phdthesis final

phdthesis final
1
Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for
Mathematics
of the Ruperto-Carola University of Heidelberg,
Germany
for the degree of
Doctor of Natural Sciences
presented by
Oliver Zahn
born in Munich
2
.
3
.
Illuminating the Universe: New Probes of
Reionization and Cosmology
Referees: Prof. Dr. Matthias Bartelmann
Prof. Dr. Matias Zaldarriaga (Harvard)
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.
5
Zusammenfassung: Wir modellieren die Epoche der Reionisation des
Universums, mittels analytischer und numerischer Methoden. In einer detailierten Analyse unserer Ergebnisse stellen wir eine gute Übereinstimmung
zwischen den alternativen Beschreibungen der Morphologie der ionisierten
Gebiete fest. Wir verwenden unsere Simulationen, um Vorhersagen fuer
Beobachtungsgrößen aufzustellen, die innerhalb weniger Jahre zur Verfügung
stehen sollten: der kinetische Sunyaev-Zel’dovich Effekt und Schwankungen
in der 21 cm Strahlung aufgrund der Hyperfeinstruktur des neutralen Wasserstoffes. Wir schlagen auch vor, die 21 cm Strahlung zur Einschränkung kosmologischer Parameter (mittels Bestimmung des Materie-Leistungsspektrums)
zu verwenden. Desweiteren benützen wir diese Observable als Hintergrund
für den Gravitationslinseneffekt aufgrund großskaliger Strukturen im Universum, und entwickeln einen Formalismus, um die Linsenverteilung aus den
charakteristischen Eigenschaften des beobachteten 21 cm Feldes zu rekonstruieren.
Abstract: We model the epoch of hydrogen reionisation of the universe,
using analytic as well as numerical methods. In a detailed statistical analysis of our results, we find good agreement in the alternative descriptions
of the morphology of ionized regions. We use the simulations to make predictions for reionisation observables that should be accessible within a few
years years: the kinetic Sunyaev-Zel’dovich effect and fluctuations in the 21
cm spin flip transition of neutral hydrogen. We also propose to use the 21
cm signal to constrain cosmological parameters by probing the matter power
spectrum. We also make use of the observable as s source screen for gravitational lensing by large scale structure, and develop a formalism to extract
the lens distribution from the characteristics of the lensed 21 cm field.
6
Chapter 1
Introduction
Within less than two decades, cosmology has progressed from a rather speculative science to one of the most successful fields of physics, through being
based on an exemplary interplay between experiment and theory. The measurement of fluctuations at the level of 10−5 in the cosmic microwave background (CMB) (e.g. [1, 2, 3]) has suggested a ‘standard model’ that has stood
up against a number of other observations based on independent physics. The
challenge of the dawning cosmological paradigm is that it is fundamentally
puzzling, and comes with the calling to develop new parts for our scientific
toolbox: to explain the fact that the universe looks the same on average in
all directions, we need to invoke an epoch of superluminal expansion (‘inflation’) following the big bang [4, 5]; to understand the haze through which we
see the primordial CMB [6, 7], the absorption pattern of emission lines from
distant luminous quasars (e.g. [8]), as well as other observations, the epoch
of reionization of the universe has to have been more complex than simple
models require; to do justice to the observed luminosity-distance relationship
of distant Supernovae Ia [9, 10], the clustering of galaxies (e.g. [11, 12], and
further observables, one has to postulate a contribution of roughly 75% of
negatively gravitating ‘dark energy’ to the total energy budget of the present
universe. Addressing these puzzles directly will require fundamentally new
ideas and specifically designed observations, to try to give us more insight
into their nature.
This thesis introduces a number of new ways of cosmological exploration.
Its central topic, the epoch of reionization (EoR), is a pivotal stage in the
process of cosmological structure formation, marking the birth of the first
luminous objects, a key landmark as the universe transforms from the relatively smooth state probed by the cosmic microwave background (CMB), to
its present day complexity.
First we will establish more accurate predictions for the ionized regions of
7
8
Introduction
hydrogen (HII) produced by the first radiative sources. We will achieve this
goal in two different ways, using numerical simulations, as well as modeling
based on analytic considerations. The close agreement we find between both
methodologies gives us confidence that we are beginning to understand the
complex physical processes guiding the EoR.
A second goal of this thesis will be to use the models we develop to make
concrete predictions for observables that will likely become important probes
of the reionization process within the next few years. Current observational
constraints on the EoR offer an incomplete picture. They come from Lyα
forest absorption spectra towards high redshift quasars (e.g. [8]), from measurements of the high redshift galaxy luminosity function from narrow-band
Lyα-emission searches [13], and from measurements of the large scale CMB
E-mode polarization [7, 14]. The claimed size of HII regions surrounding
individual quasars has also been used to infer limits on the neutral fraction
[15]. There has also been an interpretation of the relatively high temperature
of the Lyα forest at z ' 2 − 4 as evidence of an order unity change in the
ionized fraction at z < 10 [16, 17], although this depends on the properties
of He II reionization [18].
While valuable, each of these observational probes has its limitations,
and some of the current constraints are relatively meager. Quasar absorption spectra are limited in part by the high Lyα absorption cross section:
by z ∼ 6, even a highly ionized IGM completely absorbs quasar flux in
the Lyα forest. The constraints from narrow-band Lyα searches are subtle
to interpret (e.g. [19]), and restricted to narrow redshift windows around
z = 5.7 and z = 6.5, where Lyα falls in the observed optical band, and
avoids contamination from bright sky lines (e.g. [20]). These observations do
not currently allow the interpretation that the ionization state of the IGM
is evolving between these windows. The CMB polarization measurements
constrain only an integral over the ionization history, and are potentially
sensitive to polarized foreground contamination [7].
The study of the EoR may be revolutionized by experiments aimed at
detecting 21 cm emission from the high redshift IGM when the phase transition between neutral and ionized occurred. These experiments should provide
three-dimensional information regarding the distribution of high redshift neutral hydrogen (HI), constraining the topology of reionization, and its redshift
evolution (e.g. [21, 22]). Several low frequency radio telescopes are presently
ramping up to detect this signal: the Mileura Wide Field Array (MWA) 1 ,
the PrimeavAl Structure Telescope (PAST), and the Low Frequency Array
1
http://web.haystack.mit.edu/arrays/MWA/
9
(LOFAR) 2 , while another second generation experiment, the Square Kilometer Array (SKA)3 , is in the planning stage. These measurements will be
dominated by foreground contamination, but in contrast to the IGM signal,
the foregrounds are expected to be smooth in frequency, facilitating their
removal [22]. One of our goals in this thesis will be to establish accurate
predictions for the 21 cm signal to be expected in these observations.
A different new generation of cosmology experiments is being constructed
to target the so-called ‘secondary anisotropies’ (SA) in the CMB. Its ‘primary
anisotropy’ was created 380,000 years after the Big Bang, when the universe
was just 0.1% of its present size. When it had cooled down enough so that
most of its atoms had become neutral, it became transparent to the CMB
photons while expanding by a large factor. There are two extensively studied ways in which this primordial pattern can get altered: 1) relativistic
bending of light rays caused by massive structures such as clusters of galaxies (gravitational lensing); and 2) scattering off hot gas inside dense regions
changes the primordial spectrum and Doppler-shifts the photons into the
line of sight depending on the motion of the gas (the thermal and kinetic
Sunyaev-Zel’dovich (SZ) effects respectively). In this thesis we will predict
a third way: regions of ionized gas during the epoch when the first radiative sources were created led to inhomogeneous re-scatterings of the CMB
photons.
We will make predictions for how well the upcoming CMB experiments
will be able to distinguish different reionization scenarios. In order to do so,
we also need an accurate model for the signal component from the nearby
universe. Because this contains high density peaks, the z < 3 signal turns out
to be very large and dominates the overall signal (we will find this component
to make up 70-90% of the total). To model this accurately we resort to large
volume high resolution gas-dynamical simulations to model the kinetic SZ
effect. We will also use our simulations to calculate the thermal component
of the effect, which vanishes at 218 GHz, and can be subtracted by multifrequency fitting.
We will largely assume familiarity of the reader with the basic cosmological paradigm throughout most of this work, but provide basic definitions
where they seem crucial to the flow of our argument. We will assume a flat
ΛCDM cosmology parameterized by contributions to the total energy density
of matter Ωm , dark energy4 ΩΛ , and baryons Ωb . The local expansion rate
will be parametrized by h in H0 = 100 h km/s/Mpc, and the shape of the
2
http://www.lofar.org
http://www.skatelescope.org/
4
assumed to be a cosmological constant with equation of state weos =
p is pressure, ρ is density.
3
p
ρ
= −1, where
10
Introduction
primordial scalar perturbation power spectrum by its slope ns , as well as a
normalization to present day fluctuations on a comoving5 8 Mpc/h (Mpc)
scale6 of σ8 . The values of these parameters will be specified at the beginning of each individual chapter, each time in rough agreement with recent
experimental constraints (e.g. [23]).
The detailed structure of this thesis is as follows.
Chapter 2 is divided into two parts. We will first discuss the challenges
involved in modeling reionization, in Section 2.1. We will review approaches
to the problem taken in the past. We will describe a model of the morphology
of HII regions based on considerations reminiscent of the extended PressSchechter/excursion set formalism that has been used to predict the fraction
of collapsed objects in the universe.
In the second part of Chapter 2, we will describe the cosmological 21 cm
signal from the high redshift IGM. The physics of the underlying spin-flip
transition, and its evolution through different cosmological regimes, will be
the topic of Section 2.2. The power spectrum of high redshift 21 cm fluctuations will be described in Section 2.3. Because it is observed in redshift
space, we find that it can be used to distinguish its astrophysical components
from those due to the linear density field. This means that we could improve
constraints on the cosmological parameter budget substantially. We will derive cosmological parameter constraints based on planned 21 cm experiments
in Section 2.3.
In Chapter 3 we will use the 21 cm signal from the early stages of reionization as a way to measure gravitational structure formation at low redshift.
We will make use of the large number of data points provided by the 21 cm
signal as a background for gravitational lensing by large scale structure in
the universe. Lensing correlates different lines of sight in a unique way, and
one can use this to statistically reconstruct the lens distribution from the
observed field. We will generalize a quadratic estimator of the lensing field
developed for the CMB to this three dimensional observable. We will apply
this to survey areas and depths as they should be seen by conceived radio
observatories with large collecting areas. We will discuss benefits and disadvantages of 21 cm lensing reconstruction in comparison with the CMB. As
the 21 cm signal potentially contains orders of magnitude more information
than the CMB, in theory it could be more useful for the reconstruction. In
5
A comoving observer is one who experiences the expanding cosmos as homogeneous
and isotropic
6
We will employ the parsec (pc) as standard measure of cosmological distance throughout this thesis. 1 Parsec = 3.26163626 lightyears =3.08568025 × 1016 m. To describe the
scales relevant to reionization, we will mostly be using comoving Megaparsec (Mpc)=106
parsec
11
practice the effort will be limited by costs for collecting area and antenna
correlation to combat the large foregrounds.
In Chapter 4 we will first utilize the semi-analytic reionization model
introduced in Chapter 2 to make predictions for the kinetic SZ effect it produces. The processes driving formation and survival of the first luminous
sources are not well constrained at the moment, and we will in this Chapter
account for this uncertainty by implementing two different reionization scenarios, which vary in the duration of the partially reionized phase. To gauge
the relative importance of the effect of patchy reionization on the CMB, we
model the traditional secondary anisotropies (SZ effects and lensing) using
a large scale, high resolution cosmological simulation that models the adiabatic gas physics between redshift of reionization and today. This simulation
is one of the most sophisticated simulations of the SZ effects to date. We
will calculate sensitivities and make predictions for the detectability of the
kinetic SZ signals with upcoming ground based CMB experiments. We will
also show that neglecting reionization can lead to substantial biases in cosmological parameter constraints if derived from the Planck satellite, which
through its precision will be sensitive to the small peak amplitude changes
caused by the reionization signal.
In Chapter 5 we will attempt to gain deeper insight into the large scale
morphology of the reionizing phase by running a large volume radiative transfer simulation. This simulation takes place in a 100 Mpc volume, representing the cosmological background in which reionization takes place. It is the
reionization simulation with the largest number of resolved sources performed
to date. We will scrutinize the semi-analytic reionization model introduced
earlier in more quantitative detail by comparing it to the radiative transfer
simulation. We will find that the analytic model predictions are surprisingly
accurate, and agree remarkably well with full simulations, statistically as well
as on a side-by-side basis.
With the level of precision increasing, the field of cosmology will become
more sensitive to careful modeling of different observables, combining various
methods available to us. Analytic models guide our intuition of the seemingly complex processes that drove the reionization of the universe. The
robustness of their predictions can on the one hand be seen as confirmation
of convergence of our understanding of the process. It also suggests that
the methods proposed here will prove useful in future analyses of data from
reionization experiments, when the challenge will be to compare a variety of
possible parametrizations to the observations.
