted Hi CDDFs are compared to observations in Figure 3.10. We
note that the Hi CDDF predicted by our reference simulation is not converged
with respect to the size of the simulation box (see Appendix B in Rahmati et al.,
2013). Since the photoionization caused by local stellar radiation only exceeds
the UVB photoionization rate close to galaxies (see Figure 3.4), we can assume
that the impact of local stellar radiation on the Hi CDDF is independent of the
size of the simulation box. Therefore, for all the curves shown in Figure 3.10
we have corrected the box size effect by multiplying the CDDF of the fiducial
L = 6.25h−1Mpc box by the ratio of the CDDF in a converged simulation (with
L = 50h−1Mpc) and in our reference simulations with L = 6.25h−1Mpc, both
in the absence of local stellar radiation (i.e., with the UVB and recombination
radiation).
As we showed in Rahmati et al. (2013), the simulated Hi CDDFs in the presence of the UVB and recombination radiation is in a reasonable agreement with
observations (see also Altay et al., 2011) with a small deviation of ∼ 0.2 dex. But
as the top section of Figure 3.10 illustrates, the addition of local stellar radiation
increases this deviation to & 0.5 dex. While the difference between the predicted
Hi CDDFs and observations is larger at the highest Hi column densities (i.e.,
NHI & 1021 cm−2 ), we have seen that in this regime the effect of local sources is
sensitive to the complex physics of the ISM, which our simulations do not capture. These very high Hi column densities are also sensitive to the strength and
the details of different feedback mechanisms (see Altay et al. in prep.). The small
but significant discrepancy in the LL and weak DLA regime is therefore more
interesting. It is important to confirm this discrepancy with larger simulations,
so that a correction for box size will no longer be necessary, but this requires
more computing power than is presently available to us.
3.5 Discussion and conclusions
The column density distribution function (CDDF) of neutral hydrogen inferred
from observations of quasar absorption lines is the most accurately determined
observable of the distribution of gas in the high-redshift Universe. Moreover,
the high column density absorbers (NHI > 1017 cm−2 ) arise in gas that is either
already part of the ISM or will soon accrete onto a galaxy (van de Voort et al.,
2012) and hence these systems directly probe the fuel for star formation.
To predict the distribution of high column density absorbers, it is necessary
to combine cosmological hydrodynamical simulations with accurate radiative
transfer (RT) of ionizing radiation. Because of the cost and complexity associated with RT calculations that include many sources, nearly all studies have only
87
The impact of local stellar radiation on the Hi CDDF
Figure 3.10: The Hi CDDF at different redshifts. Curves show predictions for the reference simulation in the presence of the UVB, diffuse recombination radiation and local stellar radiation after correcting for the box size (see the text). For calculating the Hi CDDF,
the neutral gas is assumed to be fully atomic (i.e., no H2 ). The observational data points
represent a compilation of various quasar absorption line observations at high redshifts
(i.e., z = [1.7, 5.5]) taken from Péroux et al. (2005) with z = [1.8, 3.5], O’Meara et al. (2007)
with z = [1.7, 4.5], Noterdaeme et al. (2009) with z = [2.2, 5.5] and Prochaska & Wolfe
(2009) with z = [2.2, 5.5]. The orange filled circles show the best-fit based on the lowredshift 21 cm observations of Zwaan et al. (2005). The top section shows the ratio between
the Hi CDDFs at different redshifts and the predicted CDDF at z = 3. While simulations
without local stellar radiation are in reasonable agreement with the observed Hi CDDF
(not shown here, but see Rahmati et al., 2013), in the presence of local stellar radiation the
simulated Hi CDDFs of LLSs deviate from observations by a factor of ≈ 2. Local stellar
radiation weakens the evolution of the Hi CDDFs at 1017 . NHI . 1020 cm−2 .
88
Discussion and conclusions
considered ionization by the ultraviolet background radiation (UVB), whose intensity can be inferred from observations of the Lyα forest. However, analytic arguments suggest that the radiation field to which high column density absorbers
are exposed is typically dominated by local stellar sources (Miralda-Escudé,
2005; Schaye, 2006). It is therefore important to investigate whether the remarkable success of simulations that consider only the UVB, such as the agreement
with the observed CDDF over 10 orders of magnitude in column density at z = 3
(Altay et al., 2011) as well as its evolution down to z = 0 (Rahmati et al., 2013),
is compromised when local sources are included. Here we addressed this question by repeating some of the simulations of Rahmati et al. (2013) with our RT
code TRAPHIC (Pawlik & Schaye, 2008, 2011; Rahmati et al., 2013), but this time
including not only the UVB and diffuse recombination radiation, but also local
stellar sources.
In agreement with the analytic predictions of Miralda-Escudé (2005) and
Schaye (2006), we found that local stellar radiation is unimportant for NHI ≪
1017 cm−2 and dominates over the UVB for high column density absorbers. For
all redshifts considered here (i.e., z ≤ 5), local sources strongly reduce the
abundance of systems with NHI & 1021 cm−2 . The impact of local sources increases with redshift for 1018 < NHI < 1021 cm−2 . At z = 5 the CDDF is substantially reduced for NHI ≫ 1017 cm−2 , but at z = 0 the effect only becomes
significant for NHI & 1021 cm−2 . As a result, the remarkable lack of evolution
in the CDDF that we found in Rahmati et al. (2013) for z = 0 − 3, and which
is also observed, extends to z = 4 if local sources are taken into account. On
the other hand, the agreement with the observed z ∼ 3 CDDF is not quite as
good as before, with the simulations underpredicting the rates of incidence of
1019 < NHI < 1021 cm−2 absorbers by factors of a few. However, because of the
large corrections that we had to make because of the small size of the simulation box used to study the effect of local sources (6.25h −1Mpc), this discrepancy
will have to be confirmed with larger simulations. Moreover, we did not account
for possible hydrodynamical effects that might be caused by extra heating due to
ionizing radiation from local sources. This process might change the distribution
of gas that is affected by local stellar radiation and requires further investigation.
We found that the average photoionization rate due to young stars in highdensity gas is weakly dependent on the gas density and is ∼ 10−13s−1 . We
showed analytically that this rate follows directly from the imposed (and observed) Kennicutt-Schmidt star formation law if we assume that most of the
ionizing photons that are produced by star-forming gas are absorbed on scales
. kpc. However, in reality we expect the photoionization rate in the ISM to
fluctuate more strongly than predicted by simulations like ours, which lack the
resolution required to model the cold, interstellar phase.
Indeed, the spatial resolution that is required for accurate RT of ionizing radiation through the ISM is several orders of magnitude higher than the smallest
scales accessible in current cosmological simulations. This makes tackling this
problem in the near future hardly feasible and poses a difficult challenge for
89
The impact of local stellar radiation on the Hi CDDF
studying the impact of local stellar radiation on the distribution of Hi in and
around galaxies. Fortunately, one can circumvent part of the problem by tuning
the production rate of ionizing photons (which is equivalent to adjusting the escape fraction of ionizing photons from the unresolved ISM) such that the models
reproduce the observed mean photoionization rate in the IGM (after subtracting
the contribution from quasars). In other words, if one knows the amount of
ionizing radiation that is required to reach the IGM, its ionization impact on
the intervening gas can be determined even if we cannot predict what fraction
escapes from the immediate vicinity of the young stars.
We adopted this approach but found that tuning was unnecessary for our
reference simulation at z ∼ 3. Moreover, since our simulations also yield star
formation histories that are in good agreement with observations (Schaye et al.,
2010), we used them to constrain the implied star formation rate weighted mean
escape fraction that relates the predicted star formation rate density to the intensity of the UVB. We found that the average escape fraction in our simulations
is 10−2 − 10−1 at z = 2 − 5, which agrees with previous constraints on the escape fraction from observations (e.g., Shapley et al., 2006) and theoretical work
(e.g., Schaye, 2006; Gnedin et al., 2008; Kuhlen & Faucher-Giguère, 2012).
The limited spatial resolution of cosmological simulations mandates the use
of simplified models for the structure of the ISM. To estimate the impact of such
subgrid models on the CDDF and on the effect of local sources, we varied some
of the underlying assumptions. In particular, we considered a porous ISM model
which assumes that molecular hydrogen is confined to clouds with such small
covering fractions that we can ignore them when performing the RT. We also
considered a model which assumes that all molecular clouds are completely
dissociated, but not ionised, by the absorption of stellar radiation. Although
none of these models are realistic, we used them to estimate the potential impact
of our simplified treatment of the ISM. We found that provided that we rescale
the source luminosities so that the different models all reproduce the observed
background radiation, the models predict nearly the same Hi CDDFs in the
regime where the absorbers are well resolved. We therefore concluded that our
results on the effect of local sources are relatively robust for NHI ≪ 1021 cm−2 ,
but that their predicted impact is highly sensitive to the assumptions about the
ISM for NHI & 1021 cm−2 .
Different studies have found qualitatively different results for the impact of
local stellar radiation on the CDDF. For instance, Fumagalli et al. (2011) used relatively high-resolution zoomed simulations of individual objects to demonstrate
that local stellar radiation significantly reduces the abundance of high column
density absorbers. On the other hand, some studies using cosmological simulations similar to those presented here (with roughly the same resolution) found
that local stellar radiation has a negligible impact on the CDDF (Nagamine et al.,
2010; Yajima et al., 2012).
We found that difference in the resolutions of the simulations that were used
in these previous studies may explain their inconsistent findings. Using star
90
Acknowledgments
particles as sources, as was done by (Nagamine et al., 2010; Yajima et al., 2012),
we also found that local sources have a negligible impact on the abundance of
strong Hi systems. However, we demonstrated that it is possible to dramatically improve the sampling of the distribution of ionizing sources by using starforming gas particles (i.e., gas with densities at least as high as those typical of
the warm ISM), thus effectively increasing the resolution without modifying the
time-averaged production rate of ionizing photons. We adopted this strategy in
our fiducial models and found, as summarized above, that the radiation from
local sources significantly affects the high column density end of the CDDF. This
result is in agreement with Fumagalli et al. (2011), Gnedin (2010) and analytic
estimates of Miralda-Escudé (2005) and Schaye (2006), and confirms that poor
sampling of the distribution of ionizing sources can lead to an under-estimation
of the impact of local stellar radiation.
Further progress will require higher resolution simulations and, most importantly, more realistic models for the ISM. In the near future it will remain
unfeasible to accomplish this in a cosmologically representative volume. Until
this challenge is met, predictions for the escape fractions of ionizing radiation
averaged over galaxy populations should be considered highly approximate.
Predictions for the abundances of LL and weak DLA systems based on models that neglect local sources of stellar radiation should be interpreted with care,
particularly for z > 2. Predictions for the CDDF in the strong DLA regime
(NHI & 1021 cm−2 ) must be considered highly approximate at all redshifts.
Acknowledgments
We thank the anonymous referee for a helpful report. We also would like to
thank Dušan Kereš, J. Xavier Prochaska and Tom Theuns for valuable discussions. The simulations presented here were run on the Cosmology Machine
at the Institute for Computational Cosmology in Durham (which is part of the
DiRAC Facility jointly funded by STFC, the Large Facilities Capital Fund of
BIS, and Durham University) as part of the Virgo Consortium research programme. This work was sponsored with financial support from the Netherlands Organization for Scientific Research (NWO), also through a VIDI grant
and an NWO open competition grant. We also benefited from funding from
NOVA, from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement 278594GasAroundGalaxies and from the Marie Curie Training Network CosmoComp
(PITN-GA-2009-238356). AHP receives funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number
301096-proFeSsOR.
91
REFERENCES
References
Aguirre, A., Dow-Hygelund, C., Schaye, J., & Theuns, T. 2008, ApJ, 689, 851
Altay, G., Theuns, T., Schaye, J., Crighton, N. H. M., & Dalla Vecchia, C. 2011,
ApJL, 737, L37
Bigiel, F., Leroy, A., Walter, F., et al. 2008, AJ, 136, 2846
Bird, S., Vogelsberger, M., Sijacki, D., et al. 2013, MNRAS, 514
Blitz, L., & Rosolowsky, E. 2006, ApJ, 650, 933
Bolton, J. S., Haehnelt, M. G., Viel, M., & Springel, V. 2005, MNRAS, 357, 1178
Bolton, J. S., & Haehnelt, M. G. 2007, MNRAS, 382, 325
Cen, R., Ostriker, J. P., Prochaska, J. X., & Wolfe, A. M. 2003, ApJ, 598, 741
Chabrier, G. 2003, PASP, 115, 763
Croft, R. A. C. 2004, ApJ, 610, 642
Dalla Vecchia, C., & Schaye, J. 2008, MNRAS, 387, 1431
Faucher-Giguère, C.-A., Lidz, A., Hernquist, L., & Zaldarriaga, M. 2008, ApJ,
688, 85
Fumagalli, M., Prochaska, J. X., Kasen, D., et al. 2011, MNRAS, 418, 1796
Fumagalli, M., O’Meara, J. M., Prochaska, J. X., & Worseck, G. 2013,
arXiv:1308.1101
Gardner, J. P., Katz, N., Hernquist, L., & Weinberg, D. H. 1997, ApJ, 484, 31
Gnedin, N. Y., Kravtsov, A. V., & Chen, H.-W. 2008, ApJ, 672, 765
Gnedin, N. Y. 2010, ApJL, 721, L79
Haardt F., Madau P., 2001, in Clusters of Galaxies and the High Redshift Universe Observed in X-rays, Neumann D. M., Tran J. T. V., eds.
Haehnelt, M. G., Steinmetz, M., & Rauch, M. 1998, ApJ, 495, 647
Haehnelt, M. G., Madau, P., Kudritzki, R., & Haardt, F. 2001, ApJL, 549, L151
Hopkins, P. F., Richards, G. T., & Hernquist, L. 2007, ApJ, 654, 731
Katz, N., Weinberg, D. H., Hernquist, L., & Miralda-Escude, J. 1996, ApJL, 457,
L57
Kennicutt, R. C., Jr. 1998, ApJ, 498, 541
Kim, J.-h., Krumholz, M. R., Wise, J. H., et al. 2012, arXiv:1210.3361
Kim, T.-S., Carswell, R. F., Cristiani, S., D’Odorico, S., & Giallongo, E. 2002,
MNRAS, 335, 555
Komatsu, E., et al. 2011, ApJS, 192, 18
Kroupa, P. 2001, MNRAS, 322, 231
Kuhlen, M., & Faucher-Giguère, C.-A. 2012, MNRAS, 423, 862
Leitherer, C., Schaerer, D., Goldader, J. D., et al. 1999, ApJS, 123, 3
McQuinn, M., Oh, S. P., & Faucher-Giguère, C.-A. 2011, ApJ, 743, 82
Miralda-Escudé, J. 2005, ApJL, 620, L91
Nagamine, K., Springel, V., & Hernquist, L. 2004, MNRAS, 348, 421
Nagamine, K., Choi, J.-H., & Yajima, H. 2010, ApJL, 725, L219
Noterdaeme, P., Petitjean, P., Ledoux, C., & Srianand, R. 2009, A&A, 505, 1087
Noterdaeme, P., Petitjean, P., Carithers, W. C., et al. 2012, arXiv:1210.1213
O’Meara, J. M., Prochaska, J. X., Burles, S., et al. 2007, ApJ, 656, 666
92
Appendix A: Hydrogen molecular fraction
O’Meara, J. M., Prochaska, J. X., Worseck, G., Chen, H.-W., & Madau, P. 2013,
ApJ, 765, 137
Osterbrock, D. E., & Ferland, G. J. 2006, Astrophysics of gaseous nebulae and
active galactic nuclei, 2nd. ed. by D.E. Osterbrock and G.J. Ferland. Sausalito,
CA: University Science Books, 2006,
Paardekooper, J.-P., Pelupessy, F. I., Altay, G., & Kruip, C. J. H. 2011, A&A, 530,
A87
Pawlik, A. H., & Schaye, J. 2008, MNRAS, 389, 651
Pawlik, A. H., & Schaye, J. 2011, MNRAS, 412, 1943
Péroux, C., Dessauges-Zavadsky, M., D’Odorico, S., Sun Kim, T., & McMahon,
R. G. 2005, MNRAS, 363, 479
Pontzen, A., Governato, F., Pettini, M., et al. 2008, MNRAS, 390, 1349
Prochaska, J. X., & Wolfe, A. M. 2009, ApJ, 696, 1543
Prochaska, J. X., Worseck, G., & O’Meara, J. M. 2009, ApJL, 705, L113
Rahmati, A., Pawlik, A. H., Raičevic̀, M., & Schaye, J. 2013a, MNRAS, 430, 2427
Razoumov, A. O., Norman, M. L., Prochaska, J. X., & Wolfe, A. M. 2006, ApJ,
645, 55
Robbins, D. 1978, Amer. Math. Monthly 85, 278
Salpeter, E. E. 1955, ApJ, 121, 161
Schaye, J. 2001, ApJ, 559, 507
Schaye, J. 2004, ApJ, 609, 667
Schaye, J. 2006, ApJ, 643, 59
Schaye, J., & Dalla Vecchia, C. 2008, MNRAS, 383, 1210
Schaye, J., Dalla Vecchia, C., Booth, C. M., et al. 2010, MNRAS, 402, 1536
Shapley, A. E., Steidel, C. C., Pettini, M., Adelberger, K. L., & Erb, D. K. 2006,
ApJ, 651, 688
Songaila, A., & Cowie, L. L. 2010, ApJ, 721, 1448
Springel, V. 2005, MNRAS, 364, 1105
Vanzella, E., Giavalisco, M., Inoue, A. K., et al. 2010, ApJ, 725, 1011
van de Voort, F., Schaye, J., Altay, G., & Theuns, T. 2012, MNRAS, 421, 2809
Wiersma, R. P. C., Schaye, J., Theuns, T., Dalla Vecchia, C., & Tornatore, L.
2009a, MNRAS, 399, 574
Wiersma, R. P. C., Schaye, J., & Smith, B. D. 2009b, MNRAS, 393, 99
Yajima, H., Choi, J.-H., & Nagamine, K. 2012, MNRAS, 427, 2889
Zuo, L. 1992, MNRAS, 258, 36
Zwaan, M. A., van der Hulst, J. M., Briggs, F. H., Verheijen, M. A. W., & RyanWeber, E. V. 2005, MNRAS, 364, 1467
Appendix A: Hydrogen molecular fraction
To account for the effect of molecular hydrogen, we adopt the same H2 conversion relation that Altay et al. (2011) used to successfully reproduce the Hi CDDF
93
The impact of local stellar radiation on the Hi CDDF
high-end cut-off. We follow Blitz & Rosolowsky (2006) and adopt an observationally inferred scaling relation between the gas pressure and the ratio between
molecular and total hydrogen surface densities:
Pext α
,
(3.15)
Rmol =
P0
where Rmol ≡ ΣH2 /ΣHI , Pext is the galactic mid-plane pressure, α = 0.92 and
P0 /k b = 3.5 × 104 cm−3 K. Furthermore, if we assume Rmol gives also the local
mass ratio between molecular and atomic hydrogen, the fraction of gas mass
which is in molecular form can be written as
f H2 =
MH2
MH2
1
.
=
=
−1
MTOT
MH2 + MHI
1 + Rmol
(3.16)
The last equation also assumes that at high densities ionization is not dominant
and the gas is either molecular or neutral. In our simulations we model the
multiphase ISM by imposing an effective equation of state with pressure P ∝ ρ γeff
⋆ , where n⋆ = 0.1 cm−3 which is normalized to P /k =
for densities nH > nH
⋆ b
H
3
−
3
1.08 × 10 cm K at the threshold. Therefore we have:
nH γeff
.
(3.17)
Pext = P⋆
⋆
nH
To make the Jeans mass and the ratio of the Jeans length to the SPH kernel
independent of the density, we use γeff = 4/3 (Schaye & Dalla Vecchia, 2008).
Consequently, after combining equations (3.15), (3.16) & (3.17) the fraction of
gas mass which is converted into H2 , f H2 can be written as:
f H2 =
1+A
nH
⋆
nH
− β ! −1
,
(3.18)
where A = ( P⋆ /P0 )−α = 24.54 and β = α γeff = 1.23.
Appendix B: Resolution effects
B1: Limited spatial resolution at high densities
At high densities, ionizing photons are typically absorbed within a short distance from the sources. The propagation of ionizing photons in these regions is
therefore controlled by distribution of gas on scales that are comparable to the
very short mean free path of ionizing photons. In Figure 3.11, the mean free path
of stellar ionizing photons, λmfp = 1/(nHI σ̄⋆ )4 , is plotted as a function of density for our reference simulation at z = 3. The blue dashed curve in Figure 3.11
4
σ̄⋆ is the spectrally averaged hydrogen photoionization cross section for stellar ionizing photons.
For the spectral shape of the stellar radiation, we adopt a blackbody spectrum with Tbb = 5 × 104 K.
94
Appendix B: Resolution effects
shows the result when only the UVB and recombination radiation are included.
The effect of adding local stellar radiation is shown by the red dot-dashed curve.
For comparison, also the result for a constant UVB radiation (i.e., optically thin
gas) is illustrated with the green dotted curve. The ionization by local stellar
radiation decreases the hydrogen neutral fraction at nH & 10−2 cm−3 which in
turn increases the mean free path of ionizing photons making it comparable to
the optically thin case.
The simulations cannot provide any reliable information about the spatial
distribution of gas and ionizing sources on scales smaller than the resolution
limit (i.e., the typical distance between SPH particles) which is shown by the
black dotted line in Figure 3.11. At densities for which the mean free path of
ionizing photons is shorter than the spatial resolution, the RT results may not
be accurate since all the photons that are emitted by sources are absorbed by
their immediate neighbors. This happens at densities nH & 10−2 cm−3 without
local stellar radiation (blue dashed curve in Figure 3.11) and at densities nH &
10−1 cm−3 with local stellar radiation (red dot-dashed curve in Figure 3.11). On
the other hand, Rt is irrelevant if the gas is highly neutral, which is predicted to
be the case at slightly higher densities (see Figure 3.3).
B2: The impact of a higher resolution on the RT
In Rahmati et al. (2013) we showed that the self-shielding limit is not very sensitive to the resolution of the simulation. On the other hand, the small scale structure of the ISM may significantly change the propagation of stellar radiation
and their impact on the Hi distribution. While this suggests that the impact
of local stellar radiation might be sensitive to the resolution of the underlying
simulation, our approach of tuning the escaped stellar radiation such that it
can generate the desired UVB, circumvent most resolution effects. To study
the impact of resolution on the propagation of stellar ionizing photons in the
ISM and their escape to the IGM, we performed a simulation similar to our reference simulation but using 8 times more dark matter and SPH particles (i.e.,
using 2563 dark matter particles and the same number of SPH particles whose
masses are 8 times lower than in our reference simulation). Figure 3.12 shows
that at z = 3, the photoionization rate of stellar radiation in the higher resolution simulation (red dashed curve) is qualitatively similar to that obtained from
our reference simulation (blue solid curve). As expected, the stellar photoionization rate at the highest densities is in agreement with our analytic estimate
in §3.2. However, because of the shorter inter-particle distances in the higher
resolution simulation, the stellar photoionization rate peaks at slightly higher
densities. The biggest difference between the stellar photoionization rates of
the high-resolution and our reference simulation occurs at the lowest densities.
Despite having a ∼ 2 times higher total star formation rate, the high-resolution
simulation results in a UVB which has a ∼ 5 times lower photoionization rate.
This means that the effective escape fraction in the high-resolution simulation is
95
The impact of local stellar radiation on the Hi CDDF
Figure 3.11: The mean free path of stellar ionizing photons as a function of hydrogen
number density for the reference simulation at z = 3. The red dot-dashed curve shows
the result when all radiation sources, i.e., UVB, local stellar radiation (LSR) and recombination radiation (RR), are included and the blue dashed curve shows the result when
local sources of radiation are not present. The green dotted curve shows the mean free
path in a simulation that assumes a completely optically thin gas. The typical distance
between SPH particles as a function of density is shown by the black dotted line. The
Jeans length for a given density is illustrated by the black dashed line. The colored lines
indicate the medians and shaded regions around them show the 15% − 85% percentiles.
Figure 3.12: Comparison between the stellar photoionization rate profiles in the reference
simulation and a simulation with 8 times higher resolution at z = 3. The colored lines indicate the medians and shaded regions around them shows the 15% − 85% distributions.
96
Appendix C: Calculation of the escape fraction
∼ 10 times lower than for the reference simulation which has f esc ∼ 10−2.
Finally, we emphasize that due to our simplified treatment of the ISM, it is not
even clear whether the results become more realistic with increasing resolution.
Appendix C: Calculation of the escape fraction
Using the equilibrium photon production rate per unit star formation rate from
equation (3.2), the total production rate of ionizing photons in the simulation
box, Ṅγ,⋆ , can be calculated as a function of the comoving star formation rate
density within the simulation box, ρ̇⋆ , :
!
ρ̇⋆ L3box
53
Ṅγ,⋆ = 2 × 10
.
(3.19)
1 M⊙ yr−1
Noting that the typical time the ionizing photons spend inside the simulation
box is ∼ αLbox /(1 + z)/c, and letting f esc be the star formation rate-weighted
mean fraction of ionizing photons that escape into the IGM, one can calculate
the comoving equilibrium photon number density in the IGM:
n̄γ ∼
f esc α Ṅγ,⋆
,
2
Lbox c (1 + z)
(3.20)
where we ignored absorptions outside the host galaxies because Lbox ≪ λmfp .
The comoving number density of ionizing photons in the simulation box is
related to the effective hydrogen photoionization rate they produce in the IGM,
Γin
⋆:
Γin
⋆
n̄γ =
,
(3.21)
c σ̄⋆ (1 + z)3
where σ̄⋆ is the effective hydrogen ionization cross section for stellar photons.
Combining equations (3.20) and (3.21) yields the effective escape fraction of ionizing photons from stars into the IGM:
f esc ∼
2
Γin
⋆ Lbox
σ̄⋆ α Ṅγ,⋆ (1 + z)2
.
(3.22)
After putting numbers in equation (3.22) and using equation (3.19), the escape
fraction becomes:
−1
Γin
σ̄⋆
⋆
−2
f esc ∼ 10
10−14 s−1
2.9 × 10−18 cm2
−1
ρ̇⋆
×
0.15 M⊙ yr−1 cMpc−3
−1 α −1 L
1 + z −2
box
.
(3.23)
×
0.7
10 cMpc
4
97
4
Predictions for the relation
between strong Hi absorbers and
galaxies at redshift 3
We combine cosmological, hydrodynamical simulations with accurate radiative transfer corrections to investigate the relation between strong Hi absorbers
(NHI > 1016 cm−2 ) and galaxies at redshift z = 3. We find a strong anticorrelation between the column density and the impact parameter that connects
the absorber to the nearest galaxy. The median impact parameters for Lyman
Limit (LL) and Damped Lymanα (DLA) systems are ∼ 10 and ∼ 1 proper kpc,
respectively. If normalized to the size of the halo of the nearest central galaxy,
the median impact parameters for LL and DLA systems become ∼ 1 and ∼ 0.1
virial radii, respectively. At a given Hi column density the impact parameter increases with the mass of the closest galaxy, in agreement with observations. We
predict most strong Hi absorbers to be most closely associated with extremely
low-mass galaxies, M⋆ . 108 M⊙ . We also find a correlation between the column
density of absorbers and the mass of the nearest galaxy. This correlation is most
pronounced for DLAs with NHI > 1021 cm−2 which are typically close to galaxies with M⋆ & 109 M⊙ . Similar correlations exist between column density and
other properties of the associated galaxies such as their star formation rates, halo
masses and Hi content. The galaxies nearest to Hi absorbers are typically far too
faint to be detectable with current instrumentation, which is consistent with the
high rate of (often unpublished) non-detections in observational searches for the
galaxy counterparts of strong Hi absorbers. Moreover, we predict that the detected nearby galaxies are typically not the galaxies that are most closely associated
with the absorbers, thus causing the impact parameters, star formation rates and
stellar masses of the counterparts to be biased high.
