Simulations of Cosmic Reionization

Simulations of Cosmic Reionization
Simulations of Cosmic Reionization
Simulations of Cosmic Reionization
Shapes & Sizes of H II regions around Galaxies and
Quasars
by
Martina M. Friedrich
Coverimage:
Photomontage showing me standing on a computer harddisk painting the
temperature map from test4 from PAPER II looking at the code algorithm.
The background shows a photo of Stockholm with the City hall on the left
hand side.
The design element used in the chapter headings is a position-redshift slice
provided by Garrelt Mellema. It should be noted that the redshift direction is
streched in this representation.
c Martina M. Friedrich, Stockholm 2012
ISBN 978-91-7447-448-0
Printed by Universitetsservice, US-AB, Stockholm 2012
Distributor: Department of Astronomy, Stockholm University
Abstract
After the era of recombination, roughly 360 000 years after the big
bang (redshift 1100), the universe was neutral, continued to expand and
eventually the first gravitationally collapsed structures capable of forming
stars, formed. Observations show that approximately 1 billion years later
(redshift 6), the Universe had become highly ionized. The transition from a
neutral intergalactic medium to a highly ionized one, is called the epoch of
Reionization (EoR). Although quasar spectra and polarization power-spectra
from cosmic microwave background experiments set some time-constrains
on this epoch, the details of this process are currently not known.
New radio telescopes operating at low frequencies aim at measuring directly the neutral hydrogen content between redshifts 6 - 10 via the HI spinflip line at 21cm. The interpretation of these first measurements is not going
to be trivial. Therefore, simulations of the EoR are useful to test the many illconstrained parameters such as the properties of the sources responsible for
reionization. This thesis contributes to such simulations.
It addresses different source models and discusses different measures to
quantify their effect on the shapes and sizes of the emerging H II regions.
It also presents a new version of the widely used radiative transfer code C2 R AY which is capable of handling the ionizing radiation produced by energetic sources such as quasars. Using this new version we study whether 21cm
experiments could detect the signature of a quasar.
We find that different size measures of ionized regions can distinguish between different source models in the simulations and that a topological measure of the ionized fraction field confirms the inside-out (i.e. overdense regions
ionize first) reionization scenario. We find that the HII regions from luminous
quasars may be detectable in 21cm, but that it might not be possible to distinguish them from the largest HII regions produced by clustered galaxies.
E NJOY...
Contents
List of publications . . . . . . . . . . . . . . . .
List of acronyms . . . . . . . . . . . . . . . . . .
Preface . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction to Cosmic Reionization
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1.1 Background and observational constrains . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Future observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Modelling the EoR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Solving radiative transfer in one dimension . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Ray-tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Quasars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Quasar luminosity – Halo mass relation . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Quasar lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 The spectrum of quasars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Summary: Quasars in our simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Summary of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
xiii
1
3
4
8
13
17
19
22
25
28
32
33
34
37
5.1 Summary of “Topology and sizes of H II regions during cosmic reionization”
(PAPER I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
5.2 Summary of “Radiative transfer of energetic photons: X-rays and helium ionization in
C2 -R AY” (PAPER II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Summary of “Prospects of observing a quasar H II region during the EoR with
redshifted 21cm” (PAPER III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
- ix -
39
41
47
List of Publications
This thesis is based on the following publications, which are referred to in the
text by their Roman numerals.
I
II
III
M. M. Friedrich, G. Mellema, M. A. Alvarez, P. R. Shapiro & I. T.
Iliev (2011) Topology and sizes of H II regions during cosmic
reionization, MNRAS, 413, 27
M. M. Friedrich, G. Mellema, I. T. Iliev & P. R. Shapiro (2012)
Radiative transfer of energetic photons: X-rays and helium
ionization in C2 -R AY, MNRAS, 2385
M. M. Friedrich, K. K. Datta, G. Mellema & I. T. Iliev (2012)
Prospects of observing a quasar HII region during the EoR
with redshifted 21cm, to be submitted to MNRAS
The reprints of this publications can be found at the end of this thesis. My contribu-
tion to these publications is as follows:
PAPER I: I performed the analysis presented, produced all the figures and
wrote the initial paper draft which was revised by the co-authors and me.
PAPER II: I developed the code-extensions subject to this publication, performed the test simulations presented in the paper (except the cosmological
simulation test 3 without helium and the hydrogen only simulation of test 4),
produced all figures and wrote the initial draft which was revised by the coauthors and me.
PAPER III: I performed the radiative transfer simulations including helium
(i.e. not the hydrogen only simulations), made the analysis presented in Section 3 and in the conclusions. I wrote the first draft of Sections 1-3, the abstract
and most of the conclusions. I participated in the discussion and editing but
not in the analysis and writing of Sections 4–7. The whole draft was revised
by the co-authors and me.
Publications not included in this thesis:
- xi -
M. M. Friedrich, G. Mellema, M. A. Alvarez, P.R. Shapiro,I. T. Iliev (2011)
The Euler Characteristic as a Measure of the Topology of Cosmic Reionization 2011, Revista Mexicana de Astronomia y Astrofisica Conference Series
K. Weltecke, M. M. Friedrich, T. Gaertig (2012) Non-invasive in situ measurements of top soil gas diffusivity at urban soils, submitted to SSSAJ
- xii -
List of acronyms and
abbreviations
21CMA
21 Centimeter Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
ΛCDM
flat, cold dark matter model including dark energy . . . . . . . . . 4
AGN
active galactic nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
bb
big bang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
BB
black body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
BH
black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
CDM
cold dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
CMB
cosmic microwave background radiation . . . . . . . . . . . . . . . . . . 1
DM
dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
EDGES
Experiment to detect the global Epoch of Reionization
signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
EE
E-mode polarization power spectrum . . . . . . . . . . . . . . . . . . . . . 6
EM
electro magnetic radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
EoR
Epoch of Reionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
GMRT
Giant Metrewave Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
GP
Gunn-Peterson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
GRB
gamma ray bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
HST
Hubble Space Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
IGM
intergalactic medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
IMF
initial mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
JWST
James Webb Space Telescope. . . . . . . . . . . . . . . . . . . . . . . . . . . .9
OTS
on the spot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
LOFAR
Low Frequency Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
- xiii -
mfp
mean free path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
MWA
Murchison Widefield Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
PAPER
Precision Array to Probe the Epoch of Reionization . . . . . . 11
POP II
Population II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
POP III
Population III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
PS
power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
QSO
quasi stellar object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
RJ
Rayleigh-Jeans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
SDSS
Sloan Digital Sky Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
SED
spectral energy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
SKA
Square Kilometre Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
TE
temperature-E-mode cross correlation power spectrum . . . . . 6
UV
ultraviolet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
WMAP
Wilkinson Microwave Anisotropy Probe . . . . . . . . . . . . . . . . . . 6
i.e.
id est (Latin: that is)
e.g.
exempli gratia (Latin: For Example)
l.h.s.
left hand side
- xiv -
Preface
The topic of this thesis is the Epoch of Reionization (EoR). More specific, it
deals with different aspects of modelling this epoch. Little is known to date
about the details of this time interval: After recombination, the universe was
neutral and continued to expand, the density perturbations grew to form dark
matter (DM) halos, first stars and galaxies formed therein and eventually the
intergalactic medium (IGM) became ionized (again). The latter we know from
quasar spectra and the polarization power spectrum of the cosmic microwave
background radiation (CMB).
Chapter 1 places the EoR into a cosmological context and gives an overview
of the current observational constrains on the EoR. It also gives some basic
background to the planned 21 cm experiments to which the results of EoR
simulations can be hopefully compared in the near future.
In Chapter 2, I illustrate how we are modelling the EoR with simulations.
Here I also point out at which stage the three publications which form the
basis for this thesis contribute to the process of modelling. Chapter 3 serves
as in introduction to the methodology PAPER II that describes the changes
made in the radiative transfer code C2 R AY. In Chapter 4, I give some useful
background and justifications for the quasar model used in the simulations of
PAPER III. I give a summary of the papers in Chapter 5.
The results from the appended publications (PAPER I – III) are not repeated
in the introductory part of this thesis. However, the last chapter comprises a
summary of the appended papers. Some of the figures in PAPER I and PAPER II are originally in colour but appear here in black/white.
-1-
C HAPTER 1
I NTRODUCTION TO C OSMIC
R EIONIZATION
The Epoch of Reionization (EoR) usually refers to the timespan in between the
following two events: (1) The formation of the first sources of light forming
in the first halos that collapse in a neutral Universe under the influence of selfgravity and decouple from the Hubble expansion. (2) The time when most
of the intergalactic hydrogen has become ionized by the ionizing radiation
emitted from these sources of light and escaping into the IGM. Very roughly,
in terms of the age of the universe since the big bang (bb), this corresponds
to 0.1 Gyr (1) and 1 Gyr (2). These times dependent on the definition of the
“beginning” and “end” of the EoR and the cosmology.
Before going into more details about this epoch and the observational constrains we have about it, we need to define some cosmological foundations.
In this thesis, I assume a flat Λ cold dark matter (CDM) model, ΛCDM to be
the underlying cosmological model. Here, flat indicates that the curvature is
0, Λ indicates that the model includes dark energy and cold refers to the nonrelativistic speed of the dark matter particles at the time of matter decoupling
from radiation.
There are good reasons to trust this so called standard model of cosmology: structure formation simulations based on this model agree well with the
observed large scale structure of the universe ( e.g. Springel et al. 2005), the
anisotropies in the CMB observed with WMAP1 can be explained with this
model (e.g. Spergel et al. 2003, 2007; Page et al. 2007; Jarosik et al. 2011;
Larson et al. 2011), the acceleration of the expansion of the universe caused
by Λ is actually observed (Riess et al. 1998; Perlmutter et al. 1999; Perlmutter
& Riess 1999)2 and measurements of the deuterium fraction at redshifts z ∼ 3
are in agreement with the predicted ones from bb nucleosyntheis (Burles &
Tytler 1998a,b).
1 http://map.gsfc.nasa.gov/
2 Both
Saul Perlmutter and Adam G. Riess together with Brian P. Schmidt received the Nobel
Prize in physics 2011 for this discovery.
-3-
CHAPTER 1. INTRODUCTION TO COSMIC REIONIZATION
However, there are also potential problems with the model, for example
the missing satellite problem3 and unsolved questions (e.g. what is the nature
of dark matter and dark energy? ). This thesis is not about the underlying
cosmology but deals with the transport of ionizing radiation through a matter
(dominated) universe evolving by means of a fixed cosmology, the flat, cold
dark matter model including dark energy (ΛCDM) cosmology. Therefore, I
will not provide an introduction to cosmology but follow a more descriptive
approach and explain notations when needed, concentrating on the parameters
directly related to the study of the EoR.