The measurement of the 21 cm and small scale CMB signals we predict in
this thesis, will in the future be supplemented by further quasar absorption
spectra (including clues from metal absorption lines: [20, 24]), high redshift
12
Introduction
gamma ray bursts (e.g. [25, 26]), high redshift galaxy surveys (e.g. [27]). Analyzed in combination, these measurements will provide us with a wealth of
data on the EoR. The analytic and numerical models for reionization we developed, will then prove helpful to convert these data into information about
the physical mechanisms underlying reionization. Cosmology is awaiting an
exciting time of breaking through new frontiers of knowledge, and we hope
the results presented here will be useful tools on the way.
13
Acknowledgments
At this point I would like to thank some of the people who have accompanied me throughout the last three years, and played, directly and indirectly,
major roles in accomplishing my work.
First of all I want to thank Matias Zaldarriaga, for being the most fun
to work with, and at the same time the most challenging, advisor I could
have wished for. Since those months I spent as undergraduate at NYU, he
encouraged me to work on quite a diverse range of topics in cosmology, and
for sure my contribution to each one of them would have been much smaller
without the continuous help of his intuition and knowledge. I am eagerly
looking forward to my remaining months at the CfA, and hope that little
Marina will let her daddy get enough sleep!
I now want to express my gratitude to Matthias Bartelmann, for the
unconditional support he has given me in following my interest to go as a
Visiting Fellow to Harvard and collaborate with ‘the other Mat(th)ias’, for
almost the entire PhD phase. My time here has been very fulfilling, and
perhaps the biggest drawback was that it did not allow me to interact as
much with, and learn from, Matthias. My great hope is that there will be
more opportunities to work together in the future.
The major reason, of course, why I did not spend more time in Germany,
has not been directly science related! This is were I want to thank Breeze,
my love, my wife, for the inspiration and fulfillment she has contributed on
a daily basis to my life since the summer of 2004. That our ‘world lines’
converged, is perhaps the most lucky thing that has happened to me, and
astonishingly I didn’t even have to leave the CfA to have it happen!
I next want to thank Adam Lidz, for being a close and inspiring coworker.
His experience with many of the topics I had just begun to work on sped
up and improved my research (and the english language presentation of it!)
immensely. It was a great benefit to always be able to peek through his door
across the hallway, and discuss one of those stubborn questions that just
wouldn’t get answered by my student’s brain.
I furthermore want to thank Lars Hernquist and Matt McQuinn for their
involvement in the projects we worked on together so far, and their readiness
to talk about science whenever I wanted to.
Finally I want to thank my parents, Elke and Jochen, for being supportive
and helping me advance to this stage of my life. I hope the long distance
between us, although it may continue to last for a while, will not hinder us
from being close, and partake in each others lives.
14
Introduction
Contents
1 Introduction
7
2 Reionization models and 21 cm radiation from high redshift
2.1 Modeling Reionization . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The view from simulations . . . . . . . . . . . . . . . .
2.1.2 Analytic Modeling . . . . . . . . . . . . . . . . . . . .
2.1.3 Monte-Carlo type implementation of the analytic model
2.2 21 cm radiation from high redshifts . . . . . . . . . . . . . . .
2.2.1 Compton Heating and Collisional Coupling . . . . . . .
2.2.2 X-ray heating . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Wouthuysen-Field effect (Ly-α pumping) . . . . . . . .
2.3 The 21 cm Power Spectrum . . . . . . . . . . . . . . . . . . .
2.4 Using the 21 cm power spectrum to constrain cosmological
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
18
18
21
25
31
33
34
37
40
3 Lensing Reconstruction using redshifted 21 cm fluctuations1
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Weak Lensing Reconstruction . . . . . . . . . . . . . . . . . .
3.2.1 Quadratic Estimator, General Consideration . . . . . .
3.2.2 Extension to a three dimensional signal . . . . . . . . .
3.3 Antenna Configuration, Sensitivity Calculation . . . . . . . . .
3.4 Results and a Comparison with the CMB . . . . . . . . . . . .
3.5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . .
3.6 Appendix A: Quadratic estimator applied to a three dimensional observable . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Appendix B: Quadratic estimator lensing reconstruction in
practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
52
56
56
60
64
67
76
44
78
83
4 The influence of inhomogeneous reionization on the CMB1 87
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2 Simulation of secondary anisotropy and patchy reionization . . 91
15
16
CONTENTS
4.3 Results for various time dependence of ζ . . . . . . . . . . .
4.4 Observability of patchy reionization with future CMB experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Power spectral constraints from ACT and SPT . . .
4.4.2 Expected bias in cosmological parameter determination from Planck . . . . . . . . . . . . . . . . . . . .
4.5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . .
4.6 Appendix: Simulations of the thermal Sunyaev-Zel’dovich effect and comparison to other authors . . . . . . . . . . . . .
. 94
. 98
. 98
. 106
. 109
. 110
5 Simulations and Analytic Calculations of Reionization Morphology 1
115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.1 N-body simulations . . . . . . . . . . . . . . . . . . . . 118
5.2.2 Ionizing Sources . . . . . . . . . . . . . . . . . . . . . . 120
5.2.3 Radiative Transfer . . . . . . . . . . . . . . . . . . . . 121
5.3 Numerical scheme based on analytic considerations . . . . . . 124
5.4 Statistical Description . . . . . . . . . . . . . . . . . . . . . . 128
5.4.1 The Bubble PDF . . . . . . . . . . . . . . . . . . . . . 128
5.4.2 Power Spectra of the ionized fraction . . . . . . . . . . 130
5.5 Improved numerical scheme . . . . . . . . . . . . . . . . . . . 134
5.6 21 cm signal and power spectra . . . . . . . . . . . . . . . . . 138
5.7 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . 141
5.8 Appendix: Photon Conservation in our approximate simulation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Chapter 2
Reionization models and 21 cm
radiation from high redshift
In this chapter we will begin by describing the difficulties encountered in
the modeling of reionization. We will discuss different methodologies, including numerical modeling by radiative transfer simulations, and analytic
approaches, in Section 2.1. We will introduce a model to describe the morphology of HII regions that is reminiscent of the extended Press-Schechter
theory of the halo mass function. In later chapters we will expand upon the
version of the model described here. We suggest a novel approach to describe
the complex non-spherical clustering of HII regions likely to be expected in
the real universe. This approach is based on implementing the analytic model
in a Monte-Carlo way on top of realizations of the large scale cosmic web.
An advantage of this implementation that we will exploit in chapter 5 to test
this model, is that it allows side-by-side comparison of the analytic model
with purely numerical methods of modeling reionization.
Perhaps the most promising future probe of reionization is radiation from
the 21 cm spin-flip transition of neutral hydrogen in the high redshift universe. In Section 2.2 we will introduce the physics of this cosmological 21 cm
signal. The brightness temperature of this line transition depends on a number of physical interactions which make it appear in emission or absorption
against the CMB, depending on the epoch. These considerations will be the
basis for predictions of the 21 cm power spectrum we will make based on the
numerical and analytic reionization models developed in this thesis.
In Section 2.3 we will discuss properties of the 21 cm power spectrum.
Redshift space distortions introduce a peculiar line-of-sight anisotropy that
can be used to distinguish astrophysical contributions (due to homogeneous
radiative processes) from cosmological contributions (imprinted by the linear velocity field). The sensitivity of upcoming reionization interferometers
17
18
Reionization models and 21 cm radiation from high redshift
should allow us to use the pre-reionization 21 cm signal to extract cosmological parameters.
2.1
Modeling Reionization
2.1.1
The view from simulations
Simulations of reionization have a history dating back almost a decade (e.g.
[28, 29, 30, 31, 32, 33, 34, 35, 36]). Many of the efforts were inspired after
the WMAP collaboration announced [3], in 2003, that the total integrated
optical depth due to Thomson scattering of the CMB due to reionization was
about 2 times larger than previously expected, τrei ' 0.17, corresponding in
the WMAP cosmology to a reionization redshift of zrei ' 20. On the other
hand, extended gaps of transmission seen in the spectra of quasars at redshifts z=5.5-6.5, led to the interpretation that there is a strongly fluctuating
ionization background at z ' 6 [8]. In this light the WMAP result suggested
a quite complex and extended reionization scenario. This lead to a surge of
theories, including interest in high redshift ‘Population III’ stars in leading
to an early phase of reionization (e.g. [33, 37]). 1 .
The reason for the early reionization claim of the first year WMAP release
(WMAP-1) was a large feature found in the polarization signal on the scale
of tens of degrees (in WMAP-1 this feature was only detected in cross correlation with the CMB temperature beause of its larger S/N). The theoretical
motivation for this interpretation [38] is re-scattering of the primordial CMB
quadrupole off free electrons created during reionization. The height and
angular scale of the large scale feature reflect the Hubble scale (size of the
universe) during reionization.
The large scale polarization bump was three years later (2006), with the
release of the WMAP-3 analysis, understood to be to a large extent sourced
by polarized foreground emission, dominated by galactic synchrotron [7].
Many of the characteristics of models developed since WMAP-1 carry over
to lower redshift reionization 2 .
To model reionization most accurately, we would want to use numerical simulations to incorporate effects of non-linear clustering and radiative
transfer. The morphology of the ionized regions and evolution of the ionized
1
These are extremely metal poor stars with a top-heavy initial mass function (IMF),
which cool through molecular hydrogen. They are to be contrasted to sun-like ‘Pop II’stars with a Salpeter IMF, which cool through atomic hydrogen.
2
and for most reionization probes, such as the 21 cm signal discussed in the next section,
the redshift of reionization needs to be lower than z=20 to make observations feasible
2.1 Modeling Reionization
19
fraction should also depend sensitively on the assumed source prescription.
To date it is not clear what the contribution of quasars to the total ionizing
flux might have been. Theoretically, this depends on the extrapolation of the
observed quasar luminosity function to low luminosities, which suggests that
the high redshift contribution is small [39, 40]. Another argument against
the importance of quasars in reionizing the universe is that the observed soft
X-ray background does not have a very large un-resolved component which
one might expect if quasars reionized the IGM [41]. In what follows, we will
assume that the influence of quasars to the formation of HII regions is of
secondary nature.
Simulations have supported theoretical claims (e.g. [42, 43]) that reionization should be a very inhomogeneous process. In them, reionization proceeds
from high to low density regions and recombinations seem to play a subdominant role to large scale bias. Strömgren spheres of neighboring protogalaxies
overlap, and, as a result, overdensities harbor large ionized regions, up to
tens of comoving Mpc across. Besides uncertainties over the physical approximations made in current radiative transfer calculations of reionization,
memory and CPU requirements pose a serious limitation, so that these simulations have until recently only been performed on scales of up to 10 Mpc/h,
corresponding to ' 6 arcminutes on the sky [44] for a typical reionization
redshift of z = 8. If the HII regions are of comparable size (as we will show
later they can indeed become several tens of comoving Mpc across), sample
variance becomes a problem for these simulations. In addition, [45] showed
that the halo mass function will be biased in a simulation that is normalized
to the mean density of the universe on the cutoff scale of the box.
Figure 2.1 shows the result of the simulation by [44]. The radiative transfer is typically performed as a post-processing step after running a simulation
of the cosmic density field, in this case a smoothed particle hydrodynamical
simulation (SPH) with 3243 particle resolution (for each dark matter and
gas). Simulation snapshots are then searched for compact systems (halos),
that are in turn assigned an ionizing luminosity. Then rays are cast from
those sources, and propagated through the hydrogen fluctuation field given
by the cosmological simulation. This procedure can quickly become very
CPU and memory intensive, when we have to account for the joint influence
of many thousand sources on the ionization state of many volume elements
in the box. Each HII region in this simulation contains many sources, whose
strong clustering makes the bubbles grow large (comparable to the box scale)
quickly. We can see that the 10 Mpc/h simulation size used here, although a
substantial step at the time, only allows to model the very beginning stages
of the process, after which bubbles quickly merge and we are left in the dark
about the actual clustering properties of the ionization field.
20
Reionization models and 21 cm radiation from high redshift
Figure 2.1 21 cm signal (see Section 2.2 resulting from a radiative transfer
simulation by [44]. The comoving size of this box (shown are 9 snapshots at
different evolution stages) was 10 Mpc/h on a side. We can see that already
at small ionization fractions this simulation box size is insufficient to sample
the bubble size distribution evenly. Permission to print granted from Aaron
Sokasian.