Alireza Rahmati, Joop Schaye
to be submitted
Strong Hi absorbers and galaxies at z = 3
4.1 Introduction
Studies of high-redshift galaxy are often based on the light emitted by stars
and hot/ionized gas. This limits the observations to the small fraction of galaxies that are bright enough to be detected in emission. Given the large number
density of faint galaxies, one may speculate that most high-redshift galaxies are
missing from observational studies. The analysis of absorption features in the
spectra of background QSOs, provides an alternative probe of the distribution
of matter at high redshifts. The large distances that separate most absorbers
from their background QSOs make it unlikely that there is a physical connection between them. This opens up a window to study an unbiased sample of
matter that resides between us and background QSOs. The rare strong Hi Lyα
absorbers which are easily recognizable in the spectra of background QSOs due
to their Lyα damping wings, for which they are called Damped Lyman-α (DLA)
systems1 , are of particular interest. DLAs are likely to be representative of the
cold gas in, or close to, the interstellar medium (ISM) in high-redshift galaxies (Wolfe et al., 1986). Because of this, DLAs provide a unique opportunity
to define an absorption-selected galaxy sample and to study the ISM, particularly at early stages of galaxy formation, and they have therefore been studied
intensely since their discovery (see Wolfe et al., 2005 for a review).
Based on the observed velocity width of metal lines associated with DLAs, it
was initially suggested that large, massive galactic disks are responsible for the
observed DLAs at z ∼ 3 (Prochaska & Wolfe, 1997, 1998). However, it has been
shown that (collections of) smaller systems are also capable of having high velocity dispersions as a result of infall of material during structure formation
(Haehnelt et al., 1998) or galactic winds (McDonald & Miralda-Escudé, 1999;
Schaye, 2001a). Nevertheless, reproducing the observed velocity width distribution remains a challenge for hydrodynamical simulations (e.g., Razoumov et al.,
2006; Pontzen et al., 2008).
Some recent studies suggest that at z ∼ 2 − 3, a large fraction of strong
Hi absorbers like Lyman Limit Systems (LLS; NHI > 1017 cm−2 ) and DLAs
are primarily associated with galaxies similar to Lyman-Break Galaxies (e.g.,
Steidel et al., 2010; Rudie et al., 2012; Font-Ribera et al., 2012), which have stellar and total halo masses ∼ 1010 and ∼ 1012 M⊙ , respectively. If such
massive galaxies were indeed the prime hosts of strong Hi absorbers, then
many of the galaxy counterparts of strong absorbers should be detectable with
current surveys. However, observations that aim to find galaxies close to
DLAs often result in non-detections (e.g., Foltz et al., 1986; Smith et al., 1989;
Lowenthal et al., 1995; Bunker et al., 1999; Prochaska et al., 2002; Kulkarni et al.,
2006; Rahmani et al., 2010; Bouché et al., 2012) or find galaxies that are at
unexpectedly large impact parameters from DLAs (e.g., Yanny et al., 1990;
Teplitz et al., 1998; Mannucci et al., 1998). Those findings suggest that strong Hi
1
The official column density limit of a DLA is somewhat arbitrarily defined to be NHI >
1020.3 cm−2 .
100
Introduction
systems such as DLAs are more closely associated to low-mass galaxies which
are too faint to be observable with the detection thresholds of the current studies.
Because observational studies are limited by the small number of known
DLAs and are missing low-mass galaxies, we have to resort to cosmological
simulations to help us understand the link between DLAs and galaxies. Many
studies have used simulations to investigate the nature of strong Hi absorbers
and particularly DLAs (e.g., Gardner et al., 1997, 2001; Haehnelt et al., 1998;
Nagamine et al., 2004; Razoumov et al., 2006; Pontzen et al., 2008; Tescari et al.,
2009; Fumagalli et al., 2011; Cen, 2012; van de Voort et al., 2012a). To maximize
the numerical resolution required for accurate modelling of the high Hi column
densities, most previous studies have used small simulation boxes or zoomed
simulations. Those studies often try to compensate for the lack of a full cosmological distribution of absorbers by combining the results from their small-scale
simulations with analytic halo mass functions to predict the properties of the
DLA population (e.g., Gardner et al., 1997, 2001) or to determine what kinds of
galaxies dominate the cosmic DLA distribution (e.g., Pontzen et al., 2008). This
approach requires some preconceptions about the types of environments that
can give rise to DLA absorbers and cannot easily account for the large scatter in
the distribution of absorbers in halos with similar properties. Moreover, zoom
simulations cannot fully capture the possibility of absorber-galaxy pairs being
found outside of the considered regions. As a result, the statistical properties
found using zoomed simulations may be biased. Finally, the impact of finite
detection thresholds on the observed relation between strong Hi absorbers and
galaxies cannot be studied with simulations that do not contain a representative
sample of Hi absorbers and galaxies.
In this work, we use cosmological hydrodynamical simulations that contain a representative sample of the full distribution of strong Hi systems
(Rahmati et al., 2013a). Similar to what is done observationally, we connect each
absorber to its nearest galaxy. A significant improvement in this work is the
use of photoionization corrections that are based on accurate radiative transfer
simulations and that account for both the uniform ultraviolet background (UVB)
radiation and recombination radiation. In addition we show that our main conclusions are insensitive to the inclusion of local sources and to variations in the
subgrid physics. The ionization corrections we use have been shown to reproduce the observed Hi column density distribution function over a wide range of
redshifts (Rahmati et al., 2013a).
We predict correlations between the column density of strong Hi absorbers,
their impact parameters, and the properties of the associated galaxies. While the
fraction of Hi absorbers that are linked to relatively massive galaxies increases
with Hi column density, most LLS and DLAs are closely associated with very
low-mass galaxies, with typically M⋆ . 108 M⊙ , that are generally undetectable
with current instruments. We show that our predictions are nevertheless in good
agreement with existing observations, including those of Rudie et al. (2012) who
found that a large fraction of strong Hi absorbers at z ∼ 3 are within 300 proper
101
Strong Hi absorbers and galaxies at z = 3
kpc radius from massive Lyman-Break galaxies.
The structure of this chapter is as follows. In §4.2 we discuss our numerical
simulations and ionization calculations for obtaining the Hi column densities
and describe our method for connecting Hi systems to their host galaxies. We
present our results in §4.3 and compare them with observations. In this section we also investigate how the distribution of Hi absorbers varies with the
properties of their host galaxies. Finally, we conclude in §4.4.
4.2 Simulation techniques
In this section we describe different parts of our simulations. We briefly explain
the details of the hydrodynamical simulations that are post-processed to get the
Hi distribution by accounting for various ionization processes. Then we explain
our halo finding method, our Hi column density calculations, and the procedure
we use to connect Hi absorbers to their host halos.
4.2.1 Hydrodynamical simulations
We use cosmological simulations performed using a significantly modified and
extended version of the smoothed particle hydrodynamics (SPH) code GADGET-3
(last described in Springel, 2005). The simulations are part of the Overwhelmingly Large Simulations (OWLS) described in Schaye et al. (2010). For our reference model, we use a subgrid pressure-dependent star formation prescription of Schaye & Dalla Vecchia (2008) which reproduces the observed KennicuttSchmidt law. The chemodynamics is based on the model of Wiersma et al.
(2009b) which follows the abundances of eleven elements assuming a Chabrier
(2003) IMF. These abundances are used for calculating radiative cooling/heating
rates, element-by-element and in the presence of the uniform cosmic microwave
background and the Haardt & Madau (2001) UVB model (Wiersma et al., 2009a).
Galactic winds driven by star formation are modeled using a kinetic feedback
recipe that assumes 40% of the kinetic energy generated by Type II SNe is injected as outflows with initial velocity of 600 kms−1 and with a mass loading
factor η = 2 (Dalla Vecchia & Schaye, 2008). To bracket the impact of feedback,
we also consider simulations with different feedback and sub-grid models. We
found that our results are not sensitive to the variations in feedback and sub-grid
physics (see Appendix B).
We adopt cosmological parameters consistent with the WMAP year 7 results: {Ωm = 0.272, Ωb = 0.0455, ΩΛ = 0.728, σ8 = 0.81, ns = 0.967, h =
0.704} (Komatsu et al., 2011). Our reference simulation has a periodic box
of L = 25 comoving h−1 Mpc and contains 5123 dark matter particles with
mass 6.3 × 106 h−1 M⊙ and an equal number of baryons with initial mass
1.4 × 106 h−1 M⊙ . The Plummer equivalent gravitational softening length is
set to ǫcom = 1.95 h−1 kpc and is limited to a minimum physical scale of
102
Simulation techniques
ǫprop = 0.5 h−1 kpc. In addition to our reference simulation explained above,
we use simulations with different resolutions and box-sizes to investigate numerical effects (see Appendix D).
4.2.2 Finding galaxies
For identifying individual galaxies in our cosmological simulations, we assume
that galaxies are bound to their dark matter haloes. This assumption implies that
any given baryonic particle belongs to its closest dark matter halo. We use the
Friends-of-Friends (FoF) algorithm to identify groups of dark matter particles
that are near each other (i.e., FoF haloes), using a linking length of b = 0.2. Then,
we use SUBFIND (Dolag et al., 2009) to connect gravitationally bound particles as
part of unique structures (halos) and to identify the center of each halo/galaxy
as the position of the most bound particle in that halo. We take the radius within
which the average density of a given halo reaches 200 times the mean density
of the Universe at a given redshift, R200 , as the size of that halo. The galaxy
that sits in the center of each halo is considered as a central galaxy and all the
other gravitationally bound structures in that FoF halo are considered as satellite
galaxies. Note that we do not require satellite galaxies to be within the R200 of
their central galaxy.
In our analysis, we use all the simulated galaxies that have star formation
rates SFR > 4 × 10−3 M⊙ yr−1 . By using this SFR threshold, more than 99%
of our selected galaxies are resolved with > 100 resolution elements (i.e., dark
matter particles and/or baryonic particles). We test the impact of different SFR
thresholds on our results, which provides useful insights for observational studies with finite detection threshold (see §4.3).
4.2.3 Finding strong Hi absorbers
The first step in identifying Hi absorbers in the simulations is to accurately
calculate the hydrogen neutral fractions. To accomplish this, the main ionization
processes that shape the distribution of neutral hydrogen must be accounted
for. In this context, photoionization by the metagalactic UVB radiation is the
main contributor to hydrogen ionization at z & 1 while collisional ionization
becomes more important at lower redshifts (Rahmati et al., 2013a). Although the
photoionization from local stellar radiation is the dominant source of ionization
at high Hi column densities (e.g., Rahmati et al., 2013b), our tests show that it
does not have a significant impact on the results we present in this work (see
Appendix C).
We use the UVB model of Haardt & Madau (2001) to account for the largescale photoionization effect of quasars and galaxies. The same UVB model is
used for calculating heating/cooling in our hydrodynamical simulations. It has
been shown that this UVB model is consistent with metal absorption lines at
103
Strong Hi absorbers and galaxies at z = 3
Figure 4.1: The simulated Hi column density distribution around a massive galaxy with
M⋆ = 1010 M⊙ and SFR = 29 M⊙ yr−1 at z = 3. In each panel, galaxies with different
SFRs are at the center of circles. The size of dark circles indicates the virial radius of the
central galaxies (R200 ) while the small white circles show satellite galaxies. From top-left
to bottom-right, panels show galaxies with SFR > 10 M⊙ yr−1 , SFR > 1 M⊙ yr−1 , SFR >
0.1 M⊙ yr−1 and SFR > 0.01 M⊙ yr−1 , respectively. As the SFR threshold decreases, more
galaxies are detected and the typical impact parameter between galaxies and absorbers
decreases.
104
Simulation techniques
z ∼ 3 (Aguirre et al., 2008) and the observed Hi column density distribution
function and its evolution in a wide range of redshifts (Rahmati et al., 2013a).
We characterize the UVB by its optically thin hydrogen photoionization rate,
ΓUVB , and the effective hydrogen absorption cross-section, σ̄νHI (see equations
3 and 4 and Table 2 in Rahmati et al., 2013a). In self-shielded regions, ΓUVB is
attenuated to an effective total photoionization rate, ΓPhot , which is decreasing
with density. In Rahmati et al. (2013a) we performed radiative transfer simulations of the UVB and recombination radiation in cosmological density fields
using TRAPHIC (Pawlik & Schaye, 2008, 2011). We showed that the effective photoionization rate at all densities can be accurately reproduced by the following
fitting function:
"
1.64 #−2.28
−0.84
nH
nH
ΓPhot
= 0.98 1 +
,
(4.1)
+ 0.02 1 +
ΓUVB
nH,SSh
nH,SSh
where nH is the hydrogen number density and nH,SSh is the self-shielding density
threshold given by
nH,SSh = 6.73 × 10
−3
cm
−3
σ̄νHI
2.49 × 10−18 cm2
−2/3 ΓUVB
10−12 s−1
2/3
.
(4.2)
We use the photoionization rate from equations (4.1) and (4.2) together with
the hydrogen number density and temperature of each SPH particle in our hydrodynamical simulations to calculate the equilibrium hydrogen neutral fraction
of that particle in post-processing (see Appendix A2 in Rahmati et al., 2013a). It
is also worth noting that in our hydrodynamical simulations, ISM gas particles
(which all have densities nH > 0.1 cm−3 ) follow a polytropic equation of state
that defines their temperatures. Since these temperatures are not physical and
only measure the imposed pressure (Schaye & Dalla Vecchia, 2008), in our calculations we set the temperature of the ISM particles to TISM = 104 K, the typical
temperature of the warm-neutral phase of the ISM.
At very high Hi column densities, where the gas density and the optical
depth for H2 -dissociating radiation is high, hydrogen is expected to be mainly
molecular. This process has been suggested as an explanation for the observed cut-off in the abundance of absorbers at NHI & 1022 cm−2 (Schaye, 2001c;
Krumholz et al., 2009; Prochaska & Wolfe, 2009). It has been also shown that accounting for H2 formation can produce a good agreement between cosmological
simulations and observations of the Hi column density distribution function
(Altay et al., 2011; Rahmati et al., 2013a). To test the impact of H2 formation on
the spatial distribution of Hi absorbers, we adopted the observationally inferred
pressure law of Blitz & Rosolowsky (2006) to compute the H2 fractions in postprocessing (see Appendix A in Rahmati et al., 2013b). Once the H2 fractions
have been calculated, we exclude the molecular hydrogen from the total neutral
gas for calculating the Hi column densities. We note that the adopted empirical
relation for calculating the H2 fractions is calibrated based on observation of
105
Strong Hi absorbers and galaxies at z = 3
low redshift galaxies and may not be accurate in very low metallicity regimes.
However, due to the tight mass-metallicity-SFR relation observed out to z & 3
(Mannucci et al., 2010; Lara-López et al., 2010), exceptionally low metallicities
are not very likely to be typical at z = 3, the redshift on which we focus here.
We calculate Hi column densities by projecting the Hi content of the simulation box along each axis onto a grid with 100002 pixels2 , using SPH interpolation.
While the projection may merge distinct systems along the line of sight, this is
not expected to affect very high Hi column density systems that are rare in the
relatively small simulation boxes that we use. We tested the impact of projection
effects by performing projections using multiple slices instead of the full box.
In fact, our numerical experiments show that at z = 3 and for simulations with
box sizes comparable to that of our simulation, the effect of projection starts to
appear only at NHI < 1016 cm−3 . Since the focus of our study is to characterize
the properties of strong Hi absorbers with NHI & 1017 cm−3 , our results are not
sensitive to the above mentioned projection effect. The top-left panel of Figure
4.1 shows an example for the distribution of the Hi column density around a
galaxy with stellar mass of M⋆ = 1010 M⊙ in our simulation at z = 3 .
In addition to Hi column densities, we calculate the HI-weighted velocity
along each line of sight (LOS), hVLOS iHi and use it to constrain the position of
the strongest absorber along the projection direction. That this is the same procedure used in observations to associate absorbers and galaxies. Compared to
the actual LOS velocity of absorbers, calculating the HI-weighted velocity is less
expensive but using it as a proxy for the position of the strongest absorber along
each line of sight might be prone to projection effects. However, we found that
both the hVLOS iHi velocity and the actual velocity of absorber along the LOS
produce nearly identical results. For calculating hVLOS iHi we use the combin−
→
ation of local peculiar velocity of each SPH particle, V peculiar , and its Hubble
−
→
velocity, V Hubble :
D −
E
→
−
→
hVLOS iHi = ( V peculiar + V Hubble ) . x̂LOS
Hi
,
(4.3)
where x̂LOS is the unit vector along the projection direction. The Hubble velocity
−
→
of each SPH particle is given by the position vector of that particle, X , and the
Hubble parameter, H (z),
−
→
−
→
V Hubble = H (z) X .
(4.4)
We take the Cartesian coordinates of the simulation box as the reference frame.
As we describe in the next section, we use the same reference frame to calculate
the velocities of galaxies along the projection direction. The choice of the origin
is not important since we use the velocity differences between Hi absorbers and
galaxies for our analysis and not the absolute velocities.
2
Using 100002 cells produces converged results. The corresponding cell size is similar to the
minimum smoothing length of SPH particles at z = 3 in our simulation.
106
Simulation techniques
Figure 4.2: The predicted impact parameter of absorbers (in proper kpc) as a function of
Hi column density at z = 3. The color of each cell (in the 2D grid) shows the median
stellar mass of the galaxies associated with the Hi absorbers in that cell. The median
impact parameter as a function of NHI is shown using blue solid curve while the dotted
curves indicate the 15% − 85% percentiles. The gray cells show the region where the H2
formation drains the atomic gas. The gray dashed and dotted curves show the median
impact parameter and the 15% − 85% percentiles as a function of NHI for a fully atomic
Hi where the conversion of high pressure gas into H2 is neglected.
4.2.4 Connecting Hi absorbers to galaxies
An example of how galaxies and Hi absorbers are distributed in our simulation is shown in Figure 4.1. The colored map, which is repeated in all four
panels, shows the Hi column density distribution in a 5002 proper kpc2 region
which is centered on a galaxy with M⋆ = 1010 M⊙ . Galaxies are shown with
circles while the star formation rate cut for illustrating galaxies is decreasing
from SFR > 10 M⊙ yr−1 in the top-left panel to SFR > 0.01 M⊙ yr−1 in the
bottom-right panel. The dark circles, with sizes proportional to the virial radius
of galaxies, are centered on the central galaxies and the white small circles show
satellite galaxies. Galaxies with different SFRs are shown in the bottom panels
of Figure 4.1. As this figure shows, galaxies and LLSs and DLAs (that are shown
using green and red colors, respectively) are strongly correlated. In addition, it
seems that the Hi column density of absorbers is increasing as they get closer
to the center of the galaxies. For a quantitative study, a well defined connection between absorbers and galaxies must be established. This connection can
be made in two ways: by linking any given absorber to its closest galaxy (i.e.,
absorber-centered) or by finding absorbers that are closest to a given galaxy (i.e.,
107
Strong Hi absorbers and galaxies at z = 3
galaxy-centered). In the present work, we use the absorber-centered matching to
connect the simulated Hi absorbers to their neighboring galaxies. This approach
is particularly efficient for associating rare strong Hi absorbers to galaxies. The
galaxy-centered approach, on the other hand, is more suited for studying the
properties of absorbers around certain classes of galaxies. As we discuss in
§4.3.6, these two approaches are closely related, but not identical.
The projected distances between Hi absorbers and galaxies, together with
their LOS velocity differences, can be used to associate them with each other.
We use this method for a direct comparison between simulations and observational studies that employ the same approach. First, we calculate the velocity of
each simulated galaxy along the LOS by adding its peculiar and Hubble velocities along the projection direction (see equation 4.3). Then, for every simulated
absorber we define a galaxy counterpart that has the shortest projected distance
(i.e., the impact parameter) among galaxies with the LOS velocity differences
less than a chosen maximum value, ∆VLOS, max, with respect to the LOS velocity
of the absorber, hVLOS iHi . With this approach, each galaxy can be connected
to more than one absorber, but each absorber is connected to one and only one
galaxy.
We note that the difference between the LOS velocities of absorbers and
galaxies includes not only the distance between the absorbers and galaxies
along the LOS, but also their relative peculiar velocities along the LOS. Therefore, choosing values of ∆VLOS, max that are less than the expected peculiar
velocities around galaxies results in unphysical associations between Hi absorbers and neighboring galaxies. We know that accretion of the gas into
halos together with galactic outflows produces typical peculiar velocities of a
few hundreds of kilometers per second. Similar velocity differences have been
observed between the LOS velocity of absorbers and their host galaxies (e.g.,
Fynbo et al., 1999; Rakic et al., 2012; Rudie et al., 2012) in addition to being common in our simulations (van de Voort & Schaye, 2012b). For this reason, we
chose ∆VLOS, max = 300 km s−1 , which is consistent with recent observations
(Rudie et al., 2012). However, as we show in Appendix A, our results are not
sensitive to this particular choice, for ∆VLOS, max & 100 km s−1 .
4.3 Results and discussion
Using the procedure described in the previous section, we match Hi column
density systems that have NHI > 3 × 1016 cm−2 to the galaxies with non-zero
SFRs in our simulations (i.e., ≈ 2 × 106 strong Hi absorbers and more than
10000 galaxies for every projection). In the following we use this matching to
study the relative spatial distribution of galaxies and Hi absorbers around them.
108
109
Results and discussion
Figure 4.3: Left: Predicted median impact parameters vs. NHI for different SFR thresholds at z = 3. Right: Median impact parameters
normalized to the virial radii (R200 ) as a function of NHI . Since satellite galaxies reside in the halo of central galaxies, they do not have
defined virial radius. Therefore, the result in the right panel is based on matching Hi absorbers to central galaxies only. In both panels,
the red dotted, green dashed and blue solid curves show the SFR thresholds of 1, 0.06 and 0.004 M⊙ yr−1 , respectively. The shaded areas
around the blue solid and red dotted curves show the 15% − 85% percentiles. Red data points in the left panel show a compilation of
DLAs with observed galaxy counterparts described in Table 4.1. Because of a very efficient conversion of hydrogen atoms into molecules,
absorbers with NHI & 1022 cm−2 (indicated by the gray areas) are not expected to exist.
Strong Hi absorbers and galaxies at z = 3
4.3.1 Spatial distribution of Hi absorbers
After connecting absorbers and galaxies, one can measure the typical projected
distances (i.e., impact parameters, b) separating them. The predicted distribution of impact parameters as a function of Hi column density is shown in Figure
4.2 for our simulation at z = 3. Each cell in this figure shows the position of Hi
absorbers in the b − NHI space. The color of each cell indicates the median stellar
mass of galaxies that are associated with the absorbers in that cell (see the colorbar on the right-hand side). The distribution of cells only shows the range of the
b − NHI space that is spanned by the Hi absorbers in our simulation at z = 3. To
show how the impact parameter of absorbers is distributed at any given NHI , we
plot the median impact parameter as a function of NHI using the blue solid curve
and the 15% − 85% percentiles using the blue dotted curves. This result shows
that our simulation predicts a strong anti-correlation between the Hi column
density of absorbers and their impact parameters. While the weak Lyman Limit
Systems (LLSs) with NHI ≈ 1017 cm−2 have typical impact parameters b ≈ 30
proper kpc, the impact parameter decreases with increasing the Hi column
density such that strong DLAs with NHI > 1021 cm−2 are typically within a
few proper kpc from the center of their neighboring galaxies. The increase in
the impact parameter of Hi absorbers with decreasing Hi column density is in
agreement with observations (Moller & Warren, 1998; Christensen et al., 2007;
Monier et al., 2009; Rao et al., 2011; Péroux et al., 2011; Krogager et al., 2012) and
consistent with previous theoretical studies (Gardner et al., 2001; Pontzen et al.,
2008).
Despite the strong anti-correlation between the impact parameter and NHI ,
there is a large scatter around the median impact parameter at any given Hi
column density, as dotted curves in Figure 4.2 show. Since galaxies actively exchange material with their surroundings through accretion and outflows, the Hi
distribution around them has a very complex geometry (see the top-left panel of
Figure 4.1 for an example). This complexity is a major contributor to the scatter
in the impact parameters. In addition, part of this scatter is due to the fact that,
at any given NHI , there is a large number of host galaxies with different sizes that
contribute to the total distribution of absorbers. This is also consistent with the
color gradients in Figure 4.2: at any given impact parameter, the mass of galaxies that are linked to Hi absorber is increasing with their NHI and at any given
Hi column density (particularly for DLAs), the mass of galaxy counterparts is
increasing with the impact parameter of the Hi absorbers. We will discuss this
further in §4.3.5.
To show the impact of H2 formation on the distribution of Hi absorbers, in
Figure 4.2 we also show the regions where the Hi gas is fully converted into
molecules using gray cells (see §4.2.3 for the details of H2 calculation). The
median impact parameters and 15% − 85% percentiles for the fully atomic gas
(i.e., no Hi to H2 conversion) are shown with the gray dashed and gray dotted
curves, respectively. The comparison between the colored and gray areas (and
110
Results and discussion
curves) in Figure 4.2 shows that H2 formation only affects Hi column densities NHI > 1022 cm−2 . This is consistent with the sharp cut-off in the observed
Hi column density distribution at NHI > 1022 cm−2 as shown in Rahmati et al.
(2013a) (see also Altay et al., 2011; Erkal et al., 2012). The formation of H2 thus
only drains the atomic gas at very high Hi column densities and does not significantly affect the impact parameters of the Hi absorbers with NHI < 1022 cm−2 .
4.3.2 The effect of a finite detection threshold
As seen from the colors in Figure 4.2, our simulation predicts that most strong
Hi absorbers with 1017 < NHI . 1021 cm−2 are closely associated with lowmass galaxies, with typical stellar masses of M⋆ ∼ 108 M⊙ . The typical SFR
for those galaxies is ∼ 10−1 M⊙ yr−1 . On the other hand, the typically accessible sensitivity of observations only allows the detection of galaxies that have
SFR & 1 − 10 M⊙ yr−1 (at z ≈ 3)3. Because of this relatively high detection
threshold, most galaxy counterparts are not detectable and the chance of observing galaxies that host LLSs and DLAs is slim. In fact, this could be the
main reason why observational surveys that are aiming to find galaxies close
to DLAs, often result in non-detections (e.g., Foltz et al., 1986; Smith et al., 1989;
Lowenthal et al., 1995; Bunker et al., 1999; Prochaska et al., 2002; Kulkarni et al.,
2006; Rahmani et al., 2010; Bouché et al., 2012). Moreover, the finite and relatively low sensitivity of observational surveys might cause biases in the measured
typical impact parameter of absorbers by mis-associating them to the closest detectable galaxy in their vicinity, instead of their real hosts that are likely to fall
below the detection limit.
Galaxies in our simulation are resolved down to SFRs that are much lower
than the typical limited detection threshold of observations. Therefore, we are
able to analyze the impact of varying the detection limit on the impact parameter
of strong Hi absorbers. Figure 4.1 shows the distribution of the Hi column
densities and positions of galaxies in a simulated region of size 500 proper
kpc around a randomly selected massive galaxy at z = 3. The top-left panel
of Figure 4.1 shows the distribution of Hi column density and galaxies that
have SFR > 10 M⊙ yr−1 . With this detection threshold, only the central galaxy
(shown with the dark circle whose size is proportional to the virial radius of the
central galaxy) and one of its satellites (shown with the small white circle) are
detectable. Other panels in this figure show that as the SFR threshold for detecting galaxies decreases, more galaxies show up in the field, which decreases the
typical impact parameter.
The impact of varying detection threshold on the impact parameter of strong
Hi absorbers (i.e., NHI & 1017 cm−2 ) is shown more quantitatively in the left
panel of Figure 4.3. Different curves show the median impact parameter as a
3 For a Chabrier IMF, a star formation rate of 1 M yr−1 corresponds to a Lyα luminosity of
⊙
7.3 × 1041 erg s−1 , which translates into an observed flux of ≈ 3 × 10−17 erg s−1 cm−2 and ≈ 1 ×
10−17 erg s−1 cm−2 at redshifts z = 2 and z = 3, respectively.