1.1
Background and observational constrains
The ΛCDM model is a model whose current day energy content is dominated
by some yet unknown form of energy Λ. In the model, Λ contributes roughly
70% to the total energy content in the universe today. In the following, the
energy content will be given in units of the critical density ρc,0 (where the subscript 0 indicates here and henceforth, time t = today), which in a flat universe
model is equal the total energy content. So we write ΩΛ ≈ 0.7. In analogy, Ωm
and Ωb are the total and baryonic mass contents of the universe, respectively.
The currently best estimates are (ΩΛ , Ωm , Ωb ) = (0.728, 0.272, 0.0455) (Komatsu et al. 2011).4 Although these values are only valid at t = today, we skip
the subscript 0.
The ΛCDM model is a model in which the universe emerges from a singularity and has been expanding since (a bb model). The most direct evidences5
for a bb are Hubble expansion diagrams that locally (i.e. for a small redshift
z) show a linear relation between the distance d of objects and their recession speed v (e.g. Hubble 1929; Freedman et al. 2001; Freedman & Madore
2010). However, due to the rather large uncertainties connected to the distance measurements, current estimates of the Hubble parameter H0 = v/d
using this method have rather large errors (Freedman & Madore 2010, give
H0 = 73 ±random 2 ±systematics 4 km s−1 Mpc−1 ). In general, H is not a constant,
3 According
to numerical ΛCDM, DM only, simulations, there should be many more dwarf
galaxies than are observed (e.g. Moore et al. 1999; Klypin et al. 1999), which might or might
not be explained by including detailed gas and radiation physics in the simulations
4 The values given above are the maximum likelihood values from considering WMAP and
baryonic acoustic oszillations. The mean and 1σ errors are (ΩΛ , Ωm h2 , Ωb ) = (0.725 ±
0.016, 0.1352 ± 0.0036, 0.0458 ± 0.0016), here h is the hubble parameter in units of 100
km/s/Mpc, see below and h = 0.702 ± 0.014.
5 however not a proof since other models are not excluded by this
-4-
and defined as the ratio of the rate of change of the scale factor6 , ȧ over the
scale factor a where the scale factor today a0 is define as 1. This relationship
makes it possible to use the Doppler shift of photons emitted from a source at
a certain distance as a measure of distance, named the redshift
z=
λobserved − λemitted
.
λemitted
Since the speed of light is finite, redshift also serves as a measure of look-back
time.
A more accurate estimate of H0 (or h := H0 /[100 km s−1 Mpc−1 ], which I
will use from now on for convenience) can be derived from the six primary
ΛCDM model parameters (for details, see any of the references following
in this paragraph) fitted to the different statistics of the measurements of the
anisotropies of the CMB. The best estimate today is h = 0.702 ± 0.014 (Komatsu et al. 2011). The bare existence of the CMB is another evidence for
a bb: it emerges from the time when the temperature of the universe cooled
down to a value where the number of photons energetic enough to ionize hydrogen fell short of the number of protons. This process of (re-)combination
is not an instantaneous change but occurred during a finite redshift interval
(∆z about several hundreds) and ended at z ∼ 1100 when the temperature was
roughly 3000 K. The matter inhomogeneities present at that time can be observed today in the temperature fluctuations (due to gravitational redshift from
this time of last scattering, Sachs & Wolfe 1967) in the CMB. However, fluctuations on scales smaller than the scales corresponding to ∆z are suppressed
due to the finite width of the last scattering. Thanks to the accurate measurements of temperature and polarization anisotropies of the CMB (Spergel et al.
2003, 2007; Page et al. 2007; Jarosik et al. 2011; Larson et al. 2011) and their
statistics, there is not only qualitative confirmation of the ΛCDM model, but
the model parameters can be fitted to great accuracy (Komatsu et al. 2011).
One of the model parameters that are directly fitted to the CMB data is the
so called reionization optical depth
Z t(zrec )
τ = cσT
ne (z(t))dt .
0
Here, c = 2.998 × 108 m/s is the speed of light, σT = 6.65 × 10−29 m2 is the
Thomson scattering cross-section and ne (z(t)) is the mean number density of
free electrons at redshift z (e.g. Page et al. 2007). This parameter is derived
6 As well known, given the different energy contents of the universe, the first Friedman equation
2
8πGρ0,matter
8πGρ
ȧ(t) 2
describes the change of scale factor: a(t) = 3a(t)
+ 3a(t)0,rad
+ Λ3c . Here, the curvature
3
4
term is omitted since in the standard model k = 0. It should be noted that in most representations,
the radiation energy content is omitted due to the fact that it becomes negligible for z 3400.
-5-
CHAPTER 1. INTRODUCTION TO COSMIC REIONIZATION
primarily from the E-mode polarization power spectrum (EE)7 : The scattering
off of photons of free electrons at redshift z introduces additional polarization
at the scale of horizon at z, so a bump at low multipole moments l (for not
too small l , l = π/θ , where θ is the angular scale) in the EE power spectrum is the result. In theory, from the shape of the bump, more information
than just an integrated optical depth can be extracted. However, the measurement errors are too large wherefore the power spectrum is integrated over a
range of l values, see Larson et al. (2011). The currently best estimate for
the optical depth is τ = 0.088 ± 0.014 (Komatsu et al. 2011). Assuming an
instantaneous reionization (H I −→ H II and He I −→ He II) and assuming helium became doubly ionized at z ∼ 3.5, this results in an ionization redshift
of zreion = 10.5 ± 1.2 (Komatsu et al. 2011; Larson et al. 2011). The inclusion of electrons from He II −→ He III ionization is a new feature in the code
used to make model fits, CAMB (Lewis et al. 2000; Lewis 2008), this resulted in slightly lower zreion for the WMAP7 results than those published for
the WMAP5 results (Spergel et al. 2007; Page et al. 2007).
To summarize EoR results from the CMB: By statistics of the CMB measurements, one can acquire information about the time-integrated electron
density and therefore, assuming an instantaneous reionization, a redshift of
reionization zreion . The rather large change in the best fit for τ from WMAP1
to WMAP3 is partly due to a change in strategy (using mainly the EE power
spectrum instead of the TE power spectrum). The change in best fit zreion between WMAP5 and WMAP7 are mainly due to the inclusion of helium ionization electrons. Therefore, sceptics arguing that WMAP results have not been
consistent through the years regarding the EoR do not have a strong point.
The CMB results constitute one of two main constraints for the EoR available today. The other comes from spectra of high redshift quasars. Gunn & Peterson (1965) found the Lyα line of a quasar at z ∼ 2.01 to show “no obvious
asymmetry” and a maximum depression on the blue side of the line, of 40%.
They converted this into a number density for neutral hydrogen of 6 × 10−11
cm−3 in proper (i.e. not comoving) physical units at that redshift. This meant
either that the total mass density of hydrogen is much smaller than expected
or that the IGM at those redshifts must be very highly ionized. They ruled
out a collisionally ionized IGM since the timescales (collisional ionization
time scale and necessary ionization time scale) do not match for realistic IGM
temperatures (which should not violate the X-ray background from free-free
emission). They also ruled out quasi stellar object (QSO) and normal galax7 As
Spergel et al. (2007) mention, the Wilkinson Microwave Anisotropy Probe (WMAP) 1st
year results were mainly based on the temperature-E-mode cross correlation power spectrum
(TE) and τ was degenerate with the power law spectral index of primordial density fluctuations
ns , where the likelihood for τ varied over a large range [0.05 – 0.3] only slightly, which resulted
in a rather high value for τ as the best fit parameter
-6-
ies as sources of reionization. Their best bet was black body (BB) radiation
from the IGM itself, assuming a temperature of TIGM ∼ 2.5 × 105 K. While
the latter is excluded today, the question of the main sources of reionization
still remains.
As time went by, QSO spectra at higher and higher redshifts were taken and
the spectra blueward of the Lyα emission line were examined for the existence
of complete absorption, called the Gunn-Peterson (GP) trough, expected from
a moderately neutral IGM, as shown by Gunn & Peterson (1965). The spectra
of QSOs around redshift 6 found in the Sloan Digital Sky Survey (SDSS)8
were the first to indicate a low, but rapidly rising neutral fraction (Fan et al.
2006; Willott et al. 2007), see Fig.1.1 for a reproduction of a plot showing the
calculated effective GP optical depth τ eff .9 This suggests that the EoR ended
around a redshift z ∼ 6.
There are other observations which set more indirect limits on the EoR,
such as the measurement of the ultraviolet (UV) background photoionization
rate Γ from 2 ≤ z ≤ 6 ( see for example Faucher-Giguère et al. 2009„ who
confirm by integrating the luminosity functions of galaxies and quasars the
values measured by H I Lyα forest10 data of Γ ∼ (5 − 10) × 10−13 /s /neutral
atom) or direct observations of high redshift galaxies constraining the luminosity function at high redshifts. The problem with the former is that the UV
background is observed at a time where reionization is (almost) completed
and therefore the information on the sources that ionized the universe some
redshift units earlier, is not clear. The problem with the latter is that only the
brightest galaxies can be observed at redshifts relevant for reionization. Those
galaxies are very rare and most likely not the main contributor to cosmic reionization. However, gravitational lensing might help, as Hall et al. (2011) point
out: as is well known, the area probed by gravitational lensing decreases with
increasing magnification. This means that the probed area in space decreases
with the minimum intrinsic luminosity of background galaxies to be still observable. (Galaxies with lower intrinsic luminosity need to be magnified more
and therefore, the area of investigation is smaller) However, the luminosity
function, which describes the number of galaxies in a certain volume of space
as a function of luminosity, is believed to be very steep at the faint end and
therefore the number of small galaxies increases rapidly with decreasing intrinsic luminosity. Hall et al. (2011) found that the probed area (as function of
8 http://www.sdss.org/
9 Fan
et al. (2006) define the effective GP optical depthD as the
E natural logarithm of the average
of the ratio observed flux over intrinsic flux, τ eff = ln( ffobs
).
int
10 The
Lyα forest is the part in a spectrum of a distant quasar/galaxy (bluewards of its Lyα
emission line) where many thin absorption lines, originating from neutral hydrogen clouds in
the IGM between us and the quasar/galaxy, are visible.