To fully satisfy the requirements on dynamical range, we would on one
hand like to reliably simulate the large scales on which bubbles occur, i.e.
roughly 100 comoving Megaparsec (Mpc)3 . On the other hand we would
need to resolve the smallest dwarf galaxies potentially contributing to the
reionization process. The question of what the lowest mass sources are is
essentially a requirement to the temperature of virialized systems. For cooling through atomic hydrogen4 , the required temperature is about 104 Kelvin.
3
We will justify this assertion in Chapter 5
In contrast, molecular hydrogen cooling is the mechanism allowing very massive and
metal poor high redshift stars to cool, so called Population III systems. The LymanWerner photons produced by these stars have a tendency to photo-dissociate the molecular
4
2.1 Modeling Reionization
21
This corresponds to a virial mass of [43]
mmin ' 3.0 × 109 (1 + z)−1.5 M .
(2.1)
corresponding to 108 M , where M is the mass of our sun, at redshifts z = 8.
To determine the number of particles necessary to resolve this mass within a
100 Mpc simulation volume, notice that the critical density is measured [14]
to be
g
M
ρcrit ' 10−29 3 ' 1.4 × 1017
(2.2)
cm
(100 Mpc)3
To resolve the minimum mass required for atomic cooling, we therefore require
1.4 × 1017
= 4.6 × 1010 ' 35003
(2.3)
3 × 106
particles! This number is out of reach to the largest computer clusters we
are aware of. In Chapter 5 we will try to come closer to this goal by running
a simulation based on 30 times more particles, in a two orders of magnitude
larger volume than the simulation described. This will allow us to model the
bubble morphology in a representative volume even into the late stages of
reionization, when the bubbles become very large. Our simulation will still
fall short of the above mentioned resolution goal, however we will argue for
why its minimum mass is sufficient to simulate the large scale morphology
of reionization.
Even if the dynamical range requirements in a simulation of reionization
were fulfilled, there would still be a large uncertainty in many of the input
parameters of these simulations. Even within the paradigm of reionization by
Population II hosting galaxies, the prescriptions used for the star formation
efficiency, the escape fraction, the number of photons emitted on average per
baryon, and the number of recombinations encountered, are at present all
uncertain by at least an order of magnitude.
2.1.2
Analytic Modeling
In the light of these difficulties, it would be helpful if we could come up
with an analytic prescription to guide our intuition for the ways in which
the reionization morphology depends on a number of parameters. If found
hydrogen they need for their cooling. It is a matter of present debate whether this negative
feedback makes Pop III stars a realistic possibility. In most of this thesis we will assume
Pop II stars, although the models we develop can also be applied for Pop III hosting halos,
see Chapter 4.
22
Reionization models and 21 cm radiation from high redshift
successful, such models would be useful because of their larger dynamical
range, making them less prone to the resolution problems of the simulations.
The most important ingredients for a model of reionization are the number density of sources as a function of mass, their clustering strength, and
their correlation with the underlying density field. Because HII regions extend to larger radii than the correlation length of individual galaxies, building
an analytic model purely from first principles based on local galaxy properties
can only succeed before clustering beomes important[46]. A straightforward
generalization is then to assume that HII regions trace dark matter halos
but to include source clustering in an approximate way. One possibility is
to specify the global neutral fraction x̄i (z) and the characteristic bubble size
Rchar (z). The number density of bubbles nb (m) does then follow automatically, and the bubbles can be associated with dark matter halos of the same
number density [47]. This allows the model to make use of the well established halo-clustering formalism [48] to describe the bubble pattern, however
it requires Rchar (z) to be arbitrarily specified.
It would be more satisfying to motivate Rchar (z) physically. The central
quantity is the clustering of reionization sources so that overdense regions,
which harbor more sources, will be ionized before underdense regions [43] 5 .
Let us begin by associating an HII region with a single galaxy. The size
distribution of ionized regions then follows directly from the halo mass function when the ansatz mion = ζmgal is made, where mgal is the mass in a
collapsed object. The parameter ζ is the efficiency factor for ionization, for
example composed as ζ = fesc f∗ Nγ/b n−1
rec , where fesc is the escape fraction of
ionizing photons from the object, f∗ is the star formation efficiency, Nγ/b the
number of ionizing photons produced per baryon by stars, and nrec is the typical number of times a hydrogen atom has recombined. The efficiency factor
is a rough combination of uncertain source properties, but encapsulating a
variety of reionization scenarios in it can be regarded as a starting point to
gain insights into the morphological properties of the partly ionized phase.
According to the extended Press-Schechter model [49], the collapsed fraction (or the fraction of baryons that lie in galaxies) in a region of size r
depends on the mean overdensity of that region, δ r , as
"
#
δc (z) − δ r
fcoll (mmin ) = erfc p
.
(2.4)
2[σ 2 (rmin ) − σ 2 (r)]
Here, σ 2 (r) is the linear theory rms fluctuation on scale r and rmin is taken to
5
A scenario in which the higher level of clumpiness associated with the overdense regions completely reverses this trend [42], leading to ‘outside-in’ reionization, can not be
confirmed in present reionization modeling
2.1 Modeling Reionization
23
be the radius that encloses the mass mmin (at average density ρ) corresponding to a virial temperature of 104 K, at which atomic hydrogen line cooling
becomes efficient and δc (z) is the numerical factor 1.686 scaled to today using linear theory. The redshift dependence of the minimum mass is given by
Equation 2.1. The fraction of mass in galaxies required to ionize all hydrogen
atoms is inversely proportional to the ionizing efficiency, so one requires that
fcoll ≥ ζ −1 .
(2.5)
Combining equations 2.4 and 2.5, we can define a barrier which fluctuations have to cross for their baryonic content to become ionized [50]
√
δr ≥ δx (m, z) ≡ δc (z) − 2 erfc−1 (ζ −1 )[σ 2 (rmin ) − σ 2 (r)]1/2 .
(2.6)
Because Press-Schechter theory assumes Gaussian fluctuations on the mass
scale m, this formalism can be applied only to mass scales larger than the
size of individual collapsed objects.
Let us try to compute the size-distribution of HII regions fulfilling the
above condition, as a function of mass. We cannot treat each HII bubble in
isolation, because even if a region does itself not contain many sources, it
might receive ionizing radiation from sources outside of it. In combination
the incident radiation could - especially if the region is a low density part of
the universe - lead to reionization. This consideration is reminiscent of the
excursion set formalism [49, 51], which was used to derive the Press-Schechter
[52] mass function. In that case the critical overdensity δc (z) describes a condition for virialization. Any region with δ > δc (z) is assigned to be part of
a halo. The Press-Schechter formalism suffers from a problem similar to our
problem of a region anemic of ionizing sources. There is has been called the
“cloud-in-cloud” problem, meaning that a point can be part of many regions
with δ > δc (z) on different mass scales. For halos, as for HII regions, only
the largest scale is physically important, because it incorporates all of the
smaller scales. The excursion set formalism treats this as a diffusion problem
in the (σ 2 (m), δ) space with an absorbing barrier (δc (z)). As the rms density
fluctuation increases monotonically toward smaller scales, σ 2 plays the role
of time, while δ plays the role of space. One can use this as a basis to compute the distribution of crossing times, or masses, from which the halo mass
function follows. The cloud-in-cloud problem is solved by following trajectories from large to small mass scales, or increased σ 2 (m), while assigning each
diffusion trajectory to the largest halo (‘absorbing’ smaller ones) which it is
embedded in.
The problem of finding the mass function of HII regions is only different
from the problem of finding the halo mass function in that, instead of being a
24
Reionization models and 21 cm radiation from high redshift
constant, the absorbing barrier δx follows from the condition fcoll (δ, σ(m)) =
ζ −1 , i.e. it is a function of mass. As can be guessed from Figure 2.3, the
ionization barrier can however be approximated by a linear function in σ 2 ,
δx (σ 2 ) ≈ B(σ 2 ) ≡ B0 + B1 σ 2 , which allows us to find an analytic solution for
the ‘bubble mass function’ [53, 50, 54]
r
m nb (m) dm =
2 ρ̄ d lnσ B0
B 2 (m, z)
exp −
dm.
π m d lnm σ(m)
2σ 2 (m)
(2.7)
As a convenient result of this, many of the tools used for halo mass functions,
clustering, etc. can be carried over to HII regions.
The solid curves in the left panel of Figure 2.2 show the size distribution
following from Equation 2.7 for different x̄i at z = 18 − 12. The ordinate is
the fraction of the ionized volume filled by bubbles of a given size. The model
suggests that bubbles grow large toward the middle stages of reionization,
with characteristic sizes exceeding 10 Mpc/h during the middle and final
stages. It would be satisfying if we could confirm this picture with numerical
simulations of reionization performed on an appropriate scale (as we will do
in Chapter 5).
The right panel of Figure 2.2 shows the evolution of the global ionized
fraction of the IGM, ζfcoll in the analytic model, for three different values of
the ionization efficiency.
Note that we have only incorporated homogeneous recombinations as a
background into the overall efficiency parameter ζ (the intrinsic photon output of the sources being largely unknown). In reality, dense systems, such
Lyman-limit systems or minihalos (if these survive the photo-ionizing background inside the first HII regions) might play an important contributing
role. A property of those systems would be that the larger the bubble they
reside in, in other words the higher the photoionizing background, the deeper
photons penetrate into them, and the higher the resulting recombination
rate becomes. Because these photons are lost to reionizations and to further growth of the bubble, at some point a saturation will be reached, at
which the total photon output of sources just cancels with recombinations,
and the bubble growth could be effectively stalled. Given a prescription of
the small scale density structure of the IGM (e.g. calibrated off simulations),
this scenario can be incorporated in analytic models [55] 6
6
We are currently actively working on including recombinations in the semi-analytic
scheme proposed in the next section, validating them against simulations, and using our
generalizations of the model to efficiently explore the degeneracies in reionization parameter space.
2.1 Modeling Reionization
25
Figure 2.2 Left panel: bubble size distribution according to Equation 2.7.
The lines represent the redshifts and ionization fractions (from left to right)
z=18 (xi = Q̂=0.037), 16 (0.11), 14 (0.3), 13 (0.5), and 12 (0.74). The
efficiency parameter ζ assumed here was 40. Right panel: the evolution of
the global ionization fraction ζfcoll according to Equation 2.4 for the values of
ζ = 500, 40, 12. The dashed lines are the exact result, the solid lines are from
the linear barrier approximation described in the text. From [50], permission
to re-print from M. Zaldarriaga.
2.1.3
Monte-Carlo type implementation of the analytic model
We would like to use the formalism described in the previous section to
predict non-spherical bubble shapes. In particular, we would also like to
be able to compare the model directly to simulations of the reionization
morphology that use radiative transfer. In this section we propose a new
way of establishing the reionization morphology which amounts to a MonteCarlo realization of the analytic model described in the previous section.
As the analytic reionization model provides us with a barrier prescription
for assigning ionized regions, Equation 5.2, we can apply it to any three
dimensional realization of the density field. A convenient option for the
relatively high redshifts of interest is to generate Gaussian random fields
from a linear theory power spectrum (the amplitude and shape of which we
think we understand very well). The field is then smoothed with a top-hat
window, describing spheres of influence from ionizing sources. The window
26
Reionization models and 21 cm radiation from high redshift
convolution becomes a simple multiplication with the Fourier space analogue
of the spherical top-hat, given by
WT H (k, RTH ) =
(sin x − x cos x)
,
x3
x = kRTH .
(2.8)
To find the smoothing scale at which a given cell is ionized, we have to
keep in mind (recall the discussion in the previous section) that it could
also be ionized by photons originating from neighboring regions. Thus we
should smooth the density on all possible scales to see whether a given point
was above the ionization threshold of equation (5.2) for some smoothing
scale. In practice, we start at large radii (comparable to the simulation box
size) and record the smoothed overdensity as we smooth logarithmically on
progressively smaller scales. When a cell first crosses the barrier (which
depends only on redshift and ionization efficiency), it is deemed ionized. If
later, as a function of decreasing scale, it crosses the barrier downwards, this
means that the region had been initially ionized by a neighboring overdensity.
Figure 2.3 shows the ‘random walk’ of δ0 , the density fluctuation scaled to
the present using linear theory, for three different regions inside a simulation
box (where the efficiency parameter was chosen such that the universe is half
ionized at redshift z = 14.2) with increasing σ 2 (m) (decreasing smoothing
radius r). It also shows the ionization efficiency dependent barrier, from
equation (5.1), as the solid curve. The short-dashed curve describes a region
of high overdensity that self-ionizes. In the dotted curve at σ 2 (m) ' 2.2 the
barrier is crossed downwards, so the volume element was ionized by sources
in neighboring cells. The long-dashed curve corresponds to an element that
did not ionize at this redshift.