111
Table 4.1: A compilation of the confirmed DLA-galaxy pairs from the literature. The columns from left to right show respectively: the ID
of each DLA, their redshifts, the Hi column densities, the impact parameters in arc seconds, the impact parameters in proper kpc, the star
formation rates of the host galaxies and finally the references from which these values are extracted. We note that the star formation rate
estimates are often based on Lyα emissions, which provides a lower limit for the SFR since this is difficult to account for the dust reddening
and incomplete flux measurements. The SFR estimates that are corrected for the dust reddening are indicated with bold number.
zDLA
log NHI
b
bp
SFR
M⋆
Reference
( cm−2 ) (arcsec) (pkpc) (M⊙ yr−1 ) (109 M⊙ )
Q2206-1958
1.92
20.65
0.99
8.44
3
[1]
Q0151+048A
1.93
20.45
0.93
7.93
71
[3] & [4]
PKS 0458-02
2.04
21.65
0.31
2.63
6
[2]
Q1135-0010
2.21
22.10
0.10
0.84
25
[6]
Q0338-0005
2.22
21.05
0.49
4.12
[5]
Q2243-60
2.33
20.67
2.28
23.37
36
[7]
Q2222-0946
2.35
20.65
0.8
6.67
13
2
[8] & [9]
Q0918+1636
2.58
20.96
2.0
16.38
27
12.6
[ 10] & [5] & [15]
Q0139-0824
2.67
20.70
1.6
13.01
[11]
J073149+285449 2.69
20.55
1.54
12.50
12
[12]
PKS 0528-250
2.81
21.27
1.14
9.15
17
[1]
2233.9+1318
3.15
20.00
2.3
17.91
20
[13]
Q0953+47
3.40
21.15
0.34
2.58
[14]
[1]- Möller et al. (2002); [2]- Möller et al. (2004); [3]- Moller & Warren (1998); [4]- Fynbo et al. (1999); [5]- Krogager et al. (2012);
[6]- Noterdaeme et al. (2012); [7]- Bouché et al. (2012); [8]- Fynbo et al. (2010); [9]- Krogager et al. (2013); [10]- Fynbo et al.
(2011); [11]- Wolfe et al. (2008); [12]- Fumagalli et al. (2010); [13]- Djorgovski et al. (1996); [14]- Prochaska et al. (2003); [15]Fynbo et al. (2013)
112
Strong Hi absorbers and galaxies at z = 3
ID
Results and discussion
function of NHI assuming different SFR detection thresholds. The blue solid
curve assumes a SFR detection threshold identical to that of Figure 4.2, where
all galaxies with SFR > 4 × 10−3 M⊙ yr−1 are considered as galaxy counterparts. The green dashed and red dotted curves, which respectively indicate
SFR thresholds of > 6 × 10−2 M⊙ yr−1 and > 1 M⊙ yr−1 , show the impact
of increasing the SFR threshold on the resulting impact parameter distribution. The shaded areas around the blue solid and red dotted curves show the
15% − 85% percentiles, and the overlap region between the two shaded areas is
shown in purple. The gray area at NHI > 1022 cm−2 shows the region affected
by the formation of H2 . The comparison between the three curves in the left
panel of Figure 4.3 shows that the impact parameters increase with the detection threshold. Almost all absorbers that are associated to galaxies using the
highest SFR threshold (i.e., red dotted curve) find fainter galaxies close the them
(i.e., blue solid curve). Moreover, the anti-correlation between the impact parameter and NHI is sensitive to the detection threshold. As the green dashed and
red dotted curves show, for detection threshold SFR & 10−1 M⊙ yr−1 the strong
anti-correlation between the impact parameter and NHI becomes insignificant at
NHI < 1020 cm−2 . The main reason for this is that most galaxy counterparts that
are detectable with relatively low sensitivities (i.e., high detection thresholds)
are not physically connected to the strong Hi absorbers. As a result, the probability distribution of the impact parameters for those systems is controlled by
the average projected distribution of the detectable galaxies. As the red dotted
curve in the left panel of Figure 4.3 shows, our simulation predicts that with a
detection threshold of SFR > 1 M⊙ yr−1 the typical impact parameters between
strong Hi absorbers and their nearest galaxies vary from several tens of kpc to
a few hundred kpc. This result is in excellent agreement with the measured
impact parameters between DLAs and galaxies in observational surveys that
used similar detection thresholds at z ≈ 2 − 3 (not shown in this plot but see
e.g., Teplitz et al., 1998; Mannucci et al., 1998). It is also worth noting that the
anti-correlation between b and NHI remains in place for absorbers with highe
Hi column densities (i.e., NHI > 1021 cm−2 ), even for a relatively high detection threshold like SFR > 1 M⊙ yr−1 . However, the impact parameters of those
systems are increasingly over-estimated as the galaxy detection threshold is increased.
We show the observed impact parameters of a compilation of confirmed
DLA-galaxy pairs using red symbols with error-bars in the left panel of Figure 4.3 (see Table 4.1 for more details). Given the large scatter in the b − NHI
relation, there is a broad agreement between the observations and simulations.
We note, however, that the observed impact parameters are generally above the
blue solid curve. Before interpreting this systematic difference, it is important
to keep in mind that there are many more non-detections of close galaxy-DLA
pairs in the literature compared to those that are detected (e.g., Foltz et al., 1986;
Smith et al., 1989; Lowenthal et al., 1995; Bunker et al., 1999; Prochaska et al.,
2002; Kulkarni et al., 2006; Rahmani et al., 2010; Bouché et al., 2012). Therefore,
113
Strong Hi absorbers and galaxies at z = 3
the existing observational sample is most likely unrepresentative and cannot be
used for statistical analysis. Moreover, relatively high SFR detection thresholds
used in the observations strongly bias the measured impact parameters towards
larger values. In other words, it not easy to rule out the presence of fainter
and therefore non-detected galaxies at smaller impact parameters, for DLAs that
have observed bright galaxies around them. Finally, we note that SFRs of the observed galaxies that are associated with DLAs are typically & 10 M⊙ yr−1 (see
Table 4.1), which implies that those galaxies are massive and therefore large
systems. As we show in §4.3.5, the observed impact parameters of those DLAgalaxy pairs are consistent with the distribution of impact parameters around
similar galaxies in our simulation.
4.3.3 Distribution of Hi absorbers relative to halos
It is useful to compare the impact parameters that connect absorbers to their
neighboring galaxies with the size of their halos. We therefore define the normalized impact parameter of absorbers as the ratio of the impact parameter
and the virial radius (R200 ) of the host galaxies. Since the virial radius is well
defined only for central galaxies, we only consider those objects when associating absorbers to galaxies. As shown by the blue solid curve in the right panel of
Figure 4.3, strong Hi absorbers tend to be located closer to the center of haloes
as their NHI increases. Most LLSs are found within one virial radius of their host
galaxies and the majority of DLAs is found within a few tenths of viral radius
from their associated central galaxies. There is, however, a large scatter in the
normalized impact parameters at given NHI which is shown by the shaded area
around blue solid and red dotted curves in the right panel of Figure 4.3. As
implied by this figure, a non-negligible fraction of DLAs are expected at impact
parameters comparable to, or even larger than the virial radius of their associated central galaxies. This is, in part, due to neglecting satellite galaxies in
matching absorbers and galaxies (because they do not have a well defined virial
radius) which effectively associates the strong Hi absorbers that are near satellites to their closest central galaxies4 . For the same reason, the median impact
parameter at a given NHI increases by leaving the satellite galaxies out of the
analysis (not shown). Moreover, the complex and highly structured distribution
of strong Hi absorbers which often extends to distances beyond the virial radius
of central galaxies (see Figure 4.1) also contributes to the large scatter around
the median normalized impact parameter at a given NHI .
There is also an anti-correlation between the normalized impact parameter
of absorbers and their Hi column density, which is steeper than the relation we
see between the absolute impact parameters and NHI (shown in the right panel
of Figure 4.3). This difference is most pronounced at NHI & 1021 cm−2 where
4 Note that satellite galaxies (and hence their associated absorbers) are not necessarily within the
virial radius of their host galaxies, due to the non-spherical distribution of FoF structures. This is
shown in Figure 4.1.
114
Results and discussion
the impact parameter flattens with increasing NHI but the normalized impact
parameters still decrease steeply with NHI . This trend can be explained by the
contribution of very massive (and hence large) galaxies becoming increasingly
more dominant at very high Hi column densities (see §4.3.5). Therefore, there
are two competing trends: on the one hand, systems with higher Hi column
densities tend to be closer to the center of their neighboring (host) haloes. On
the other hand, the size of the haloes that are linked to those absorbers becomes
larger with increasing column density. These two effects neutralize each other
at very high Hi column densities, forming a less steep b − NHI relation at NHI >
1021 cm−2 compared to that of the lower column densities. By normalizing the
impact parameter of absorbers to the virial radii of their associated haloes, the
latter effect (i.e., the increasing size of the dominant haloes with increasing NHI )
is canceled out.
The trends we discussed earlier for the effect of varying the SFR threshold on
the impact parameter of absorbers hold qualitatively for the normalized impact
parameters as well. As the green dashed and red dotted curves in the right panel
of Figure 4.3 show, increasing the SFR threshold for galaxies that are considered
in matching absorbers to galaxies, results in larger normalized impact parameters. Despite the qualitatively similar trends, the differences between the normalized impact parameters for different SFR thresholds are smaller than the differences between absolute impact parameters. At 1017 < NHI < 1021 cm−2 , both the
green dashed and red dotted curves in the right panel of Figure 4.3 are nearly
flat. This implies that absorbers at those Hi column densities are several virial
radii away from their closest central galaxies (with SFR > 6 × 10−2 M⊙ yr−1 and
SFR > 1 M⊙ yr−1 , respectively). This supports our earlier statement that the association between bulk of strong Hi absorbers and galaxies that have relatively
high SFRs is not physical (particularly at NHI < 1021 cm−2 ), if fainter galaxies be
omitted.
4.3.4 Resolution limit in simulations
The simulated Hi distribution and its connection to galaxies are both sensitive
to the high resolution adopted in our simulations. The Hi column density distribution function is converged for LLSs and most DLAs at the resolution we use
in this work (Rahmati et al., 2013a), but this does not rule out that their distribution relative to galaxies is also resolution independent. The relative position of
galaxies that are identified as bound structures in a cosmological simulation is
mainly determined by the distribution of cosmological overdensities, and therefore, is not expected to be highly resolution dependent. However, by increasing
the resolution, more and more low-mass galaxies start to be identified which
might affect the typical distances between strong Hi absorbers and galaxies. Because of this, it is important to adopt a lower limit for galaxies that are taken into
account in our analysis to make sure our findings are not driven primarily by
the resolution we use. At the same time, this lower limit should be such that it
115
116
Strong Hi absorbers and galaxies at z = 3
Figure 4.4: The predicted median impact parameters for subsets of Hi absorbers associated with galaxies in different star formation
rate bins (left) and stellar mass bins (right) as a function of NHI , at z = 3. The shaded area around the solid green curves in the left
(right) panel shows the 15% − 85% percentile in the distribution of absorbers that are linked to galaxies with 1 < SFR < 16 M⊙ yr−1
(2 × 108 < M⋆ < 1010 M⊙ ). The dashed curves in both panels show the median impact parameter of all absorbers as a function of NHI . The
data points show the observed impact parameters for the confirmed DLA-galaxy pairs. The colored circles and squares around the data
points show the SFR/mass bin to which they belong. Note that the squares and circles, which show the two bins with the highest values
(of SFR or mass) respectively, are in agreement with our results. Because of a very efficient conversion of hydrogen atoms into molecules,
absorbers with NHI & 1022 cm−2 (indicated with the gray areas) are not expected to exist.
117
Results and discussion
Figure 4.5: The top panels show the predicted median impact parameters for subsets of Hi absorbers associated with galaxies in different
star formation rate bins (left) and stellar mass bins (right) as a function of NHI , at z = 3. The fraction of absorbers that are associated with
galaxies in different star formation rate bins (left) and stellar mass bins (right) to the total number of strong Hi absorbers as a function of
NHI , at z = 3. The star formation rate bins and stellar mass bins are identical to that of Figure 4.4. Because of a very efficient conversion of
hydrogen atoms into molecules, absorbers with NHI & 1022 cm−2 (indicated with the gray areas) are not expected to exist.
Strong Hi absorbers and galaxies at z = 3
Figure 4.6: The top-left panel shows the predicted cumulative distribution of normalized
impact parameter of absorbers in different NHI bins at z = 3, where only central galaxies
are taken into account. The other five panels show the cumulative distribution of different properties where all galaxies (with SFR > 0.004 M⊙ yr−1 ) are taken into account.
The top-right panel shows the cumulative distribution of halo masses and other panels
from middle-left to bottom right respectively show the cumulative distribution of impact
parameters, star formation rates (of associated galaxies), stellar masses and Hi masses.
Except for strong DLAs with 1021 < NHI < 1022 cm−2 , bulk of Hi absorbers reside
close to galaxies with similar properties (i.e., virial mass, stellar mass, gas mass and star
formation rate). However, their typical (normalized) impact parameters are significantly
different.
118
Results and discussion
allows for the inclusion of as many galaxies as possible, without throwing away
useful information.
We use SFR as a quantity to limit galaxies that are considered in our analysis,
which is the same quantity we used in the previous section as a proxy for the observational detectability of galaxies. The advantage of using the SFR of galaxies
to identify them is the better sampling it provides compared to other reasonable
quantities such as the stellar mass (see the discussion in §4.2 in Rahmati et al.,
2013b). We adopt SFR > 4 × 10−3 M⊙ yr−1 as a threshold and include only
galaxies that have SFRs above this threshold in the analysis. Most bound (sub)
structures (i.e., ≈ 98%) that are found in our simulation have SFRs above this
threshold.
It is reasonable to expect that an increase in the number density of galaxies
would decrease the typical distance between absorbers and galaxies. This would
mean that the cumulative number density of galaxies in the simulation is the
key factor that changes the typical distances between strong Hi absorbers and
galaxies. We show in Appendix D that this is indeed the case, and the b −
NHI relation is converged with resolution if the cumulative number density of
galaxies that are considered is the same. Unlike the position of galaxies, the
lowest SFRs in simulations are sensitive to the resolution and tend to decrease
with increasing the resolution for a fixed number density. As a result, we do
not expect that our result to remain the same if the resolution of our simulation
increases for the same adopted SFR threshold (i.e., SFR > 4 × 10−3 M⊙ yr−1 ).
Instead, we expect our results to be insensitive to the resolution if the adopted
threshold be such that the cumulative number density of galaxies be identical
to the corresponding value in our simulation. We note that the number density
of galaxies in our simulation that satisfy the adopted criterion is 0.5 galaxy per
comoving Mpc3 (i.e., equivalent to 31.5 galaxy per proper Mpc3 ).
Nonetheless, we found that the b − NHI relation for various SFR threshold is
converged if SFR > 6 × 10−2 M⊙ yr−1 . In other words, the total number density
of galaxies with SFR > 6 × 10−2 M⊙ yr−1 is not expected to change by increasing
the resolution of our simulation. This is another way of stating that for galaxies
that satisfy the above mentioned condition, the SFR is converged with respect to
the resolution.
4.3.5 Correlations between absorbers and various properties of
their associated galaxies
The gas content of galaxies is correlated with their other properties like stellar
mass, size and star formation rate. This implies also correlations between the
abundance and distribution of strong Hi absorbers and properties of galaxies
that are associated with them. For instance, as mentioned earlier, at a given
impact parameter, the typical stellar mass of host galaxies increases with increasing Hi column density. Similarly, at a fixed NHI , the typical host stellar
mass increases with increasing the impact parameter (see Figure 4.2). Moreover,
119
Strong Hi absorbers and galaxies at z = 3
Figure 4.7: The impact of SFR threshold on the cumulative distributions that are shown in
Figure 4.6. In each panel the blue and green solid curves show the cumulative distribution
of LLSs (i.e., 1018 < NHI < 1020 cm−2 ) and strong DLAs (i.e., 1021 < NHI < 1022 cm−2 ),
respectively. While the solid curves, that are identical to those shown in Figure 4.6, indicate the SFR threshold of 4 × 10−3 M⊙ yr−1 , the dotted curves show the result obtained
by imposing a SFR threshold of 6 × 10−2 M⊙ yr−1 . Panels from top-left to bottom-right
show the cumulative distribution of normalized impact parameters, halo masses, impact
parameters, SFRs, stellar masses and Hi masses, respectively. The detection of galaxies
with SFRs as low as SFR ∼ 10−2 M⊙ yr−1 future instruments such as MUSE.
120
Results and discussion
the LLSs are closely linked to low-mass galaxies with typical stellar masses of
M⋆ ∼ 108 M⊙ , in contrast to strong DLAs which tend to be associated with more
massive galaxies.
The above mentioned trends are shown more clearly in Figure 4.4. These
plots illustrate the b − NHI relation for subsets of Hi absorbers that are linked
to galaxies with different star formation rates and stellar masses (respectively
shown in the left and right panels). The colored curves in the left (right) panel
which show different bins of SFR (stellar mass), increasing by going from red to
blue indicate that the impact parameter of absorbers is increasing with the SFR
and mass of their associated galaxies. Note that there is a relatively large scatter
around the median impact parameter of each subset of absorbers, as shown with
the shaded area around the green curves. This result explains the color gradient
that we see in Figure 4.2 and is in good agreement with observations that are
shown with the red diamonds in Figure 4.4. The observed DLA-galaxy pairs
with reliable SFR or mass estimate are shown with the green circles and blue
cubes, respectively for the highest two bins. Note that the curves are close to the
observational results (see also Table 4.1).
We also show the median impact parameters of all absorbers using the
dashed curve in Figure 4.4. The comparison between this curve and colored solid
curves confirms our earlier conclusion (based on the color gradient in Figure 4.2)
that the impact parameter of absorbers with higher Hi column densities indicates that they are more likely to be around more massive galaxies with higher
SFRs. To illustrate this more clearly, we show also the fraction of Hi absorbers,
at any given column density, that are associated to galaxies with a particular
property. The result of this analysis for different SFR and mass bins is shown in
the left and right panels of Figure 4.5, respectively. These results clearly show
that the fraction of Hi absorbers associated with massive galaxies (which also
have high SFRs), is decreasing rapidly with decreasing their Hi column densities. Also, it is clear from the left panel of Figure 4.5 that most absorbers with
NHI < 1021 cm−2 are closely linked to galaxies with SFR < 6 × 10−2 M⊙ yr−1 , or
equivalently M⋆ < 108 M⊙ . While there are only about 20 − 30% of those systems associated with more massive galaxies with higher SFRs (i.e., M⋆ > 108 M⊙
and SFR > 6 × 10−2 M⊙ yr−1 ), the same galaxies are associated with a large
fraction of strong DLAs with NHI > 1021 cm−2 . We reiterate that these results are not changing by increasing the resolution and only the lowest SFRs are
expected to be reduced at higher resolutions (shown with the red regions in
the left panel of Figure 4.5), which is not expected to affect the fraction of absorbers that are associated with well-resolve galaxies that have M⋆ > 108 M⊙
and SFR > 6 × 10−2 M⊙ yr−1 .
We note that our results are in agreement with Tescari et al. (2009) and
van de Voort et al. (2012a). Namely, those authors found that the larges fraction of Hi absorbers with NHI < 1021 cm−2 are in very low-mass haloes with
M200 < 1010 M⊙ (i.e., M⋆ . 108 M⊙ ; see the orange regions in the right panel
of Figure 4.5). Moreover, van de Voort et al. (2012a) found that at higher Hi
121
Strong Hi absorbers and galaxies at z = 3
column densities, the contribution of more massive haloes with M200 > 1011 M⊙
(i.e., M⋆ & 109 M⊙ ; see the green regions in the right panel of Figure 4.5) rapidly
increases.
To investigate further the distribution of galaxy properties for absorbers with
different Hi column densities, we split absorbers in different NHI bins. Then,
for galaxies that are associated with the absorbers in that bin, we construct the
cumulative distribution of different properties. The result of this exercise is
shown in Figure 4.6. The top-left panel of this figure shows the cumulative
distribution of normalized impact parameters, where in contrary to the other five
panels, only central galaxies are taken into account (because satellite galaxies do
not have well defined viral radius required for the normalization). The top-right
panel shows the cumulative distribution of halo masses. Since satellite galaxies
are in the halo of their centrals, they do not have well defined halo masses and as
the starting point of the cumulative distribution functions in the top-right panel
indicates, ≈ 30% of absorbers are associated to satellite galaxies. The other four
panels from middle-left to the bottom-right respectively show the cumulative
distribution of impact parameters, star formation rates, stellar masses and Hi
masses.
Comparing the cumulative distribution of (normalized) impact parameters
with other four panels in this figure indicates that the Hi column density of
absorbers is mostly sensitive to their projected distance from their associated
galaxies (i.e., the impact parameter), compared to other properties, such as the
stellar mass, SFR, Hi mass or halo size of their associated galaxies. As shown in
the top-left panel, while more than 50% of strong DLAs are within R . 0.1 R200 ,
most weak LLSs with NHI . 1018 cm−2 are likely to be beyond the virial radius
of their host galaxies.
Another result that is shown in Figure 4.6 is that the fraction of absorbers that
are linked to more massive galaxies is increasing by increasing their Hi column
densities. The same trend is visible for the SFRs, halo masses and Hi masses.
However, as the distinction between strong DLAs NHI > 1022 cm−2 (blue solid
curves) and lower Hi column densities show, galaxies associated with strong
DLAs have distinct distributions in SFR, Hi mass, stellar mass and halo mass.
For instance, the median SFR of galaxies that are associated with other strong
Hi absorbers is ≈ 10 times lower than the same quantity for galaxies associated
with strong DLAs. Similar trend also holds for Hi masses, stellar and halo
masses. Comparing the median properties (i.e., at CDF = 0.5) shows that only
≈ 10% of strong DLAs are linked to galaxies that are typically associated with
weaker Hi absorbers. This suggests that strong DLAs with NHI > 1021 cm−2 are
preferentially linked to the most massive galaxies while other Hi absorbers are
distributed among more abundant galaxies with lower masses.
The middle-right panel of Figure 4.6 is particularly useful to understand
the difficulty of finding the observed galaxy counterparts of strong Hi absorbers. The distribution of SFRs shown in this plot can be used to predict the chance by which a galaxy counterpart can be detected above a cer122
Results and discussion
123
Figure 4.8: Fraction of strong Hi systems that are associated to galaxies with SFR > 10 M⊙ yr−1 (left) and SFR > 1 M⊙ yr−1 (right) in our
simulation at z = 3. Curves with different line styles and colors show results that are obtained using different methods for associating Hi
absorbers to galaxies that are above the imposed SFR threshold: blue solid curves are obtained by associating Hi absorbers to their closest
galaxies, where all the simulated galaxies are taken into account (see Figures 4.4 and 4.5); green long-dashed curves show the fraction of
absorbers that reside within the virial radius of the selected galaxies, and finally, red short-dashed curves show the fraction of absorbers
that are within 300 proper kpc from the selected galaxies. For comparison, the colored bars show Rudie et al. (2012) findings for fraction
of absorbers within 300 proper kpc from galaxies with similar SFRs that are shown here. The dark orange part of the bar is the observed
fractions and the light orange part shows the correction for missing galaxies in their spectroscopic sample. Predicted fractions are in
excellent agreement with observations and show a large fraction of strong Hi absorbers are less than 300 proper kpc away from galaxies
with SFR > 1 M⊙ yr−1 . However, most of those systems are far beyond the virial radius of those galaxies and are associated with less
massive objects with lower SFRs which are below the detection limit of observations.
Strong Hi absorbers and galaxies at z = 3
tain detection threshold. For instance, if the detection threshold is equivalent to SFR = 1 M⊙ yr−1 , the chance of detecting the galaxy counterpart of a
strong DLA with NHI ≈ 1022 cm−2 is 3 in 10. In other words, using this detection threshold results in non-detection rate of 70%. For weak DLAs (i.e.,
1020 < NHI < 1021 cm−2 ) the non-detection rate would be even higher at 90%.
This explains the large number of non-detections in observational studies (e.g.,
Foltz et al., 1986; Smith et al., 1989; Lowenthal et al., 1995; Bunker et al., 1999;
Prochaska et al., 2002; Kulkarni et al., 2006; Rahmani et al., 2010; Bouché et al.,
2012).
As mentioned before, with the typical detection threshold of observations at
z ∼ 3, only galaxies that have SFR > 1 − 10 M⊙ yr−1 can be identified. On the
other hand, galaxies are resolved to much lower SFRs in our simulation and it
is not straightforward to compare current observations and the results shown in
Figure 4.6. With the advent of future surveys like MUSE (Bacon et al., 2010), the
accessible Lyα detection thresholds would be pushed to lower SFRs which allows
the identification of galaxies with SFRs as low as SFR ∼ 10−2 − 10−1 M⊙ yr−1
at z ∼ 3. These deep observations can be used to identify faint galaxies associated with strong Hi absorbers and analyze the cumulative distribution of
their properties. However, the results would still depend on the accessible detection threshold and might be different from what is shown in Figure 4.6. To
address this issue, in Figure 4.7 we compare the cumulative distributions for two
different detection thresholds. Panels in this figure are identical to Figure 4.6,
but show the result for only two Hi column density bins. The green and blue
curves, respectively, represent the Hi absorbers (or galaxies associated to them)
that have 1018 < NHI < 1020 cm−2 (i.e., LLSs) and 1021 < NHI < 1022 cm−2
(i.e., strong DLAs). The solid curves show our fiducial detection threshold of
SFR > 4 × 10−3 M⊙ yr−1 , which has been used in Figure 4.6. The dotted curves
indicate a higher detection threshold of SFR > 6 × 10−2 M⊙ yr−1 , which is comparable to what would be accessible using deep MUSE observations. As the
difference between the solid and dotted curves show, the predicted/observed
distributions are sensitive to the detection threshold. In other words, the bias
introduced by the limited detection threshold should be taken into account for
interpreting/modeling the observed distributions.
4.3.6 Are most strong Hi absorbers at z ∼ 3 around LymanBreak galaxies?
In this work, we adopted the absorber-centered point of view in which each absorber is associated with its closest galaxy. This approach is particularly efficient
to establish a relationship between rare absorbers and their galaxy counterparts.
As we showed in the previous sections, our simulation predicts that for most Hi
absorbers, nearby galaxies are low-mass objects (i.e., M⋆ ≈ 108 M⊙ ) with low
SFRs. We showed that this prediction is consistent with the impact parameters
of observed DLA-galaxy counterparts, and more importantly, the high incident
124
Results and discussion
rate of finding no detectable galaxy close to DLAs. There is, however, an alternative approach which is to search for Hi absorbers around galaxies (i.e., galaxycentered point of view) (e.g., Steidel et al., 1995; Adelberger et al., 2003, 2005;
Hennawi & Prochaska, 2007; Steidel et al., 2010; Rakic et al., 2012; Rudie et al.,
2012; Prochaska et al., 2013). While the two approaches are complementary, their
results might seem inconclusive. For instance, using galaxy-centered approach,
Rudie et al. (2012) found that at z ≈ 2 − 3 most Lyman Limit absorbers (i.e.,
NHI ≈ 1017 cm−2 ) are within an impact parameter of 300 proper kpc, and within
300 km/s LOS velocity difference with respect to rest-frame UV-selected starforming galaxies (see their Figure 30). Given that the typical galaxy mass in
their sample is M⋆ ≈ 1010 M⊙ , one might conclude that their result is in conflict
with our finding that most strong Hi absorbers (i.e., LLSs and DLAs) are closely
associated with galaxies with M⋆ ≈ 108 M⊙ .
To understand the source of this discrepancy, first we note that the 300 proper
kpc transverse distance and the 300 km/s LOS velocity difference that are adopted by Rudie et al. (2012), are, respectively, 300 and 50 per cent larger. than
the virial radius and the circular velocity which is expected for the host haloes
of their galaxies (Rakic et al., 2012, 2013). In other words, the region Rudie et al.
(2012) define as the “circumgalactic medium” lies well beyond the virial radius
of the haloes that are thought to host their galaxies.
In addition, it is important to note that galaxies are strongly clustered and
low-mass galaxies prefer to live around more massive galaxies. This implies that
most strong Hi absorbers are also likely to be found close to massive galaxies,
while they actually belong to their low-mass parent galaxies. As a result, searching for Hi absorbers within a reasonably large radius around massive, and hence
easily observable galaxies, recovers a large fraction of the existing strong Hi absorbers. This effect can be seen in the bottom-right panel of Figure 4.1, which
shows the Hi column density distribution around a galaxy with M⋆ = 1010 M⊙ .
While the maximum projected distance between the galaxy and Hi absorbers
that are shown around it is less than 300 proper kpc, most of those absorbers
have low-mass galaxies very close to them.