-7-
CHAPTER 1. INTRODUCTION TO COSMIC REIONIZATION
Figure 1.1: Reproduction of figure 4 from Goto et al. (2011) by permission of John
Wiley and Sons (original in colour). Effective GP optical depth τ eff from several high
redshift quasars. The limits from the spectrum of CFHQS J2329-0301, a z=6.4 quasar
are from Lyα, Lyβ and Lyγ, as indicated in the legend, the black triangles are from Fan
et al. (2006) and the small squares are from Songaila (2004). The solid line is the best
power-law fit to the data at z < 5.5 by Fan et al. (2006) τ eff = 0.85 ((1 + z)/5)4.3 (their
equation 5) including more low-redshift quasars from Songaila (2004). The lower limits on the effective optical depth come from no-flux-detections.
magnification and hence minimum intrinsic luminosity) decreases slower than
the expected number of galaxies with a certain minimum luminosity. Therefore, number counts should still be possible. Measurements of the soft X-ray
background can be used to limit the contribution from quasars to the EoR
(Dijkstra et al. 2004, 2011).
1.2
Future observations
Although there are other promising future observations such as new constrains
on the anisotropies in the CMB from Planck11 (smaller errors on τ , see e.g.
Zaldarriaga et al. 2009), probing intergalactic Lyα absorption with gamma ray
bursts (GRB) afterglow spectra or observing Lyα emitters during the EoR (see
11 This
is not an acronym, it is named after Max Planck. www.rssd.esa.int/planck/
-8-
for example Dijkstra 2010) with the James Webb Space Telescope (JWST)12 ,
I concentrate in this section on the future 21cm observations. See McQuinn
(2010) for a more detailed overview of current and future observational constrains.
A complementary probe of the state of the IGM can be obtained by observing neutral hydrogen directly via the hyperfine structure line of ground
state neutral hydrogen at a wavelength of λ21 = 21 cm (ν21 = 1.45 GHz). This
corresponds to an energy difference between the two states (parallel and antiparallel spins of the electron and the proton, where the parallel state is more
energetic) of ∆E = 5.9 × 10−6 eV or in terms of a temperature T∗ = ∆E/kB =
0.068 K (see e.g. Furlanetto et al. 2006; Pritchard & Loeb 2011, for reviews
on physics of and with the 21cm line), where kB = 1.38 × 10−23 m2 kg s−2 K−1
is the Boltzmann constant. Although this spin-flip has a very low transition
probability (for a single atom, spontaneous emission occurs about once every
10 Myr), it can be observed because hydrogen is so abundant in the universe.
In thermal equilibrium, Kirchhoff’s law applies and there is a fixed relation
between absorption and emission coefficients dependent on the temperature
of the emitting/absorbing medium and the frequency, see e.g. Spitzer (1978).
Since the wavelength of the peak of the BB radiation (today at 1.9 mm, but
smaller by (1 + z)−1 at higher redshifts) is much smaller than the wavelength
of the spin-flip line, the Planck law for the radiation field intensity of the CMB
(and the BB of the emitting/absorbing medium) can be approximated by the
low energy Rayleigh-Jeans (RJ) limit, Iν = 2ν 2 kB T /c2 . This is used to convert
all intensities into temperatures and the resulting brightness temperature TB of
the 21cm radiation is then
TB = TCMB e−τ + TS (1 − e−τ )
(1.1)
Here, TS is the equivalent temperature of the emitting/absorbing medium (at
that frequency) and τ the optical depth at the frequency in question. What
is measured by a (single-dish) radio telescope is the differential brightness
temperature13 . For a given frequency (for a specific line this corresponds to a
certain redshift), this is
δ TB :=
=
12
TB − TCMB
= TS (1 − e−τ ) + TCMB e−τ − TCMB /(1 + z)
1+z
TS − TCMB
TS − TCMB
(1 − e−τ ) ≈
τ.
(1.2)
1+z
1+z
http://www.jwst.nasa.gov/
13 Interferometers miss an absolute scaling and measure therefore the fluctuations against a back-
ground radiation δ T . In PAPER III, we subtracted the mean of δ T to simulate this effect. The
background radiation for the IGM is the CMB
-9-
CHAPTER 1. INTRODUCTION TO COSMIC REIONIZATION
The approximation made in the last step is valid for small τ . This is a very
good assumption for the 21cm radiation.
For the 21cm line of neutral hydrogen, TS is defined through the ratio of
the number of atoms in excited state N1 over ground state N0 which is given
by the Boltzmann equation N1 /N0 = 3e−T∗ /TS (3 is the ratio of the statistical
weights between the excited triplet state and the singlet ground state). Since T∗
is so small, the ratio of excited states over ground state is almost independent
of TS and equals 3. This is used below to obtain the factor 1/4 (one out of 4
hydrogen atoms is in the ground state and contributes to the optical depth).
Since hν kB TS , the optical depth can be expressed as
τ(ν) =
NHI hν
hν
B jk
φ (ν)
4 c
kTS
(1.3)
(Spitzer 1978). To relate the absorption coefficient B jk to the spontaneous
emission coefficient Akj , thermodynamic equilibrium can be assumed to yield
c3
B jk = 3 8πhν
3 Ak j . NHI is the column density of neutral hydrogen and φ (ν) is
the line shape. In the cosmological context (Furlanetto et al. 2006) one can
approximate φ NHI(z) by (cnHI ) / (H(z)ν).14 In a matter dominated universe
√
(3400 z 0.3), H(z) = H0 Ωm (1 + z)3/2 and therefore (using the definition of T∗ )
nHI (z) 3λ21 T∗
τ(ν) = p
(1 + z)−3/2 Akj
(1.4)
H0 Ωm,0 32π TS
Whether this line is measured in absorption (δ TB negative) or in (stimulated)
emission (δ TB positive), depends on the spin temperature TS and the
temperature of the cosmic microwave background TCMB . In the redshift
range interesting for reionization, the spin temperature is believed to be
coupled to the gas temperature Tg by Lyα coupling through the so called
Wouthuysen-Field effect (e.g.Wouthuysen 1952; Field 1959; Furlanetto et al.
2006) or in very high density regions also through collisions. Therefore, it
depends on the temperature of the gas if the line is seen in absorption or
emission. However, in case the coupling processes (Wouthuysen-Field effect
and collisional coupling) are not sufficient, the spin temperature would be
radiationally coupled to the CMB and the neutral hydrogen would not be
observable.
Taking Eq.1.2 and Eq.1.4 together, the differential brightness temperature
can be expressed as
14 Here
we neglect the peculiar velocity contribution.
- 10 -
δ Tb =
TS − TCMB 3λ03 A10 T∗
√ (1 + z)−5/2 nHI (z)
TS
32πH0 Ωm
| {z } |
{z
}
≈1
(1.5)
=K
Under the assumptions that the the spin temperature is coupled to
the gas temperature and not to the CMB temperature and that the gas
temperature is much larger than the spin temperature, the first factor in
Eq.1.5 is 1. The second factor is a constant which can be evaluated to
approximately K = 4.6 × 104 K cm3 . Further, considering the comoving
density (ncomoving = nphysical /(1 + z)3 ) instead of the physical density, Eq.1.5
reads
√
δ Tb = K nHI (z, comoving) z + 1
(1.6)
This approximation was used in PAPER III to convert the ionization fraction
and density fields into a differential brightness temperature field15 . However,
we note that the signal could be much stronger if the gas would be cooler than
the CMB temperature, since the first term in Eq.1.5 could then reach large
negative values with absolute values much larger than 1.
Given the above constrains on the redshift range on reionization from
WMAP measurements and QSO spectra, the interesting frequency range for
observing this transition is roughly ν = ν21 × [1/(1 + 6) − 1/(1 + 14)] ≈ [200− 100] MHz (which corresponds roughly to λ = 1.5 – 3 m). Existing
and future radio telescopes capable of measuring at such low frequencies
(Giant Metrewave Telescope (GMRT)16 , 21 Centimeter Array (21CMA)17 ,
Low Frequency Array (LOFAR)18 , Murchison Widefield Array (MWA)19 ,
Precision Array to Probe the Epoch of Reionization (PAPER)20 Square
Kilometre Array (SKA)21 ) should be able to detect the signal of the neutral
hydrogen during the EoR. Luckily, the atmosphere is transparent for electro
magnetic radiation (EM) of wavelength between several cm to tens of meters
(radio window). However, at these frequencies, the contribution from galactic
and extragalactic foregrounds (e.g. radio galaxies) is much stronger than the
signal from the EoR itself. Therefore, a careful modelling of the foregrounds
is needed in order to extract the signal (e.g. Jelic 2010). Since we are dealing
15 We
are aware that some groups (e.g. Baek et al. 2010) follow the heating and inclusion of
X-rays and find non global heating. However, we note that they lack low mass sources.
16 http://gmrt.ncra.tifr.res.in
17 http://21cma.bao.ac.cn
18 http://www.lofar.org
19 Murchison Widefield Array, http://www.mwatelescope.org
20 http://astro.berkeley.edu/˜dbacker/eor
21 http://www.skatelescope.org/
- 11 -
CHAPTER 1. INTRODUCTION TO COSMIC REIONIZATION
with line radiation, one could in theory map the spatial time-dependent
distribution of neutral hydrogen in the Universe, but the line of sight direction
and time-information are mixed. Such measurements would require a higher
sensitivity than can be achieved with e.g. LOFAR. Instead, what will be
measured first are power spectra.
Two such experiments using the 21cm line already gave first results: Bowman & Rogers (2010) used a broadband radio spectrometer, Experiment to
detect the global Epoch of Reionization signature (EDGES) (Rogers & Bowman 2008; Bowman et al. 2008) designed to measure the global signal in
the above mentioned frequency range. They did not detect any signature that
would be introduced (an edge) by a rapid reionization, they were able to set a
lower limit on the duration of reionization of ∆z > 0.06. Paciga et al. (2011)
use the GMRT to constrain the morphology of the ionized fraction for the
case of a cold IGM22 at z ∼ 8 − 9 by a non detection of features in the power
spectrum (PS) of 21cm radiation.
22 as
mentioned above, the absorption signal can be much higher than the emission signal
- 12 -
C HAPTER 2
M ODELLING THE E O R
In this section I describe a generic way of modelling the EoR numerically.
There are other approaches, such as semi-numerical modelling (e.g. Mesinger
& Furlanetto 2007; Santos et al. 2010), on which I will not expand here. Instead, I concentrate on the path we took in the publications this thesis is based
on. A pictorial description of this path is given in Fig. 2.1. While describing,
I will refer to the individual steps by the Latin upper case letters as indicated
in that figure.