We show the resulting ionization field for an ionized volume fraction of
xi,V = 0.48 in Figure 2.4. As underlying realization of the cosmic web we
here used a Gaussian random field realization of the linear redshift 8 matter
power spectrum generated with CMBFAST [56]. The box has a side length
of 100 comoving Mpc/h. The slice shown is 10 Mpc/h deep. There is a wide
range of different bubble sizes, mirroring the complex clustering behavior of
sources, which themselves are of course missing in our simulation, which is
solely based on the collapse fraction given a linear overdensity. In Chapter 5
we will study in more detail how the bubble morphology changes at different
ionization fractions.
The ionization threshold according to Equation 5.2 can also be varied
across a given box to show the redshift evolution of the signal and provide
the reionization scenario on the light cone. The results of this procedure is
shown for a 500 Mpc/h box in Figure 2.5, where we plot comoving extend of
the field on the abscissa, and redshift on the ordinate. This implementation
2.1 Modeling Reionization
27
Figure 2.3 The behavior of the overdensity scaled to today, δ0 , for four regions
from a 100 Mpc/h, 2563 , calculation. The abscissa in this plot is the rms
fluctuation, dependent on the applied smoothing scale. The solid curve is
the barrier given from equation (5.1). The region corresponding to the short
dashed curve crosses the barrier for ionization at σ 2 (m) ' 1.8. The region
described by the dotted curve wanders below the barrier again, but has been
ionized by sources in a neighboring region with higher density. Finally the
long dashed curve represents a region in which the gas stays neutral at this
particular redshift (z = 14.2)
.
28
Reionization models and 21 cm radiation from high redshift
Figure 2.4 We show the result of the fast numerical scheme suggested here,
based on Gaussian random field of extension 100 Mpc/h with a resolution of
5123 cells. The (volume weighted) ionized fraction achieved at z=8 with an
efficiency of ζ = 12 is xi,V = 0.48. Clearly the bubble morphology produced
by the semi-analytic scheme we proposed is very complex. There is also a
wide distribution of bubble sizes (compare the black curve in Figure 2.2),
and the largest HII regions can be several tens of comoving Mpc across.
2.1 Modeling Reionization
29
illustrates nicely how small ionized regions form around a few most massive
sources initially, and how the HII regions formed later merge to clusters that
can be (at the final stages of reionization) of the order 100 comoving Mpc/h
in size. For orientation, the angular scale probed by this box is given at the
500
angular diameter distance7 corresponding to z = 8 by θ = 6260
= 4.2◦ . This
simulation also re-iterates that within our analytic model, reionization takes
place over a redshift interval of ∆z ' 3.
This ‘Fourier space radiative transfer’ may seem as a bold simplification
of the problem of describing propagation of radiation from sources through
the IGM. In Chapter 5 we will expose our model to a number of detailed
tests, and show that it in fact performs surprisingly well.
Our poor man’s solution, by exploiting the speed of Fast Fourier transforms (FFT), has a number of advantages over brute force numerical methods
of simulating reionization. To improve our understanding of reionization, various parametrizations of ζ and its time evolution may be compared to data
from the next generation of experiments. In addition it is straightforward
to generalize the analytic model to a mass-dependent ionizing efficiency [57].
In contrast to radiative transfer simulations, our implementation is not limited by similar CPU or memory related problems. For each box we need to
Fourier transform ' 50 times back and forward to do the smoothing. For
a box with the dynamical range from 0.4-100 Mpc, this takes ' 10 minutes
per model on a Xeon 3.2 GHz computer. The implementation may be parallelized easily, hence the realization of a large number of parametrizations
of reionization is feasible.
We should clarify our usage of the term ‘Monte-Carlo’. The numerical
scheme suggested in this section is not random in assigning the ionization
state of any IGM cell. For a given realization of a density field it is deterministic in that it decides whether a region is ionized based on the density
field smoothed on different scales. In this sense the criterion is non-local, because the ionization degree of a region does depend not simply on the linear
overdensity of this region, but on the fluctuation of the surroundings. The
procedure we suggested here is Monte-Carlo-like only because an underlying density field might be produced with random modes, using the power
spectrum of matter fluctuations, which we think we know well from linear
7
Given in our flat cosmology by
Z
D(a) =
a
1
da0
a0 H(a0 )
,
(2.9)
where a is the scale factor and H(a) is the expansion rate, ȧ/a. In our cosmology H(a) =
1/2
H0 a−3 Ωm + ΩΛ
.
30
Reionization models and 21 cm radiation from high redshift
sv512_z8.0.zeta10.0.ME.fts_0
z
9
8.5
8
7.5
7
100
200
300
400
500
Mpc/h
Figure 2.5 Shown is a 10 Mpc/h deep cut through an ionization box of
comoving sidelength 500 Mpc/h. The ionization fraction is averaged over
voxels and is color schemed between white (ionized) and black (neutral). Here
we evolved the ionization threshold throughout the simulation, converting
comoving distance to redshift along the y-axis. Reionization in this scenario
takes place between redshift z ' 9.5 and z ' 7.
2.2 21 cm radiation from high redshifts
31
theory.
This is the advantage of the method suggested here over others: we can
produce well-motivated reionization fields which carefully trace the large scale
density fluctuations in a non-spherical manner. As we will show in Chapter
5, the resulting ionization morphology resembles that found in simulations,
passing a number of quantitative tests. It will also be shown to guide our
intuition beyond numerical simulations, because it does not have the same
limitation in low mass source resolution.
2.2
21 cm radiation from high redshifts
In this section we will describe the physics behind a new potential probe of
the reionization epoch. The 21 cm radiation we are interested in arises from
from the spin-flip transition in clouds of neutral hydrogen at high redshift,
with a rest frequency of ν0 = 1420.4 MHz. The incident intensity Iν can
be described by its equivalent brightness temperature Tb (ν), which is what
would be required of a blackbody with spectrum Bν such that Iν = Bν (Tb ).
The Rayleigh-Jeans approximation is appropriate for the frequencies and
temperatures of interest, so that Tb (ν) ' Iν c2 2 kB ν 2 .
Following the radiative transfer equation along a line of sight through a
cloud of uniform excitation temperature Tex , the emergent brightness at a
frequency ν is given by
Tb (ν) = Tex (1 − e−τν ) + TR (ν)e−τν
(2.10)
R
where the optical depth τν ≡ ds αν is the integral of the absorption coefficient (αν ) along the ray through the cloud, TR0 is the brightness of the
background radiation field incident on the cloud along the ray, and s is the
proper distance.
The excitation temperature Tex in this equation is also called the spin
temperature TS of the 21 cm transition. It quantifies the relative populations,
ni , of atoms in the two hyperfine levels (n=1 for triplet, n=0 for singlet) of the
electronic ground state. The excitation temperature is then defined through
g1
n1
= e−E10 /kB TS = 3 e−T? /TS
(2.11)
n0
g0
where gi is the statistical weight (here g0 = 1 and g1 = 3), E10 = 5.9×10−6 eV
is the energy splitting, and T? ≡ E10 /kB = 0.068 K is the temperature corresponding to the energy difference between the levels. In all astrophysical
applications TS T∗ , hence approximately three of four atoms find themselves in the excited state. Therefore the absorption coefficient must include
a correction for stimulated emission, and it depends on TS as well.
32
Reionization models and 21 cm radiation from high redshift
Let us next consider how the brightness temperature of the spin-flip transition evolves throughout different cosmological regimes. The spin temperature is a result of three processes whose relevance changes with time:
1. absorption and stimulated emission of CMB photons
2. collisions with other hydrogen atoms, electrons, and protons
3. scattering of UV photons.
In other words, the spin temperature is governed by (e.g. [58])
n1 (C10 + P10 + A10 + B10 ICMB ) = n0 (C01 + P01 + B01 ICMB ) ,
(2.12)
where C10/01 and P10/01 are the de-excitation/excitation rates from collisions
and UV induced transitions, B01 and B10 are the Einstein transition rates,
and ICMB is the intensity of incident CMB radiation. It can be shown that
all relevant timescales are shorter than the typical expansion time, so the
assumption of equilibrium is justified. In the Rayleigh-Jeans regime, equation
(2.12) can be rewritten as [59]
TS−1 =
Tγ−1 + xc TK−1 + xα Tc−1
,
1 + xc + xα
(2.13)
where xc and xα are coupling coefficients for collisions and UV scattering,
respectively, and TK is the gas kinetic temperature. Here we have assumed
detailed balance by requiring
g1 −T? /TK
T?
C01
≈3 1−
= e
.
C10
g0
TK
(2.14)
We furthermore introduced an effective color temperature of the UV radiation
field Tc via
T?
P01
≡3 1−
.
(2.15)
P10
Tc
We then need to determine xc , xα , and Tc . It can be shown that in most
situations of interest Tc → TK (see Section 2.2.3), and equation (2.13) may
be rewritten
xc + xα
Tγ
Tγ
=
1−
.
(2.16)
1−
TS
1 + xc + xα
TK
2.2 21 cm radiation from high redshifts
2.2.1
33
Compton Heating and Collisional Coupling
At high redshifts, Compton scattering in the IGM between CMB photons
and free electrons left over from recombination control the gas temperature.
The heating rate is given by (e.g. [60])
2 comp
x̄i
8σT uγ
=
(Tγ − TK ) .
3 kB n
1 + fHe + x̄i 3me c
(2.17)
In this equation, fHe is the primordial helium fraction, uγ ∝ Tγ4 is the energy
density of the CMB, and σT is the Thomson scattering cross-section. The
first factor appears as CMB photons scatter off free electrons, while the
heat must be shared with all particles. The efficiency of Compton heating
decreases with time when electrons recombine and the CMB energy density
drops. The IGM thermally decouples from the CMB at a redshift given by
[61]
1 + zdec ≈ 150(Ωb h2 /0.023)2/5 .
(2.18)
The precise decoupling history can be calculated with the publicly available code RECFAST [62]. The black solid line in Figure 2.6 shows the CMB
temperature evolution, the blue dashed line the gas temperature evolution
resulting from RECFAST, leading to decoupling from the CMB at z ' 200.
At high redshift the spin temperature is coupled to the gas temperature
through collisions. The collisional coupling strength is quantified through
the coefficients in Equation 2.16, which are given by
xic ≡
i
T?
ni κi10 T?
C10
=
.
A10 Tγ
A10 Tγ
(2.19)
Here, κi10 is the rate coefficient for spin de-excitation in collisions with that
species in units of cm3 s−1 . We must take into account coupling coefficients
between various species: H-H collisions, H-e− collisions and collisions of neutral atoms with protons, deuterium atoms, as well as Helium in its neutral or
ionized states. We will not go in much detail here but refer the reader to e.g.
[63] for details. It can be shown that H − H collisions dominate throughout
all regimes, and that collisions are able to couple the spin temperature effectively to the kinetic temperature of the gas between redshifts z ' 150 and
z ' 70.
Once Compton heating becomes less important, as the free electron density decreases, the gas cools adiabatically, its temperature following a 1/a2
scaling of pressure-less dust. The CMB continues its 1/a cooling.
At even lower redshift the gas dilutes, the Hubble expansion makes the
collision rate subdominant relative to the radiative coupling rate, and the
34
Reionization models and 21 cm radiation from high redshift
spin temperature converges again toward the CMB temperature, driven by
Compton scatterings. We can get a rough estimate for when collisions become
unimportant by looking at the critical overdensity δcoll at which xc = 1 (e.g.
[63])
2
0.024
κ10 (88K)
70
1 + δcoll = 1.04
,
(2.20)
κ10 (TK )
Ωb h2
1+z
By z ' 30 the spin temperature again traces the CMB temperature so
that the IGM becomes invisible. This is only valid as long as there are no
early radiative processes from first sources, such as metal-poor Population
III stars.
In Figure 2.6 we show the spin temperature evolution resulting from initial
Compton heating and later adiabatic cooling. The CMB temperature is
shown in the black solid line. The evolution of the gas temperature (blue
dashed line) has been obtained by integrating over equation 2.17. The spin
temperature (red, dot-dashed line) initially follows the gas temperature, but
once collisional couplings become less important is drawn towards the CMB
temperature through Compton scattering.
2.2.2
X-ray heating
Once the first sources of radiation turn on, the neutral IGM will be heated
by their radiation. The most important heating channel is thought to be
X-rays because of their long mean free path. X-ray sources are thought to
homogeneously fill the universe almost as soon as collapsed objects form.
There are two main candidates for their sources: supernovae and high mass
binary systems. In the first case, radiation can inverse-Compton scatter of
the relativistically ejected ionized particles to become highly energetic. This
mechanism could be more important [64] at the high redshifts we are interested in, than in the local universe, because the energy density of radiation
uγ ∝ (1 + z)4 , so that the radiation field is substantially larger than in the
starbursts observed locally. The second class of sources, high-mass binary
systems, should be more abundant. In them, material from a massive main
sequence star accretes onto a compact neighbor. The systems should be
present within a few million years after the first stars are formed.