The aforementioned arguments are illustrated more quantitatively in Figure
4.8, which shows the fraction of strong Hi absorbers in our simulation at z =
3 that are in the vicinity of galaxies with star formation rates comparable to
that of LBG galaxies at the similar redshifts5 . The red dashed curve in the left
(right) panels shows the fraction of Hi absorbers that are within the impact
parameter of 300 proper kpc from galaxies with SFR > 10 M⊙ yr−1 (SFR >
1 M⊙ yr−1 ), as a function of NHI . This result is in excellent agreement with the
fractions Rudie et al. (2012) found (shown with the colored bars) and predicts
that the fraction of absorbers that are within 300 proper kpc from these galaxies
is roughly constant for 1016 < NHI < 1021 cm−2 . However, only a small fraction
of LLSs that are within the impact parameter of 300 proper kpc from those
5
The star formation rate of galaxies used in Rudie et al. (2012) is between several to a few hundreds of M⊙ yr−1 .
125
Strong Hi absorbers and galaxies at z = 3
galaxies are also within their virial radii (shown with the green long-dashed
curves). As the blue solid curves show, even smaller fractions of LLSs remain
associated to those galaxies if we account for galaxies with lower SFRs, which
are typically closer to LLSs but are too faint to be observed.
4.4 Summary and conclusions
We have used cosmological simulations that have been post-processed using accurate radiative transfer corrections that account for photoionization by the UVB
and recombination radiation to investigate the relation between strong Hi absorbers (i.e., LLSs and DLAs) and galaxies at z = 3. The simulation we used for
our study has been shown to reproduce the observed Hi column density distribution function accurately (Altay et al., 2011; Rahmati et al., 2013a). After identifying sight-lines with high Hi column densities (i.e., NHI > 3 × 1016 cm−2 ) and
calculating the line-of-sight velocity of absorbers, we used a procedure similar
to that used in observational studies to associate absorbers with nearby galaxies. Namely, we associated each strong Hi absorber to the galaxy which has the
shortest transverse distance to the absorber and a line-of-sight velocity difference within ±300 km s−1 . Associating all strong Hi absorbers in the periodic
simulation box to galaxies close to them allowed us to predict statistical trends
between the strength of the absorbers and the distance to, and properties of, the
galaxies.
Among the various dependencies we studied in this work, we found that
the anti-correlation between the Hi column density of absorbers and the transverse distance that connects them to their neighbouring galaxies (i.e., the impact
parameter) to be the strongest. While LLSs have impact parameters & 10 proper
kpc, DLAs are typically within a few proper kpc from the nearest galaxies. Relative to the virial radius of the halo that hosts the nearest central galaxy, LLSs
have typical impact parameters & 1 R200 , while DLAs are typically ∼ 10 times
closer to the center of the nearest halo (i.e. . 0.1 R200 ). The predicted strong
anti-correlation between the impact parameter of strong Hi absorbers and their
Hi column densities is in agreement with observations and previous work. We
also found a relatively large scatter around the median impact parameter of absorbers, at a given NHI , due to the complex geometry of gas distribution around
galaxies, and also the variation in the size and gas content of the galaxies that
are producing the absorbers.
We predict that most strong Hi absorbers are closely associated with relatively low-mass galaxies, M⋆ . 108 M⊙ , but that the fraction of strong Hi
absorbers that are linked to more massive galaxies increases rapidly with the
Hi column density. This correlation between column density and galaxy mass
is particularly pronounced for strong DLAs, NHI > 1021 cm−2 , the majority of
which are associated with galaxies with M⋆ & 109 M⊙ . We analyzed different properties of galaxies that are linked to strong Hi absorbers with different
126
Summary and conclusions
Hi column densities and found similar trends as we found for stellar mass:
most strong Hi absorbers are closely associated with galaxies that have relatively low halo masses, low SFRs and low Hi masses, but strong DLAs (i.e.,
NHI > 1021 cm−2 ) are typically linked to more massive galaxies with significantly higher halo masses, SFRs and Hi masses.
By analyzing subsets of strong Hi absorbers for which the associated galaxies have specific properties, we found that observationally confirmed DLAgalaxy pairs that have measured mass or SFR, have impact parameters that
are in good agreement with our predictions. We stress, however, that the majority of DLAs are predicted to be more closely associated with galaxies that
are at smaller impact parameters, but are too faint to be detected with current surveys. Hence, the masses and impact parameters of the observed galaxy
counterparts of DLAs are both biased high. This is consistent with the large
number of non-detections in observational campaigns that searched for galaxies
close to DLAs (e.g., Foltz et al., 1986; Smith et al., 1989; Lowenthal et al., 1995;
Bunker et al., 1999; Prochaska et al., 2002; Kulkarni et al., 2006; Rahmani et al.,
2010; Bouché et al., 2012).
In order to facilitate the comparison between cosmological simulations and
observations, we provides statistics on DLA-galaxy pairs for a few different SFR
thresholds. However, a proper comparison requires observational studies aiming to find galaxies close to DLAs, to report their detection limit, the maximum
allowed velocity separation, and either impact parameter of the nearest detectable galaxy or, in the case of non-detection (which must always be reported), the
maximum impact parameter that has been searched. For the few studies that report such information (e.g., Teplitz et al., 1998; Mannucci et al., 1998), we found
a good agreement with our simulation.
Interestingly, some recent observational studies indicate that strong Hi absorbers at z ∼ 2 − 3 are associated with surprisingly massive galaxies. In
particular, Rudie et al. (2012) studied the distribution of Hi absorbers around
a sample of rest-frame UV-selected Lyman-Break galaxies (LBG) with typical
masses of M⋆ > 1010 M⊙ at z ∼ 2 − 3 and found that nearly half of absorbers
with 1016 < NHI < 1017 cm−2 reside within a line-of-sight velocity difference
of 300 km s−1 and a transverse seperation of 300 proper kpc from a LBG, a
region they labelled the circumgalactic medium. This result appears to contradict our finding that most strong Hi absorbers are associated with galaxies with
M⋆ . 108 M⊙ . We demonstrated, however, that even though the absorbers are
physically most closely associated with low-mass galaxies, these galaxies cluster
strongly around galaxies as massive as LBGs, which is sufficient to reproduce the
observations of Rudie et al. (2012). Moreover, we noted that the clustering does
not even need to be that strong: Since 300 proper kpc and 300 km s−1 exceed
the virial radius and the circular velocity of the haloes thought to host LBGs, by
more than 300 and 50 per cent, respectively (e.g., Rakic et al., 2012, 2013), nearly
all the volume of the “circumgalactic medium” lies beyond the virial radius if
we employ the definition of Rudie et al. (2012).
127
REFERENCES
Future deep observational surveys using new instruments (e.g., MUSE;
Bacon et al., 2010) will improve this situation by detecting fainter galaxies. However, missing faint galaxies is a generic feature for any survey that has a finite
detection limit and that takes an absorber-centered point of view. The incompleteness problem can be overcome by taking a galaxy-centered point of view,
but this approach is inefficient for rare absorbers such as the interesting strong
Hi systems we studies here. Moreover, while galaxy-centered surveys can measure the statistical distribution of absorbers (such as their covering factor), we still
need to avoid interpreting the selected galaxy as the counterpart to any absorber
that is detected. Absorber-centered surveys will probably remain the most efficient way to build up large numbers of galaxy-DLA paris. Even with modest
detection limits, such surveys provide highly valuable constraints on the relation between absorbers and galaxies, provided all non-detections are reported
and that the detection limits and the maximum possible impact parameter are
clearly specified.
Acknowledgments
We would like to thank A. Pawlik, X. Prochaska, M. Raičevic̀ and M. Shirazi
for useful discussion, reading an earlier version of this chapter and providing
us with comments that improved the text. The simulations presented here were
run on the Cosmology Machine at the Institute for Computational Cosmology in
Durham (which is part of the DiRAC Facility jointly funded by STFC, the Large
Facilities Capital Fund of BIS, and Durham University) as part of the Virgo Consortium research programme. This work was sponsored with financial support
from the Netherlands Organization for Scientific Research (NWO), also through
a VIDI grant and an NWO open competition grant. We also benefited from
funding from NOVA, from the European Research Council under the European
Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement 278594-GasAroundGalaxies and from the Marie Curie Training Network
CosmoComp (PITN-GA-2009-238356).
References
Adelberger, K. L., Steidel, C. C., Shapley, A. E., & Pettini, M. 2003, ApJ, 584, 45
Adelberger, K. L., Shapley, A. E., Steidel, C. C., et al. 2005, ApJ, 629, 636
Aguirre, A., Dow-Hygelund, C., Schaye, J., & Theuns, T. 2008, ApJ, 689, 851
Altay, G., Theuns, T., Schaye, J., Crighton, N. H. M., & Dalla Vecchia, C. 2011,
ApJL, 737, L37
Altay, G., Theuns, T., Schaye, J., Booth, C. M., & Dalla Vecchia, C. 2013,
arXiv:1307.6879
Bacon, R., Accardo, M., Adjali, L., et al. 2010, Proceedings of the SPIE, 7735
Blitz, L., & Rosolowsky, E. 2006, ApJ, 650, 933
128
REFERENCES
Booth, C. M., & Schaye, J. 2009, MNRAS, 398, 53
Bouché, N., Murphy, M. T., Péroux, C., et al. 2012, MNRAS, 419, 2
Bunker, A. J., Warren, S. J., Clements, D. L., Williger, G. M., & Hewett, P. C.
1999, MNRAS, 309, 875
Cen, R. 2012, ApJ, 748, 121
Chabrier, G. 2003, PASP, 115, 763
Christensen, L., Wisotzki, L., Roth, M. M., et al. 2007, A&A, 468, 587
Dalla Vecchia, C., & Schaye, J. 2008, MNRAS, 387, 1431
Di Matteo, T., Springel, V., & Hernquist, L. 2005, Nature, 433, 604
Djorgovski, S. G., Pahre, M. A., Bechtold, J., & Elston, R. 1996, Nature, 382, 234
Dolag, K., Borgani, S., Murante, G., & Springel, V. 2009, MNRAS, 399, 497
Erkal, D., Gnedin, N. Y., & Kravtsov, A. V. 2012, ApJ, 761, 54
Foltz, C. B., Chaffee, F. H., Jr., & Weymann, R. J. 1986, AJ, 92, 247
Font-Ribera, A., Miralda-Escudé, J., Arnau, E., et al. 2012, JCAP, 11, 59
Fumagalli, M., O’Meara, J. M., Prochaska, J. X., & Kanekar, N. 2010, MNRAS,
408, 362
Fumagalli, M., Prochaska, J. X., Kasen, D., et al. 2011, MNRAS, 418, 1796
Fynbo, J. U., Møller, P., & Warren, S. J. 1999, MNRAS, 305, 849
Fynbo, J. P. U., Laursen, P., Ledoux, C., et al. 2010, MNRAS, 408, 2128
Fynbo, J. P. U., Ledoux, C., Noterdaeme, P., et al. 2011, MNRAS, 413, 2481
Fynbo, J. P. U., Geier, S., Christensen, L., et al. 2013, arXiv:1306.2940
Gardner, J. P., Katz, N., Hernquist, L., & Weinberg, D. H. 1997, ApJ, 484, 31
Gardner, J. P., Katz, N., Hernquist, L., & Weinberg, D. H. 2001, ApJ, 559, 131
Haardt F., Madau P., 2001, in Clusters of Galaxies and the High Redshift Universe Observed in X-rays, Neumann D. M., Tran J. T. V., eds.
Haas, M. R., Schaye, J., Booth, C. M., et al. 2012, arXiv:1211.1021
Haehnelt, M. G., Steinmetz, M., & Rauch, M. 1998, ApJ, 495, 647
Hennawi, J. F., & Prochaska, J. X. 2007, ApJ, 655, 735
Krogager, J.-K., Fynbo, J. P. U., Møller, P., et al. 2012, MNRAS, 424, L1
Krogager, J.-K., Fynbo, J. P. U., Ledoux, C., et al. 2013, arXiv:1304.4231
Komatsu, E., et al. 2011, ApJS, 192, 18
Krumholz, M. R., Ellison, S. L., Prochaska, J. X., & Tumlinson, J. 2009, ApJL,
701, L12
Kulkarni, V. P., Woodgate, B. E., York, D. G., et al. 2006, ApJ, 636, 30
Lara-López, M. A., Cepa, J., Bongiovanni, A., et al. 2010, A&A, 521, L53
Lowenthal, J. D., Hogan, C. J., Green, R. F., et al. 1995, ApJ, 451, 484
Mannucci, F., Thompson, D., Beckwith, S. V. W., & Williger, G. M. 1998, ApJL,
501, L11
Mannucci, F., Cresci, G., Maiolino, R., Marconi, A., & Gnerucci, A. 2010,
MNRAS, 408, 2115
McDonald, P., & Miralda-Escudé, J. 1999, ApJ, 519, 486
Moller, P., & Warren, S. J. 1998, MNRAS, 299, 661
Möller, P., Warren, S. J., Fall, S. M., Fynbo, J. U., & Jakobsen, P. 2002, ApJ, 574,
51
129
REFERENCES
Möller, P., Fynbo, J. P. U., & Fall, S. M. 2004, A&A, 422, L33
Monier, E. M., Turnshek, D. A., & Rao, S. 2009, MNRAS, 397, 943
Nagamine, K., Springel, V., & Hernquist, L. 2004, MNRAS, 348, 421
Noterdaeme, P., Laursen, P., Petitjean, P., et al. 2012, A&A, 540, A63
Pawlik, A. H., & Schaye, J. 2008, MNRAS, 389, 651
Pawlik, A. H., & Schaye, J. 2011, MNRAS, 412, 1943
Péroux, C., Bouché, N., Kulkarni, V. P., York, D. G., & Vladilo, G. 2011, MNRAS,
410, 2237
Pontzen, A., Governato, F., Pettini, M., et al. 2008, MNRAS, 390, 1349
Prochaska, J. X., & Wolfe, A. M. 1997, ApJ, 487, 73
Prochaska, J. X., & Wolfe, A. M. 1998, ApJ, 507, 113
Prochaska, J. X., Gawiser, E., Wolfe, A. M., et al. 2002, AJ, 123, 2206
Prochaska, J. X., Gawiser, E., Wolfe, A. M., Cooke, J., & Gelino, D. 2003, ApJS,
147, 227
Prochaska, J. X., & Wolfe, A. M. 2009, ApJ, 696, 1543
Prochaska, J. X., Hennawi, J. F., & Simcoe, R. A. 2013, ApJL, 762, L19
Rahmani, H., Srianand, R., Noterdaeme, P., & Petitjean, P. 2010, MNRAS, 409,
L59
Rahmati, A., Pawlik, A. H., Raičevic̀, M., & Schaye, J. 2013a, MNRAS, 430, 2427
Rahmati, A., Schaye, J., Pawlik, A. H., & Raičevic̀, M. 2013b, MNRAS, 431, 2261
Rakic, O., Schaye, J., Steidel, C. C., et al. 2013, arXiv:1306.1563
Rakic, O., Schaye, J., Steidel, C. C., & Rudie, G. C. 2012, ApJ, 751, 94
Rao, S. M., Belfort-Mihalyi, M., Turnshek, D. A., et al. 2011, MNRAS, 416, 1215
Razoumov, A. O., Norman, M. L., Prochaska, J. X., & Wolfe, A. M. 2006, ApJ,
645, 55
Rudie, G. C., Steidel, C. C., Trainor, R. F., et al. 2012, ApJ, 750, 67
Schaye, J. 2001a, ApJL, 559, L1
Schaye, J. 2001c, ApJL, 562, L95
Schaye, J., & Dalla Vecchia, C. 2008, MNRAS, 383, 1210
Schaye, J., Dalla Vecchia, C., Booth, C. M., et al. 2010, MNRAS, 402, 1536
Smith, H. E., Cohen, R. D., Burns, J. E., Moore, D. J., & Uchida, B. A. 1989, ApJ,
347, 87
Springel, V. 2005, MNRAS, 364, 1105
Steidel, C. C., Pettini, M., & Hamilton, D. 1995, AJ, 110, 2519
Steidel, C. C., Erb, D. K., Shapley, A. E., et al. 2010, ApJ, 717, 289
Tescari, E., Viel, M., Tornatore, L., & Borgani, S. 2009, MNRAS, 397, 411
Teplitz, H. I., Malkan, M., & McLean, I. S. 1998, ApJ, 506, 519
van de Voort, F., Schaye, J., Altay, G., & Theuns, T. 2012a, MNRAS, 421, 2809
van de Voort, F., & Schaye, J. 2012b, MNRAS, 423, 2991
Wiersma, R. P. C., Schaye, J., Theuns, T., Dalla Vecchia, C., & Tornatore, L.
2009a, MNRAS, 399, 574
Wiersma, R. P. C., Schaye, J., & Smith, B. D. 2009b, MNRAS, 393, 99
Wolfe, A. M., Turnshek, D. A., Smith, H. E., & Cohen, R. D. 1986, ApJS, 61, 249
Wolfe, A. M., Gawiser, E., & Prochaska, J. X. 2005, ARA&A, 43, 861
130
Appendix A: The maximum LOS Velocity difference
Wolfe, A. M., Prochaska, J. X., Jorgenson, R. A., & Rafelski, M. 2008, ApJ, 681,
881
Yanny, B., York, D. G., & Williams, T. B. 1990, ApJ, 351, 377
Appendix A: Choosing the maximum allowed LOS
Velocity difference
As discussed in §4.2.4, for associating the Hi absorbers to galaxies that are close
to them we take into account their relative LOS velocities. This allows us to
take out systems that appear to be close in projection but are apart by large
physical distances. In analogy to observational studies, we calculate LOS velocities of Hi absorbers and galaxies. If the difference between these two LOS
velocities be larger than a minimum value, then we do not associate them as
counterparts even though they are very close in projection. One might think
that the above mentioned allowed minimum velocity difference between Hi absorbers and galaxies, should be as small as possible to minimize the projection
effect. In reality, however, the peculiar velocity of the Hi gas around galaxies
is not negligible and is controlled by complex processes like accretion and outflows. If the allowed LOS velocity differences are too small, peculiar velocity
of Hi absorbers around galaxies would be miss-interpreted as large LOS sight
distances. As shown in Figure 4.9, our exercise show that the median impact
parameter of absorbers as a function of NHI is converged for maximum LOS
velocity difference of ∆VLOS, max > 100 km/s and the scatter around the median
impact parameters is converged for ∆VLOS, max > 300 km/s. Therefore, we adopt ∆VLOS, max = 300 km/s which is also consistent with recent observational
studies (e.g., Rakic et al., 2012; Rudie et al., 2012).
Appendix B: Impact of feedback
The evolution of gas and stars is determined by complex baryonic interactions that are often modeled in cosmological simulations by combining various physically motivated and empirical ingredients. In this context, different
feedback mechanisms can change both the distribution of gas around galaxies
with a given mass and the abundance of galaxies with different masses (e.g.,
van de Voort & Schaye, 2012b; Haas et al., 2012). As a result, the strength and
details of various feedback mechanisms change the distribution of the Hi absorbers (Altay et al., 2013) and may also affect the relative distribution of Hi absorbers and galaxies. To quantify the impact of feedback on our results, we compare the relation we found in our reference model between the impact parameter
of absorbers and their NHI to the same relation in similar simulations that use
different feedback prescriptions. Figure 4.10 shows this comparison between the
131
Strong Hi absorbers and galaxies at z = 3
Figure 4.9: The impact of changing the maximum allowed LOS velocity difference
between Hi absorbers and galaxies to associate them with each other. The blue solid
curve in all panels shows the median impact parameter of absorbers in our simulation at
z = 3 where our fiducial value of ∆VLOS, max = 300 km/s is adopted and the shaded area
around the blue solid curve shows the 15% − 85% percentiles. In each panel, the same
result but using a different ∆VLOS, max is shown by red dashed and dotted curves which,
respectively, show the median and the 15% − 85% percentiles. The ∆VLOS, max that is
compared with our fiducial choice of 300 km/s is varying from 50 km/s from the top-left
panel to 500 km/s in the bottom-right panel. While the median impact parameters are
converged for ∆VLOS, max > 100 km/s, both the median impact parameters and scatter
around it are converged for ∆VLOS, max > 300 km/s.
reference simulation at z = 3 and two bracketing models that use AGN feedback
and do not have SNe feedback and metal cooling. The solid green curve in the
left panel of Figure 4.10 shows the reference model while the red dashed curve
and blue dot-dashed curves respectively indicate the simulation with AGN and
the simulation without SNe feedback and metal cooling (NOSN_NOZCOOL).
The relation between the normalized impact parameter and NHI is shown in the
right panel of Figure 4.10, where only central galaxies are taken into account for
the matching process.
The absorbers with NHI . 1021 cm−2 in the AGN simulation have typical
impact parameters which are larger than in the reference model. At very high
Hi column densities however, the two models are similar except that there are
fewer very high NHI systems in the AGN simulation. The same trend is seen
132
Appendix B: Impact of feedback
Figure 4.10: The (normalized) impact parameter as a function of NHI at z = 3 for simulations with different feedback models is shown in the (right) left panel. The green solid,
red dashed and blue dot-dashed curves respectively show the reference simulation, the
impact of adding AGN feedback and the result of turning off SNe feedback and metal
cooling. While the impact parameters are sensitive to the adopted feedback prescription, the differences are much smaller than the intrinsic scatter caused by the complex
geometry of gas distribution around galaxies with a wide range of sizes.
Figure 4.11: The (normalized) impact parameter as a function of NHI at z = 3 for the
REFL06N128 simulation with different photoionization models is shown in the (right) left
panel. The red dashed curve shows the result when only the UVB and recombination
radiations (RR) are present while the blue solid curve indicates the result of including
local stellar radiation (LSR). The data points in the left panel are observations. Photoionization from local stellar radiation reduces the impact parameter of DLAs by up to 50%
and increases the typical impact parameter of Lyman Limit systems by similar amount.
133
Strong Hi absorbers and galaxies at z = 3
in the normalized impact parameters and is consistent with previous studies
implying that AGN feedback pushes the gas away from the center of haloes by
preventing the rapid cooling of the accreted gas and moderates the formation
of the most massive galaxies which are the main contributors to very high Hi
column densities (e.g., Di Matteo et al., 2005; Booth & Schaye, 2009; Schaye et al.,
2010; van de Voort & Schaye, 2012b; Haas et al., 2012). We also found that the
contribution of galaxies with different stellar masses and SFRs as a function of
NHI in the AGN simulation is very similar to the reference model (not shown).
The only different between the two simulations in this context is that the number
of strong Hi absorbers in very massive galaxies (with M⋆ & 1010 M⊙ ; SFR &
1 M⊙ yr−1 ) becomes smaller while the contribution of less massive galaxies(with
M⋆ . 109 M⊙ ; SFR . 10−1 M⊙ yr−1 ) remains intact.
In contrast, turning off the SNe feedback and metal cooling reduces the
typical impact parameters of Hi absorbers at almost all Hi column densities.
This can be understood by noting that the absence of metal cooling reduces the
amount of cold gas available for accretion into galaxies, forcing the dense gas
to be found closer to the center of galaxies. Moreover, the lack of SNe feedback
allows the gas to falls more regularly along well defined filaments without being
redistributed by SNe driven outflows. The lack of SNe feedback also allows the
formation of dens cores in the center of lower mass galaxies which are interrupted easier by SNe explosions. As a result, the contribution of smaller galaxies
(with M⋆ . 109 M⊙ ; SFR < 1 M⊙ yr−1 ) to the abundance of high NHI systems
increases. This results in smaller impact parameter at high Hi column densities compared to the reference simulation, while leave their normalized impact
parameters nearly identical.
All the above mentioned differences in the impact parameters of strong Hi
absorbers due to variations in feedback are much smaller than the intrinsic scatter in the expected impact parameter at a given NHI which is caused by the
complex geometry of gas distribution around galaxies with different sizes. The
strong anti-correlation between the impact parameter of absorbers and their Hi
column densities is present and remains similar despite large variations in feedback mechanisms. Therefore, we conclude that feedback variation has a minor
impact on our main results, although observations agree better with models that
have stronger feedbacks.
Appendix C: Impact of local stellar radiation
After the reionization of the Universe at z ∼ 6, ionizing background radiation
produced by the radiation coming from all sources in the Universe keeps hydrogen atoms mostly ionized. While on large scale, e.g., in the intergalactic
medium, this radiation is uniform, it becomes highly non-uniform close to radiation sources. In particular, as was shown in Rahmati et al. (2013b), for absorbers
with NHI & 1017 cm−2 the radiation from local stellar radiation becomes import134
Appendix D: Resolution tests
ant. Since we study the distribution of such strong Hi absorbers, it is important
to investigate the impact of local stellar radiation on the results of this work,
where we neglect it.
The impact of local stellar radiation on the spatial distribution of strong Hi
absorbers for the REFL06N128 simulation at z = 3 is shown in Figure 4.11. The
blue solid curve shows the result of radiative transfer calculation by taking into
account the photoionization of the UVB, recombination radiation and local stellar radiation as explained in Rahmati et al. (2013b), while the red dashed curve
indicates the result of including only the photoionization of the UVB and recombination radiation in the same simulation. As can be seen by comparing
the two curves in the left panel of Figure 4.11, local stellar radiation can change
the impact parameters of strong Hi absorbers by up to 50%. The median impact parameters of DLAs with NHI ∼ 1021 cm−2 is reduced because their Hi
column densities decreases due to additional photoionization by local stellar radiation. For Lyman Limit systems on the other hand, the ionization from local
stellar radiation mainly affect systems that are closer to the galaxies compared
to absorbers that are at larger distances. This results in an increase in the median impact parameter of absorbers at a given NHI by decreasing the Hi column
density of absorbers at shorter impact parameters. At lower Hi column densities (i.e., NHI . 1017 cm−2 ) where the effect of local stellar radiation is negligible
(Rahmati et al., 2013b), the impact parameters remain intact 6 . We conclude that
local stellar radiation is not changing the results we present in this work.
Appendix D: Resolution tests
We use three different cosmological simulations that have identical box sizes,
but different resolutions. These simulations are part of the OWLS project
(Schaye et al., 2010) and have cosmological parameters that are consistent with
WMAP year-3 values (i.e., {Ωm = 0.238, Ωb = 0.0418, ΩΛ = 0.762, σ8 =
0.74, ns = 0.951, h = 0.73}), slightly different from our fiducial cosmology
in this work. The simulation with the highest resolution (REFL25N512) has
identical mass resolution and box size compared to the simulation we use in
this work and the other two simulations have 8 times (REFL25N256) and 64 time
(REFL25N128) lower resolutions (see Table 4.2 for more details).
As we showed in Rahmati et al. (2013a), the Hi column density distribution
function is converged for LLS and most DLAs at the resolution that we use in
this work. The position of galaxies in cosmological simulations is determined
by the distribution of overdensities and therefore, is not expected to be highly
sensitive to the resolution. This is not true for the number of galaxies that are
6 We note that using a small simulation box results in underproducing the strong Hi absorbers
due to missing very massive galaxies (Rahmati et al., 2013a). Given that the contribution of very
massive galaxies to the total abundance of absorbers increases with increasing the NHI (see Figure
4.4 and 4.5), missing them in the small simulation box allows the smaller galaxies to be the main
DLA counterparts and hence decreases the typical impact parameters of strong Hi absorbers.
135
Strong Hi absorbers and galaxies at z = 3
Simulation
REFL06N128
REFL25N512-W7
REFL25N512
AGN
NOSN_NOZCOOL
REFL25N256
REFL25N128
L
(h−1 Mpc)
6.00
25.00
25.00
25.00
25.00
25.00
25.00
N
1283
5123
5123
5123
5123
2563
1283
mb
( h −1 M ⊙ )
1.4 × 106
1.4 × 106
1.4 × 106
1.4 × 106
1.4 × 106
1.1 × 107
8.7 × 107
mdm
( h −1 M ⊙ )
6.3 × 106
6.3 × 106
6.3 × 106
6.3 × 106
6.3 × 106
5.1 × 107
4.1 × 108
ǫcom
(h−1 kpc)
1.95
1.95
1.95
1.95
1.95
3.91
7.81
ǫprop
(h−1 kpc)
0.50
0.50
0.50
0.50
0.50
1.00
2.00
zend
Model
0
2
1
2
2
2
0
REF, WMAP7 cosmology, RT
REF, WMAP7 cosmology
REF
with AGN
w/o SN, w/o metal cooling
REF
REF
136
Table 4.2: List of cosmological simulations used in this work. The detailed description of the model ingredients are discussed in Schaye et al.