Figure 2.1: Schematics of modelling the EoR. For details, see text
We start by performing large scale cosmological DM only simulations (A).
For this, we use the C UBE P3 M code (Particle-Particle, P-Mesh) (see Iliev
et al. 2008, for a short description of the code) which was developed from the
- 13 -
CHAPTER 2. MODELLING THE EOR
PMFAST (Particle-Mesh) code (Merz et al. 2005). This particle distribution
representing the density field, say, at time t1 , is used to extract halos at that
time t1 (B). The particle data is converted into a density field on a grid by
smoothing the DM particle distribution using a kernel and integrating in each
cell of the grid the parts of the kernel that intersect with it. In fact, what I show
in (A) is already the density field on the grid. However, the halos are extracted
from the particle data.
We assume the gas to follow closely the DM density distribution and we
assume a constant DM/ baryonic matter fraction everywhere. For the scales of
interest, this is a good approximation23 . So, after (A) and (B) we have the gas
density fields and halo lists at times ti .
Next, we need to illuminate the halos (C). In the simulations of PAPER I,
this is solely done by assuming stellar sources: We assume some fraction of
the gas in each halo is being converted into stars (a star formation efficiency).
Multiplied with the total baryonic halo mass, this gives a total stellar mass
(or number of baryons in stars) in the halo. Next, we set the number of ionizing photons produced per baryon in a star. For a single star, this depends on
two things: How effective is the nuclear fusion to convert rest-mass energy
into EM energy (about 0.7% of the rest mass energy)? And: How much of
this EM energy is in form of photons more energetic than 13.6 eV? The latter
depends on the spectrum, that is, on the effective temperature of the star and
therefore on the mass of the star. The former depends on the details of the nuclear reactions and the amount of energy carried away by neutrinos. Both are
connected to the metallicity. For galaxies, Iliev et al. (2005) give the following numbers: Adopting a Salper initial mass function (IMF) and Population
II (POP II) stars24 , yields around 3000 ionizing photons/stellar baryon and
for POP III stars (first stars, no metals, massive and hot) values above 25 000
would be appropriate. Of course, this has to be seen as an average over the
population lifetime, approximately 10 Myr.25 Some groups working in the
23 During
reionization, where the ionization fronts move supersonically through the IGM, the
mechanical feedback to the gas by pressure forces can be neglected (Shapiro & Giroux 1987).
Therefore, doing the radiative transfer (RT) as post-processing on the N-body density field is a
valid approximation.
24 low metallicity stars, less massive than Population III (POP III). Today, the only remainings of
this population of stars are the lightest long-lived ones in the population. Therefore, POP II stars
today are older, lower luminosity stars, typically found in the nucleus of galaxies. However, if I
refer to POP II stars this is to indicate that they are not metal free and not extremely massive.
25 Sparke & Gallagher (2007) give as an estimate for stellar lifetimes τ as function of their mass
l
−2.5
τl ∼ 1010 MM
yr, so a 15 M corresponds to a lifetime of roughly 10 Myr. The effective
temperature of such a star is about 30 000 K. Sternberg et al. (2003) give for such a star an
ionizing photon output QH ∼ 1048 s−1 which translates into roughly 10 000 ionizing photons
per 10 Myr per baryon.
- 14 -
field include a chemical enrichment model and therefore have time-dependent
photons/stellar-baryon rates (e.g. Trac & Cen 2007).
Not all the produced photons escape from the galaxy into the IGM. To take
this into account, one introduces an escape fraction. In principle, this escape
fraction can be dependent both on the direction and on the source distribution and density distribution inside the galaxy (see for example galaxy sized
(semi-) numerical studies of Ciardi et al. 2002; Fujita et al. 2003). It may also
be redshift dependent and differ for galaxies of different mass (see Razoumov
& Sommer-Larsen 2006; Gnedin et al. 2008, for large scale cosmological numerical studies that investigate among other things the dependence of the escape fraction on galaxy-mass). Observational estimates of the escape fraction
of lower redshift galaxies give mostly upper limits due to non detections (e.g.
Deharveng et al. 2001; Leitherer et al. 1995; Malkan et al. 2003), but Bergvall
et al. (2006) reported a detection and estimate the escape fraction to be around
4 – 10 %. Since this thesis is not about escape fractions, I will not expand on
this but note that the escape fraction is a rather unconstrained parameter.
By including the escape fraction, we already account for the ionization of
the gas in the galaxy. Therefore we subtract it from the density field before
doing the radiative transfer. For reasonable values of (1) the star formation
efficiency (2) the number of ionizing photons produced per stellar baryon (3)
the escape fraction, the resulting conversion factor between halo mass and
emitted ionizing photons in 10 Myr is some tens to hundreds of photons per
halo baryon. These three ingredients are degenerate for reionization and the
only important number for our simulations is the product of these, not the individual multiplicands. This is quantified in PAPER I26 . In Section 4, I outline
the line of thinking for converting halo mass into quasar luminosity, this is
quantified in PAPER III.
In the next step (D), we transfer the ionizing radiation through the IGM until
the next ti (next N-body/source list output). How we do the radiative transfer
is outlined in Section 3 and described in Mellema et al. (2006) and PAPER II.
Here, one has the possibility of including sub-grid physics, for example, one
can introduce a clumping factor, C which will affect the recombinations: the
recombination rate depends on the square of the number density, n2 . However,
the density value in the cell is in fact an average over the cell. Therefore, the
recombination in the cell is calculated on the basis of the square of the average
density, hni2 . If the density varies much on scales smaller than the cell, the recombination varies much, and its average in the cell would be proportional to
26 Note
however that Eq.1 in PAPER I contains an error and an inconsistency in the naming: The
mean molecular weight in the denominator should not be there since we are considering total
number of baryons. Ω0 should really be named Ωm , the total current mass content in units of
critical density. See PAPER II, Eq. 21 for the correct equation.
- 15 -
CHAPTER 2. MODELLING THE EOR
the average of the squares of the density, n2 . To correct for this, one can in
clude a factor C = n2 / hni2 in the recombination (and collisional ionization)
calculations. C as a function of density can be fitted to the N-body simulation
and can be included in the RT simulation. However, this clumping is based on
the dark matter density field since the underlying N-body simulations are dark
matter only. Due to the heating of the gas, the gas clumping is expected to be
smaller than the clumping of dark matter and therefore, the gas recombination
is expected to be somewhere between the cases without inclusion of a clumping factor and the cases with including a clumping factor based on the N-body
results. McQuinn et al. (2007) investigate several different clumping models
and find that the effect on the large scale structure of H II regions is small but
that it adds small scale structure at the edges of H II regions.
The result of such a simulation (A-D) is a time dependent ionization fraction field (E) which can be statistically analysed (G), as we did in PAPER I to
test the effect of different source models (C). However, the important quantity related to observations is actually not the ionization fraction but the neutral density. Therefore, we multiply the neutral fraction of each cell (i.e. 1ionization fraction) with the density of each cell (at each time ti ) to receive the
neutral density field (F). This can be transformed into a differential brightness
temperature (assuming a global heating) as outlined at the end of Section 1.2
which in theory is measurable at the interesting redshifts. In PAPER III, we
present the prospects of detecting quasar H II regions in redshifted 21cm maps
by using a method developed by Datta et al. (2007) and Datta et al. (2008) (H).
It should be mentioned that in all practical cases we have been studying, the
mean free path for the vast majority of the photons is smaller than the lighttravel distance during one timestep. Therefore, we do not need to deal with
remaining photons in timestep i + 1 that were not absorbed in timestep i.
- 16 -
C HAPTER 3
R ADIATIVE T RANSFER
The equation of radiative transfer in an expanding universe in comoving coordinates is (e.g. Norman et al. 1998; Abel et al. 1999; Gnedin & Abel 2001)
1 ∂ Iν n · ∇Iν H(t)
∂ Iν
+
−
ν
− 3Iν = jν − κν Iν ,
c ∂t
ā
c
∂ν
(3.1)
where Iν is the specific intensity at frequency ν , n is the unit vector in the
direction of light ray propagation, H(t) = ȧ(t)/a(t) is the Hubble constant at
1+zem
time t , c is the speed of light, ā = 1+z(t)
is the ratio of cosmic scale factors
at emission and present time t , jν is the emission coefficient and κν is the
absorption coefficient.
Norman et al. (1998) show that in the case of local sources, i.e. the mean
free path (mfp) of photons, λmfp , is small against the simulation box scale L
(the scale of interest), and if L is small against the horizon scale, c/H(t), the
third term on the l.h.s. of Eq. 3.1 is negligible27 . This means that the cosmological redshift of the photons between emission and absorption and the
dilution due to the expansion of the universe is negligible.
Furthermore, since zem = z(t + λmfp /c) ∼ z(t), 28 it follows that ā ∼ 1. This
implies that the change of path length along a ray due to cosmic expansion
is negligible. Consequently Eq. 3.1 reduces to the classical transfer equation
(e.g. Peraiah 2001)
1 ∂ Iν
+ n · ∇Iν = jν − κν Iν .
(3.2)
c ∂t
27 However,
as Abel et al. (1999) point out, this is strictly only valid if the radiation has a rather
smooth spectrum, see there for details on fixes for line radiation.
28 For very high energetic photons, this might not hold since λ
mfp can be very large, Furlanetto
(2009) give the comoving mean free path of X-rays with energy E as λmfp = 4.9 hxHI i−1/3 ((1 +
z)/15)−2 (E/300eV)3 Mpc. However, we currently do not follow photons for distances longer
than the simulation box size L. Also, as noted at the end of Chapter 2, long mean free paths
might force to implement explicitly the limited speed of light which in turn means that not
absorbed photons have to be stored with their current position. This is not implemented in the
code at the moment.
- 17 -
CHAPTER 3. RADIATIVE TRANSFER
In a further approximation one neglects the time dependence in the absorption and emission coefficients, which is equivalent to assuming that the light
travel time through the box is much shorter than the time scale on which the
absorption and emission coefficients change. This reduces the equation to:
n · ∇Iν = jν − κν Iν
(3.3)
A common way of reducing the dimensionality of the equation further is
to separate the anisotropic (local point sources) from the isotropic (diffuse
radiation due to recombination) contribution of Iν = Iνdiff + Iνps . This results in
two equations that are coupled to each other via the absorption coefficient κν ,
(see e.g. Abel et al. 1999, for details).