In studies of the high redshift X-ray background, the dependence of the
luminosity for 0.2-10 keV photons on the star formation rate is usually extrapolated from low redshift observations with a conversion factor fX to
account for the different physics at high redshift (e.g. [65, 66]). Figure 2.7,
panel (a) demonstrates the effect of varying this extrapolation normalization
between 0.2 (in the blue dot-dashed curves) and 1 (in the solid curves).
2.2 21 cm radiation from high redshifts
35
Figure 2.6 The evolution of the global spin temperature in the absence of
collapsed objects. At z=200 the gas adiabatically cools away from the CMB
and so does the spin temperature, coupled to the kinetic temperature through
collisions. At z ' 100, the dilution of the gas leads to a ceasing of the
collisional coupling. Compton scattering with remaining free electrons then
brings the spin temperature in equilibrium with the CMB again. The bottom
panel decomposes the brightness temperature fluctuation in fluctuations in
the spin temperature (red dot-dashed), and those in the neutral hydrogen
density (blue dashed). From [60], printed with permission by M. Zaldarriaga.
X-rays deposit their energy in the IGM by photoionizing hydrogen and
helium. The hot primary electron then distributes its energy through collisional ionizations, producing more secondary electrons, collisional excitation
of He, which produces a photon capable of ionizing H, and through Coulomb
collisions with thermal electrons. The relative cross-sections of these processes determine the fraction of X-ray energy available for heating (fX,h ) and
for ionization (fX,ion ). These rates depend on x̄i and the initial photon energy. It can be shown [67, 68] that X-ray heating should be quite rapid, and
the fraction of photons going into ionization decreases quickly once there are
36
Reionization models and 21 cm radiation from high redshift
Figure 2.7 Spin temperature history in the presence of collapsed objects which
lead to a X-ray background (heating the gas) and UV photons (coupling spin
and kinetic temperatures) is shown in panel a). The CMB temperature is
shown in the dotted, the gas kinetic temperature in the thin, and the spin
temperature in the thick curve, respectively. Here reionization by Pop II
star forming galaxies was assumed. The dot-dashed curves assume that a
larger fraction of photons goes into reionizing the hydrogen atoms, fX = 0.2.
In this case the resulting spin temperature fluctuations could be substantial
even during reionization, adding a layer of complexity. Panel b) shows the
evolution of the ionized fraction. The long and short dashed curves assume a
model for photo-heating feedback, limiting the influence of low mass sources.
Panel c) then shows the differential 21 cm brightness temperature against
the CMB. The two dotted lines show δTb if shock heating is ignored. From
[67], printed with permission by P. Oh.
sufficient free electrons. The formalism was first developed by [68].
The result of those calculations is shown in Figure 2.7. The thin solid and
dot-dashed lines in panel (a) represent the gas ‘kinetic’ temperature, which
is raised above the CMB temperature by the X-ray background. The dotted
line is the CMB temperature The thick solid and dot-dashed lines are for
the spin temperature. To understand the evolution of the spin temperature
(thick lines) we have to discuss the role played by the UV background.
2.2 21 cm radiation from high redshifts
2.2.3
37
Wouthuysen-Field effect (Ly-α pumping)
Let us focus on the regime below z=30, which should be accessible to 21 cm
experiments planned for the foreseeable future (the brightness temperature of
galactic synchrotron, which is also called the system temperature, rises with
falling radio frequency roughly as ν −2.6 , while Tsys = 440K at z=8.). During
this regime, collisional coupling has ceased to be important because of the
dilution of the IGM. As was suggested by Field and Wouthuysen [69, 59] at
this time coupling of the spin temperature to the gas kinetic temperature
by the UV background created by first collapsed objects becomes important.
The way this process works is illustrated in Figure 2.8, where the hyperfine
sublevels of the 1S and 2P states of hydrogen are drawn. The convention
nF LJ was used to denote an atom with radial quantum number n, orbital
angular momentum L, and total angular momentum J. F = I + J is the
quantum number obtained from the nuclear spin I and J.
Let us imagine an incident Lyα photon being absorbed by a hydrogen
atom in the hyperfine singlet state. Transitions in which ∆F = 0, 1 are
allowed except for F = 0 → 0. So the atom state can jump to either of the
central 2P states.The dipole selection rules then allow this state to decay to
the 1 S1/2 triplet level. The net effect is a change in hyperfine state by the
absorption and subsequent re-emission of a Lyα photon. In principle any
Lyman series photons can contribute through cascading effects, however in
practive the higher level decays have only small importance [70].
Let us understand how effectively the Wouthuysen-Field effect will equilibrate kinetic and spin temperatures. The Lyα coupling depends on the
effective temperature Tc of the UV radiation field, defined in equation (2.15).
This is determined by the shape of the photon spectrum at the Lyα resonance. The color temperature of the radiation field matters because there
is an energy difference between the hyperfine splittings of the Lyα transition. Hence the mixing process is sensitive to the gradient of the background
spectrum near the Lyα resonance.
The medium is optically thick and a large number of Lyα scatterings will
tend to bring the Lyα profile to a blackbody of temperature Tk near the line
center. This is thought to be the case [69] in the high-redshift IGM, where in
our cosmology the mean optical depth of a Lyα photon that redshifts across
the entire resonance is given by [71]
3/2
1+z
χα nHI (z) c
5
τGP =
≈ 3 × 10 xHI
.
(2.21)
H(z)να
7
The primary mechanism responsible for establishing this equilibrium has
been shown to be atomic recoil during scattering, tilting the spectrum to
38
Reionization models and 21 cm radiation from high redshift
Figure 2.8 Level diagram illustrating the Wouthuysen-Field effect. We show
the hyperfine splittings of the 1S and 2P levels. The solid lines label transitions that mix the ground state hyperfine levels, while the dashed lines label
complementary transitions that do not participate in mixing.
the red [72]. For this reason the color temperature will essentially be equal
to the kinetic gas temperature, Tc ' Tk . Finally the WF effect will couple the
color temperature to the spin temperature so during this period we expect
Ts ' Tk overall.
Again we refer to Figure 2.7, a continuation of Figure 2.6 to lower redshift.
The thin lines are the kinetic gas temperatures and the CMB temperature.
Two additional pieces of physics have been added to the calculations shown
here. The thick curves in panel (a) show the evolution of the spin temperature
in the interval z < 25 for two different efficiencies of X-ray production. In
the case with fX = 0.2 (dot-dashed line), the hard X-ray luminosity is only
20% of that seen at low redshift for a given star formation rate [65]. In
this case a larger fraction of radiation presumably goes into ionizations and
some spin temperature fluctuations might survive into the beginning stages
of reionization at z ' 11 (compare panel (b)), but quite soon we are in
the regime Ts TCMB . In panel (c), which shows the differential 21 cm
brightness temperature, the two dotted lines show the case where heating by
shocks in the IGM is ignored. This makes clear that X-ray heating is the
most relevant mechanism in most cases.
2.2 21 cm radiation from high redshifts
39
Summary
We find that there are three important regimes for 21 cm cosmology. The
line transition is first visible in absorption against the CMB when the gas
cools adiabatically and couples the spin temperature to the gas temperature
through collisions. Second, the collisions cease to be important and at redshift 30-50 there might be a dark gap where Compton scattering with residual free electrons re-establishes equilibrium of the spin temperature with the
CMB. Once the first sources turn on, and distribute UV photons throughout
the IGM, Wouthuysen-Field coupling can again lower the spin temperature
toward the gas. Once the gas is heated by injection of X-ray radiation from
inverse Compton scattering off the photons of electrons accelerated in Supernovae and accreting binaries, the spin temperature is driven (still coupled
efficiently by the UV photons present to the gas kinetic temperature) to
temperatures far above the CMB, before the universe undergoes substantial
ionization.
Summing up the results from the previous sections, we may therefore
express the 21 cm brightness temperature fluctuation as a combination of
fluctuations in the following way
δ21 = Aδb + Ax δx + Aα δα + AT δT − δ∂v .
(2.22)
Here δi describes the fractional variation in quantity i where: i = b for
the baryonic density, i = α for the Lyα coupling coefficient xα , i = x for
the neutral fraction (note that using the ionized fraction would cause a sign
change), i = T for TK . Finally we have a term i = ∂v for the line-ofsight peculiar velocity gradient, that arises because the signal is observed in
redshift space. It is the only anisotropic term in the above expansion. The
expansion coefficients can be shown to have simple physical explanations [67].
In chapters 3 and the power spectrum calculation of chapter 5 we will
operate under the assumption that, within an extended interval before and
during reionization, we can assume that the Wouthuysen-Field coupling and
gas heating by X-rays are homogeneous. This assumption is based on the long
mean free path expected by those photons with energies above 10.2 eV. In the
next section, we will discuss the 21 cm power spectrum in this regime, which,
as theoretical arguments suggest, should be limited to a redshift interval of
roughly ∆z ' 5 [73, 74, 67] directly proceeding reionization.
We will show how the presence of large scale velocity streams breaks the
spherical symmetry of the observed (i.e. in redshift space) 21 cm power
spectrum. We can in principle use this fact to distinguish cosmological parts
of the power spectrum, i.e. those due to fluctuations in the distribution of
40
Reionization models and 21 cm radiation from high redshift
matter) from astrophysical fluctuation modes (i.e. those due to the spin
temperature and ionization fraction fluctuations).
2.3
The 21 cm Power Spectrum
The difference between the 21 cm brightness temperature at the observed
frequency ν and the CMB temperature is [72]
3c2 h A10 nH (x) a3 [Ts (x) − TCMB (z)] ∂r Tb (x) =
(2.23)
∂ν ,
32π kB Ts (x) ν0
where A10 = 2.85 × 10−15 s−1 is the spontaneous 21 cm transition rate, Ts
is the spin temperature, ν0 = 1420 M Hz, and nH is the number density of
neutral hydrogen. The factor |∂r/∂ν| accounts for the Hubble flow as well
as peculiar velocities.
Ignoring peculiar velocities, the 21 cm brightness temperature, relative
to the CMB, at observed frequency, ν, and redshift, z, is then (e.g. [75]):
Ωb h2
TS − TCMB
δT (ν) ≈ 26 xH (1 + δρ )
TS
0.022
1/2
0.15
1+z
×
mK.
(2.24)
2
Ωm h
10
In this equation, xH is the hydrogenic neutral fraction, 1 + δρ is the gas
density in units of the cosmic mean, TS is the spin temperature, and TCMB is
the CMB temperature. The other symbols have their usual meanings. In this
section, we will make the simplifying assumption that TS TCMB globally
during reionization, implying δT ∝ (1 + δρ )xH [73, 67, 74].
One way to quantify fluctuations δTb (x) in the brightness temperature is
through the power spectrum, the Fourier transform of the two point correlation function, defined through
hδ(k)δ ∗ (k0 )i = (2π)3 δD (k − k0 )Pδ (k)
(2.25)
where δ(k) is the Fourier transform of the real space contrast δ(x). To the
extend that fluctuations are Gaussian, the power spectrum offers a complete
statistical description of the field. In the case of 21 cm fluctuations, we can
at best expect this to be true in the epoch directly before reionization, where
fluctuations in the spin temperature might be negligible compared to fluctuations in the density field. During reionization the extended HII regions with
their sharply defined fronts will render the assumption of Gaussianity invalid
2.3 The 21 cm Power Spectrum
41
and we can hope to gain complementary information from statistics beyond
the power spectrum. We can further define the dimensionless power spectrum ∆2 (k) = k 3 Pδ (k)/(2π 2 ), which gives the contribution to the variance of
the field per ln(k).
Ignoring peculiar velocities for now, the 21 cm power spectrum can be
decomposed into the sum of several terms (generalizing the formula in [76]):
∆221 (k) = hTb i2 hxH i2
[∆2δx ,δx (k) + 2∆2δx ,δρ (k) + ∆2δρ ,δρ (k)
+
2∆2δx δρ ,δx (k) + 2∆2δx δρ ,δρ (k)
+
∆2δx δρ ,δx δρ (k)]
(2.26)
In this equation δx = (xH − hxH i)/hxH i is the fractional fluctuation in the
hydrogenic neutral fraction, and hTb i is the average 21 cm brightness temperature relative to the CMB. Here and throughout ∆2a,b (k) indicates the
dimensionless cross-power spectrum between two random fields, a and b.
The terms on the first line of Equation (2.26) are the usual low-order terms,
representing the power spectrum of neutral hydrogen fluctuations, the cross
power spectrum between neutral hydrogen and gas over-density, and the density power spectrum, respectively (e.g. Furlanetto et al. 2006c).
The terms on the following lines, which we can be referred to as ‘higher
order’, were the focus of a paper not discussed in this thesis [77]. They reflect
mode-couplings due to non-linear growth of structure, and can amount to
large corrections, of order 100%, to the 21 cm power spectrum.