(2010). From left to right the columns show: simulation identifier; comoving box size; number of dark matter particles (there are equally
many baryonic particles); initial baryonic particle mass; dark matter particle mass; comoving (Plummer-equivalent) gravitational softening;
maximum physical softening; final redshift; remarks about the used model, cosmology and the use of explicit radiative transfer calculations
instead of fitting function for the Hi calculations (RT).
Appendix D: Resolution tests
Figure 4.12: Cumulative number of galaxies that are resolved in simulations at z = 3
with different resolutions as a function of their SFR. Blue solid, green long-dashed and
red dashed curves show the REFL25N512, REFL25N256 and REFL25N128 simulations,
respectively.
Figure 4.13: The resolution dependence of b − NHI relation at different SFR thresholds.
The blue solid and green dashed curves represent the REFL25N512 and the REFL25N256
simulations, respectively. Panels show the impact parameter of absorbers as a function NHI by including only galaxies that have SFR above a certain threshold. The SFR
thresholds from top-left to bottom-right are 1, 0.63, 0.4 and 0.2 M⊙ yr−1, respectively.
The results are similar in the two simulations for SFR > 0.4 M⊙ yr−1 .
137
Strong Hi absorbers and galaxies at z = 3
Figure 4.14: Impact parameter of Hi absorbers as a function of their NHI for different SFR
thresholds. In the left panel, the blue solid curve shows the result for the REFL25N512
simulation where only galaxies with SFR > 0.16 M⊙ yr−1 are taken into account. The
green long-dashed curve shows the result for the REFL25N256 simulation where the SFR
threshold is such that the cumulative number density of galaxies is matched to that of
galaxies with SFR > 0.16 M⊙ yr−1 in the REFL25N512 simulation. In the right panel, the
blue solid curve shows the b − NHI relation for the REFL25N512 simulation where only
galaxies with SFR > 1.6 M⊙ yr−1 are taken into account. The green long-dashed and red
dashed curves respectively show the result in the REFL25N256 and REFL25N128 simulations where the SFR thresholds are chosen such that the total number density of galaxies
that are taken into account is matched to that of galaxies with SFR > 1.6 M⊙ yr−1 in
the REFL25N512 simulation. This shows that for a fixed total number density of galaxies the relation between impact parameters and the Hi column density of absorbers is
independent of the resolution.
resolved in simulations and as the resolution increases, the number of structures
that are resolved in a simulation also increases. This is shown in Figure 4.12
which shows the cumulative distribution of number of objects that are identified
as bound structures in our simulations7 . Comparing the blue solid and green
long-dashed curves in this figure shows that the number of galaxies that have
SFR > 1 M⊙ yr−1 is identical in the two simulations with highest resolutions.
Together with the converged Hi column density distribution function, this result implies that the relation between the impact parameter of Hi absorbers and
galaxies with SFR & 1 M⊙ yr−1 is also converged in the REFL25N256 simulation. Indeed, as Figure 4.13 shows, this is the case for SFR > 0.4 M⊙ yr−1 . This
suggests that the b − NHI relation is converged in the REFL25N512 simulation
for SFR > 6 × 10−2 M⊙ yr−1 . By increasing the number density of galaxies in
a simulation, one would expect a decrease in the average distance between Hi
absorbers and galaxies. This suggests that the distribution of Hi absorbers relative to galaxies to be primarily sensitive to the cumulative number density of
7 We note that contrary to the REFL25N512 simulation in which cosmological parameters are
based on WMAP year-3 values, our reference simulation in this work is based on WMAP year-7
cosmological parameters. As a result its cumulative distribution of galaxies flattens at lower SFR
compared to what is shown by the blue solid curve in Figure 4.12.
138
Appendix D: Resolution tests
all galaxies. As a result, one might expect to retrieve the same relation between
impact parameters and NHI of absorbers if the total number density of galaxies
be the same in simulations with different resolution. As Figure 4.14 shows, this
is indeed true in our simulations where in each panel we choose different SFR
thresholds depending on the resolution to match the total number(density) of
galaxies above the SFR threshold for all the simulations. This result suggests
that if one keeps the total number density of galaxies we use in our study, increasing the resolution is not expected to change the b − NHI relation and other
conclusions we derived in this work. In this work we used the relatively low SFR
threshold of SFR > 4 × 10−3 M⊙ yr−1 to include as many bound (sub) structure
as possible in our analysis. The total number density of galaxies that are selected
in our reference simulation (i.e., REFL25N512-W7) by this criterium is 0.5 galaxy
per comoving Mpc3 (i.e., equivalent to 31.5 galaxy per proper Mpc3 ). We note,
however, that the SFR threshold that correspond to this total number density
at higher resolutions is expected to be lower than 4 × 10−3 M⊙ yr−1 . This can
be seen in Figure 4.12 which shows that at a fixed cumulative number density,
the SFR of galaxies in the REFL25N256 simulation that have SFR < 1 M⊙ yr−1
decreases by increasing the resolution.
139
5
Genesis of the dusty Universe:
modeling submillimetre source
counts
We model the evolution of infrared galaxies using a phenomenological approach
to match the observed source counts at different infrared wavelengths. In order
to do that, we introduce a new algorithm for reproducing source counts which is
based on direct integration of probability distributions rather than using MonteCarlo sampling. We also construct a simple model for the evolution of the luminosity function and the colour distribution of infrared galaxies which utilizes
a minimum number of free parameters; we analyze how each of these parameters is constrained by observational data. The model is based on pure luminosity
evolution, adopts the Dale & Helou Spectral Energy Distribution (SED) templates, allowing for evolution in the infrared luminosity-colour relation, but does
not consider Active Galactic Nuclei as a separately evolving population. We find
that the 850 µm source counts and redshift distribution depend strongly on the
shape of the luminosity evolution function, but only weakly on the details of the
SEDs. Based on this observation, we derive the best-fit evolutionary model using
the 850 µm counts and redshift distribution as constraints. Because of the strong
negative K-correction combined with the strong luminosity evolution, the fit has
considerable sensitivity to the sub-L∗ population at high redshift, and our bestfit shows a flattening of the faint end of the luminosity function towards high
redshifts. Furthermore, our best-fit model requires a colour evolution which
implies the typical dust temperatures of objects with the same luminosities to
decrease with redshift. We then compare our best-fit model to observed source
counts at shorter and longer wavelengths which indicates our model reproduces
the 70 µm and 1100 µm source counts remarkably well, but under-produces the
counts at intermediate wavelengths. Analysis reveals that the discrepancy arises
at low redshifts, indicating that revision of the adopted SED library towards
lower dust temperatures (at a fixed infrared luminosity) is required. This modification is equivalent to a population of cold galaxies existing at low redshifts, as
also indicated by recent Herschel results, which are underrepresented in IRAS
sample. We show that the modified model successfully reproduces the source
counts in a wide range of IR and submm wavelengths.
Alireza Rahmati, Paul van der Werf
Monthly Notices of the Royal Astronomical Society
Volume 418, Issue 1, pp. 176-194 (2011)
Introduction
5.1 Introduction
Although dust is an unimportant component in the mass budget of galaxies, its
presence radically alters the emergent spectrum of star forming galaxies. Since
stars are born in dusty clouds, most of the energy of young stars is absorbed
by dust particles, which are heated by the absorption process and radiate their
energy away by thermal emission at infrared (IR) and submillimetre (submm)
wavelengths. As a result, the infrared radiation of star forming galaxies is a
useful measure of the massive star formation rate. Mainly due to the atmospheric opacity, the thermal radiation from dust could not be systematically
studied until the launch of IRAS in the mid-1980s, which provided the first
comprehensive view of the nearby dusty Universe. Ten years later COBE discovered the Cosmic Infrared Background (CIB) (Puget et al., 1996; Fixsen et al.,
1998) and it turned out the observed power of the CIB is comparable to what
can be deduced from the optical Universe. This was in contrast to the observations of local galaxies which suggest only one third of the energy output of
galaxies is in IR bands (Lagache et al., 2005). Moreover, the relatively flat slope
of the CIB at long wavelengths indicated a population of dusty galaxies which
are distributed over a wide range of redshifts (Gispert et al., 2000). Thanks to
various large surveys performed with different satellites and ground based observatories, this background radiation is now partly resolved into point sources
at different wavelengths. While at shorter wavelengths the emission is mainly
coming from local and low redshift galaxies, longer wavelengths contain information about larger distances.
The shape of emission spectrum of warm dust particles resembles a modified blackbody spectrum with a peak varying with the typical dust temperature
which is observed to be around T ∼ 30 − 40K in galaxies; therefore the far-IR
(FIR) and submm spectrum of a typical dusty galaxy consists of a peak at restframe wavelengths around λ ∼ 100 − 200µm which drops on both sides (see
Figure 5.1). While the presence of other emitters like polycyclic aromatic hydrocarbons (PAHS) and Active Galactic Nuclei (AGNs) complicates the shape of
spectra at shorter wavelengths, at submm wavelengths the spectra behave like
modified blackbodies and their amplitudes drop steeply. In fact, because of this
steep falloff in the Rayleigh-Jeans tail of the Spectral Energy Distribution (SED),
the observed flux density at a fixed submm wavelength can rise by moving the
SED in redshift space (the so called K-correction), which counteracts the cosmological dimming. It turns out that the interplay between these two processes
enables us to observe galaxies at submm wavelengths out to very high redshifts
(see also the discussion in §5.6 and Figure 5.7).
After the first observations of SCUBA at 850 µm confirmed the importance of submm galaxies at high redshifts (Smail et al., 1997), there were
many subsequent surveys using different instruments which explored different cosmological fields (Smail et al., 2002; Webb et al., 2003; Borys et al., 2003;
Greve et al., 2004; Coppin et al., 2006; Bertoldi et al., 2007; Austermann et al.,
143
Genesis of the dusty Universe
2010; Scott et al., 2010; Vieira et al., 2010) and also some surveys which used
gravitational lensing techniques to extend the observable submm Universe to
sub-mJy fluxes (Smail et al., 1997; Chapman et al., 2002; Knudsen et al., 2008;
Johansson et al., 2011).
While low angular resolution makes individual identifications and spectroscopy a daunting task, the surface density of sources as a function of brightness (i.e., the source counts) can be readily analysed and contains significant
information about the population properties and their evolution. One can assume simple smooth parametrized models for the evolution of dusty galaxies and relate their low redshift observed properties (e.g., the total IR luminosity function, which we simply call the luminosity function, or in short LF
hereafter) to their source counts (Blain & Longair, 1993; Guiderdoni et al., 1997;
Blain et al., 1999; Chary & Elbaz, 2001; Rowan-Robinson, 2001; Dole et al., 2003;
Lagache et al., 2004; Lewis et al., 2005; Le Borgne et al., 2009; Valiante et al.,
2009; Bethermin et al., 2011). Such backward evolution models usually combine SED templates and the low redshift properties of IR galaxies which are
allowed to change with redshift based on an assumed parametric form, together
with Monte-Carlo techniques to reproduce the observed source counts. These
models are therefore purely phenomenological, and suppress the underlying
physics. Their power lies exclusively in providing a parametrized description
of statistical properties of the galaxy population under study, and the evolution of these properties. The results of such modeling provides a description
of the constraints that must be satisfied (in a statistical sense) and could be explored further by more physically motivated models such as hydrodynamical
simulations embedded in a ΛCDM cosmology, with subgrid prescriptions for
star formation and feedback (e.g. Schaye et al. (2010)). It is important to realize
that backward evolution models are constrained only over limited observational
parameter space (e.g., at particular observing wavelengths, flux levels, redshift
intervals, etc.), and are ignorant of the laws of physics. Therefore it is dangerous
to use them outside of their established validity ranges; in other words, these
models have descriptive power but little explanatory power. As a result, one
of the most instructive uses of these models is in analyzing where they fail to
reproduce the observations, and how they can be modified to correct for these
failures which implies that the underlying assumptions must be revised. This is
the approach that we use in the present chapter.
Since nowadays many and various observational constraints exist (e.g. source
counts at various IR and submm wavelengths, colour distributions, redshift distribution), backward evolution models can reach considerable levels of sophistication. However, early backward evolution models used only a few SED
templates (or sometimes even only one SED) to represent the whole population
of dusty galaxies. This approach neglects the fact that dusty galaxies are not
identical and span a variety of dust temperatures and hence SED shapes. As a
result, such models cannot probe the possible evolution in the SED properties
of submm galaxies, for which increasing observational support has been found
144
Model ingredients
during the last few years (Chapman et al., 2005; Pope et al., 2006; Chapin et al.,
2009; Symeonidis et al., 2009; Seymour et al., 2010; Hwang et al., 2010).
In addition, existing backward evolution models typically use only “luminosity” and/or “density” evolution which respectively evolves the characteristic
luminosity (i.e., L⋆ ) and the amplitude of the LF without changing its shape. In
other words, they do not consider any evolution in the shape of the luminosity
function. Finally, the intrinsically slow Monte-Carlo methods used in many of
the models make the search of parameter space for the best-fit model laborious
and tedious.
In this chapter, we use a new and fast algorithm which is different from
conventional Monte-Carlo based algorithms, for calculating the source counts
for a given backward evolution model. We also use a complete library of IR SED
templates which form a representative sample of observed galaxies, at least at
low redshifts. This will enable us to address questions about the evolution of
colour distribution of dusty galaxies during the history of the Universe. We also
consider a new evolution form for the LF which allows us to constrain evolution
in the shape of the LF in addition to quantifying its amplitude changes.
The structure of the chapter is as follows: in §5.2 we present different ingredients of our parametric colour-luminosity function (CLF) evolution model
and introduce a new algorithm for the fast calculation of the source counts for a
given CLF. Then, after choosing our observational constraints for 850 µm objects
in §5.3 we try to find a model consistent with those constraints by studying the
sensitivity of the produced source counts to different parameters in §5.4. After
finding a 850 µm-constrained best-fit model, we test its performance in producing source counts at other wavelengths in §5.6 which leads us to a model capable of producing source counts in a wide range of IR and submm wavelengths.
Finally, after discussing the astrophysical implications of our best-fit model in
§5.7, we end the chapter with concluding remarks. Throughout this chapter, we
use the standard ΛCDM cosmology with the parameters Ωm = 0.3, ΩΛ = 0.7
and h = 0.75.
5.2 Model ingredients
Although luminosity is the main parameter to differentiate between different
galaxies, it is an integrated property of SED and cannot distinguish between different SED shapes corresponding to different physical processes taking place in
different objects with similar outgoing integrated energy fluxes. Using colour
indicators can resolve this degeneracy. Dale et al. (2001) demonstrated that the
R(60, 100) which is defined as log(S60 µm /S100 µm ), is the best single parameter
characterization of IRAS galaxies. Moreover, it was shown that IRAS galaxies exhibit a slowly varying correlation between R(60, 100) and luminosity such that
objects with larger characteristic luminosities have warmer characteristic colours
(Dale et al., 2001; Chapman et al., 2005; Chapin et al., 2009). Based on these facts,
145
Genesis of the dusty Universe
we construct a model for the Colour-Luminosity Function (hereafter CLF). This
model consists of the observed CLF in the local Universe and a parametric evolution function. Then, by adopting a suitable set of SED models, we calculate the
source count of IR objects at different wavelengths and compare the results with
observations.
5.2.1 CLF at z = 0
The local volume density of IRAS galaxies, Φ0 ( L, C ), can be parametrized
as a function of total IR luminosity, L, and R(60, 100) colour, C, where the
total infrared luminosity is calculated by integrating over the SED from 3 to
1100 µm (Dale et al., 2001). Furthermore, it is possible to express the local colourluminosity distribution as the product of the local luminosity function, Φ0 ( L),
and the local conditional probability of a galaxy having the colour C given the
luminosity L, P0 (C | L),
Φ0 ( L, C ) = Φ0 ( L) P0 (C | L).
(5.1)
Since the IRAS galaxies represent an almost complete sample of IR galaxies
out to the redshift ∼ 0.1 (Saunders et al., 1990), we use the luminosity and colour functions found by analyzing a flux limited sample of S60 µm > 1.2Jy IRAS
galaxies (Chapman et al., 2003; Chapin et al., 2009). Chapman et al. (2003) and
Chapin et al. (2009) analyze this sample which covers most of the sky, using an
accessible volume technique for finding the LF and fit a dual power law function to the observed luminosity distribution. The parametric form of luminosity
function based on Chapin et al. (2009) is given by
Φ0 ( L ) = ρ ∗ (
L 1− α
L
) (1 + ) − β ,
L∗
L∗
(5.2)
where L∗ = 5.14 × 1010 L⊙ is the characteristic knee luminosity, ρ∗ = 1.22 ×
1
10−14Mpc−3 L−
⊙ is the number density normalization of the function at L∗ , and
α = 2.59 and β = 2.65 characterize the power-laws at the faint (L < L∗ ) and
bright (L > L∗ ) ends, respectively.
Chapman et al. (2003) and Chapin et al. (2009) also found a Gaussian representation for the colour distribution of IRAS galaxies
P0 (C | L) = √
1
2πσc
C − C0 2
1
) ],
exp[− × (
2
σc
(5.3)
where C0 can be represented by a dual power law function
C0 = C∗ − δ log(1 +
L′
L
) + γ log(1 + ′ ),
L
L
(5.4)
with C∗ = −0.48, δ = −0.06, γ = 0.21 and L′ = 3.2 × 109 L⊙ , and the distribution
width, σc , is expressed as
′
′
σc = σf (1 − 2− L /L ) + σb (1 − 2− L/L ),
where σf = 0.2 and σb = 0.128 (Chapin et al., 2009).
146
(5.5)
Model ingredients
5.2.2 CLF evolution
The CLF introduced in the previous section, contains information about the
distribution of IR galaxies only in the nearby Universe. For modeling the IR
Universe at higher redshifts, its evolution must be modeled. The necessity of
the CLF evolution with redshift is shown directly by the fact that the observed
power of the the CIB is comparable to what can be deduced from the optical
cosmic background. This cannot be explained by our understanding of the local
Universe which indicates the infrared output of galaxies is only one third of their
optical output (Lagache et al., 2005). Furthermore, several studies have shown
for a fixed total IR luminosity the typical temperature of infrared sources is lower
at higher redshifts (Chapman et al., 2005; Pope et al., 2006; Chapin et al., 2009;
Symeonidis et al., 2009; Seymour et al., 2010; Hwang et al., 2010; Amblard et al.,
2010), which advocates an IR colour evolution. Therefore, we need to adopt a
reasonable form of evolution in the luminosity and colour distributions to reproduce correctly the CLF evolution of IR galaxies with redshift.
There are three different ways to evolve the local luminosity function of IR
galaxies: (i) changing ρ∗ with redshift (i.e., density evolution); (ii) changing L∗
with redshift (i.e., luminosity evolution) and (iii) changing the bright and faint
end slopes (i.e., α and β) with redshift. While the density and luminosity evolution change the abundances of all sources independent of their luminosities,
they leave the shape of the LF unchanged. Such models are called translational
models since they amount to only a translation of the LF in parameter space.
Going beyond translational models, a variation of the shape of LF with redshift
(i.e., varying α and β slopes in equation 5.2) can be used to change the relative
contributions of bright and faint sources at different redshifts.
As discussed by Blain et al. (1999), luminosity and density evolution affect
the CIB predicted by the models similarly, but luminosity evolution has a much
stronger effect on the source counts. Combining source counts and integrated
background can therefore distinguish between luminosity and density evolution.
The result of this analysis shows luminosity evolution must strongly dominate
over density evolution, since pure density evolution consistent with the observed
850 µm source counts would overpredict the integrated background by a factor
50 to 100 (Blain et al., 1999). We therefore assume negligible density evolution.
However, as we will show in §5.4, luminosity evolution is not sufficient to reproduce the correct source count and redshift distribution of submm galaxies,
which forces us to drop the assumption of a purely translational LF evolution
model. Moreover, there is no reason to assume that the slopes of LF at faint and
bright ends remain the same at all redshifts. Therefore, we allow them to change
in our model and introduce a redshift dependent LF which can be written as
Φ( L, z) = ρ∗ (
L
L
)1− αz (1 +
)− βz ,
g ( z) L∗
g ( z) L∗
(5.6)
which is similar to equation (5.2) but now αz and β z are changing with redshift
147
Genesis of the dusty Universe
and L∗ is multiplied by g(z), which we refer to as the luminosity evolution
function.
Since the average total-IR-luminosity density and its associated star formation density closely follow the luminosity evolution function, we choose a form
of luminosity evolution which is similar to the observed evolution of the cosmic star formation history (see Hopkins & Beacon (2006)); in other words, we
assume that the luminosity evolution is a function which is increasing at low
redshifts and after reaching its maximum turns into a decreasing function of
look-back time at high redshifts:

if z ≤ z a
 (1 + z ) n
(5.7)
(1 + z a ) n
if z a < z ≤ zb
g(z) =

(1 + z a )n (1 + z − zb )m if zb < z
where n, m, z a and zb are constants. As we will show in §5.4.2, the source
count is not strongly sensitive to the model properties at high redshifts. In other
words, the sensitivity of the source count calculation to the difference between
z a and zb is much less than its high sensitivity to the value of z a itself. Moreover,
the model outputs are not highly sensitive to the slope of g(z) at high redshifts
if it remains negative (i.e., m < 0). Therefore, we can simplify the model by
assigning suitable fixed values to zb − z a and m, without any significant change
in its flexibility. We choose zb − z a = 1 and m = −1, knowing that any different
choice for those values can be compensated by a very small change in z a or/and
n. we will discus this issue in more detail in §5.4.2.
We also adopt linear forms for changing αz and β z with redshift
αz = 2.59 + aα z,
β z = 2.65 + a β z,
(5.8)
(5.9)
z a +zb
2
where aα and a β are constants. We apply the slope evolution only for z <
noting that the observed source count is not very sensitive to the exact properties
of our model at very high redshifts and extending the evolution of LF slopes to
even higher redshifts does not change our results.
As mentioned earlier, some studies have shown that high redshift IR galaxies
have lower dust temperatures than low redshift galaxies, at a fixed total IR luminosity. In our model, in order to evolve the colour distribution accordingly, we
adopt a relation similar to Valiante et al. (2009) to shift the centre of the colour
distribution for a given luminosity towards colours associated with lower dust
temperatures (i.e., smaller R(60, 100) values) at higher redshifts (see Figure 5.12).
In this way, the colour distribution of IR galaxies can be written as
P( C | L) = √
C − C0′ 2
1
) ],
exp[− × (
2
σc
2πσc
1
(5.10)
where σc is given by equation (5.5) and C0′ is
C0′ = C∗ − δ log(1 +
148
L
L′ (1 + z ) w
),
) + γ log(1 + ′
L
L (1 + z ) w
(5.11)
Model ingredients
Figure 5.1: Some SED samples from the template library of Dale & Helou (2002) which
we use in our model. The blue, green, orange and red lines (from top to bottom) are
respectively associated with colours C = 0.16, 0.04, −0.27 and −0.54. The normalization
is arbitrary.
where w is a constant. Although for colour evolution we adopt a similar formalism to Valiante et al. (2009), unlike them we allow w to vary as a free parameter
which enables us to constrain the colour evolution as well.
Based on the above formalism, the general CLF of IR galaxies can be written
as
Φ( L, C, z) = Φ( L, z) P(C | L),
(5.12)
where Φ( L, z) and P(C | L) are defined by equations (5.6)-(5.11).
5.2.3 SED Model
Several studies have shown that AGNs do not dominate the FIR energy output of the Universe (Swinbank et al., 2004; Alexander et al., 2005;
Chapman et al., 2005; Lutz et al., 2005; Valiante et al., 2007; Pope et al., 2008;
Menendez-Delmestre et al., 2009; Fadda et al., 2010; Jauzac et al., 2010). Therefore, in order to avoid unnecessary complexity in our model we adopt a single
population of galaxies, modeled with only one family of SEDs representing star
forming galaxies.
This choice is justified for long wavelengths, for instance at 850 µm, by considering the fact that at those wavelengths the AGN contribution to the observed
flux is negligible. In other words, to be able to change the 850 µm fluxes and
source counts, AGN should be the dominant contributor to the SED at restframe wavelengths longer than ∼ 200 µm which is highly unlikely.
149
Genesis of the dusty Universe
On the other hand, excluding AGN contribution could be potentially important at shorter wavelengths: if we adopt a simple assumption where the AGN
continuum is well represented by a simple torus model (Efstathiou et al., 1995;
Valiante et al., 2009), then a strong enough AGN could create a bump, on top of
the starburst SED, at rest-frame wavelength ranges ∼ 10 − 50 µm (see Figure 2
in Efstathiou et al. (1995) and Figure 9 in Valiante et al. (2009)). This feature, together with increasing AGN luminosity with redshift could modify the observed
source counts at observed wavelengths shorter than 200 µm but still is unlikely
to affect submm counts.
However, Mullaney et al. (2011) recently used the deepest available Herschel
survey and showed that for a sample of X-ray selected AGNs up to z ∼ 3,
the observed 100 µm and 160 µm fluxes are not contaminated by AGN. Consequently, even if all the galaxies which contribute to the FIR and submm source
counts host AGN, their observed fluxes is driven by star formation activity at
wavelengths around 100 µm or longer.
In order to get accurate SED templates for star-forming galaxies, we use the
Dale et al. (2001) and Dale & Helou (2002) SED models which are produced by a
semi-empirical method to represent spectral energy distribution of star-forming
galaxies in the IR region of spectrum. Dale et al. (2001) add up emission profiles
of different dust families (i.e. large grains, very small grains and polycyclic
aromatic hydrocarbons) which are exposed to a range of radiation field strengths
in a parametrized form to generate the SED of different star-forming systems.
In this way, it is possible to produce the SED corresponding to each R(60, 100)
colour and scale it to the desired total infrared luminosity. In our model, we use
spectral templates taken from the Dale & Helou (2002) catalog which provides 64
normalized SEDs with different R(60, 100) colours, ranging from −0.54 to 0.21.
Some SED examples taken from this template set are illustrated in Figure 5.1.
5.2.4 The algorithm
Given the distribution of objects in the Universe, and their associated SEDs, it
is possible to calculate the number of sources which have observed fluxes above
some detection threshold, Sth , at a given wavelength λ = λobs :
N (> Sth ) =
Z Z Z
Q × Φ( L, C, z)
dV
dzdLdC,
dz
(5.13)
where V is the volume and Q is the probability that a source with luminosity L, colour C and redshift z has an observed flux density greater than Sth at
that wavelength (i.e., its detection probability). At each point of redshift-colourluminosity space, Q is either 1 or 0 for any particular galaxy, but it is useful to
think of Q as the average probability of detection for galaxies in each cell of that
space.
In order to calculate Q, we first note that for each source with a given luminosity, L, and colour, C, there is a redshift, zmax , at which the observed flux is
150
850 µm Observational Constraints
equal to the detection threshold
S( L, C, λ0 , zmax ) = Sth =
(1 + zmax ) Lν ( L, C, λ0 )
,
4πD2L (zmax )
(5.14)
where D L is the luminosity distance and Lν is the rest-frame luminosity density
(W Hz−1 ) of the object with total luminosity L and colour C at wavelength λ0 =
λobs /(z + 1).
Now consider a cell defined by redshift interval ∆z = z2 − z1 (where z1 < z2 ),
luminosity interval ∆L = L2 − L1 and colour interval ∆C = C2 − C1 . Assuming
negligible colour and luminosity evolution between z1 and z2 , the average detection probability in the cell is equal to the fraction of detectable objects in that
cell which is:

if zmax > z2

 13
D ( zmax )− D 3 ( z1 )
Q=
(5.15)
if z1 ≤ zmax ≤ z2
3
3

 D ( z2 )− D ( z1 )
0
if zmax < z1
where D (z) is the proper distance equivalent to redshift z. For writing equation
5.15, we assumed galaxies to be distributed uniformly in space between redshifts
z1 and z2 . Consequently, based on equation 5.13, the number of objects which
are contributing to the source count in each cell of the redshift-colour-luminosity
space is
∆N (> Sth ) = Q × Φ̄( L, C, z)∆L∆C∆V,
(5.16)
where ∆V is the volume corresponding to the redshift interval ∆z and Φ̄ ( L, C, z)
is the average CLF in the cell, which for small enough ∆L, ∆C and ∆z can be
written as
Φ̄( L, C, z) = Φ( L̄, C̄, z̄).