Since we assume that λmfp is small, we ignore the contribution from sources
outside the box. Furthermore, we treat the diffuse photons from recombinations in an on-the-spot (OTS, see below) manner and incorporate their effect
on κν in this way in the equation for the local point sources. Therefore, we
are only left with one equation. In one dimension (i.e. in the spherically symmetric case) it can be written as (dropping the super-script ps and the subscript
ν)
∂I
(r) = −κ(r)I(r)
(3.4)
∂r
A formal solution to Eq. 3.4 is then
Zr
I(r) = I0 exp − κ(s)ds ,
(3.5)
0
where I0 is the intrinsic intensity of the source assuming a point source such
that
L = 4πI0 . Introducing as usual the optical depth as the integral, τ(r) =
Rr
κ(s)ds
, gives
0
I(r) = I0 exp (−τ(r))
(3.6)
For ionizing radiation, τ (at each frequency ν ) is given by the sum of
the products of the column densities Ni and ionization cross section σi (ν)
of species i: τ(ν) = ∑i Ni σi (ν). I(r) can be related to the flux F(r) going
through a unit surface at distance r by F = I/r2 (assuming a point like emission source). Instead of evaluating the flux, in the case of ionizing radiation,
one is interested in the ionization rate which can be locally written as (e.g.
Osterbrock 1989)
Γ(r) =
1
4πr2
Z
ν
L(ν)σ (ν)e−τ(ν,r)
dν.
hν
(3.7)
This ionization rate alters the ionization fraction at any instance in space and
changes therefore the column density and therefore the optical depth. How
- 18 -
to solve simultaneously for the ionization rate and the ionization fraction is
explained in the next section.
3.1
Solving radiative transfer in one dimension
Mellema et al. (2006) described how to solve the ionizing radiation transport
problem (i.e. to solve Eq. 3.4 combined with the change of κ due to photo
ionisation) in a photon conserving fashion. In the following, I will sketch the
basic ideas since it is this code that I extended to include helium (see PAPER
II). For details on the original C2 R AY code, I refer the reader to Mellema et al.
(2006). For details on the inclusion of helium, I refer the reader to PAPER II.
This section serves solely as a conceptual introduction.
The basic idea of C2 R AY is to equal the number of ionizations in a given
cell to the number of absorptions in that cell. The latter is given by the difference between photons entering the cell and photons leaving the cell per unit
time. The number of photons entering the cell per unit time is dependent on
the optical depth τin to the cell. The number of photons leaving the cell is a
function of this τin and the optical depth over the cell ∆τ . The ∆τ changes
due to the effect of ionizing radiation (assuming that the incoming ionizing
radiation/ optical depth has been solved for already). The iteration procedure
can be schematically represented as in Fig. 3.1.
Figure 3.1: Schematic iteration for finding the photon conserving outgoing ionization
rate and optical depth of a single cell ∆τ. The index i counts the iteration. n without
any index is the neutral number density from the last timestep. Γin (out) is the ingoing
(outgoing) photoionisation rate.
- 19 -
CHAPTER 3. RADIATIVE TRANSFER
In any practical application, time and space are discretised in finite
timesteps ∆t and widths of cells ∆x. These discretisations introduce two
problems: (a) During one timestep, the neutral fraction in a given cell can
change substantially (with respect to time). (b) the optical depth over a cell
can vary substantially because the neutral fraction can vary substantially
within a cell (with respect to space) and because of the intrinsic dependence
of τ on the path length. Since we are only interested in the ionization rate that
comes out of the cell, (b) is not a problem since any algorithm following the
sketch above would give immediately a spatial average of the neutral fraction
and therefore a correct ∆τ for the cell. Analogously, (a) is not a problem (we
are only interested in the ionization rate at the end of the timestep) if we use
a time averaged neutral fraction in the cell to calculate the outgoing optical
depth.
Assuming a constant electron density, and knowing the ionizing flux, the set
of rate equations for the hydrogen only case can be represented as in Fig. 3.2.
These are ordinary linear differential equations that can be solved analytically.
Therefore, a time averaged fraction over a timestep can be calculated easily.
Figure 3.2: Left: Schematic ionization diagram of hydrogen only. 4 symbolize recombinations, symbolizes ionizations (photo- and collisional ionizations) and 4
γ
symbolizes recombination photons which are taken into account by using αB recombination (sum of all recombination rates to all states but the ground state, symbolized
by the white frame around the triangle) rates; Right: Symbolic rate equations with the
same meaning of the symbols. 5 means negative contribution from recombination.
The arrow in the left hand panel of Fig. 3.2 pointing back from the recombinations to the ionizations represent the recombinations to the ground level.
Here, a photon is emitted that itself again is able to ionize a hydrogen atom.
As mentioned above, those are treated on-the-spot, assuming that they ionize
close to their origin. This means in practice that the recombinations to the
ground state are not counted, instead one uses the so called B-recombination
rate, αB .29 Therefore, in the symbolic notation of the rate equation, these recombination photons are not explicitly included since they have been already
subtracted from the recombination rate.
29 This is the sum of the recombination rates to all levels but the first, see for example Osterbrock
& Ferland (2006).
- 20 -
Figure 3.3: Left: Schematic ionization diagram of hydrogen and hydrogen coupling.
4 symbolize recombinations, symbolizes ionizations (photo- and collisional ionizations) and 4
γ symbolizes recombination photons; Right: Symbolic rate equation
with the same meaning of symbols, 5
γ means negative contribution from recombination photons. Note the symmetry.
When helium is included, dealing with the recombination photons gets
slightly more complicated. Using a similar symbolic representation of the processes involved, we illustrate the situation with helium in Fig. 3.3.
The photons from helium recombination that are energetic enough to at
least ionize hydrogen now have to be included explicitly. Those that are energetic enough to ionize at least two species have to be split between the species
in question depending on their relative optical depths. This is explained in
detail in PAPER II. The set of equations for coupled helium and hydrogen
(helium only) can be reduced to 3 (2) equations by taking into account that
the ionization fractions for each of the two species individually add up to 1.
- 21 -
CHAPTER 3. RADIATIVE TRANSFER
3.2
Ray-tracing
In two or three dimensions one needs a method to compute the optical depths
to all the cells in every direction from the source as well as over every cell.
In one dimension, the latter is trivial and the former is simply the sum over
the optical depth in all cells between the source and the cell. For multiple
dimensions, there are two so-called ray-tracing approaches, which can also
be combined: the long-characteristic approach and the short-characteristic approach.
In the long-characteristic approach, a ray from the source is cast through
every cell. The optical depth to the cell is the sum of the optical depths through
the cells that the rays crosses on the way, weighted by the path length of
the ray through each cell. Several steps have to be taken in this approach:
1) Choosing direction angles for casted rays. 2) Determining which cells are
crossed by each ray and 3) finding the path lengths in each cell (e.g. Abel
et al. 1999). The advantage of this method is that each ray is independent of
the others. Therefore, they can be calculated in parallel. Obviously the density
of rays decreases with the distance to the source. Since every cell has to be
reached by at least one ray, this results in an oversampling of the cells near to
the source if the accuracy at larger distances should be maintained. To solve
this problem, Abel & Wandelt (2002) split rays into child-rays as a function
of distance to the source.
Another way of avoiding the redundant calculations near the source is the
method of short characteristics. Rays are cast from the source to the centre
of each cell, but only the ray-segment in the last cell is retained. The actual
way to the source, i.e. the optical depth between the source and each cell, is
approximated by the cells which are nearest to the point where the ray enters
the cell in question. Mellema et al. (2006) describe in Appendix A the raycasting method used for C2 R AY and motivate the choice of the weighting
functions for the cells contributing to the optical depth to the cell. The optical
depth over the cell is just twice the optical depth from the point where the ray
enters the cell to the cell centre.
The disadvantage of the short characteristics approach is that the optical
depths from the neighbouring cells have to be known, setting constraints on
the domain decomposition in the case of a parallel computation. Also, since
the optical depth to a given source cell is calculated via interpolation, there is
some diffusion of radiation.
Rijkhorst et al. (2006) use a combination of long- and short-characteristic
ray-tracing schemes, where the simulation volume is divided into
patches containing a number of cells. Inside every patch, the method of
- 22 -
short-characteristics is used, but long-characteristics is used for the patches,
which enables parallel calculation for the patches.
When more than one source is present, the problem arises that a ray from
one source (“a”) may alter the neutral hydrogen fraction in a cell on the way of
a ray from another source (“b”). The calculation of the ionization fractions in
cells of the ray from source (“b”) that are located behind the crossing point of
the cells now depends on the order of calculation if the contributions would be
calculated independently. To circumvent this problem, the iteration loop displayed in Fig. 3.1 should not be done for every source independently. Instead,
in each cell, the ionization rates from all sources have to be added before the
ionization fractions in the cells are updated. In the picture of Fig. 3.1, this
means that (Γin − Γout ) has to be replaced by (∑i=sources (Γiin − Γiout )). First after all sources have contributed to the ionization rate in each cell (calculated
on the basis of the optical depth from the last iteration), this ionization rate is
applied to the cell to calculate a new ionization fraction in each cell. See figure 2 of PAPER II for an iteration flow chart for the case of three dimensions.
Ray-tracing methods typically scale as the product of number of grid cells and
number of sources.
Alternatives to ray-tracing methods are Monte Carlo approaches (e.g. Ciardi et al. 2001; Maselli et al. 2003) and Moment methods (e.g. Gnedin &
Abel 2001; Norman et al. 1998). The latter have the advantage of not scaling with the number of sources. However, they are less accurate for very
anisotropic and heterogeneous intensity distributions and tend to be rather diffusive. Several codes used for reionization simulations participated in a comparison project, see Iliev et al. (2006) for a description of the results.
- 23 -
C HAPTER 4
Q UASARS
This chapter serves as an introduction to PAPER III were we investigate the
detectability of a quasar H II region during the EoR. It provides the motivation
and more background for the parameter choices of the quasar properties in that
paper.
Quasars (quasi stellar radio source) were first observed in radio (e.g.
Bolton et al. 1949) and matched with their optical counter parts later when
more precise position measurements at radio wavelength became possible
(e.g. Schmidt 1963; Matthews & Sandage 1963). Due to their point-like
appearance (angular extends of less than an arcsecond), they were also
named quasi stellar object (QSO). The prefix quasi- was added because
of their unusual spectrum consisting of many emission lines. From the
redshift of these emission lines it became clear that these sources were in
fact extragalactic, which meant that they must be extremely luminous given
their rather low apparent magnitudes. Time-variability in the emission (e.g.