Going to redshift space
Starting from equation 2.24, we can ask what the effect of redshift space
distortions will be. To calculate ∂r/∂ν, we relate comoving distance to frequency [78]
Z 1
c da
r=
,
(2.27)
2
ν
a H(a)
ν (1−v /c)
0
r
where vr is the l.o.s. peculiar velocity. Differentiating this expression, we
find
∂r
c
1 ∂vr
=− 2
1−
,
(2.28)
∂ν
a ν0 H
Ha ∂r
where we have dropped terms of order [(Ha)−1 ∂vr /∂r]2 and [vr /c]. In the
limit Ts TCMB , fluctuations in the 21 cm brightness temperature at x can
42
Reionization models and 21 cm radiation from high redshift
be expressed as
∆ Tb (x)
= (1 − x̄i [1 + δx (x)]) (1 + δ(x))
T̃b
1 ∂vr (x)
− x̄H ,
×
1−
Ha ∂r
(2.29)
where x̄i ≡ 1 − x̄H is the global ionized fraction, δx is the overdensity in the
ionized fraction and δ is the dark matter overdensity (on the scales and redshifts of interest, the baryons trace the dark matter), and we define the
normalized temperature T̃b ≡ T̄b /x̄H . In Fourier space, since the linear
theory velocity at redshifts where dark energy is unimportant is v(k, z) =
−i H a k δL /k 2 , the peculiar velocity term is δv ≡ (Ha)−1 ∂vr /∂r = −µ2 δL
where µ ≡ k̂ · n̂, the cosine of the angle between the wavevector and the
l.o.s., and L denotes the linear theory value.8 Keeping terms to second order
in {δ, δ L }, the brightness temperature power spectrum is
T̃b−2 P∆T (k) = x̄2H Pδδ + Pxx − 2x̄H Pxδ + Pxδxδ
+ 2µ2 x̄2H PδL δ − x̄H PxδL + µ4 x̄2H PδL δL
+ [2Px δ δv x + Px δv δv x ] ,
(2.30)
noting that Pxx = x̄2i Pδx δx and Pxδ = x̄i Pδx δ . In our calculations, we drop
2
the connected part and set Pxδxδ = Pxδ
+ Pxx Pδδ . In equation 2.30, we have
decomposed the power spectrum into powers of µ; the last bracket in this decomposition has a non-trivial dependence on µ. For notational convenience,
we refer to the k-dependent coefficients in equation (2.30) as Pµ0 , Pµ2 and
Pµ4 and the terms in the last bracket as Pf (µ,k) . The above decomposition
should allow one to extract the “physics” – PδL δL – from the “astrophysics”–
Pxx and Pxδ [80, 76]. The terms in the last bracket in equation (2.30) were
omitted in their analysis, but must be included if reionization is patchy because δx ∼ 1 on scales at or below the bubble size. The disconnected part
was calculated in [80].
The evolution of the ionized fraction over a mode can also affect the
spherical symmetry of P∆T , since time is changing in the l.o.s direction but
not in the angular directions. The magnitude of this effect depends strongly
on the morphology of reionization and we do not discuss it in this thesis (but
see [81]).
The 21 cm background directly measures the baryonic density field δb (or
even more precisely, the hydrogen density field). For most purposes, this
The velocity field at z ∼ 10 is in the linear regime for k ≤ 5 Mpc−1 . See [79] for
a discussion of the effect of the non-linear velocity field on the 21 cm signal. Upcoming
interferometers are most sensitive to scales where the velocity field is linear.
8
2.3 The 21 cm Power Spectrum
<xi> = .1
<xi> = .7
2
10
[k3 P(k)/2 π2] (mK2)
43
µ0
µ4
4
µ
1
10
µ2
µ2
0
10
µ0
−
+
−1
10
−2
10
−1
10
0
10
k (Mpc−1)
1
10
−1
10
0
10
1
10
k (Mpc−1)
Figure 2.9 The µ decomposition of the signal (see equation 2.30) for x̄i = 0.1
and 0.7, corresponding to z = 13.5 and 9 in the ζ = 12 model. The thick
solid, dashed and dot-dashed are Pµ0 , Pµ2 , and Pµ4 , respectively. The three
thin solid curves are the connected part of the terms in the last bracket of
2.30, calculated with µ2 = 0.0, 0.5 and 1.0 (in order of increasing amplitude).
is equivalent to the total matter density δ and in the following we will set
δb = δ throughout. However, note that on small scales the finite pressure of
the baryons introduces a cutoff absent from the dark matter [82]; in detail,
galaxy formation processes and feedback can also work on the two separately.
To the extent that the matter fluctuations are probed we may try to extract
cosmological information from the 21 cm power spectra. We will present
results of such an analysis in the next section.
In Figure we show the different power spectrum components in equation
2.30, labeled by their characteristic dependence on µ. We used the model
described in Section 2.1 to describe fluctuations in the ionized fraction. The
µ4 term is simply proportional to the density power spectrum, and we will use
this feature below to constrain cosmological parameters. At high ionization
fractions, the astrophysical terms, in particular the µ0 -term, can become
large, as shown in the left panel.
44
Reionization models and 21 cm radiation from high redshift
2.4
Using the 21 cm power spectrum to constrain cosmological parameters
We can now turn to predictions of cosmological parameter constraints with
future 21 cm experiments. A number of groups are ramping up to measure the
reionization signal, among them the Mileura Wide Field Array (MWA) 9 , the
PrimeavAl Structure Telescope (PAST), and the Low Frequency Array (LOFAR) 10 , while another second generation experiment, the Square Kilometer
Array (SKA)11 is being conceived. We will here focus on the MWA, LOFAR,
and the SKA, for which experimental specifications are available 12 . Some
specifications that are relevant for calculating sensitivities are shown in 2.1.
The parameters we adopt come from [83] for MWA, [84] and www.lofar.org
for LOFAR, and [85] for SKA.
LOFAR will have 77 large “stations,” each of which combines the signal
from thousands of dipole antennae to form a beam of ≈ 10 square degrees.
Each station is able to simultaneously image Np regions in the sky. We set
Np = 4 in our estimates, but this number may be higher. The signal from
these stations is then correlated to produce an image. In contrast, MWA will
have 500 correlated 4 m × 4 m antenna panels, each with 16 dipoles. This
amounts to a total collecting of 7000 m2 at z = 8, or 15% of the collecting
area in the core of LOFAR. While correlating such a large number of panels
is computationally challenging, this design gives MWA a larger field of view
(f.o.v.) than LOFAR, which is an advantage for a statistical reionization
survey. The properties of the SKA have not yet been finalized, and it is quite
possible that the EOR science driver for SKA may form a distinct array from
the other, higher-frequency drivers. In addition, the successes of MWA and
LOFAR will likely influence the final design of SKA. The collecting area for
SKA is projected to be roughly 100 times larger than that of MWA. There
are a number of competing designs for SKA’s antennae. In one case, SKA
will have roughly 5000 smaller antennae (like a much larger MWA).
Details of the sensitivity calculation are discussed in [80]. There we also
derive the optimal antenna distribution for measurements of the EoR 21 cm
signal. Figure 2.10 shows the result of this calculation, as statistical error
in [k 3 P∆T (k)/2π 2 ], for MWA (dashed), LOFAR (dash-dotted), and SKA
(solid). Even though P∆T (k) is not spherically symmetric, we spherically
average P∆T as well as the errors for the purpose of this plot. Because of
9
http://web.haystack.mit.edu/arrays/MWA/
http://www.lofar.org
11
http://www.skatelescope.org/
12
In the case of SKA, the specifications we use should be seen as tentative etimates
10
2.4 Using the 21 cm power spectrum to constrain cosmological
parameters
45
this averaging, the interferometers will be slightly more sensitive to some
modes than this plot implies. At z = 6, the trend is as expected: SKA is
more sensitive than LOFAR and LOFAR is more sensitive than MWA. Still,
LOFAR’s gains over MWA are not proportional to the square of the collecting
area, a result of its smaller field of view. At higher redshifts, LOFAR and
MWA are comparably sensitive on most scales. We also plot the sensitivity of
MWA at z = 8 for a flat rather than the fiducial r−2 distribution of antennae.
In this case, MWA is substantially less sensitive at all scales. This contrasts
with angular power spectrum measurements, where a flat distribution of
antennae is always more sensitive at larger k than a tapered distribution.
For these three arrays, the system temperature is dominated by the sky
temperature. In our calculations, we set Tsys = Tsky = 250 K at z = 6,
Tsys = 440 K at z = 8 and Tsys = 1000 K at z = 12 [83], and we set B = 6
MHz bandwidth, which translates to a conformal distance of 100 Mpc at
z = 8.
All the significant foreground contaminants should have smooth powerlaw spectra in frequency. Known sources of radio recombination lines are
estimated to contribute to the fluctuations at an insignificant level [20]. Before fitting a model to the cosmological signal, it is necessary to clean the
foregrounds from the data. The idea is to subtract out a smooth function
from the total signal prior to the parameter fitting stage [86]. Such preprocessing is common with CMB data sets, and [47] showed that this procedure can also be used in handling 21 cm observations. We will not go into
detail here but refer the reader to [80]. The upshot is that in the Fisher
analysis, the results of which are shown in Table 2.2, we only used modes
, where B = 6 MHz is
larger then (scales smaller then) a minimum kmin = 2π
B
the frequency depth of the observation. This has been chosen to minimize
foreground contamination and evolution effects of the signal (the density field
and ionization fraction) throughout the data volume.
To obtain an estimate, for how well our set of reionization parameters
can be determined with planned experiments, we employ the Fisher matrix
formalism. The Fisher or curvature matrix is defined to be expectation value
of the second derivative of the natural logarithm of the likelihood function
around its maximum for small parameter deviations:
Fij = h
δ 2 (lnL)
i,
δθi δθj
(2.31)
It provides us with a rough guess for the shape and width of the likelihood
function in the multi-dimensional parameter space. The covariance matrix
between the parameters M is given by the inverse of the Fisher matrix. This
46
Reionization models and 21 cm radiation from high redshift
2
10
0
k3 P
∆T
(k)/2 π2 (mK2)
10
−2
z=6
10
2
10
0
10
−2
10
2
10
z=8
−2
−1
0
1
0
10
−2
z = 12
10
−2
10
−1
0
10
10
1
10
−1
k (Mpc )
Figure 2.10 Detector noise plus sample variance errors for a 1000 hour observation on a single field in the sky, assuming perfect foreground removal, for
MWA (thin dashed curve), LOFAR (thin dot-dashed curve), and SKA (thin
solid curve) using the specifications given in Table 2.1 and for bin sizes of
∆k = 0.5 k. These errors are for the spherically averaged signal. The hashed
line in the middle panel is for MWA with a flat distribution of antennae
rather than the fiducial r−2 distribution. The detector noise dominates over
sample variance for these sensitivity curves on almost all scales. The thick
solid curve is the spherically averaged signal for x̄i 1 and Ts Tcmb . We
use this curve to calculate the sample variance error. For comparison, the
thin dashed curves are the signal from the analytic model [50] when x̄i is
equal to 0.20, 0.55 and 0.75 for z = 12, 8 and 6, respectively. The foreground
cutoff is not shown here, instead some experiments such as LOFAR have a
sharp cutoff because of the size of their minimum baseline (compare Table
2.1).
2.4 Using the 21 cm power spectrum to constrain cosmological
parameters
47
Array
Nant
MWA
500
LOFAR 64
SKA
5000
Nant Ae (m2 )
at z = 8
7000
4.2 × 104
6.0 × 105
f.o.v. (deg2
at z=8)
π 162
4 × π 2.02
π 5.62
min. baseline (m)
4
100
10
Cost
(106 $)
∼ 10
∼ 102
∼ 103
Table 2.1 Survey parameters for upcoming 21 cm experiments: MWA, LOFAR, and SKA. Here we optimized the design for SKA for observations of
the EOR, while keeping the current gross specifications for this array. Values
for 103 hours of observation with B = 6 MHz at 150 MHz. Due to its high
survey speed, MWA will observe the largest field of these three experiments.
means that the one sigma error in the determinability of the parameters can
be approximated by
−1/2
(2.32)
σi = Fij
and that the correlation coefficient between the different parameters is
rij = Mij /(σi σj )
(2.33)
where the Mij are the elements of the parameter covariance matrix. Given
an estimate for the 21 cm power spectrum and measurement sensitivity, 1σ errors in the cosmological parameters λi are given in terms of the Fisher
matrix
X
∂P∆T (k) ∂P∆T (k)
1
.
(2.34)
Fij =
2
(δP
∂λ
∂λ
∆T (k))
i
j
k
The calculation in Table 2.2 assumes Gaussian variance of the signal,
which is appropriate before reionization where we expect to encounter a
Gaussian random field on large scales. During reionization, the ionization
fraction fluctuations on the scale of the HII bubbles will not be Gaussian.