(5.17)
Then, the total source count is obtained by summing over the contribution from
all cells (see Appendix A).
As a result of using Q and its special form as expressed in equation 5.15,
our algorithm computes the source count for continuously distributed sources
in the Universe, independent of the size of ∆z which is used in equation 5.13.
Thanks to this property, the computational cost required for the source count
calculation is reduced significantly, and the small size of ∆z is only necessary
for accurate calculation of K-corrected SEDs and CLFs at different redshifts and
not to guarantee a uniform distribution of sources (see also Appendix A).
Now that we have all the tools ready, we can test the capability of our model
in reproducing the observed properties of 850 µm sources and find which particular choices of model parameters are implied. In the following sections, we first
discuss the observational constraints that we want to reproduce with our model
and then we will proceed with finding the best-fit model which can reproduce
those properties.
151
Genesis of the dusty Universe
Figure 5.2: A compilation of some observed 850 µm source counts. Blue circles, green
triangles and red squares are respectively data points taken from Knudsen et al. (2008),
Coppin et al. (2006) and Johansson et al. (2011).
5.3 850 µm Observational Constraints
5.3.1 Observed 850 µm source count
Among several existing extragalactic submm surveys which provide source
counts at 850 µm (Coppin et al., 2006; Weiss et al., 2009; Austermann et al., 2010),
the SCUBA Half-Degree Extragalactic Survey (SHADES) (Coppin et al., 2006)
is the largest one which has the most complete and unbiased sample. However, this survey and other surveys which use JCMT and the same blank field
method are restricted by the JCMT confusion limit of ∼ 2mJy at 850 µm and
cannot probe the source counts of the fainter population. Using a complementary method, the lensing technique has been used to probe 850 µm source
counts to flux thresholds as low as 0.1mJy (Smail et al., 2002; Knudsen et al.,
2008; Johansson et al., 2011). As we will show later in this section, the sensitivity
of bright and faint submm source counts to the evolution of dusty galaxies is
completely different and it is essential to incorporate a large dynamic range to
constrain possible evolutionary scenarios. Therefore, we use the best available
observational information at both faint and bright tails of 850 µm source count,
by combining all of the SHADES data points with those of Knudsen et al. (2008)
at flux thresholds < 2mJy to assemble our reference source count which is also
in agreement with other observations (see Figure 5.2).
152
Finding 850 µm best-fit Model
Figure 5.3: The sensitivity of the model to various parameters. While the left panel
illustrates how much the variation of different parameters could change the amplitude of
the 850 µm source count curve, the right panel shows their effect in changing the shape
of the source count curve. To measure the amplitude changes, we varied each parameter
by 10% around its best-fit value and measured the difference in the total source counts.
The shape sensitivity is probed by measuring the relative change each varying parameter
can cause in the ratio between two source counts at faint and bright flux thresholds (i.e.
N1mJy
N1Jy ).
The lengths of the coloured segments on each axis shows how large those changes
are. The blue (red) segments which are connected by solid (dash-dotted) lines represent
the result as we increased (decreased) the value of each parameter. We show parameters
aα and a β on one axis since their effect in changing the source count properties is similar.
5.3.2 Redshift distribution of bright 850 µm sources
Although the observed number counts at 850 µm, especially at faint fluxes, are
sensitive to the redshift distribution of infrared galaxies, they do not constrain it
directly. As we will discuss later, different models with different redshift distributions can reproduce the 850 µm source counts with the same accuracy. Therefore, it is important to impose an additional constraint on redshift distribution
of those objects.
Unfortunately, studying the redshift distribution of submm galaxies is extremely difficult and there are only a few works on spectroscopically confirmed
redshift distribution of bright submm galaxies with observed fluxes grater than
∼ 4 − 5 mJy (Chapman et al., 2005; Wardlow et al., 2011). Those studies show
a redshift distribution peaking around z ∼ 2 (see the right panel in Figure 5.6).
We use this redshift distribution as one of our observed constraints and force
our model to reproduce a redshift distribution which peaks at the same redshift.
153
Genesis of the dusty Universe
Figure 5.4: The likelihood (i.e. exp(− 12 χ2 )) distribution of different parameters for the
best-fit model. For each panel, we fixed all the parameters at their best-fit values and
changed only one parameter at a time. By varying each parameter, the quality of the fit
and hence the likelihood is changing. We normalized the amplitude of curves such that
the area under each curve (i.e. the total probability) be unity.
5.4 Finding 850 µm best-fit Model
In order to be able to extract meaningful trends and differentiate between them,
it is important to keep our phenomenological model as simple as possible. It
is therefore desirable to find the minimum number of free parameters without
which the model cannot produce an acceptable result. Furthermore, the role of
different parameters are not identical: while some parameters are not strongly
constrained, small changes in others can change the result significantly. It is also
important to look at degeneracies between various parameters; some parameters
are not independent and varying one may be compensated by varying the others. In this section we investigate our model to understand those issues before
presenting our best-fit model.
5.4.1 The source count curve: Amplitude vs. Shape
To match the observed source counts, our model should be able to reproduce
both the typical number of sources (i.e., the amplitude of the source count curve
as a function of flux threshold) and the ratio between the source counts at faint
and bright flux thresholds (i.e., the shape of the source counts curve). Following
this line of argument, we can categorize our model parameters into two different
154
Finding 850 µm best-fit Model
groups: those which play a stronger role in forming the amplitude of the curve
and those which mainly affect its shape. As mentioned earlier, the luminosity
evolution (see equation 5.7) has the dominant role in determining the amplitude
of the source count curve while its shape is controlled by other ingredients of
our model which are colour evolution and the evolution of LF slopes.
These trends are illustrated in Figure 5.3. In the left panel, the relative contribution of each parameter in changing the amplitude of the source count curve
is visualized; the length of the coloured segments along each axis (coloured segments along different axis are connected to each other) represents the sensitivity
of source count amplitude to that parameter. The length of coloured segments
is computed by changing all the model parameters (one at a time) by the same
fraction, for instance 10%, and measure the change caused in the total source
count. The segments which are connected by dot-dashed (red) and solid (blue)
lines correspond respectively to a decreasing and an increasing parameter. In
the right panel, the relative contribution of different parameters in controlling
the shape of the source counts curve is shown. In this figure, the length of
coloured segments represents the change in the ratio between faint and bright
source counts. This time, we measured how much the ratio between a very faint
source count, like 1 mJy, and a very bright one, like 1 Jy, is changing due to a
fixed change in different parameters (e.g., 10%). The dot-dashed (red) and solid
(blue) lines connect the coloured segments which correspond to an increase or a
decrease in the values of the parameters.
As is evident from these diagrams, the amplitude of the source count curve
is strongly sensitive to the parameters of the luminosity evolution, particularly
z a and n, while its shape is determined mainly by an evolution in the shape of
LF and/or the strength of the colour evolution.
5.4.2 The luminosity evolution
The luminosity evolution is the backbone of our model and has the most important role in producing the observed source counts and the redshift distribution
of 850 µm objects. However, it is not surprising that the calculated source count
is not strongly sensitive to the model properties at high redshifts. This is mainly
because objects at very high redshifts have decreasing fluxes (despite the advantageous K-correction). Consequently, the chosen value of m has a negligible
effect on the amplitude and shape of the source count curve (see Figure 5.3).
Nevertheless, one should note that this argument would not necessarily work
if the population of bright IR galaxies continued to evolve with look-back time
for all redshifts. In other words, the exact value of m is not strongly constrained
by the observed source counts and all the negative values can produce similar
results (see the middle panel in the bottom row of Figure 5.4).
Contrary to m, the source count curve is very sensitive to the exact values
of n and z a which respectively control the rate by which the characteristic luminosity of IR galaxies increases with redshift and up to which redshift this
155
Genesis of the dusty Universe
Figure 5.5: Map of likelihoods for models with different n and z a where [ w, aα , a β ] =
[0, 0, 0]. The region which contains the best-fit models is in the middle of the coloured
band and surrounded by the dashed blue contours which indicate the likelihood of 0.85.
growth continues. However, there are two solutions for reproducing the same
source counts: one is to increase the characteristic IR luminosity rapidly up to
a relatively low redshift and the other one is to increase the characteristic IR
luminosity by a moderate rate but for a longer period of time (i.e., up to higher
redshifts). Therefore, as illustrated in Figure 5.5, there is a degeneracy between
n and z a and one cannot constrain them individually by looking at the observed
source counts. However, additional information about the redshift distribution
of submm galaxies or the slope by which the characteristic IR luminosity is growing at low redshifts could resolve this degeneracy. As we will discuss later, the
former constraint is used in finding our best-fit model.
Unlike n and z a , the length of the redshift interval during which the luminosity evolution remains constant before starting to decline, zb − z a , is not an
essential part of our model. We only introduced this feature to have a smoother
transition between growing and declining characteristic IR luminosity and also
redshift distribution of submm galaxies. As Figures 5.3 and 5.4 show, variations
in the adopted value for zb − z a do not change the source count significantly.
Moreover, any change in its value could be compensated by a very small change
156
Other necessary model ingredients
in n and/or z a .
Based on these considerations and as we already mentioned in §5.2.2, we
reduce the number of free parameters we use in our model by choosing m = −1
and zb − z a = 1. Furthermore, the observed redshift distribution of submm
galaxies which peaks around z ∼ 2 (Chapman et al., 2005; Wardlow et al., 2011),
limits the acceptable values of z a to ∼ 1.6 (see the right panel in Figure 5.6).
5.5 Other necessary model ingredients
Although the luminosity evolution is necessary for producing correct number of
observable sources, it is not sufficient to provide a correct shape for the source
count curve. Based on our experiments by varying different parameters of the
luminosity evolution, one can either get a good fit at the faint number counts
and under-produce the bright end or produce correctly the bright source counts
and over-estimate the faint objects. In other words, other model ingredients like
the colour evolution and/or the evolution of LF slopes are required to adjust the
shape of the source count curve appropriately in order to fit the faint and bright
source counts at the same time. For instance, the colour evolution can be used
to help the model with under-production of bright sources and the evolution of
LF slopes can compensate for the over-production of the faint sources.
As a starting point, we have first explored fits using either the colour evolution or the evolution of the LF slopes, in order to keep the model as simple as
possible. We did this by trying to fit the 850 µm source counts once using a combination of the luminosity evolution together with the colour evolution when
the slopes of LF do not evolve (i.e., No-a-Evol) and once using the luminosity
evolution combined with the evolving LF slopes, without evolving the colour
distribution (i.e. No-C-Evol). The results are shown in Figure 5.6 and are compared with the best-fit result when the colour evolution and the evolution of LF
slopes are both active (i.e., 850-model). The quality of the fit for “850-model”
is better than both of the other cases which may be attributed to the additional
free parameter used in the “850-model”. However, both “No-a-Evol” and “No-CEvol” models have unfavourable implications. The best-fit for the “No-C-Evol”
case (see Table 5.1) requires a large increasing slope in the luminosity evolution function which gives rise to a violation of the counts of IR sources at short
wavelengths (e.g. at 70 µm, see §5.6). The “No-a-Evol” fit on the other hand,
requires a steep colour evolution which is too extreme to be acceptable, since it
would imply that at redshifts z > 1 all infrared sources, independent of their
luminosities, have the same dust temperatures as low as T ∼ 20K. However,
when both colour evolution and evolution of the bright and faint-end slopes of
the LF are allowed simultaneously, these problems disappear which makes this
solution preferable.
157
158
Genesis of the dusty Universe
Figure 5.6: Left panel: The best-fit model which is constrained by 850 µm source count and the redshift distribution of submm galaxies
(the blue solid line) is illustrated next to the reference observed data points (see §5.3). For comparison, three other best-fit models are also
plotted: the best-fit model without colour evolution shown by the dashed orange line (i.e., “No-C-Evol”), a model without evolving LF
slopes shown by the dot-dashed green curve (i.e., “No-a-Evol”) and finally the best-fit model which is only constrained by the source count
is shown using the dotted blue curve. Right panel: the comparison between the observed redshift distribution of 850 µm galaxies which
are brighter than 5 mJy (Chapman et al., 2005) and what our best-fit model implies. The histogram shown in red with dashed line is the
observed probability distribution and the histogram with solid blue line shows the probability distribution of similar objects in our model.
For comparison, the best-fit model without constrained redshift distribution, is shown with dotted blue histogram ("No-za -Const"). The
other two models, “No-C-Evol” and “No-a-Evol” are respectively shown by dashed orange and dot-dashed green lines.
Other necessary model ingredients
Table 5.1: Parameters which define different best-fit models constrained to reproduce
the observed source counts at 850 µm. All the models are using zb − z a = 1 and m =
−1 and except the "No-z a -Const" model, all of them are constrained to reproduce the
redshift distribution of submm galaxies and therefore use z a = 1.6. The predicted source
count each of those models and their implied redshift distribution for submm sources is
illustrated in Figure 5.6
Model
850-model
No-C-Evol
No-a-Evol
No-z a -Const
n
2.0
2.8
1.6
2.4
za
1.6
1.6
1.6
3.6
w
2.0
0
5.6
2.4
aα
0.6
0.6
0
1.0
aβ
0.4
0.6
0
2.2
5.5.1 The 850 µm best-fit model
As we showed, it is necessary to have an evolutionary model which incorporates
a luminosity evolution, colour evolution and also evolving LF slopes, in order to
fit the observed 850 µm source counts. We also argued that we can fix some of the
initial parameters of the model since they have no significant effect on the results
and chose proper values for them (i.e., m = −1 and zb − z a = 1). Moreover, we
showed that we need to set the redshift at which the luminosity evolution peaks
to z a = 1.6 to reproduce the redshift distribution of submm galaxies correctly
which also resolves the degeneracy between z a and n. After taking into account
all of those considerations, we end up with 4 free parameters in our model which
are needed to be adjusted properly to reproduce our observed sample of 850 µm
source counts; those parameters are the slope of the colour evolution, w, the rate
by which the faint and bright end slope of LF is changing with redshift, aα and
a β , and finally the growth rate of the characteristic IR luminosity, n.
Since our algorithm for calculating the source count is fast and accurate,
contrary to much more cumbersome Monte-Carlo-based approaches, we can
perform a comprehensive search in the parameter space for the best-fit model
instead of choosing it “by hand”. We split each dimension of relevant regions
of parameter space into equally spaced grids and calculate the source count
for modes associated with each node of our grid structure. Then we calculate the likelihood of each model for reproducing the observed source counts
(i.e. ex p(− 12 χ2 )). Finally, we choose the model with maximum likelihood as our
best-fit model. The parameters which define the best-fit model for 850 µm source
counts, “850-model”, is shown in Table 5.1 together with those of "No-C-Evol"
and "No-a-Evol" which we discussed in the previous section. Those models are
also compared with observational data sets used for constraining them, in Figure 5.6 where also the redshift distribution implied by each model is illustrated
(in the right panel).
It is also interesting to inspect the properties of a best-fit model in which the
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Genesis of the dusty Universe
peak of the luminosity evolution, z a , is not fixed. The parameters which define
this model are presented in the last row of Table 5.1 and its source count and
redshift distribution are shown by the blue dotted lines in Figure 5.6. The quality
of the fit for this model is even better than “850-model” except at very low flux
thresholds (unsurprisingly, since it has an additional free parameter, namely
z a ). The rising slope of luminosity evolution, n, and the colour evolution, w,
in this model are not drastically different from the “850-model” but its peak of
luminosity evolution happens at higher redshifts and the model requires much
steeper slope evolutions to compensate for too many observable sources. Not
only is this model unable to produce the observed redshift distribution ( see
the right panel in Figure 5.6), it predicts too few observable sources at shorter
wavelengths which makes it unfavorable.
5.6 Other wavelengths
In the previous section, we discussed our model parameters and their role in
producing the observed 850 µm source counts and the redshift distribution of
submm galaxies. We used those observed quantities as constraints to find our
best-fit model, “850-model”. If we assume the evolution scenario that our bestfit model suggests is correct, and also the set of SEDs we used for reproducing
the 850 µm source count are good representatives for real galaxies, then we expect the same model, together with the same set of SEDs, to reproduce the
observed number counts of IR sources at other wavelengths. Moreover, one
can use the source count at other wavelengths to find a best-fit model for that
specific wavelength and again expects to reproduce the source count at other
wavelengths correctly.
Before examining those expectaions, it is important to note that there are
some fundamental differences between the source counts at shorter wavelengths
and in the submm. Most importantly, as illustrated in Figures 5.7 and 5.11,
objects observed at short wavelengths (e.g., 70 µm) are mainly at low redshifts
while the observed submm galaxies are distributed in a wider redshift interval and at higher typical redshifts. Also, the brightest objects at 70 µm are the
closest ones which is not the case for brightest submm galaxies1 (Berta et al.,
2011). Moreover, there are additional physical processes, like AGNs, which
can change the energy output of galaxies at shorter wavelengths. Since we do
not take into account AGNs as a separate population, our model is not expected
to reproduce necessarily good results for wavelengths shorter than 60 − 70 µm
(see also the discussion in §5.2.3). Going from longer to shorter wavelengths,
the observed sources are typically at lower and lower redshifts. Consequently,
the source counts at short wavelengths are only sensitive to the very low redshift
1 In fact the wide distribution of observed submm galaxies in redshift space, combined with the
availability of a measured redshift distribution are the primary reasons we chose their observational
properties to constrain our model.
160
Other wavelengths
Figure 5.7: The modeled redshift distribution of objects which are observed with different
flux thresholds at 70 µm (in the left panel) and 850 µm (in the right panel). Three different
flux thresholds which are 0.1, 1 and 10mJy are shown respectively using solid(blue), dotdashed (green) and dashed (red) lines.
Table 5.2: Parameters which define different best-fit models constrained to reproduce
the observed source counts at different wavelengths. The first column, λ, indicates the
wavelength for which the model is constrained to produce the best fit to the observed
source counts. All the models are forced to reproduce the redshift distribution of submm
galaxies and therefore use z a = 1.6, zb − z a = 1 and m = −1. The predicted source count
each of those models is implying for different wavelengths is illustrated in Figure 5.8 and
Figure 5.9
λ
850 µm
500 µm
350 µm
250 µm
160 µm
70 µm
n
2.0
2.6
3.0
3.0
2.2
2.2
w
2.0
3.8
4.4
4.4
4.2
2.0
aα
0.6
0.0
-0.2
0.0
0.8
0.6
aβ
0.4
0.6
1.4
1.4
0.2
0.6
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Genesis of the dusty Universe
properties of our model. In fact, the 70 µm source count is mainly sensitive to the
growth rate of the luminosity evolution, n, and the evolution in LF slopes; but by
going to longer wavelengths, the colour evolution and the redshift at which the
luminosity evolution is peaking, z a , become more important. In other words, if a
model reproduces the observed source counts at short wavelengths, this forms a
confirmation that the evolution at low redshifts is represented correctly but for a
best-fit model constrained by observations at shorter wavelengths, the properties
of model at intermediate and high redshifts are not strongly constrained.
To investigate the performance of different best-fit models constrained by
source count observations of wavelengths other than 850 µm, we use the same
procedure we incorporated in finding the “850-model” and only vary the effective parameters of the model, namely n, w, aα and a β , to fit the observed data. The
parameters which define best-fit models at different wavelengths are shown in
Table 5.2 and the source counts they produce at different wavelengths are illustrated next to the observational data points in Figure 5.8 and Figure 5.9. Different panels are for source count at different wavelengths ranging from 1100 µm
on the top left to 70 µm on the bottom right. We do not show the results for
350 µm to maintain the symmetry of figures since for this wavelength models
and their comparison with observed data are in many respects identical to the
case of 250 µm. In Figure 5.8 and Figure 5.9, the source counts produced by
different models are also shown using lines with different styles and colours:
purple short-dashed for 70 µm, green long-dashed for 160 µm, orange dot-dotdashed for 500 µm and finally solid blue lines for 850 µm (i.e the "850-model").
In the following we discuss those results by categorizing them in different
wavelength ranges, namely 850 µm and 1100 µm as long submm wavelengths,
500 µm, 350 µm and 250 µm as SPIRE or intermediate wavelengths and finally
70 µm and 160 µm as short wavelengths.
5.6.1 Long submm wavelengths: 850 µm and 1100 µm
As we mentioned earlier, the SED of star forming galaxies at long submm
ranges is essentially controlled by the Rayleigh-Jeans tail of the dust emission
which simply falls off smoothly (see Figure 5.1). This means that if a model
reproduces the observed counts at 850 µm, a good fit to the observed counts
at similar and longer wavelengths is guaranteed that given the distribution of
sources which produce those counts have a similar redshift distributions, which
in turn makes those counts equally sensitive to different parameters in our
model. As illustrated in the top panels of Figure 5.8, both models which are
constrained by observed 70 µm and 850 µm counts and produce a good fit to
850 µm data, also produce a good fit to 1100 µm source counts. Our experiments with other models also confirm that the quality of the fit they produce
for observed counts at 850 µm and 1100 µm is highly correlated. Both of the
models constrained by 160 µm and SPIRE (e.g. 500 µm) source counts over produce the long submm wavelength source counts: the "160-model" over-produces
162
Other wavelengths
Figure 5.8: In different panels, the best-fit model predictions for cumulative source counts
at 850 µm and differential source counts at 1100, 500 and 250 µm are plotted next to the
observed data (see also the same for 160 and 70 µm in Figure 5.9). The 1100 µm observational data sets are from Scott et al. (2010) (red diamonds) Austermann et al. (2009,
2010) (orange circles and green squares respecitvely) and Hatsukade et al. (2011) (blue triangles); data at 850 µm is from Coppin et al. (2006)(green triangles), Knudsen et al. (2008)
(red circles) and Johansson et al. (2011) (orange squares). The data for 500 and 250 µm
source counts are from the BLAST experiment taken from Patanchon et al. (2009) (red diamonds), Herschel data taken from Oliver et al. (2010) (green squares) and Clements et al.
(2010) (orange circles). We also used one data point at 350 µm from the SHARC2 survay
(green triangle)(Khan et al., 2007). Different lines correspond to the source counts produced by various models which are constrained to fit the observed source counts at different wavelengths: blue solid line shows the Ò850-modelÓ, purple dashed line shows the
"70-model" and green long-dashed and orange dot-dot-dashed lines are for "160-model"
and "500-model" respectively (see Table 5.2)
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Genesis of the dusty Universe
Figure 5.9: The same as Figure 5.8 for 160 and 70 µm. The data at 160 and 70 µm are
based on Spitzer observations taken from Bethermin et al. (2010) where the data points
shown by filled squares represent stacking results.
submm source counts mainly at bright flux thresholds due to its extreme colour
evolution (see Table 5.2) but fits the observations at faint fluxes, thanks to its
luminosity evolution which is similar to those of "850-model" and "70-model".
However, SPIRE constrained models (e.g. "500-model") have stronger evolutions
both in terms of luminosity and colour and over-produce the data in all observed
fluxes.
One should note that the 1100 µm data points are highly incomplete below
3mJy (Hatsukade et al., 2011) and are prone to large field to field variations and
relatively large errors in the whole range of observed fluxes. Moreover, the
observed counts are at bright flux thresholds which makes them sensitive to a
narrower redshift interval in comparison to the fainter flux thresholds (see the
right panel of Figure 5.7) and because of those reasons they cannot constrain
models better than what 850 µm source counts are capable of. Therefore, we
only consider the "850-model" as a model constrained by long submm source
counts.
5.6.2 SPIRE intermediate wavelengths:
250 µm
500 µm, 350 µm and
The three 500 µm, 350 µm and 250 µm wavelengths are close together and besides being in the middle of wavelength ranges we study, they have intermediate
properties with respect to the redshift range each wavelength is mostly sensitive
to: as it is shown in Figure 5.11, while at the bright flux thresholds the source
counts mainly consist of low redshift sources, at fainter fluxes they are sensitive
to the intermediate (i.e. 1 < z < 2) and high redshifts (i.e. 2 < z). Natur164
Other wavelengths
ally the general behavior of models for 500 µm is closer to 850 µm while 250 µm
is close to shorter wavelengths. However, those intermediate wavelengths are
closely similar to each other more than being similar to other wavelengths. Consequently, the best-fit models constrained by SPIRE wavelengths have similar
parameters ( 250-model and 350-model have almost identical parameters) and
any of those models agrees with the observed source counts of the other two.
However, SPIRE-constrained models all require steep luminosity evolution and
strong colour evolution in addition to a steepening bright end slope of the LF
with redshift (i.e. positive aα together with almost zero a β , see Table 5.2). The
first two mechanisms produce too many objects in comparison to what is needed
for the source counts at other wavelengths while the steeper bright end of the LF
partially compensate the over-production of bright objects: as is shown in Figure 5.8 and Figure 5.9, SPIRE-constrained models over-produce the faint source
counts both at longer and shorter wavelengths but the steep bright end of LF
produces results which are close to the observed faint source counts at very
long wavelengths (i.e. 850 µm and 1100 µm) which in turn is too strong to leave
enough sources required for the bright 160 µm sources. As we will discuss later,
it is not surprising that SPIRE-models agree with bright 70 µm counts since they
are essentially z ∼ 0 objects for which evolutionary mechanisms in the model
barely have any effect.
While the SPIRE models fail to agree with observations at shorter and longer
wavelengths, the two models which are successful at long submm ranges (i.e. the
"850-model" and "70-model") also have a reasonable agreement with the SPIRE
data. However, they under-produce the counts at Sth < 100mJy by a factor of
∼ 2. As we will show later, assuming the "850-model" to be correct, this underproduction is a hint for a population of cold luminous IR galaxies residing only
in low and intermediate redshifts (i.e. z ∼ 1). This is also consistent with the
steep coluor evolution which is implied by best-fit models at those wavelengths
which produces enough cold galaxies at lower redshifts.
5.6.3 Short wavelengths: 160 µm and 70 µm
At short wavelengths like 70 µm, the K-correction is not strong enough to counteract the effect of cosmological dimming which makes the source counts at
these wavelengths almost insensitive to the properties of IR galaxies at high redshift. Since the starting point of our model is the observed distribution of IRAS
galaxies which are selected to be local galaxies with S60 µm > 1Jy, and the SED
templates we use are extensively tested to match these galaxies, any variation of
our model by construction cannot disagree with bright 70 µm counts 2 . However,
at fainter flux thresholds (e.g. Sth < 100mJy) the sensitivity of 70 µm counts to
low and intermediate redshifts, 0.5 < z < 2, increases (see Figure 5.11) which
2 Unless the luminosity of IR galaxies increase with redshift extremely steep to produce high-z
objects which are observed in 70 µm band as bright as the brightest local IR galaxies despite the
cosmological dimming and negative K-correction.
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Genesis of the dusty Universe
makes them more sensitive to the slope of the luminosity evolution. As a result, all the models agree with 70 µm observations at bright flux threshold while
the SPIRE-model which has steeper luminosity evolution (i.e. greater n, see
Table 5.2) diverges from observed counts and over-produces them. It should be
noted that the "70-model" which is tuned to agree with mainly low-z IR sources
performs as good as "850-model" in longer wavelengths.
At 160 µm, the redshift distribution of sources is almost identical to the distribution of galaxies which are responsible for 70 µm: the bright end of the counts
is governed by local IR galaxies while for flux thresholds Sth < 100mJy the
intermediate redshifts (i.e. 0.5 < z < 2) become important. However, the flux
threshold at which the turn over between domination of local and farther sources
happens is one order of magnitude higher for 160 µm sources (see bottom row
in Figure 5.11). Moreover, at 160 µm the domination of intermediate redshift
sources in shaping the faint source counts is slightly stronger than at 70 µm. In
other words, the 160 µm source counts are slightly more sensitive to the distribution and evolution of distant IR galaxies in comparison to 70 µm counts.
The best-fit model which is constrained by 160 µm source counts has similar
slope of luminosity evolution, n, compared with the "850-model" and "70-model"
but requires much stronger colour evolution and larger fraction of bright objects
(i.e. steeper faint end slope and flatter bright end slope of LF at higher redshifts).
The SPIRE-models on the other hand, violate the observed 160 µm counts by
under-producing them at bright end and over-producing them at faint fluxes.