Matthews & Sandage 1963; Boller et al. 1997) set constrains on the spatial
extent of the sources. These observations together (high luminosity from a
very small region) put constrains on the source of energy for these objects.
In the following, I use the terms quasars, active galactic nucleus (AGN) and
QSO interchangeable. In the standard model of AGN today, the main ingredient of a quasar is an accreting black hole (BH) in the centre of a galaxy surrounded by a hot accretion disk. Gravitational energy is partly converted (via
friction) into electromagnetic energy.30 The accretion is limited by radiation
pressure. Assuming isotropic radiation and spherically symmetric accretion,
the limiting so called Eddington luminosity can be calculated by equating the
inward and outward forces. The Eddington luminosity only depends on the
mass of the accreting object since both the radiation pressure and the gravita30 The
accretion disk has increasing temperatures towards the centre. Due to the different peaks
of the BB-curves, the spectrum looks very different from a typical BB spectrum. Parts of the
emitted photons are reprocessed in the hot electron corona around the disk and boosted to higher
energies via inverse Compton scattering.
- 25 -
CHAPTER 4. QUASARS
tional attraction decrease with distance squared, hence
LEdd =
4πMBH m p c
M
≈ 1.3 × 1031
W,
σT
M
(4.1)
(e.g. Robson 1996). Here, G is the gravitational constant, MBH the mass of the
BH, m p the proton mass, c the speed of light and σT the Thompson scattering
cross section. One introduces an Eddington efficiency parameter, commonly
named λ , where
LQSO = λ LEdd (MBH ).
(4.2)
An AGN with λ = 1 accretes therefore at its Eddington limit. As mentioned,
the above equating of the inward and outward forces assumes spherically symmetric accretion and isotropic radiation. Heinzeller et al. (2007) show by numerical simulations that for the case of a thin accretion disc, super-Eddington
luminosities with up to λ = 20 are possible31 . However, as Steinhardt & Elvis
(2010) and Steinhardt & Elvis (2011) show by examining over 60 000 SDSS
quasars, most quasars emit below their Eddington limit. I reproduce here figure 1 from Steinhardt & Elvis (2010), see Fig. 4.1. It shows the loci of the
SDSS quasars in the mass-luminosity plane. The dashed line shows the Eddington luminosity. It can be seen that there is a large spread in λ but most
quasars are below the Eddington luminosity32 .
As mentioned above, the energy that is radiated away is basically gravitational energy. The amount of energy obtainable for radiation depends on
the lowest state of potential energy that can be reached before the accreted
matter falls into the BH: Matter of mass m at infinity, has potential energy 0
(with respect to the BH) and a rest mass energy mc2 . It can be shown (e.g.
Misner et al. 1973; Kembhavi & Narlikar 1999) that the last stable orbit of a
non-rotating BH is at 3RS where RS = 2GMBH /c2 is the Schwarzschild radius
where the escape velocity is equal to the speed of light. The total energy of
the matter at this distance to the BH is lower than the rest mass energy (since
the potential energy is negative) and has to be calculated relativistically, Mis√
ner et al. (1973) give E(last stable orbit) = 32 2mc2 . Therefore it follows that the
difference between the rest mass energy and the last stable orbit energy (the
31 Loeb
(2009) mention that theoretical models predict that radiation pressure puffs up the inner
edge of the accretion disc and therefore the inner geometry is more spherical and the concept of
an Eddington limited accretion holds. This is also more in line with observations
32 The mass of a BH can be measured my a technique called reverberation mapping. It is based
on the geometrical model of the quasar according to which the continuum radiation from the
quasar originates from very close to the BH while the line radiation originates from gas further
out at a distance d. Any luminosity change in the continuum radiation will eventually (with a
time delay ∆τ = c d) show up in the line radiation. Measuring the time delay and the line width,
the virial theorem can be used to calculate MBH .
- 26 -
Figure 4.1: This figure is adapted from Steinhardt & Elvis (2010) (figure 1) by permission of John Wiley and Sons. It shows Quasar loci in the mass-luminosity plane
for quasars in the redshift range 0.2 < z < 2.0. The colour indicates the redshift bin:
0.2 < z < 0.8 (black), 0.8 < z < 1.4 (light grey) or 1.4 < z < 2.0 (grey). For details
on how the virial masses where estimated, see Shen et al. (2008). See Richards et al.
(2006) for details on the bolometric luminosity estimation. The dashed line indicates
the Eddington luminosity.
difference is the binding energy) is lost from the BH system. It is radiated
away as
Z
t
LQSO dt = εmc2 .
(4.3)
Here t is the time for accreting the mass m. √
In the case of a non-rotating BH,
the upper limit for the efficiency is ε = 1 − 2 2/3 ≈ 0.053. But one can show
(e.g. Misner et al. 1973) that for an extreme rotating BH, ε ∼ 0.42, the last
stable orbit is closer√to the BH. By accreting the matter, the non-rotating BH
grows its mass by 2 2/3m, or more generally by (1 − ε)m. Therefore,
LQSO = ε ṁc2
and
ṀBH = (1 − ε)ṁ
(4.4)
where ṀBH is the BH mass growth rate. So a higher ε means that the actual
mass accretion rate ṁ for reaching a certain luminosity LQSO can be smaller
than for a lower value of ε . However, equating Eq. (4.4) and Eq. (4.2) gives
λ (1 − ε)
LEdd .
(4.5)
εc2
This shows that a higher value of ε also implies a lower mass growth rate of
the black hole. Therefore, to grow quickly very massive black holes, the efficiency ε to convert rest-mass energy into EM should be small. But to reach
ṀBH =
- 27 -
CHAPTER 4. QUASARS
observed luminosities of quasars and to avoid unrealistically high mass accretion rates, the accretion efficiency should be high. If the mass supply is
limited, the accretion efficiency has an effect on the quasar’s luminous lifetime. Reasonable values for the accretion efficiency are thought to be several
to several tens of percent, ε ∼ 0.05 − 0.2 (Haiman & Hui 2001; Shankar et al.
2010; Martini & Weinberg 2001). As Volonteri & Gnedin (2009) point out,
a black hole with no rotation will accumulate angular momentum and in this
way increase ε . This could be a natural way for the BHs to grow relatively fast
in mass at early times and to reach high radiation efficiencies later. However, it
is still not clear how super-massive BH (such as the one in ULAS J1120+0641
which was approximated to have a mass 2 × 109 M see Bolton et al. 2011, at
z ∼ 7.1, so only roughly 750 Myr after the bb ) can grow so massive in such
short times. See for example Volonteri (2010) for a recent review on this topic.
To include quasars as sources in our reionization simulations, we need to
identify the location of the quasar (i.e. which halos host active quasars), the total luminosity of the quasar, the lifetime of the quasar and we need a spectrum
of the quasar.
In Sect. 4.1, I outline how we connect the quasar luminosity to the hosthalo mass and discuss the limitations of this relation. In Sect.4.2, I summarize
observational results that are used to estimate the lifetime of quasars. Sect.4.3
gives an overview of the observations of quasar spectra. Finally, Sect.4.4 summarizes our rather crude recipes to include quasars in our simulations.
4.1
Quasar luminosity – Halo mass relation
The general idea of this section is to connect the host halo mass to the quasar
luminosity. This can be done in three steps: (1) The quasar luminosity LQSO
can be connected to the black hole mass MBH . 33 (2) MBH can be connected to
the mass of galaxy bulges MB . (3) MB can be connected to the total mass of
the halo Mhalo . First, I will give details to each of those steps. Afterwards I will
give an overview of observations which give rise to doubts to the existence of
such a relation.
Assuming an Eddington limited accretion, what is needed for connecting
the quasar luminosity to the host halo mass, is a relation between black hole
mass and host halo mass. For low (here, z ≤ 3, or so) redshifts, Magorrian
et al. (1998) observed a relation between the masses of the central massive
33 In
the introductory section to this chapter, see Fig. 4.1, it was shown that there is a correlation
between MBH and LQSO , however with a large scatter.
- 28 -
dark object (BH) and bulge (B) (this corresponds to step 2 from above):
MBH /MB = 0.006
(4.6)
This relation is known as the Magorrian relation, or as the M-σ relation, where
σ refers to the central velocity dispersion, the velocity dispersion of the bulge.
This is the actual measured quantity.
Newer formulations of this are MBH ≈ 0.002MB (Marconi & Hunt 2003,
their measurements also use bulge luminosities34 in addition to the velocity
MB 1.12
dispersion) or slightly non-linear relations such as MMBH
≈ 7.6 × 10−5 ( M
)
35
(Häring & Rix 2004) and from Tundo et al. (2007) (equation A5) :
MBH /M = 108.21
3.83
σ
200km/s
(4.7)
Further, a correlation between the outer circular velocity vc of galaxies to
this central velocity dispersion σ of the following form is observed (Ferrarese
2002, equation (2))36 :
vc /[km s−1 ] = 100.55 (σ /[km s−1 ])0.84
(4.8)
For virialized systems, the outer circular velocity vc is related to the halo velocity vvir at the virial radius rvir which is
qdetermined by the mass enclosed in
vir
the virial radius Mvir of the halo, vvir = GM
Rvir (These two relations, the connection between the velocity dispersion of the bulge to the circular velocity
of the galaxy; and the connection between the circular velocity of the galaxy
to the virial velocity of the halo comprise step 3 from above). The details depend on the assumed halo mass profile and the stellar and gaseos content, see
Mo et al. (1998). In the following I set vc = vvir . Using equation 2 from Bullock et al. (2001), the set of cosmological parameters (h=0.702, Ωb =0.0455,
Ωm =0.272) and a required overdensity of 200 for the virial radius (from the
spherical collapse model), the resulting equation relating halo mass and virial
velocity reads37
v
3
v3vir
GMvir 3
4π
vir
5
200ρc
=
≈
3
×
10
M
(4.9)
Mvir =
3
10HG
km s−1
v2vir
34 Applying
some constant mass-to-light ratio, the bulge luminosity can be converted to a bulge
mass.
35 in all cases, the values are based on averages. For information about the spread around those
averages, I refer the reader to the original publications
36 Kormendy & Bender (2011) combine the data points from Ferrarese (2002) with several other
measurements in their figure 1 and conclude that there is no such relation. I will however continue based on the relation found by Ferrarese (2002).