They will lead to a non-Gaussian signal covariance matrix through which
the connected four-point function will effectively correlate bands and will
increase the diagonal elements. However especially for the first generation of
experiments, the noise term will be dominating.
During reionization, we could also make use of the decomposition 2.30
to get at the cosmological part of the signal. We showed in [80] that this
is difficult in practice, basically because foreground contamination limits the
number of modes that can be extracted along the line of sight. However it
is these modes that contribute most to the µ4 term Pδ which we would like
to measure to get at the purely cosmological PδL . Hence we will concentrate
(in Table 2.2) on the regime before regionization, where the 21 cm power
48
Reionization models and 21 cm radiation from high redshift
spectrum is simply the density power spectrum boosted by the constant
‘Kaiser factor’ of h(1 + µ2 )2 i = 1 + 1 + 2/3 + 1/5 ' 1.87.
We find that the first generation of 21 cm observations should moderately improve existing constraints on cosmological parameters for certain lowredshift reionization scenarios, and a two year observation with the second
generation interferometer MWA5000 in combination with the CMB telescope
Planck can improve constraints on Ωw (to ±0.019, a 30% improvement over
Planck alone), Ωm h2 (±0.0011, 50%), Ωb h2 (±0.00013, 30%), Ων (±0.003,
300%), ns (±0.0034, 30%), and αs (±0.004, 100%). Larger interferometers,
such as SKA, have the potential to do even better.
Ωw
w
Ωm h2
Ωb h2
ns
δH × 105
0.7
-1.0
0.14
0.022
1.0
3.91
0.09
0.14
0.04
0.14
3.9
0.10
0.13
0.03
0.13
3.9
0.007
0.011
0.003
0.03
0.31
0.005
0.009
0.002
0.04
0.26
0.029 0.09 0.0023 0.00018 0.0047
0.026
0.019 0.07 0.0011 0.00013 0.0034
0.018
0.018 0.07 0.0009 0.00013 0.0033
0.018
αs
0.0
0.012
0.011
0.008
0.004
0.003
Ων
x̄H
0.0
1.0
0.008
0.009
0.010
0.003 0.05
0.004 0.05
Table 2.2 Errors on cosmological parameter estimates when density fluctuations dominate the 21cm signal for two year
observations with 21 cm interferometers and in combination with Planck. We assumed, unless otherwise noted, observations
of 2000 hrs on two places in the sky in a 6M Hz band which is centered at z = 8. In these calculations, we account for
foregrounds by imposing a sharp cutoff in sensitivity at k = 2π/B, where B is the width of the box, and we avoid fitting to
scales in the non-linear regime by imposing a small scale cutoff at k = 2 Mpc−1 . The calculations are for a flat universe,
1 = Ωm + Ωw , and dashes indicate parameters which are not marginalized. MWA5000 is a conceived extension of the MWA,
which builds on the design layout by increasing the number of receivers by a factor of ten. We point out that from just the 21
cm data, the parameter δH is completely degenerate with x̄H . Because of this, for 21 cm observations alone, the constraints
in this column are really for the parameter x̄H δH . Predictions for SKA are for ten locations on the sky, 400 hours each.
Fiducial param. value
LOFAR
MWA
MWA5000
SKA
Planck
Planck +MWA5000
Planck + SKA
τ
0.1
0.0050
0.0046
0.0046
2.4 Using the 21 cm power spectrum to constrain cosmological
parameters
49
50
Reionization models and 21 cm radiation from high redshift
Summary
In comparison to the other reionization probes mentioned in the introduction,
the 21 cm transition has the great advantage of offering to probe the high
redshift IGM tomographically, by making three dimensional maps. The line
transition can also be used to probe the pre-reionization IGM. The dataset
that can be acquired this way is – e.g. in comparison to the CMB – potentially
very large. Let us imagine measuring an angular mode l, corresponding to a
wave number k ' l/r. Two maps at different frequencies will be independent
if they are separated by a radial distance 1/k. An experiment covering a
spatial range ∆r can probe a total of k∆r ' l∆r/r independent maps. A
21 cm experiment taking data over a range ∆ν centered on frequency ν is
then sensitive to ∆r/r ' 0.5(∆ν/ν)(1 + z)−1/2 . The number of data points
acquired by such an experiment is (say it is centered at z=13, and measures
fluctuation in the regime z=6-27 so that ∆ν = ν)
N21cm ' 1017 (lmax /106 )3 (∆ν/ν)(z/13)−1/2
(2.35)
independent samples, where the multipole number l corresponds to a mode
of wavelength λ = (2π)/(l/r). Of course this number has been derived under
the extremely optimistic assumption of frequency and angular resolution such
that fluctuations can be observed all the way down to the Jeans smoothing
scale of l ' 106 , but it demonstrates the point (made originally by [60]). In
2
comparison, the CMB contains only Ncmb ' 2lmax
' 2 × 107 (lmax /3000)2
Fourier independent data points in both temperature and polarization. The
limit in this case is Silk damping [87] (this process is responsible for the
exponential decay of power in the CMB on scales l ' 3000; it is due to
photon diffusion in the last scattering surface, because recombination is not
a completely instantaneous process).
Even if the amount of data in 21 cm will be larger than ongoing CMB analyses, the challenge will lie in extracting as much information about physics as
possible from it. We would expect that the 21 cm signal from the considered
range of redshifts will contain – beyond information about the fluctuations
of baryons or matter – information about fluctuations in the UV background
and about inhomogeneous X-ray heating [74]. Most importantly though, it
will contain information about fluctuations in the ionized fraction which we
would like to use to infer knowledge about the nature of the sources responsible for this phase transition of the IGM.
Chapter 3
Lensing Reconstruction using
redshifted 21 cm fluctuations1
In this chapter we investigate the potential of second generation measurements of redshifted 21 cm radiation from before and during the epoch of
reionization (EOR) to reconstruct the matter density fluctuations along the
line of sight through the gravitational lensing effect. To do so we generalize
quadratic methods developed for the Cosmic Microwave Background (CMB)
to 21 cm fluctuations. We show that the three dimensional signal can be
decomposed into a finite number of line of sight Fourier modes that contribute to the lensing reconstruction. Our formalism properly takes account
of correlations along the line of sight and uses all the information contained
in quadratic combinations of the signal.
In comparison with the CMB, 21 cm fluctuations have the disadvantage
of a relatively scale invariant unlensed power spectrum which suppresses the
lensing effect. The smallness of the lensing effect is compensated by using
information from a range of observed redshifts.
We estimate the size of experiments that are needed to measure this effect. With a square kilometer of collecting area and a maximal baseline of 3
km one can achieve lensing reconstruction noise levels an order of magnitude
below CMB quadratic estimator constraints at L = 1000, and map the deflection field out to less then a tenth of a degree (L > 2000) within a season
of observations on one field. Statistical lensing power spectrum detections
will be possible to sub-arcminute scales, even with the limited sky coverage
that currently conceived experiments have. One should be able to improve
constraints on cosmological parameters by using this method. With larger
1
Based in part on O. Zahn & M. Zaldarriaga, Astrophysical Journal, 653, in press
(2006)
51
Lensing Reconstruction using redshifted 21 cm fluctuations1
52
collecting areas or longer observing times, one could probe arcminute scales of
the lensing potential and thus individual clusters. We address the effect that
foregrounds might have on lensing reconstruction with 21 cm fluctuations.
3.1
Introduction
In Chapter 2 we discussed measurements of the 21 cm radiation from before
and during reionization as a potential probe of cosmological parameters. We
found that due to the confounding effects of their unknown astrophysics it
might be difficult to compete with the constraints on cosmological parameters (see Chapter 2 and [80]) coming from future microwave background
experiments. A few very ambitious probes on the CMB side are the Planck
satellite 2 , the Atacama Cosmology Telescope (ACT) 3 [88], or the South
Pole Telescope (SPT) 4 [89]).
Here we will go a different route, and explore the cosmological information
contained in the 21 cm measurements about the intervening mass distribution
at lower redshifts through the lensing effect. Before the ionized fraction of
the IGM becomes substantial, the 21 cm emission against the CMB is a near
Gaussian random field. In this regime, which is the focus of the present
article, the quadratic estimator should be close to optimal [90].
We will look at 21 cm fluctuations at their lowest possible redshift range,
z = 6−12, where the second generation of experiments (such as the SKA and
MWA50k described in Chapter 2) might be able to measure at high signalto-noise. Using our formalism we can also estimate information losses due to
foreground contamination, once these have been described by some model.
We also explore the possibility to constrain the dark energy density with
lensing of the 21 cm background. Another application would be to measure
nonlinearities in the density field, or the contribution of neutrinos to the
energy density of the universe. We compare our results to the potential of
a future high precision observation of the CMB. The CMB damping tail is
an advantage for lensing reconstruction since reconstruction errors decrease
with increasing slope, but at the same time it leads to a small scale limitation
for CMB reconstruction, as the signal quickly falls below the noise.
Lensing reconstruction using the redshifted 21 cm radiation in absorption
against the CMB has been investigated before by [91], in particular also the
possibility to get a handle on gravity waves from inflation by gravitational
lensing cleaning of B mode polarization (lensing converts E to B modes of the
2
http://www.rssd.esa.int/index.php?project=PLANCK&page=perf top
see http://www.hep.upenn.edu/ angelica/act/act.html
4
see http://astro.uchicago.edu/spt/
3
3.1 Introduction
53
polarization and acts as a contaminant to the primordial signal) [92]. Their
work describes two types of observations, which employ a 20/200 times larger
total collecting area than we will assume here (and five times longer observation time) to observe angular fluctuations Lmax = 5000 − 105 in the 21 cm
field. The problem is that the prospect, beating the CMB level for B mode
lensing cleaning, relies on measuring the 21 cm power spectrum at very high
redshifts (they use zsource = 30), at which the galactic synchrotron contamination is a factor ' 20 larger than for example at redshift 8. The reason
such high redshift observations are needed to compete with likelihood based
lensing estimation is that there is a partial delensing bias when comparing
lenses out to different redshifts and zs = 30 turns out to be close enough
to the last scattering surface of the CMB, where gravity waves are expected
to create the B mode fluctuations. These authors furthermore make the approximation of treating their slices through the 21 cm measurement cube as
uncorrelated, which is not warranted.
Reference [93] suggested measuring the effective convergence from the
effect it has on the real space variance map of 21 cm fluctuations. The author
also presented rough sensitivity estimates for LOFAR, PAST, and SKA. We
improve on this by using all the available information in convergence and
shear in an optimal way and use more realistic errors.
In section 2 we review the quadratic estimator technique for lensing reconstruction following [94]. Then we naturally expand the formalism to the
extraction of weak lensing from an intrinsically three dimensional signal. The
decomposition of the line of sight component of the signal into modes leads
to a hierarchy of independent lensing backgrounds that can be probed with
varying precision. Although our concentration lies on applying the quadratic
estimator to the epoch immediately before substantial ionization occurs, we
scrutinize its applicability to the patchy regime by using an analytic model
for the morphology of HII regions.
In section 3 we put our investigation in the context of experiments, estimating the potential redshift range in which they will observe. Because
signal and noise are evolving with redshift, we break down the volume of the
observation in smaller boxes along the line of sight. We calculate estimates
for the lensing reconstruction future 21 cm experiments might be able to
achieve.
We present results based on current rough specifications of the Square
Kilometer Array (SKA) in Section 4 and compare them to the possibility of
constraining the matter power spectrum with the CMB temperature and polarization. The 21 cm approach turns out to have no angular scale limitation
for constraining the convergence. We address limitations due to the galactic
foreground and show how these can be incorporated into our formalism in
54
Lensing Reconstruction using redshifted 21 cm fluctuations1
a straightforward manner. We conclude with an outlook and discussion in
section 5.
We discussed the theory of the 21 cm signal in Chapter 2. Here we will
assume that in the regime of interest f ' 1, hence we obtain with Ts TCMB
that the power spectrum of the 21 cm brightness temperature fluctuation in
the neutral regime is given by
1+z
(1 + µ2k )2 Pδ (k)
(3.1)
10
In this chapter, we shall be mainly concerned with quadratic estimator lensing reconstruction during this highly neutral regime. The extension to a
patchy epoch is complicated by the presence of a connected four-point function contribution to the source field. On the level of the power spectrum this
contribution acts as a sample variance term, correlating different k-modes.
Figure 3.1 shows the angular power spectrum of 21 cm fluctuations at
redshift 8 for different values of the efficiency and therefore ionization fraction. We see that up to an ionization fraction of xi = 0.7 the signal on small
scales decreases (structures below the bubble scale are washed out) while
the ionized regions lead to a bump in the power spectrum on scales k ' 0.2
h/Mpc which corresponds to an angular multipole at the relevant redshifts of
approximately l = 1000. At even higher ionized fractions, the entire patchy
power spectrum lies below the neutral case, so it becomes more difficult to
observe the fluctuations and hence to use them for lensing reconstruction.