The "850-model" and "70-model" on the other hand, match the faintest 160 µm
count but under produce the brighter objects by a factor of ∼ 2 which is the
same factor showing up in the difference between what those two models produce and observed faint SPIRE counts. A similar discrepancy between models
and 160 µm counts has been pointed out in some recent works (Le Borgne et al.,
2009; Valiante et al., 2009); while the Le Borgne et al. (2009) best-fit model which
produces correct source counts at 70 µm and 850 µm deviates from observations
between 10 < S160 < 100 mJy by a factor of ∼ 2, the Valiante et al. (2009) model
underestimates number counts at 160 µm by a factor of ∼ 5. We discuss this
issue further in §5.6.4.
5.6.4 A best-fit model for all wavelengths
As we showed in previous subsections, the required evolution scenarios for different models which are constrained by various wavelengths is too diverse to be
reconciled in a single model; however, the 70 µm and 850 µm source counts can
be explained by the same model, either the "850-model" or "70-model" which
are almost identical in terms of implied evolution and distribution of IR galaxies and predicted source counts at different wavelengths. While 70 µm source
counts are produced mainly by local and low redshift objects, 850 µm sources
are distributed in higher redshifts and in a wide redshift interval. This means
the model which can explain the source count at both 70 µm and 850 µm gives a
166
Other wavelengths
Figure 5.10: The same set of observed source counts which is shown in Figure 5.8 and Figure 5.9 is also illustrated here. The blue solid line shows the result of "850-model" using
a modified set of SED (see §5.6.4) to correct under-production of bright 160 µm sources
and faint SPIRE source. Source count produced by two different models, Lagache et al.
(2004) and Valiante et al. (2009), are also respectively shown by orange dashed line and
gray dot-dashed lines. While Valiante et al. (2009) and "850-model" counts at 160 µm are
almost identical, we did not find Lagache et al. (2004) model predictions at 1100 µm to
include them in the top-left panel.
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Genesis of the dusty Universe
Figure 5.11: The fractional contribution of sources in different redshift bins in producing
the observed source counts at different wavelengths are illustrated. The blue Dotted, cyan
dot-dashed, orange dot-dot-dashed and red dashed lines respectively show the source
count produced by objects in 0 < z < 0.5, 0.5 < z < 1, 1 < z < 2 and 2 < z < 5 redshift
intervals while the total source count is shown using the brown solid line. In the top
section of each panel, the fraction of sources for each redshift bin in producing source
counts at different flux thresholds is plotted.
168
Other wavelengths
consistent distribution of IR galaxies from local Universe to very high redshifts.
In other words, the same evolution scenario for star-forming galaxies which we
observe locally, can also produce the observed 850 µm counts. This scenario, contrary to what is implied by the best-fit models constrained at other intermediate
wavelengths, does not strongly violate observed source counts at other bands,
though it under-produces them at some flux thresholds. On the other hand,
models which agree with observations at intermediate wavelengths require extreme color evolutions to produce larger number of cold objects in intermediate
redshift ranges which is relevant for the source count at those wavelengths (i.e.
1 < z < 2, see the left panel in the middle of Figure 5.11). However, this steep
colour evolution at higher redshifts (i.e. z > 2) produces too many observable
sources at 850 µm.
Moreover, as we mentioned earlier different authors with fundamentally different models have reported inconsistencies in their best-fit models (which fit the
long and short wavelengths) and intermediate source counts (Le Borgne et al.,
2009; Valiante et al., 2009) 3 and the underestimation of 160 µm (and SPIRE
wavelengths) seems to be model independent.
There are two main possibilities which can explain the origin of discrepancy
between the "850-model" and source counts at SPIRE band and 160 µm, besides
doubting the validity of observed counts: (i) the IR SEDs in our model are not
representative and should be modified in a certain way to produce more flux
at observed wavelengths λobs ∼ 160 - 500 µm or (ii) existence of a population of
cold galaxies at relatively low redshifts. Indeed, there is a body of evidences supporting a population of cold galaxies which are underrepresented in IRAS galaxies and hence 70 µm source counts (Stickel et al., 1998, 2000; Chapman et al.,
2002; Patris et al., 2003; Dennefeld et al., 2005; Sajina et al., 2006; Amblard et al.,
2010). However, based on available data there is no possible way to disentangle
between the two mentioned possibilities (Le Borgne et al., 2009). Therefore, due
to its simplicity, we try to find a modification in SED amplitudes which could
improve the agreement between the "850-model" and observed source counts
at SPIRE wavelengths and 160 µm, without affecting other wavelengths. In the
following we first introduce the desired SED modification and then based on
the redshift distribution of sources responsible for different source counts in our
modified model, we try to constrain different properties a population of cold IR
galaxies should possess to be equivalent to that SED modification.
Since the "850-model" is capable of fitting the observed 70 µm source counts,
any modification of SED templates should be at wavelengths longer than 70 µm
to leave this agreement intact. The 850 µm source counts that model produces should also remains the same which put an upper limit for the allowed wavelength range of any modification: since a significant fraction of
observed counts at 850 µm is produced by sources at high redshifts, typically
z > 2 (see Figures 5.7 and 5.10), SED templates should not change at rest-frame
3
Those works point out this issue only for the source counts at 160 µm since the observations at
250 µm, 350 µm and 500 µm have been available only recently.
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Genesis of the dusty Universe
wavelengths longer than ∼ 200 µm. Based on this ansatz, we searched for the
amplitude and the wavelength range of SED changes which can bring the "850model" intermediate wavelength counts close to the observed values. We found
that if the SED templates we use in our model be amplified by a factor of 1.6 in
the rest-frame range of 70 < λ < 150 µm, the "850-model" can fit the SPIRE and
160 µm source counts, without any change in already decent results at 70 µm,
850 µm and 1100 µm. Figure 5.10 illustrates the results produced by this modified "850-model" at different wavelengths (blue solid line) together with the
results from Lagache et al. (2004) (orange dashed line) and Valiante et al. (2009)
(gray dash-doted line) models. The redshift distribution of sources responsible
for those results are also shown in Figure 5.11.
The SED boost we implemented, recovers the observed 160 µm counts by
doubling the number of observable sources with S160 > 10mJy; as illustrated
in Figure 5.10, a significant fraction of those sources is distributed at redshifts
z < 1 which is also in agreement with recent observations (Jacobs et al., 2011;
Berta et al., 2011). However, for the SPIRE source counts, the SED modification increases the number of observed sources with SSPIRE < 100mJy which
are mainly at redshifts 0.5 < z < 2. Those redshift distributions together with
the fact that the emission from cold dust with temperatures T . 30K peaks
at ∼ 100 µm, make our SED modification equivalent to adding a population
of preferentially cold dusty galaxies which do not exist at high redshifts. The
width of redshift range in which those objects can exist depends on their temperature range: a warmer population can have a wider redshift distribution but
should have a typically higher redshifts to be invisible at 70 µm while a colder
population can only exist in low redshift ranges in order to not interfere with
850 µm counts. Interestingly, this is in agreement with the required steep colour
evolution implied by models constrained by 160 µm and SPIRE band counts (see
Table 5.2).
As one can see in Figure 5.10, some models like Lagache et al. (2004) and
Valiante et al. (2009) not only produce enough observable sources in SPIRE and
160 µm bands, but at some flux thresholds produce too many sources. However, we notice that both of those models need an additional mechanism to
compensate for under-production of visible sources at intermediate wavelengths
which mimic a population of cold sources only in low redshifts. For instance,
Lagache et al. (2004) use a class of "cold galaxies" in their evolutionary model
which are present only at low redshift, z < 0.5; Lagache et al. (2003) show that
this cold population is producing up to ∼ 50% of the observed 170 µm flux which
means without them the source count is reduced by the same factor, consistent
with our finding and also other models (Le Borgne et al., 2009; Valiante et al.,
2009). Similarly, Valiante et al. (2009) need to strongly modify the colour distribution of low redshift galaxies to correct for under-producing 160 µm sources by
a factor of ∼ 5; they modified the observed colour distribution of IRAS galaxies
which they use as starting point (similar to our model, see §5.2.1) only for low
redshift (i.e. z < 1) objects, by assuming an asymmetric Gaussian distribution
170
Discussion
which is 7 times broader on the "cold" side of distribution in comparison to the
"warm" side (see §4.4 in Valiante et al. (2009)); even with this extreme modification their model does not completely match the observed 160 µm counts in
addition to under-producing luminous 70 µm sources.
Finally, it is worth mentioning that our SED modification is not expected
to change the best-fit models which are based on 70 µm and 850 µm since we
used those source counts as constraints for our SED change search. However
we double checked this issue by using the modified SED set and repeating the
search in parameter space for a best-fit model which is constrained by source
counts at 70 µm, 160 µm, SPIRE bands and 850 µm and recovering the parameters
which define the "850-model", for the best-fit model.
5.7 Discussion
In this section, we discuss different properties of our best-fit model. First we
discuss different implications of our model for the evolving properties of the IR
galaxies like their colour and luminosity distributions. Then, we compare our
model with other existing models for IR and submm source counts.
5.7.1 The implied evolution scenario for dusty galaxies
Our best-fit model mimics the number density evolution of IR galaxies by employing a luminosity evolution together with changing slopes of LF with redshift. While the former changes the amplitude of LF, the latter acts to change the
shape of LF properly to reproduce a number density history which matches the
observed FIR and submm source counts. Although the good agreement between
source counts which our best-fit model produces at different wavelengths and
the observed numbers firmly supports our model, the implied results should be
treated carefully, mainly due to the simplicity of the model: for instance, in our
model we adopted the local LF of IR galaxies which is a simple dual power law
function, and evolved its characteristic luminosity and slopes with redshift to
distribute objects with different luminosities in redshift space correctly.
Although the functional forms we chose for evolution of those parameters
are rather arbitrary, the actual distribution they produce at different redshifts
is necessary to explain the properties of the observed sources. To emphasize
this point, in Figure 5.12 we illustrate how the fractions of galaxies in different
luminosity bins are evolving with redshift, together with the exact shape of the
LF our model requires at different redshifts. As can be seen in the bottom right
panel of Figure 5.12, at low redshifts fainter objects dominate the population
of IR galaxies but at higher redshifts (i.e., z > 2), objects which are in brighter
luminosity ranges take over and become more dominant. Specifically, this
diagram shows that beyond z ∼ 1, objects in the Luminous Infrared Galaxies
(LIRGs) class (with IR luminosities between 1011 and 1012 L⊙ ) dominate the
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Genesis of the dusty Universe
Figure 5.12: Top: The evolution of CF (left) and LF (right) with redshift. The lines from
thick to thin are respectively for redshifts z = 0 (blue), 0.75 (green), 1.5 (orange) and
2.25 (red). Bottom: The change of the relative contribution of different populations of
IR galaxies to their total density with redshift, as required to reproduce the observed
source counts correctly. left: The fraction of objects with luminosities 1011 ≤ L/L ⊙ ≤
1013 which have typical temperatures of T < 30 K, ∼ 30 K, ∼ 40 K and T > 40 K
shown respectively with blue, green, orange and red lines. right: The fraction of total
IR luminosity generated by objects with luminosities 1010 ≤ L/L ⊙ ≤ 1013 , which have
luminosities of 1010 ≤ L/L ⊙ < 1011 , 1011 ≤ L/L ⊙ < 1012 and 1012 ≤ L/L ⊙ ≤ 1013
shown respectively with red, green and blue lines.
172
Discussion
obscured cosmic energy production (although in terms of numbers, fainter
galaxies still dominate). The diagram also shows that UltraLuminous Infrared
Galaxies (ULIRGs, with IR luminosities exceeding 1012 L⊙ ), although increasing
in importance towards high z, never dominate the cosmic energy production. These conclusions are in agreement with analyses of Spitzer data by
Le Floc’h et al. (2005) for z < 1 and by Magnelli et al. (2011) for z < 2.3.
Our best-fit model implies a change in the shape of the LF with redshift
and we are the first to consider this possibility in a model of this type. This
slope evolution implies that the faint end slope of the LF is flattening with increasing redshift (see the top right panel of Figure 5.12). At first sight it may
seems surprising that our results are sensitive to the faint end slope at high
redshifts, but the strong luminosity evolution combined with the strong negative K-correction brings sub-L∗ galaxies well within the region of detectability
at 850 µm. Nevertheless, this result depends strongly on the sub-mJy counts
at 850 µm which currently are derived from studies of gravitationally lensing
clusters, and therefore sample limited cosmic volume. As such, these counts may
be affected by cosmic variance and establishing their levels more firmly (e.g.,
with ALMA), is necessary for confirming our conclusion. However, we note the
non-parametric estimates of the IR luminosity function at high redshifts (e.g.,
Chapman et al., 2005; Wardlow et al., 2011) do indicate flatter faint-end slopes
than the local LF. These samples were selected at 850 µm and may not be complete in IR luminosity, in particular they may be deficient in objects with high
dust temperatures (e.g., Magdis et al., 2010). In order to settle this point, highz luminosity functions of IR luminosity-limited samples are required. Redshift
surveys of Herschel-selected samples will be needed to construct such luminosity functions, and this will become feasible with facilities such as ALMA and
CCAT. If confirmed, the flattening of the faint-end slope of the LF towards
higher redshifts requires a physical explanation, which perhaps could be found
using physically motivated models and simulations. One possibility is that a
higher metagalactic ultraviolet flux at higher redshifts would suppress the development of a star-forming interstellar medium in low-mass galaxies. Another
possibility is a stellar-mass-dependent evolution in Mdust /Mstars towards higher
redshifts. Such an evolution could result from the decreasing overall metallicity
towards high z combined with the net effect over time of the buildup of dust as
a result of stellar evolution and its consumption by star formation. The latter
model can be tested observationally using a combination of Herschel data and
multi-band optical imaging (Bourne et al., in prep.).
We also showed that our model requires a colour evolution to reproduce
the observed source counts. This colour evolution implies that objects with the
same luminosities have lower typical dust temperatures at higher redshifts. In
the bottom left panel of Figure 5.12 we showed how the fraction of objects with
different temperatures is evolving with redshift for all the objects which have
intrinsic luminosities between 1011 < LL⊙ < 1013. While at low redshifts a
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Genesis of the dusty Universe
population of objects which have typical temperatures4 of 40K, dominates the
colour distribution, at higher redshifts the cooler typical temperature of 30K
is dominating. Moreover, while at low redshifts objects with warm dust temperatures are more common than the very cold objects, the situation is the
opposite at earlier times. Although it is difficult to observe the evolution of
dust towards cooler temperatures because of different selection effects, there
is observational evidence for the implied colour (i.e., temperature) evolution
(Chapman et al., 2005; Pope et al., 2006; Chapin et al., 2009; Symeonidis et al.,
2009; Seymour et al., 2010; Hwang et al., 2010; Amblard et al., 2010).
The luminosity and colour evolutions discussed above, show up in redshift
distributions of source counts at different wavelengths. Figure 5.11 illustrates the
buildup of source counts at different wavelengths by contribution from different
redshift bins. It is evident that the longer wavelengths are reflecting the distribution of higher redshift dusty galaxies while shorter wavelengths depend more on
IR galaxies at lower redshifts. For instance 850 µm number counts mainly consist
of galaxies with redshifts higher than z ∼ 2, especially at fluxes ∼ 10 − 20mJy;
in addition, as shown in the bottom right panel of Figure 5.12, at redshifts
z ∼ 2 the luminosity distribution of IR galaxies is not evolving considerably and
even at higher redshifts the fraction of fainter sources, which have lower dust
temperatures, goes up again in comparison to the brightest objects. Moreover,
for two objects with the same redshifts and intrinsic IR luminosities, the colder
one will be brighter at observed 850 µm. This means the visible 850 µm sources
at higher flux thresholds are preferentially colder than sources with fainter observed fluxes which are distributed typically at lower redshifts (see the increasing fraction of objects in 1 < z < 2 redshift bin going towards lower fluxes in the
top right panel of Figure 5.11). This is also in agreement with our experiments
which show that at the bright end, 850 µm number counts are very sensitive
to the slope of the colour evolution in our model, which is in fact the dominant mechanism to produce those counts. Moreover, this implied broader temperature distribution of observed sources at fainter flux thresholds and shorter
wavelengths is in agreement with observations which show bright 850 µm population (i.e. > 4mJy) is highly biased towards cold dust temperatures while
fainter sources (i.e. 1 − 4mJy) also contain an increasing fraction of more luminous objects with lower redshifts but warmer dust temperatures which makes
them the main contributors to the 250 µm source counts (Chapman et al., 2004,
2010; Magnelli et al., 2010; Casey et al., 2009, 2011). Consequently, at z ∼ 2 the
density of farIR selected ULIRGs is approximately 2 times higher than that of
850 µm-selected ULIRGs. This is consistent with our model: a typical ULIRG at
redshift z ∼ 2 will be visible at 850 µm with a flux brighter than a few mJys and
at 250 µm brighter than ∼ 100mJy; on the other hand, roughly ∼ 20 − 30% of objects with few mJy fluxes at 850 µm are within redshifts z ∼ 1 − 2 while around
∼ 50% of objects with ∼ 100mJy fluxes at 250 µm are in the same redshift bin
4
We assign a single temperature to each SED based on its colour and finding a modified black
body radiation with a fixed emissivity, β = 1.5, which can produce that colour.
174
Discussion
(see Figure 5.11).
While the luminosity and colour distribution our best-fit model requires is
consistent with observed 70 µm, 850 µm and 1100 µm source counts, they are not
sufficient to produce enough sources at observed wavelengths in between. The
inconsistency between models which successfully produce 70 µm and 850 µm
counts and their results at 160 µm, is also reported in other works and has
been corrected by including a population of cold galaxies at low redshifts
(Lagache et al., 2003, 2004; Le Borgne et al., 2009; Valiante et al., 2009). While
it is important to note the under-production of 160 µm counts in models,
can be corrected equally by a modifying SED templates instead of introducing a new population (see also Le Borgne et al. (2009)), there is some observational evidence for the existence of a cold population at low redshifts
which is under-represented in IRAS sample and is often associated with bright
spiral galaxies(Stickel et al., 1998, 2000; Chapman et al., 2002; Patris et al., 2003;
Dennefeld et al., 2005; Sajina et al., 2006; Amblard et al., 2010).
Recently, flux density measurements at 250, 350 and 500 µm have become
available for large samples of local galaxies from surveys with SPIRE on the Herschel Space Observatory (Eales et al., 2010; Oliver et al., 2010; Clements et al.,
2010). In addition to problems at 160 µm, we also noticed the inconsistency
between our best-fit model and the source counts provided by SPIRE observations. However, we resolved this issue by modifying our SED templates to
be able to reproduce the source count simultaneously at 70 µm, 160 µm, SPIRE
band, 850 µm and 1100 µm. There is also observational evidence for a population of cold galaxies residing at low redshifts (equivalent to modified SEDs)
in Herschel-selected samples: using a subsample with spectroscopic or reliable
photometric redshifts from Herschel ATLAS survey, Amblard et al. (2010) performed isothermal graybody fits to low-redshift galaxies detected from 70 to
500 µm, resulting in an IR luminosity-temperature relation offset to significantly lower temperatures when compared to the IRAS-based relation derived
by Chapman et al. (2003). The new relation found by Amblard et al. (2010) is
consistent with earlier work by Dye et al. (2009) based on BLAST data. It is also
important to realize that these results do not imply that the IRAS-based dust
temperature fits are incorrect (in fact, they are often supplemented with measurements at longer wavelengths) but they imply that an IRAS-based selection is
biased towards warmer objects.
5.7.2 Our best-fit model and previous models
There are several phenomenological models in the literature which try to
reproduce the properties of IR galaxies at different wavelengths, with different levels of complexity (e.g., Blain & Longair (1993); Guiderdoni et al.
(1997); Blain et al. (1999); Chary & Elbaz (2001); Rowan-Robinson (2001);
Dole et al. (2003); Lagache et al. (2004); Lewis et al. (2005); Le Borgne et al.
(2009); Valiante et al. (2009); Bethermin et al. (2011)). In general, those models
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Genesis of the dusty Universe
use an assumed form of luminosity evolution together with a density evolution
to mimic the evolution of IR galaxy distributions. However, we have limited
ourselves to a pure luminosity evolution with no density evolution, and as pointed out in §5.2.2, the amount of density evolution allowed by the integrated CIB
is small but does not have to be zero. On the other hand, we use evolving faint
and bright ends slopes in our luminosity function. At first sight this may looks
a simple substitution of free parameters; we tried to investigate this by trying
to substitute the slope evolution of our model with a density evolution. However, the resulting fit with density evolution and in absence of slope evolution is
significantly inferior to the fit obtained with pure luminosity evolution.
Another usual practice in modeling the infrared and submm source counts is
to use only one or a few SED templates to represent the whole galaxy population. This approach neglects the observed colour distribution of local IR
galaxies and do not leave any possibility for colour evolution. However, similar
to Valiante et al. (2009), in our model we use a complete set of SED templates
which are shown to be representative of local IR galaxies. This choice enabled
our model to explore the evolution in colours of IR galaxies in addition to their
luminosities.
Our model also differs from other works in the literature in calculating the
source counts for a given evolutionary scenario: we calculate the source count
for a given model by computing the probability of observing different sources
for different flux thresholds. The direct consequence of this new approach is a
fast calculation routine which enables us to calculate the source counts at very
bright observed fluxes, where Monte-Carlo based methods are very inefficient
due to the rarity of such objects, and therefore sometimes end up with very
noisy results (see the bright source counts produced by Valiante et al. (2009) in
Figure 5.10). Moreover, our fast algorithm is an important advantage when it
comes to searching the parameter space for the best-fit model.
A comparison between our best-fit model and Lagache et al. (2004), as a
model without colour distribution and evolution and Valiante et al. (2009) as a
model with colour distributions is shown in Figure 5.8 and Figure 5.9. While at
1100 µm, Valiante et al. (2009) produces ∼ 2 times more visible objects than what
our model produces, all models do reasonably well in accounting for the comulative 850 µm number counts. The results from Valiante et al. (2009) and our
model are very close at SPIRE wavelengths but the Lagache et al. (2004) model
overpredicts the bright counts at 250 µm. At 160 µm, the Valiante et al. (2009)
model over-produces the faint counts and under-produces the bright objects.
However, Lagache et al. (2004) fit the data better, while slightly over-produces
the counts for intermediate to bright flux thresholds. Finally, at 70 µm, where all
the models are expected to fit the data since they use it as a starting point, the
Valiante et al. (2009) source counts deviate from observations by over-producing
the faint counts in expense of producing too few bright objects (probably because of a too extreme modification in colour distributions which is required
in their model to compensate for a factor of ∼ 5 under-production of 160 µm
176
Conclusions
sources).
5.8 Conclusions
We have described a backward evolution model for the IR galaxy population,
with a small number of free parameters, emphasizing which parameters are
constrained by which observations. We also introduced a new algorithm for
calculating source counts for a given evolutionary model by direct integration of
probability distributions which is faster than using Monte-Carlo sampling. This
is an important advantage for searching large volumes of parameter space for
the best-fit model.
While most of the earlier works used only one or a handful of SED templates to represent the whole population of IR objects, we used a library of IR
SEDs which are able to match the IR properties of the large variety of observed
star-forming objects. This approach is necessary in order to model the colour
evolution of IR galaxies in addition to produce simultaneously the counts and
the redshift distributions at wavelengths shorter than 850 µm.
Contrary to some other models, we assumed a negligible contribution from
AGN in our SED templates, noting the inclusion of AGN is only necessary for
reproducing the properties of IR galaxies at very short IR wavelengths5 which
could also be sensitive to other modeling difficulties such as the PAH contribution to the SEDs.
We used available 850 µm source counts together with the redshift distribution of submm galaxies to constrain our best-fit model. At 850 µm, due to the
K-correction, the source count is sensitive to the evolution of IR galaxies in a
wide redshift range and out to very high redshifts. We showed that there is a
degeneracy between the rate by which the characteristic luminosity of IR galaxies should increase to reproduce the source count and the maximum redshift
out to which this increase should be continued; we resolve this degeneracy by
requiring the model to reproduce the observed redshift distribution of submm
galaxies. Moreover, we showed that our model requires a colour evolution towards cooler typical dust temperatures at higher redshifts. The employed colour
evolution is similar to that used by Valiante et al. (2009) however, our best-fit
model predicts a somewhat stronger colour evolution than that proposed by
these authors.
Another important feature of our model is that the best-fit is obtained using pure luminosity evolution but mildly evolving high-luminosity and lowluminosity slopes in the LF. Since high-luminosity sources are rare, the evolution
in the high-luminosity slope is of little consequence. However, the evolution of
the low-lumuninosity slope affects large numbers of galaxies and if confirmed,
this effect must have a physical origin, which can be addressed using numerical
5
For instance at 24 µm where our model under-produces the counts at flux thresholds 0.1 < S24 <
10mJy by a factor of ∼ 1.5 − 2 but matches the observed data at fainter and brighter fluxes
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Genesis of the dusty Universe
simulations of the evolution of the galaxy population, as well as with a combination of deep Herschel and optical imaging.
The 850 µm-constrained best-fit model is consistent with observed 1100 µm
and 70 µm source counts, which confirm the consistency of the implied colourlumionosity-redshift distribution at both low and high redshifts. However, this
model under-produces the observed source counts at intermediate wavelengths,
namely at 160 µm and SPIRE bands. To resolve this issue, we used the observed data at different wavelengths to find best-fit models which can reproduce their observed source counts. While the best-fit models constrained at
70 µm and 850 µm are consistent with each other and also 1100 µm, the implied evolutions for models capable of reproducing observed counts at other
wavelengths are too diverse to be reconciled in a single model; specifically they
need too strong colour evolutions which contradicts 850 µm observations. While
the inconsistency at 160 µm has been reported in earlier works (Le Borgne et al.,
2009; Valiante et al., 2009), we are the first to report it for 250 µm, 350 µm and
500 µm. We showed that the source counts at these wavelengths can be reproduced consistently, by adopting the best-fit model which produces correct 70 µm,
850 µm and 1100 µm source counts, together with a modification in SED templates which is equivalent to the existence of a cold population of dusty galaxies at low to intermediate redshifts which are under-represented in IRAS data.
Besides the fact that there is some observational evidence for the existence of
such galaxies (Stickel et al., 1998, 2000; Chapman et al., 2002; Patris et al., 2003;
Dennefeld et al., 2005; Sajina et al., 2006; Amblard et al., 2010), this assumption
is similar to what other models had to assume in order to reproduce adequate
160 µm sources (Lagache et al., 2003, 2004; Valiante et al., 2009).
It is important to keep in mind that phenomenological models like what we
described in this chapter, are mainly simple mathematical forms which relate
different observations consistently rather than being physical models with explanatory power. However, their performance at different wavelengths and the
distribution of sources they require for different redshifts can be used as their
main predictions which also could be used to test their validity. While we used
the redshift distribution of submm galaxies to constrain our model, we note that
the observed redshift distribution of other wavelengths, if available, are in agreement with our best-fit model predictions (Jacobs et al., 2011; Berta et al., 2011).
Additional information including tabulated data for differential and cumulative source counts at different wavelengths and their redshift distributions is
available at http://www.strw.leidenuniv.nl/genesis/
Acknowledgments
We thank the anonymous referee for valuable comments which improved the
original version of the paper this chapter is based on. We thank D. Dale, H.
Rottgering, J. Schaye and M. Shirazi for useful discussions. During the early
178
REFERENCES
stages of this work, AR was supported by a Huygens Fellowship awarded by
the Dutch Ministry of Culture, Education and Science.