37 in the second step ρ = 3H 2 was used.
c
8πG
- 29 -
CHAPTER 4. QUASARS
These relations together (using Equation 4.7 to 4.9) give a relation between
(virial) halo mass Mvir and the mass of the central black hole MBH :
σ 3.83
MBH = 108.21
200
vc 1/0.84 3.83
8.21
−3.83
= 10 200
100.55


1/3 !1/0.84 3.83
Mvir

= 108.21 200−3.83 (100.55 )−3.83/0.84 
3 × 105
1.52
≈ 3.7 × 10−12 Mvir
(4.10)
Here, all velocities are in units of km/s and masses are in solar masses M .
Shankar et al. (2010) and Ferrarese (2002) perform an analysis along the same
lines and come to similar results. The former include a redshift dependence
and the latter test for several assumptions about the density profile, among
other things. I include some of their models in Fig.4.2. Haiman & Hui (2001)
give an upper limit for the black hole mass by using the M-σ as originally by
Magorrian et al. (1998) and setting Ωb /Ωm Mhalo as an upper limit for the bulge
mass. This results in: MBH ≈ 0.006 MB ≤ 0.006 Mhalo Ωb /Ωm ≈ 0.001 Mhalo .
Zaroubi et al. (2007) use an estimate along the same lines, however they assume a factor 6 lower proportionality factor in the Maggorian relation (which
is closer to the slightly newer result from Marconi & Hunt 2003). I summarize selected results in Fig. 4.2 to give a feeling for the spread in MB − MBH
space.
As indicated at different parts in this section, there are problems with this
approach to connect halo mass to QSO luminosity: Merloni et al. (2010) find
observationally a larger scatter in the Maggorian-type relation (step 2 from
above)38 with increasing redshift. As mentioned earlier, Kormendy & Bender
(2011) find no connection between the bulge mass of galaxies and the total
host halo mass (step 3 from above). Clustering analysis shows that QSOs are
strongly clustered at all redshifts and that the clustering increases with redshift
(Shankar et al. 2010; Shen et al. 2008). While this alone is not in contradiction
to a relation between halo mass and QSO luminosity, Croom et al. (2005) and
Shanks et al. (2011) find no strong luminosity dependence of the quasar clustering. Shanks et al. (2011) argue (supported also by numerical simulations)
that this can be interpreted as all quasars residing in similar mass halos, but
being observed at different evolutionary stages. This is indicated as the grey
box spanning a wide range of BH masses, representing a wide range of quasar
luminosities. However, Porciani & Norberg (2006) report to find a luminosity
38 However,
they look at rather low redshifts z<2.2.
- 30 -
10
10
Shanks et al. 2011
λ=1, z=8.636
9
10
λ=0.333, z=8.636
λ=1, z=7.76
8
MBH /M ⊙
10
λ=0.333, z=7.76
λ=0.333, z=8.76, 607 Mpc box
7
λ=1, 607 Mpc box
10
Haiman & Hui (2001)
Zaroubi et al. (2007)
6
10
Shankar et al.(2010) z=6
Ferrarese (2002), Eq. 7
5
10
Shankar et al.(2010) z=2
Ferrarese (2002), Eq. 6
4
10
Ferrarese (2002), Eq. 4
11
12
10
10
13
10
Mh /M⊙
Figure 4.2: Different analytic estimates for the halo mass - black hole mass relation.
Note the rather large spread. Shanks et al. (2011) and Croom et al. (2005) suggest a
narrow host halo mass range, this is indicated in the figure by a grey rectangle. Also
included in this plot are the quasars we included in our simulations of PAPER III.
Since for us, the important parameter is the total luminosity and not the BH mass,
I included two possible BH masses for each quasar implementation, one assuming
λ = 1 and one assuming λ = 0.333. The halo mass corresponds to the mass of the
most massive halo in our 1643 comoving Mpc3 simulation.
dependence in the clustering and Shen et al. (2008) do find a slight redshift dependence on the minimum halo mass (larger for higher redshifts) which they
argue is a selection effect due to the need of higher intrinsic luminosities for
objects further away to be observable, which means that there is a luminositymass relation. Also White et al. (2008) find from combining quasar clustering
and quasar space-density data, that the scatter in quasar luminosity to host
halo mass-relation is small.
To conclude this section: Quasars are highly biased objects which reside
only in the most massive halos which are very rare objects. If there is a typical
mass for halos in which all active quasars reside (and what we see as different
luminosities just corresponds to different evolutionary stages) or if there is a
relation between quasar luminosity and halo mass, is not clear.
- 31 -
CHAPTER 4. QUASARS
4.2
Quasar lifetime
As mentioned above, the lifetime of a luminous quasar is in practice constrained by the amount of material it can accrete. However, without knowing
how much material is available, we cannot, on theoretical grounds, limit its
lifetime. As a characteristic timescale tc however, one can calculate the time
scale after which the mass increases by a factor of exp(1) (Haiman & Hui
2001). Inserting the value for the Eddington luminosity (Eq. 4.2) into the expression for the BH mass growth rate (Eq. 4.5) gives
ṀBH (t) ≈
(1 − ε)λ
MBH (t)
2.3 × 10−9
ε
yr
(4.11)
And therefore
−9 t λ (1 − ε)
M(t) ≈ M0 exp 2.3 × 10
yr
ε
and
tc /yr ≈ 4.3 × 108
ε
λ (1 − ε)
(4.12)
(4.13)
Choosing the typical values for λ (∼ 1) and ε (∼ 0.1) gives 50 Myr.
To find observational limits on the lifetime of quasars, one can compare the
comoving quasar space density Φ(z) with the space density of their potential
hosts nhost (z) and in this way try to match nhost (z), since Φ(z) ∝ nhost (z)tQ .
However, the uncertainties are rather large (e.g. Haehnelt et al. 1998,find
tQ ∼ 106 − 108 yr). Haiman & Hui (2001) and Martini & Weinberg (2001)
propose to use the bias of the quasar distribution instead, this has the advantage that the assumptions about the mass of host halos are less constraining. The approach of the latter can be outlined as follows: Use the observed
quasar space density to find a minimum mass for a halo hosting a quasar as
a function of quasar lifetime over halo lifetime (that is in a way going the
host (z,t ). Next,
other way around as Haehnelt et al. 1998), so Φ(z) −→ Mmin
Q
use an analytical estimate relating the halo mass to the halo bias (Mo &
White 1996), b(M halo ), to find an effective bias for halos hosting quasars, so
host −→ b(M host ). Since M
Mmin
min can be expressed as a function of quasar lifemin
time, the bias can be expressed as a function of quasar lifetime. This function
can be compared with the measured bias of quasars. The advantage here is
that one does not make a priori assumptions about the masses of halos hosting
quasars. However, this method gives information about the quasar lifetime
over the halo lifetime which is equal to the quantity called duty cycle, usually defined as the number of quasars over objects potentially hosting quasars.
Haiman & Hui (2001) take a slightly different path and find their results also
to be consistent with quasar lifetimes tQ ∼ 106 − 108 yr.
- 32 -
A very different way of constraining the lifetime of quasars comes from
measurements of their proximity zones39 (e.g. Lu & Yu 2011; Worseck &
Wisotzki 2006; Croft 2004). While this does not directly measure the lifetime
but the time for which the quasar has been shining, it sets a lower limit on the
quasar lifetime. Here, both the proximity zone measured by the quasar light
itself (so line of sight proximity zone) as well as the transverse proximity
zone measured through the background light from a nearby quasar behind
the foreground quasar, are in principle useful to set limits on the lifetime of
quasars. Croft (2004) simulate the effect of quasars with lifetimes as short
as 107 yr on the Lyα forest and compare this to observations (SDSS) and
conclude that the best solution to match observations are total quasar lifetimes
∼ 107 yr but with single burst phases lasting ≤ 106 yr. Worseck & Wisotzki
(2006) report the detection of the transverse proximity effect (however in He
II not in HI) and estimate the minimum quasar lifetime to be 1 − 3 × 107
yr. They argue that there is no need for anisotropic UV radiation from the
foreground quasar (this is how Lu & Yu 2011, try to explain contradicting
estimates for quasar lifetimes) but instead that large scale density structure
masked the transverse proximity effect. This is based on comparing the He II
and H I absorption.
4.3
The spectrum of quasars
Somewhat confusingly there are several different ways in use to express the
spectrum of astrophysical sources. While it is not uncommon in the X-ray
community to show a spectrum in terms of photons/m2 /s/keV (over frequency
ν ), in the optical it is more common to view it as spectral flux density Fλ measured in W/m2 /nm (over wavelength λ ) and at radio frequencies as spectral
flux density Fν in W/m2 /Hz (over frequency ν ). Alternatively it is expressed
as λ Fλ = νFν in W/m2 . Sometimes the latter is referred to as spectral energy
distribution (SED)40 . However, this concept is not consistently used in the literature since sometimes Fν or Fλ are called SED. To complicate things even
more, when approximating the quasar spectrum with a power law, sometimes
the power law index is defined as F(ν) ∝ ν −α and sometimes without the mi39 Originally,
proximity zones are only defined for quasars in an already ionized IGM with an
existing ionizing background flux. The quasar produces many more ionizing photons so that
the ionizing flux density in the vicinity of the quasar is enhanced and therefore the remaining
neutral fraction is lower then elsewhere, even in the high density filaments of the cosmic web
that go through this region, the neutral fraction is reduced. Therefore, the Lyα-forest originating
from this region is weakned
40 not to be confused with the meaning of SED outside astronomy: Sozialistische Einheitspartei
Deutschlands (Socialist Unity Party of Germany)
- 33 -
CHAPTER 4. QUASARS
nus in the exponent. This can lead to serious confusion. In the following, I
use the power law index α defined by: energy flux per unit area, per unit time
between frequency ν and ν + dν is proportional to ν −α , so Fν ∝ ν −α . That
means that the output number of ionizing photons in the same frequency interval is proportional to ν −1−α . This index is sometimes called spectral photon
index Γ (e.g. Yuan et al. 1998). I will not use Γ in this context since I use it for
denoting the ionization rate, see Sect. 3.