References
Alexander D.M., et al., 2005, ApJ, 632, 736
Amblard A., et al., 2010, A&A, 518, L9
Austermann J.E., et al., 2009, MNRAS, 393, 1573
Austermann J.E., et al., 2010, MNRAS, 401, 160
Barger A.J., et al., 1999, ApJ, 518, L5
Berta B., et al., 2011, A&A, arXive:1106.3070
Bertoldi F., et al., 2007, ApJS, 172, 132
Bethermin M., et al., 2010, A&A, 512, 78
Bethermin M., et al., 2011, A&A, 529, 4
Blain A.W., Longair M.S., 1993, MNRAS, 264, 509
Blain A.W., et al., 1999, MNRAS, 302, 632
Borys, C., et al., 2003, MNRAS, 344, 385
Casey C.M., et al., 2009, MNRAS, 399, 121
Chapin E.L., et al., 2009, MNRAS, 393, 653
Chapman S.C., et al., 2002, MNRAS, 335,17
Chapman S.C., et al., 2003, ApJ, 588, 186
Chapman S.C., et al., 2004, ApJ, 614, 671
Chapman S.C., et al., 2005, ApJ, 622, 772
Chapman S.C., et al., 2010, MNRAS, 409,13
Casey C.M., et al., 2009, MNRAS, 399,121
Casey C.M., et al., 2011, MNRAS.tmp.884C
Chary R., Elbaz D., 2001, ApJ, 556, 562
Clements D.L., et al., 2010, A&A, 518, 8
Coppin K., et al., 2006, MNRAS, 372, 1621
Cowie K., et al., 2006, MNRAS, 372, 1621
Dale D., et al., 2001, ApJ, 549, 215
Dale D., Helou G., 2002, ApJ, 576, 159
Dennefeld M., et al., 2005, A&A, 440, 5
Dole H., et al., 2003, ApJ, 585, 617
Dye S., et al., 2009, ApJ, 703, 285
Eales S., et al., 2010, PASP, 122, 499
Efstathiou A., et al., 1995, MNRAS, 277, 1134
Fadda D., et al., 2010, ApJ, 719, 425
Fixsen D. J., et al., 1998, ApJ, 508, 123
Gispert R., et al., 2000, A&A, 360, 1
Greve, T. R., et al., 2004, MNRAS, 354, 779
Guiderdoni B., et al., 1997, Nature, 390, 257
Hatsukade B., et al., 2011, MNRAS, 411, 102
179
Genesis of the dusty Universe
Hopkins A.M., Beacon J.F., 2006, ApJ, 651, 142
Hwang H.S., et al., 2010, MNRAS, 409, 75
Jacobs B.A., et al., 2011, AJ, 141, 110
Jauzac J.M., et al., 2010, A&A, sub.
Johansson D., et al., 2011, A&A, 527, A117
Khan S.A., et al., 2007, ApJ, 665, 973
Knudsen K.K., et al., 2008, MNRAS, 384, 1611
Lagache G., et al., 2003, MNRAS, 338, 555
Lagache G., et al., 2004, ApJS, 154, 112
Lagache G., et al., 2005, ARA&A, 43, 727
Le Borgne D., et al., 2009, A&A, 504, 727
Le Floc’h E., et al., 2005, ApJ, 632, 169
Lewis G.F., et al., 2005, ApJ, 621, 32
Lutz D., et al., 2005, ApJ, 632, 13
Magdis, G.E., et al., 2010, MNRAS, 409, 22
Magnelli, B., et al., 2010, A&A, 518, 28
Magnelli, B., et al., 2011, A&A, 528, 35
Menendez-Delmestre K., et al., 2009, ApJ, 699, 667
Mullaney J.R., et al., 2011, MNRAS, arXive:1106.4284
Oliver S.J., et al., 2010, A&A, 518, 21
Patanchon G., et al., 2009, ApJ, 707, 1750
Patris J., et al., 2003, A&A, 412, 349
Pope A., et al., 2006, MNRAS, 370, 1185
Pope A., et al., 2008, ApJ, 675, 1171
Puget J.L., et al., 1996, A&A, 308,L5
Rowan-Robinson M. 2001, ApJ, 549, 745
Saunders W., et al., 1990, MNRAS, 242, 318
Sajina A., et al., 2006, MNRAS, 369, 939
Schaye J., et al., 2010, MNRAS, 402, 1536
Scott K.S., et al., 2010, MNRAS, 405, 2260
Seymour N., et al., 2010, MNRAS, 402, 2666
Smail I., et al., 1997, ApJ, 490, L5
Smail I., et al., 2002, MNRAS, 331, 495
Stickel M., et al., 1998, A&A, 336, 116
Stickel M., et al., 2000, A&A, 359, 865
Swinbank A.M., et al., 2004, ApJ, 617, 64
Symeonidis M., et al., 2009, MNRAS, 397, 1728
Valiante E., et al., 2007, ApJ, 660, 1060
Valiante E., et al., 2009, ApJ, 701, 1814
Vieira, J. D., et al., 2010, ApJ, 719, 763
Wardlow, J. L., et al., 2011, MNRAS, tmp.917w
Webb T.M., et al., 2003, ApJ, 587, 41
Weiss A., et al., 2009, ApJ, 707, 1201
180
Appendix A: Some numerical details
Appendix A: Some numerical details
For each specific evolution model, the source count at a given flux threshold and
wavelength can be calculated based on equation (5.13), where the integration
should be performed over all possible luminosities, colours and redshifts. As
mentioned in §5.2.4, we do this by splitting possible colour, luminosity and
redshift ranges into very small bins, assuming that in each bin the related
variable is not changing significantly.
The finite number of SED models we are using automatically splits the
colour range into 64 bins between 0.29 ≤ R(60, 100) ≤ 1.64 (see §5.2.3). We
also use logarithmically spaced bins to split the possible luminosity range of
109 L⊙ ≤ L ≤ 1014 L⊙ into 100 bins in our calculation. This logarithmic scale
which makes the integration roughly insensitive to the number of luminosity
bins, is chosen to cope with the exponential nature of luminosity function where
faint objects are much more numerous than luminous ones. It is also worth
mentioning that the source count calculation is not sensitive to the minimum
or maximum luminosity which is used in integration, if the used luminosity
range covers the important 1010 − 1013 L⊙ luminosity range; for instance, using
Lmin = 107 L⊙ instead of Lmin = 109 L⊙ as the minimum possible luminosity,
does not change any of the source count calculations we are presenting in this
paper.
As discussed in §5.2.4, the uniform distribution of galaxies in redshift space
is assured by the algorithm we are using, independent of the size of redshift
bins. But, for the precise calculation of the K-correction and evolution functions,
we split the redshift range of 0 ≤ z ≤ 8 using bin sizes equal to ∆z = 0.01.
However we noted it is possible to use even bigger redshift bins (e.g. ∆z = 0.1)
without any significant change in the results.
181
Nederlandse samenvatting
De theorie die de vorming van sterrenstelsels beschrijft, omvat natuurkunde
waarin zowel de grootste als de kleinste schalen in het heelal een belangrijke rol
spelen. Het enorme dynamische bereik van de verschillende fysische grootheden (zoals lengte, massa en leeftijd) die een sterrenstelsel karakteriseren, maakt
het modelleren van deze complexe systemen tot een grote uitdaging. Niettemin
zou de vorming en evolutie van sterrenstelsels in principe te begrijpen moeten
zijn vanuit de bestaande fysische wetten. Het nauwkeurig beschrijven en verklaren van de waargenomen trends vanuit deze wetten is tot op heden één van de
belangrijkste doelen van het onderzoeksgebied. Dankzij het werk van vele grote
wetenschappers hebben we nu een tamelijk goed beeld van hoe sterrenstelsels
vormen en zich ontwikkelen en kunnen we met een redelijke nauwkeurigheid
verklaren wat we zien van deze systemen in het heelal. Toch zijn er nog tal van
fenomenen die we niet doorgronden met de huidige theorie van sterrenstelsels,
omdat, evenals in andere gebieden van de natuurwetenschap, iedere stap vooruit in ons begrip van de vorming en evolutie van sterrenstelsels onherroepelijk
leidt tot nieuwe onontrafelde problemen en uitdagingen.
Neutraal waterstof in galactische ecosystemen
Sterrenstelsels worden beïnvloed door aan de ene kant de zwaartekracht, die
verantwoordelijk is voor de vorming van halo’s, het aantrekken van nieuw materiaal (voornamelijk gas) en het bijeenhouden van de ‘baryonische’ en donkere
materie waauit een stelsel is opgebouwd, en aan de andere kant de verschillende
‘feedback’ mechanismen die juist tegen de zwaartekracht in werken en proberen
de opgebouwde structuren te ontbinden. De interactie tussen deze strijdige mechanismen creëert complexe ecosystemen in en rondom sterrenstelsels en drukt
een stempel op de verdeling van baryonen in de nabije omgeving van deze
stelsels (het zogenaamde ‘circumgalactische medium’, CGM). De verdeling van
baryonen, en de verdeling van neutraal waterstof (HI) in het bijzonder, verschaft
ons waardevolle informatie over de ontstaansgeschiedenis van deze sterrenstelsels. De reden dat HI een belangrijke rol speelt in het onderzoek, is dat HI de
voornaamste brandstof vormt voor de vorming van moleculaire gaswolken, de
geboorteplaatsen van sterren. Derhalve is onderzoek naar de verdeling van HI
en hoe deze verandert met de tijd cruciaal voor ons begrip van de verschillende
aspecten van stervorming.
We weten dat vlak na de Big Bang het, in eerste instantie hete, plasma begint
af te koelen naarmate het heelal uitdijt en dat protonen en elektronen uiteindelijk recombineren tot neutraal waterstof. Op z ∼ 6 raakt waterstof, het meest
voorkomende element in het heelal, echter opnieuw in hoge mate geïoniseerd
(‘reïonisatie’) door de vorming van de eerste generatie sterren en sterrenstelsels.
De typische afstand die fotonen kunnen afleggen voordat ze worden geabsorbeerd, neemt vanaf dat moment aanzienlijk toe naarmate de tijd vordert, omdat
aan de ene kant het heelal uitdijt en dus minder dicht wordt en aan de andere
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Nederlandse samenvatting
kant de activiteit van stervorming in het heelal toeneemt. De vele bronnen van
ioniserende straling, die op grote schaal relatief uniform verdeeld zijn, zorgen
voor een min of meer uniforme achtergrond van ultraviolette straling (afgekort
als ‘UVB’), die op z ≥ 3 wordt gedomineerd door fotonen afkomstig van sterren en op lage roodverschuiving door het licht van quasars. Als gevolg van
deze alom aanwezige bron van straling zijn de meeste waterstofatomen blijvend
geïoniseerd vanaf het moment van reïonisatie op z ∼ 6.
Hoe complex de verschillende processen zijn die de ionisatietoestand van
waterstof bepalen, hangt sterk af van de kolomdichtheid van het waterstof. Bij
lage dichtheden (d.w.z. NHI ≤ 1017 cm−2 , wat overeenkomt met het zogenaamde
‘Lyman-α forest’) is HI in hoge mate geïoniseerd door de UVB-straling en overwegend transparant voor ioniserende straling van de UVB dan wel van andere
bronnen. In dit geval kunnen we het systeem beschouwen als ‘optisch dun’ en
kunnen we werken in de ‘optisch dunne limiet’. Bij hogere dichtheden (d.w.z.
NHI ≥ 1017 cm−2 , wat overeenkomt met de zogenaamde ‘Lyman Limit’ en ’Damped Lyman-α’ systemen) is het gas echter optisch dik en daarom overwegend
neutraal. De ionisatietoestand van HI in dergelijke systemen is sterker afhankelijk van de mate waarin de verschillende aspecten, zoals de schaduwgevende
werking van optisch dik materiaal en de fluctuaties van de UVB-straling op
kleine schaal, een rol spelen bij het stralingstransport.
Het onderzoeken van de HI-verdeling in het heelal blijkt geen gemakkelijke
opgave. In het lokale heelal kan de hoeveelheid HI in sterrenstelsels worden
gemeten door gebruik te maken van 21 cm-emissie, maar op hogere roodverschuiving is dit vooralsnog niet mogelijk. Het wachten is op de komst van
krachtigere telescopen, zoals het Square Kilometer Array. Op z ≤ 6, dus na reïonisatie, is het mogelijk HI te bestuderen met behulp van 21 cm-absorptie. We
kunnen de verdeling van het gas bepalen door spectra van heldere achtergrondbronnen zoals quasars te analyseren en te kijken naar de absorptiekenmerken
van HI-systemen door zich tussen ons en de bron bevinden. Op hoge roodverschuiving is het bestuderen van deze absorptiekenmerken daarom een goed
alternatief voor metingen in emissie. De enorme afstanden tussen de meeste
absorberende objecten en hun achtergrondbronnen maken het onwaarschijnlijk
dat ze op enig fysische manier gerelateerd zijn, iets wat de absorptielijnen in
een spectrum zou kunnen beïnvloeden. Dit geeft ons de mogelijkheid om de
verdeling van HI langs de gezichtslijn nauwkeurig te reconstrueren.
De laatste tientallen jaren hebben verscheidene observationele onderzoeken
zich toegespitst op het bepalen van de statistische eigenschappen van de HIverdeling in het heelal. Dankzij een significante toename van het aantal quasars
dat is waargenomen en een verbetering van de observatietechnieken spannen
deze waarnemingen nu een groot bereik in kolomdichtheid en roodverschuiving. Het is daarom belangrijk om de eigenschappen van de HI-verdeling ook
in kosmologische simulaties te bestuderen, die ons een beter begrip van de waargenomen trends zullen geven en ons bovendien zullen helpen de verschillende
observaties in de context van de huidige theorie van sterrenstelsels te plaatsen.
184
Hier zal ik mij op richten in dit Proefschrift. Als onze simulaties overeenkomen
met de observaties, kunnen we deze gebruiken om te voorspellen hoe toekomstige waarnemingen eruit zullen zien. Deze voorspellingen kunnen dan gebruikt
worden aan de ene kant om de juistheid van de onderliggende modellen in de
simulaties te testen, en waar nodig verbeteringen aan te brengen, en aan de
andere kant om te onderzoeken welke rol de verschillende fysische processen
spelen in de vorming en evolutie van sterrenstelsels.
Dit Proefschrift
Hedendaagse observationele onderzoeken gaan verder dan het bestuderen van
de sterren in sterrenstelsels alleen, maar richten zich ook op de verschillende belangrijke, maar zeer complexe, processen die de verdeling van gas in en rondom
stelsels beïnvloeden. Ook de hydrodynamische kosmologische simulaties die
gebruikt worden om de vorming van sterrenstelsels te onderzoeken, verbeteren
aanzienlijk door het gebruik van steeds betere numerieke technieken, sterker
onderbouwde fysische modellen en hogere resoluties. Het is belangrijk, zo niet
essentieel, om observaties en simulaties met elkaar te blijven vergelijken om aan
de ene kant een beter begrip te krijgen van de observationele resultaten en aan
de andere kant de simulaties te testen en te verbeteren.
De verdeling van HI en hoe deze verandert met de tijd is nauw verbonden
met de verschillende aspecten van stervorming. Derhalve is het begrip en modelleren van de HI-verdeling van cruciaal belang bij het onderzoek naar de vorming
en evolutie van sterrenstelsels. In dit Proefschrift zal ik mij voornamelijk richten op het bestuderen van de kosmische verdeling van neutraal waterstof met
behulp van hydrodynamische kosmologische simulaties. We zullen dit doen
door simulaties die gebaseerd zijn op de OverWhelmingly Large Simulations, te
combineren met nauwkeurige beschrijvingen van stralingstransport, waarbij we
rekening houden met de verschillende foto-ionisatieprocessen.
In Hoofdstuk 2 beschrijven we de effecten van de metagalactische UVBstraling en diffuse recombinatiestraling op het stralingstransport en de ionisatie van waterstof op z = 5 − 0. We zullen ons richten op kolomdichtheden
van NHI > 1016 cm−2 , omdat hierbij de effecten van stralingstransport vooral
van belang zijn. Door ruim 12 miljard jaar van de evolutie van de HI-verdeling
te modelleren tonen we aan dat de voorspelde verdeling van kolomdichtheden
uitstekend overeenkomt met de waargenomen verdeling en slechts een matige
verandering vertoont van z = 5 tot z = 0. We vinden ook dat op z ≥ 1 de UVB
de belangrijkste bron van ioniserende straling is, maar dat op lagere roodverschuiving botsingsionisatie een steeds grotere rol begint te spelen, wat gevolgen
heeft voor de schaduwwerking van optisch dik materiaal. Op basis van onze
simulaties presenteren we een aantal afgeleide verbanden die gebruikt kunnen
worden om heel precies de fractie neutraal waterstof te kunnen berekenen zonder rekening te houden met stralingstransport. Gezien de complexiteit van de
185
Nederlandse samenvatting
stralingstransportsimulaties zulllen deze verbanden bijzonder nuttig zijn voor
de volgende generatie van kosmologische simulaties op hoge resolutie.
Stervorming vindt typisch plaats in omgevingen van zeer hoge kolomdichtheid, waar het gas optisch dik en daarom afgeschermd van externe ioniserende
straling is. De lokale sterren vormen daarom een belangrijke bron van ioniserende straling bij deze dichtheden. Het simuleren van deze effecten is echter
geen gemakkelijke opgave doordat het aantal stralingsbronnen over het gehele
simulatievolume extreem groot is. Verschillende onderzoeken vinden tot op heden dan ook verschillende, vaak geenszins overtuigende, resultaten wat betreft
de invloed van straling afkomstig van sterren op de HI-verdeling. In Hoofdstuk
3 bieden we een oplossing voor dit probleem door kosmologische simulaties te
combineren met stralingstransport door gebruik te maken van TRAPHIC, een
code die ontwikkeld is om op een efficiënte manier om te gaan met grote aantallen stralingsbronnen. We simuleren de ioniserende straling van zowel sterren als
van de UVB en recombinatieprocessen. We tonen aan dat de straling van lokale
sterren significant van invloed kan zijn bij kolomdichtheden zoals die voorkomen in ‘Damped Lyman-α’ (DLA) en ‘Lyman Limit’ (LL) systemen. We laten
bovendien zien dat de voornaamste reden dat de resultaten van eerdere werken
niet altijd overeenkwamen, het gebrek aan een voldoende hoge resolutie is. Dit
probleem lossen wij echter op door stervormende ‘deeltjes’ in de simulatie als
ionisatiebronnen te gebruiken. We tonen ook aan dat wanneer het interstellaire
medium (ISM) in kosmologische simulaties niet volledig is opgelost, dit het modelleren van de eigenschappen van sterke DLA’s (d.w.z. NHI ≥ 1021 cm−2 ) in de
weg staat.
Objecten die zorgen voor sterke HI-absorptielijnen als DLA’s zijn hoogst
waarschijnlijk representatief voor het koude gas dat zich in, of in de buurt van,
het ISM van sterrenstelsels op hoge roodverschuiving bevindt. Deze objecten
maken het daarom mogelijk een verzameling sterrenstelsels te selecteren op basis van hun absorptiekenmerken en aan de hand hiervan het ISM te bestuderen,
met name hoe dit eruit zag in de vroege stadia van de vorming en evolutie
van sterrenstelsels. Omdat observationele onderzoeken doorgaans beperkt worden door het kleine aantal sterk absorberende bronnen en bovendien voor de
lage-massa stelsels detecties missen in emissie, is het onderzoeken van de relatie
tussen de absorberende objecten en de sterrenstelsels waarin zij zich bevinden,
echter verre van gemakkelijk en is het noodzakelijk om gebruik te maken van
kosmologische simulaties om de relatie tussen deze twee te doorgronden. In
Hoofdstuk 4 gebruiken we hydrodynamische kosmologische simulaties, waarvan we in Hoofdstuk 2 hebben laten zien dat zij goed overeenkomen met wat
we zien in observaties van de HI-verdeling, om de connectie tussen sterke HIsystemen en sterrenstelsels op z = 3 te bestuderen. We tonen aan dat de sterkst
absorberende objecten overeenkomen met lage-massa sterrenstelsels die met de
huidige waarneemtechnieken te zwak zijn om gedetecteerd te worden. We laten echter zien dat onze voorspellingen in goede overeenstemming zijn met de
bestaande observaties. We wijzen op een sterke anticorrelatie tussen de kolom186
dichtheid van sterk absorberende HI-systemen en de loodrechte afstand tussen
elk object en het dichtstbijzijnde sterrenstelsel. Bovendien onderzoeken we de
correlatie tussen de kolomdichtheid en de verschillende eigenschappen van de
sterrenstelsels waar de HI-systemen mee geassocieerd worden.
Net als neutraal waterstof, de voornaamste vorm van brandstof voor stervorming, heeft ook het stof dat aanwezig is in sterrenstelsels een sterke invloed
op de stervormingsactiviteit. Het bestuderen van de verdeling en evolutie van
stof is daarom ook van groot belang voor ons begrip van de ontwikkeling van
sterrenstelsels. Door de lage resolutie van de waarnemingen, die doorgaans op
lange golflengte worden gedaan, is de identificatie en het nemen van spectra
van individuele infrarode sterrenstelsels in het verre heelal een lastige opgave.
Relevante informatie over de evolutie en statistische eigenschappen van deze
objecten wordt daarom gehaald uit de oppervlaktedichtheid van de bronnen als
functie van hun helderheid (m.a.w. het aantal bronnen per oppervlakte met
een bepaalde helderheid). In Hoofdstuk 5 presenteren we een model voor de
evolutie van stoffige sterrenstelsels dat gebaseerd is op de waargenomen brondichtheid en roodverschuivingverdeling bij 850µm. We gebruiken een simpel
formalisme voor de evolutie van de kleur- en helderheidsverdeling van infrarode stelsels. Met behulp van een nieuw algoritme om de brondichtheden te
berekenen onderzoeken we de mate waarin de verschillende vrije parameters
van ons model worden beperkt door de beschikbare observationele data. We
laten zien dat het model uitstekend in staat is de waargenomen brondichtheid
en roodverschuivingverdeling op golflengten van 70µm . λ . 1100µm te reproduceren en dat het in goede overeenstemming is met de meest recente resultaten
van Herschel en SCUBA 2.
187
Publications
1. Predictions for the relation between strong HI absorbers and galaxies at z = 3
Alireza Rahmati & Joop Schaye
2013, MNRAS, to be submitted.
2. The effect of recombination radiation on the temperature and ionization state of
partially ionized gas
Milan Raičevic̀, Andreas H. Pawlik, Joop Schaye & Alireza Rahmati
2013, MNRAS, submitted.
3. Stars were born in significantly denser regions in the early Universe
Maryam Shirazi, Jarle Brinchmann & Alireza Rahmati
2013, ApJ, submitted, arXiv:1307.4758.
4. The impact of local stellar radiation on the HI column density distribution
Alireza Rahmati, Joop Schaye, Andreas H. Pawlik & Milan Raičevic̀
2013, MNRAS, 431, 2261-2277.
5. On the evolution of the HI column density distribution in cosmological
simulations
Alireza Rahmati, Andreas H. Pawlik, Milan Raičevic̀ & Joop Schaye
2013, MNRAS, 430, 2427-2445.
6. Genesis of the dusty Universe: modeling submillimetre source counts
Alireza Rahmati & Paul van der Werf
2011, MNRAS, 418, 176-194.
7. New biorthogonal potential–density basis functions
Alireza Rahmati & Mir Abbas Jalali
2009, MNRAS, 393, 1459-1466.
189
Curriculum Vitae
I was born on 9 Feburary 1982 in Qom, Iran. When I was two years old, I started
to taste academic life as my father started his undergraduate studies in Shiraz
University. This was the beginning of moving to different parts of the country. I
grew up in Qom, Shiraz, Tehran and Yazd, and I went to several different schools
in those cities.
In September 2000, I received my pre-university diploma in physics and
mathematics from one of the branches of the National Organization for Development of Exceptional Talents (NODET) in Yazd, Iran, and started my Physics
undergraduate studies at Amir Kabir University of Technology (Tehran Polytechnic) in Tehran, Iran. For my B.Sc. thesis, I worked with Prof. dr. B. Maragechi
on studying the self-field effects in the dispersion relation of relativistic electron
beams. After a very successful performance in national graduate entrance examinations, which brought me the exemption from compulsory military service,
I started my Physics graduate studies in September 2004, at Sharif University of
Technology in Tehran, Iran. For my M.Sc. thesis, I worked with Prof. dr. M. A.
Jalali on instabilities of elliptical galaxies, which resulted in my first scientific
publication. After finishing my M.Sc. degree in 2006, I was planning to pursue
my studies as a Ph.D. student in the US. However, as an Iranian, I was denied
a student visa to enter the US and to benefit from my awarded fellowship at
Northwestern University. A year later, in September 2007, I started at Leiden
Observatory as a M.Sc. student, thanks to the Oort and Huygens scholarships.
For my minor project, I worked with Prof. dr. M. Franx before starting my
M.Sc. thesis with Prof. dr. P. van der Werf on modeling the evolution of dusty
galaxies. I graduated cum laude and I decided to stay at Leiden Observatory for
my Ph.D., and to work with Prof. dr. Schaye by joining his research group in
September 2009.
During my Ph.D., I have presented my work in several institutes and international conferences in Aix-en-Provence (France), Barcelona (Spain), Cambridge,
Durham and Edinburgh (UK), Groningen, Dwingeloo (Netherlands), Garching
and Heidelberg (Germany), Trieste (Italy), Zurich (Switzerland), Austin, Berkeley, Madison, Santa Barbara, Santa Cruz and San Diego (US).
After a brief working visit at CONICET, Universidad de Buenos Aires (Argentina) in September 2013, I will continue to live my long-standing dream of
being an astrophysicist by starting a postdoctoral fellowship at the Max-Planck
Institute for Astrophysics (Garching, Germany), in October 2013.
191
Acknowledgements
With the limited memory that I have, and also because I am not allowed to thank
my Ph.D. supervisor here, it is an impossible-to-succeed task to thank properly
everyone who helped me to happily finish this thesis and to get to this stage of
my life.
I had six years of wonderful time at the Sterrewacht. First, I should thank
people who assured me that coming to Leiden was a good idea to begin with:
Amir (my M.Sc. supervisor in Iran), Mario Juric and Scott Tremaine (through
Mario). Also, I should thank Jarle for helping me to decide to stay. Then, I
would like to thank all the past and current sterrewachters whose presence has
created one of the best research institutes in the terrestrial world for studying
the Universe. In particular, I would like to express my special gratitude to the
support staff.
It was a great pleasure to be part of Joop’s research group during the past
four years. I would like to thank all the past and current group members who
helped to establish a solid foundation. Craig, Rob, Marcel, I am glad that I was
part of the same group with you. Claudio, we did not have much overlap, but
I am grateful for your help whenever I needed it. Andreas, thanks so much for
being a great friend, mentor and collaborator. We started our regular and often
intense meetings before I officially start my Ph.D. and when you were extremely
busy with finishing yours. Thanks for always finding time for me. I am glad that
finally we are going to be at the same institute and get to see each other and talk
more often. Milan, it was very nice to be your neighbor at the Sterrewacht as I
finally had a friend next door to share my cone-related thoughts with. I always
remember the good time we had every friday morning in the group meetings
enlightening everyone with our cones and getting enlightened in the afternoons!
Freeke, I would like to thank you and Olivera for telling me about the life as
a member of the group, before I decide to join. Also many thanks for being
such a wonderful host during my very short visit at Berkeley. Olivera, it was so
nice to have you as a friend and group-mate. Thanks for being worried about
me and telling me many secrets about the group and beyond. Joki, I was very
happy to see our RT gang growing when you joined the group and I am very
sad to leave you alone. I really appreciate your warm friendship and I always
have enjoyed your relaxed attitude towards everything. Marcello and Simone,
thanks for making the preprint meetings more lively. I always have enjoyed
talking to you about science and other things. Alex, Ben, Caroline, Marcel,
Marco, Monica and Rob, thanks for being wonderful group-mates during the
last couple of years, I will miss you all. And finally, Marijke, I am very happy
that you rejoined the group and I am sad that I cannot enjoy your company as a
group-mate. I am deeply grateful that you spend your holiday time to help me
out with my Nederlandse Samenvatting.
I am grateful for having many friends who have filled my life in the Netherlands with joy. Besides my countless colleagues at the Sterrewacht, I would
like to specially thank Azadeh & Saeed, Islam & Nynke, Maryam & Mehdi, Mo193
Acknowledgements
hammad, Mahsa & Rojman for their priceless friendship. Being with you makes
me happy and I am very fortunate to have you as friends.
And finally, my deep gratitude goes to my family for their eternal support.
Mom and dad, I am lucky to have such wonderful parents. Thanks for making
me who I am. My lovely sisters, Leili and Elahe, thank you for your kindness
and your great company whenever we get together. And Maryam, you have
always been with me during ups and downs of the last four years. Without you,
it would not have been possible to go through this stage of my life so smoothly.
I am deeply grateful for your patience as I was often complaining about many
things. The most valuable reward I got by being in Leiden during the last four
years was your priceless company.
A. Rahmati,
Buenos Aires, September 2013.
194
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