With this definition, Zheng et al. (1997) find α ≈ 1 for the far ultraviolet
radiation of quasars (105 – 220 nm or 5.6 eV– 12 eV) using Hubble Space
Telescope (HST)41 data. For the extreme ultraviolet radiation (35 –105 nm or
12 eV– 35 eV), they find α ≈ 2, see also their figure 5, reproduced here in
Fig.4.3 upper curve. Earlier studies also found a double power law to be a
good fit, however, as Zheng et al. (1997) show (for the far UV), there is a
rather large spread for the fitted power law indices (see their figure 3). In a
newer measurement, Telfer et al. (2002) find α ≈ 1.76 for (50 – 120 nm or
10 eV –24 eV) using an enhanced set of quasar spectra, see also their figure
4 reproduced here in Fig.4.3 lower curve. A similar slope is measured for the
soft X-ray region (0.6 – 6 nm or 200 eV – 2000 eV) by Laor et al. (1997) who
find α ≈ 1.72.
To conclude this section, we summarize that in the frequency range interesting for reionization simulations (above 13.6 eV [below 91.2 nm]), an average
spectral index of α ∼ 1.75 seems to be a good estimate.
4.4
Summary: Quasars in our simulation
In our study, we do not follow the mass growth of even individual halos. Instead, we assume that the accretion is Eddington limited, λ = 1 in Eq. 4.2 and
that there is enough material to accrete for several Myr to several tens of Myr,
so tQSO ∼ [5 − 25] Myr.
Due to the rather large uncertainty in the measured relation between vc and
σ (if there is at all such a relation, see footnote in Section 4.1 and caption of
Fig. 4.2) as well as between the halo mass and vc (which is model dependent),
for simplicity we use a linear relation to convert halo mass to QSO luminosity
and orientate us towards the upper limit. Our model QSOs are included in
Fig. 4.2. For the spectrum, we use a single power law as defined in Sect. 4.3
with α = 1.5. This value is slightly lower than the value found by Telfer et al.
(2002) for the complete set of quasars, but corresponds to the average of the
radio-quiet sub-set. It is this value that was used by e.g. Bolton et al. (2011)
for wavelengths below 105 nm.
41 http://hubble.nasa.gov/
- 34 -
Figure 4.3: Mean composite QSO spectra. Upper curve: reproduced by permission
of the authors and AAS from Zheng et al. (1997), figure 5, 101 quasars binned to
0.2 nm. Lower curve: reproduced by permission of the authors and AAS from Telfer
et al. (2002) figure 4, 184 quasars binned to 0.1 nm. Here, the relative Flux Fλ is
shown over λ . To convert the the power law index into a power law index for Fν , use
Fν ∝ Fλ (ν)/ν 2 , so the flat part in the upper spectrum transforms into α ∼ 2. Telfer
et al. (2002) exclude the region below 50 nm in the shown fit, since only 10 QSO
contribute to this part of the spectrum, the horizontal black line segments indicate the
continuum windows used for their fit. The line segments of Zheng et al. (1997) are
omitted in this reproduction to avoid confusion in the already busy plot.
- 35 -
C HAPTER 5
S UMMARY OF PUBLICATIONS
This thesis is based on three publications. While all publications contribute to
the large field of simulating the EoR, they are very different in their nature:
while PAPER I deals with different methods for analysing ionization fraction
results from radiative transfer EoR simulations and testing the effects of different source models on the ionization fraction fields, PAPER II describes an
extension of the radiative transfer code C2 R AY. PAPER III in turn uses this
method to investigate the effect of a quasar on the ionized fraction field during the EoR and investigates prospects of the observability of such a quasar
H II region. In Chapter 2, the three publications were put into context. In this
chapter, I give a short summary of the appended papers.
5.1 Summary of “Topology and sizes of H II regions
during cosmic reionization” (PAPER I)
We analyse the ionization fraction results of several large-scale (54 and 163
comoving Mpc) simulations with different methods. The aim of this paper
is two fold: On the one hand, we investigate the usefulness of the methods;
on the other hand we use the methods to find discriminating effects on the
resulting ionization fraction fields from different source properties.
The four independent methods used to characterize the sizes of ionized regions are the friends-of-friends (FOF) method, the spherical average method
(SPA), the power spectrum (PS) and a method based on finding the total surface area and volume of all ionized bubbles (3V/A). The first three size measures mentioned above (FOF, SPA, PS) result in a distribution function of sizes
while the 3V/A results in a single size per redshift. Additionally we consider
another method to measure the size of H II regions that is similar to the one
used by Mesinger & Furlanetto (2007) in the appendix. Since we find that it
is less useful due to the dependence on several free parameters, we do not
consider this method in the main body of the paper.
- 37 -
CHAPTER 5. SUMMARY OF PUBLICATIONS
The FOF method captures the size distribution of the small scale H II regions which contribute only a small amount to the total ionization fraction.
The spherical average method provides a smoothed measure for the average
size of the H II regions constituting the main contribution to the ionized fraction. The power spectrum is more similar to the SPA but retaining more details
on the size distribution.
To characterize the topology of the ionization fraction field, we calculate
the evolution of the Euler Characteristic to which we give a basic introduction. We also point out some flaws introduced by single-cell scale structure
in the appendix. The evolution of the topology of the ionization fraction field
during the first half of reionization is consistent with inside-out reionization
of a Gaussian density field: The Euler Characteristic of the ionization fraction
field as a function of time behaves the same way as the Euler Characteristic
of the logarithm of the density field as a function of density. The inside-out
reionization is further supported by correlating density and ionized fraction
on large scales.
5.2 Summary of “Radiative transfer of energetic
photons: X-rays and helium ionization in C2 -R AY”
(PAPER II)
We present an extension to the radiative transfer and photo-ionization code
C2 R AY. The qualitative background was introduced in this thesis in Section
3.1.
We review the basic concept of C2 R AY, introduce the extensions to the
algorithm in great detail and validate our implementation by a set of tests by
comparing to results from the photo-ionization equilibrium code CLOUDY
as well as comparisons to the hydrogen only solutions. We validated the timedependent solutions through convergence studies. Although these tests were
mainly for validating the code, we could also confirm the validity of a shortcut often taken in EoR simulations, namely that it is valid to assume that for
stellar-type sources, the helium ionization follows that of hydrogen.
The extension consists of the introduction of helium as a second component using the full on the spot (OTS) approximation, multi-frequency ionization/heating and inclusion of secondary ionizations. We described the new elements to C2 -R AY, such as the linearised solution of the set of hydrogen and
helium rate equations introduced here graphically in Fig.3.3 using the coupled OTS treatment, and the calculation of the multi-frequency ionization and
heating rates for both hydrogen and helium. We also show that the often used,
- 38 -
simpler uncoupled OTS (i.e. treating each species isolated) give substantially
different results from the here implemented fully coupled OTS approximation.
We further performed the cosmological test from the earlier mentioned Cosmological Radiative Transfer Comparison Project (Iliev et al. 2006) with the
inclusion of helium. This test includes the calculation of the IGM temperature.
We find that the inclusion of helium significantly affects the temperature distribution as the heating front is found to be steeper than in the hydrogen only
case. Helium is an important absorber of extreme UV and soft X-ray photons and should therefore be taken into account when studying hard ionizing
spectra.
5.3 Summary of “Prospects of observing a quasar H II
region during the EoR with redshifted 21cm” (PAPER III)
We use the extended version of C2 R AY (see PAPER II) to simulate quasar H II
regions during the EoR. We consider three different cases: an early quasar turn
on at z ∼ 8.6 and a late42 quasar turn on at z ∼ 7.7 in a simulation box with
side length of 164 comoving Mpc and a late quasar turn on (also z ∼ 7.7) in the
much larger simulation volume of (607 comoving Mpc)3 . The motivation for
the quasar model used in this paper is described in more detail in this thesis in
Chapter 4. We produce 21cm maps making the assumption of an IGM heated
well above the CMB temperature, as outlined here in Chapter 1.2.
The method we use to measure the quasar H II region size is a technique
that works directly with the visibilities produced by radio interferometers. It
can also subtract the strong foreground emission which affect the redshifted
21cm observations. The method was developed by Datta et al. (2007).
The main aim of this paper is to answer the questions: (1) Is it in principle
possible to use LOFAR in combination with the matched filter approach to
identify single H II regions during the EoR? (2) Is it possible to tell a quasar
H II region apart from H II regions made by galaxies in 21cm maps? (3) If we
know prior to investigating the 21cm maps the position of a quasar, can we
gain more information about the quasar properties by investigating the quasar
H II region by means of analysing the “hole” in the 21cm map?
The answer to the first question is yes, but only if the H II region is large
enough. The answer to the second question is no. Even under rather extreme
conditions, low efficiency stellar sources, high efficiency quasar, the H II region blown by the quasar seems neither much larger nor much more spherical
42 “late”
refers here to late compared to very early quasar turn on at redshift above 8, not late in
a general sense since a quasar turn on at redshift above 7 should still be considered as an early
quasar turn on.
- 39 -
CHAPTER 5. SUMMARY OF PUBLICATIONS
than the largest H II regions made by clustered stellar sources. The answer to
the third question is yes if the quasar contribution to the joint H II region made
by the quasar and the stellar sources is substantial.
- 40 -
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- 46 -
Acknowledgements
As you can see in the lower right corner, I have not managed to reach the
end of the EoR in this thesis. I reached a global volume averaged ionization
fraction of almost 95 %43 . However, you reached the end of my thesis, or, at
least the end of the introductory part. And I reached the end of my stay at the
astronomy department at Stockholm University.
Many people walked beside and in front of me during this time and walking
like that, it sometimes happens that one steps on the foot of each other. That
is almost unavoidable.
While observing the reactions of people reading other peoples acknowledgements, I realized that it is very easy to again step on other peoples foots by
not properly acknowledging everybody in the desired way. Since I am sometimes a very forgetful person, I am convinced that, still by the time of printing,
I will have forgotten to write here the names of a large percentage of the people I should acknowledge. Yes, I would even say, it is dangerous for me to
write such an (incomplete) list of names. Therefore, I go the saver way and
just say: Thanks to all who feel like they have directly or indirectly contributed
to my work.
Alternatively, I could write a list with names (of persons I know and
who know me) who have definitely not contributed in any no matter how
minor way, to my work: – If you do not find yourself on this list and
you are sure that I know you, then you may feel acknowledged by this
acknowledgments.
However, it is easy to name the two persons who contributed in the most
direct way to my work. Thank you Garrelt & Kanan!
Y también es muy fácil identificar a la persona que más me ha apoyado
durante los últimos años. Muchíssimas gracias Cris!
43 Shown
here is simulation S6 from Iliev et al. (2011): It suppresses luminous galaxies in halos
below 109 M in regions more ionized than 90%. It has the good (for this purpose here) feature
of having few small-scale bubbles. The side length corresponds to 54 comoving Mpc.
- 47 -
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