In any event it seems from Figure 3.1 that a significant part of the patchy
regime can be used for the lensing reconstruction, in part because the boosted
amplitude on scales l ' 100 − 1000 might aid somewhat in this effort. We
will evaluate these estimates more carefully in Section 3.4. Our choice of
ionization efficiency lets reionization begin at redshifts around 8 and evolves
rather slowly, due to photon consumption by clumps in the IGM. It evolves
through xi ' 50% at redshift 7 to complete at redshift 6.
We will assume a bandwidth of B = 5 MHz for calculating the power spectrum at various redshifts. This corresponds to a redshift interval ∆z = 0.286.
During the neutral phase the density fluctuations evolve slowly, however the
sensitivity of the experiments change rapidly with observation frequency. Because the comoving length scale is given in terms of bandwidth through
1/2 −1/2
1+z
Ωm h2
B
Mpc/h,
(3.2)
L ≈ 1.2
0.1MHz
10
0.15
P∆Tb (k) ' (26mK)2
our window in frequency space corresponds to a depth of 50 Mpc/h at z=6
and to 70 Mpc/h at z=12. When the first extended HII regions start forming, the power spectrum evolves more rapidly, however reionization still only
3.1 Introduction
55
Cl21cm l (l+1) /2π, Clnoise l (l+1) /2π (mK2)
10
1
0.1
0.01
100
xi=0.00
xi=0.17
xi=0.27
xi=0.38
xi=0.51
xi=0.67
flat D=2 km
1000
l
Figure 3.1 Angular power spectrum of the 21 cm signal at various stages
during reionization. The thick solid line shows the expected noise for SKA
for a flat configuration of antennas inside a circle of diameter 2 km. When the
ionization fraction rises during the expansion of HII spheres, first the overall
signal decays, then bubbles cluster quickly to form large ionized regions tens
of Mpc/h across. This can lead to a significant increase in the signal on
scales that the first imaging 21 cm experiments should be sensitive to. On
the smallest scales the signal is decreased, making usage of 21 cm fluctuations
for lensing reconstruction somewhat more difficult.
56
Lensing Reconstruction using redshifted 21 cm fluctuations1
lapses over a comoving length of about 300 Mpc/h, in comparison to which
our window is small.
We will generate 21 cm brightness temperature power spectra following
Formula 3.1 using linear power spectra that contain the acoustic oscillation
amplitude generated from CMBFAST transfer functions [56]. The baryonic
wiggles included in the code might aid somewhat in our reconstruction endeavour, for reasons given in the next section. We use the same transfer
functions to implement the signal in three dimensional Gaussian random
fields and model the patchy phase.
A ΛCDM cosmology is assumed throughout all calculations, with parameters Ωm = 0.3, ΩΛ = 0.7, Ωb = 0.04, H0 = 100hkm/sec/Mpc (with h=0.7),
and a scale-invariant primordial power spectrum with n = 1 normalized to
σ8 = 0.9 at the present day.
3.2
3.2.1
Weak Lensing Reconstruction
Quadratic Estimator, General Consideration
Lensing by large scale structure can be observed whenever there is a fluctuating background field. At position n̂ we observe the field
T (n̂) = T̃ (n̂ + ∇φ)
(3.3)
where T̃ denotes the unlensed field, δθ = ∇φ is the displacement vector while
φ is the projected potential. Here and in what follows, boldface quantities
denote vectors. The projected potential is given in terms of the gravitational
potential ψ(x, D), where x is position and D, the angular diameter distance,
is used as time variable, as
Z
DA (Ds − D)
ψ(Dn̂, D) .
(3.4)
φ(n̂) = −2 dD
DDs
So the power spectrum of the displacements is CLδθδθ = L(L + 1)CLφφ .
An estimator D(n̂) for the lensing displacement field information contained in the temperature field should contain an even number of temperature terms since the expectation value for odd powers would vanish. It must
also satisfy the condition
hD(n̂)i = δθ(n̂)
(3.5)
when averaged over many realizations of the background radiation field, that
is for example the CMB or 21 cm radiation.
3.2 Weak Lensing Reconstruction
57
[95] showed that the divergence of the temperature-weighted gradient of
the map achieves maximal signal-to-noise among quadratic statistics. In
Fourier space this quadratic estimator takes the form [96]
Z
d2 l
Dest. (L) = A(L)
F (l, L − l)T (l)T (L − l)
(3.6)
(2π)2
The Filter F (l, L − l) is obtained by minimizing the variance of Dest. under
the normalization condition for D(L)
F (l, L − l) =
and the normalization is
"Z
A(L) = L
2
[Cl L · l + CL−l L · (L − l)]
tot
2C̃tot
l C̃L−l
d l [Cl L · l + CL−l L · (L − l)]
tot
(2π)2
2 C̃tot
l C̃L−l
(3.7)
#−1
(3.8)
is the sum of the lensed angular power spectrum of 21 cm flucHere C̃tot
l
tuations C̃l and the noise power spectrum which we will give in the next
section, ClN . As shown by [97], the effect of lensing on the angular 21 cm
power spectrum for an individual plane is small, so one can use the unlensed
power spectrum in place of it. The high-pass shape of the quadratic estimator gives it a property all reconstruction methods share, that they extract
most information from the smallest scales resolved by some experiment. We
emphasize that the estimator is unbiased by construction (Equation 3.5),
independent of Gaussianity of the source field.
With the definition of the lensing reconstruction noise power spectrum
NLD
hD∗ (L)D(L0 )i = (2π)2 δ(L − L0 )(CLDD + NLD )
(3.9)
evaluation of the variance of Equation 3.6, h||D(L)||2 i = (2π)2 δ D (0)NLD gives
#−1
"Z
2
2
d l [Cl L · l + CL−l L · (L − l)]
NLD (L) = L2
(3.10)
tot
(2π)2
2 C̃tot
l C̃L−l
= A(L)L
(3.11)
Roughly, when CLDD = NLD , then structures down to the angular size 2π/L
in the lensing field can be reconstructed.
We illustrate the different power spectrum characteristics of CMB and
21 cm in Figure 3.2, where we re-scaled the 21 cm power spectrum to be
of comparable amplitude. While the CMB fluctuation signal exhibits more
Lensing Reconstruction using redshifted 21 cm fluctuations1
58
10-9
Planck TT noise
10-10
Cl l (l+1)/(2π)
21cm/(1.5 107)
10-11
CMB temperature
Ref. Exp. TT/EE noises
CMB polarisation
thermal SZ, 150 GHz
10-12
kinetic SZ+OV
-13
10
10-14
100
1000
10000
l
Figure 3.2 CMB temperature and E polarization power spectrum and noise
levels of our example experiments: Planck and a futuristic polarized satellite
mission (see text for specifications). The 21 cm power spectrum has been
rescaled to fit into the same plot. The CMB exhibits more pronounced peaks
and a sharp decay on scales l ' 2500 above which the Sunyaev-Zel’dovich
effects become important.
features on the scale of the acoustic oscillations in the pre-recombination
baryonic plasma, the primordial signal decays sharply on scales above l '
2500. The 21 cm angular power spectrum, in contrast, does not decay very
strongly.
We can investigate qualitatively which properties of the power spectrum
set the effectiveness of this estimator by looking at the limit L → 0 (on
scales where the noise is small and the lensed power is not systematically
3.2 Weak Lensing Reconstruction
59
= C̃l + CN
larger than the unlensed, C̃tot
l ' C̃l ' Cl )
l
Z
1
d2 l [L · lCl + L · (L − l)CL−l ]2
2 D −1
=
(L NL )
tot
L4
(2π)2
2C̃tot
l C̃L−l
Z
dl l2 1 1
3
'
( + α + α2 ) +
l 2π 2 2
16
Z
2
dl 1
+L2
g(α, β, γ)
(2π)2 l2
(3.12)
where
g(α, β, γ) =
3(5β + 8γ) + α(α(2 + α)(14 + 5α + 20γ))
,
384
2
(3.13)
3
lnCl
d lnCl
l
, β = ddlnl
and α ≡ dlnC
2 , γ = dlnl3 . The integrand of the first order term
dlnl
has its minimum close to a value α = −2, in other words when l2 Cl has
no slope. This implies that the CMB with its exponential decay on small
scales will have a lower value of NLD if those scales are resolved. Integrating
Equation 3.12 to the maximum multipole lmax at which ClS > ClN leads to
(L2 NLD )−1 '
1 1 1
3
l2
( + α + α2 ) max
2π 2 2
16
2
L2
+ g(α, β, γ)ln(lmax /lmin ) ,
2π
(3.14)
(3.15)
where lmin is the lower bound of the integration. The second term will lead
to departures from a constant value of (L2 NLD ). The scale at which both
terms are comparable is
s
3 2
1 + 21 α + 16
α
lmax
Lcomp. '
.
(3.16)
ln(lmax /lmin )1/2
g(α, β, γ)
Evaluation of this expression shows that for a sloped power spectrum l2 Cl ,
such as that of the CMB, Lcomp. is lower than for a spectrum that is nearly
constant, such as that of 21 cm fluctuations. The second order term contributes strongly when the slope is large. If the measured power spectrum
has a small slope, as is the case with 21 cm fluctuations, L2 NLD can be expected to be nearly constant to the scale lmax where noise becomes important.
In the following section we will generalize the quadratic estimator to a
three dimensional observable that is used to reconstruct the lensing field.
60
Lensing Reconstruction using redshifted 21 cm fluctuations1
The final estimator is then simply the sum of estimators of modes of the
wave-vector kk along the line of sight. This allows one to improve the constraints above those by the CMB especially on small scales by summing over
sufficiently many lensing backgrounds.
For our analysis in Section 3.3 we will compute the angular power spectrum of the lensing displacements as the integral over line of sight
CLδθδθ
9H04 Ω20
=
L(L + 1)c2
Z
0
Ds
2
Dls
dDl
×
Ds a(Dl )
L
×Pδ (k =
, Dl )
Dl
(3.17)
over the power spectrum of mass fluctuations. Dx denote angular diameter
distances with x = l, s for lens and source.
We use the halo model fitting function to numerical simulations of [98]
to generate nonlinear Λ CDM power spectra as input for the deflection angle
integral.
3.2.2
Extension to a three dimensional signal
Different from the CMB, in the case of 21 cm brightness fluctuations we will
be able to use multiple redshift information to constrain the intervening matter power spectrum. One could imagine applying this estimator to succesive
planes prependicular to the line of sight. However the different planes would
be correlated and there is no straightforward criterion to take this correlation
into account in establishing the final estimator, since it would depend on redshift, source properties, and during reionization on the mean bubble size and
distribution. Instead we will use the knowledge of the three-dimensional information in a different way, dividing the temperature fluctuations in Fourier
space in fluctuations k⊥ perpendicular to the line of sight, and a component
kk in the frequency direction.
We devide a volume on the sky to be probed by a given 21 cm survey into
a solid angle dΩ and a radial coordinate z. Components of wavevectors along
the line of sight are described by kk and those perpendicular to the line of
sight are given by the vector k⊥ , which is related to the multipole numbers
of the spherical harmonic decomposition on the sky as l = k⊥ D, where D is
the angular diameter distance to the volume element we are probing.
Suppose we want to measure a field I(r) with power spectrum P (k) =
P (k⊥ , kk ). Converting k⊥ to angular multipole
3.2 Weak Lensing Reconstruction
61
d3 k
I(k)eik·r
(2π)3
Z
Z
dkk I(k⊥ , kk ) i(l·θ+kk z)
d2 l
=
e
2
(2π)
2π
D2
Z
I(r) =
˜ kk ) =
with I(l,
I(k⊥ ,kk )
D2
(3.18)
we have
˜ kk )I(l
˜ 0 , kk0 )i = δ D (k − k0 )(2π)3 P (k⊥ , kk )
hI(l,
D4
= (2π)2 δ D (l − l0 )(2π)δ D (kk − kk0 ) ×
P (k⊥ , kk )
×
D2
(3.19)
where in the second step a factor of D2 got absorbed into δ D (l − l0 ) because
of l = k⊥ D.
Let us discretize the z direction of a real space observed volume with
radial length L,
kk = j
so that
Z
I(r) =
L
)
2π
(3.20)
d2 l X I(k⊥ , kk ) ik·r
e
(2π)2 j
D2 L
(3.21)
2π
,
L
It makes sense to define
δ D (kk − kk0 ) = δj1 j2 (
I(k⊥ , kk )
Iˆ ≡
D2 L
(3.22)
so that on the sphere we have
hIˆj1 (l1 )Iˆj∗2 (l2 )i = (2π 2 )δ D (l1 − l2 )δj1 j2
P (k, µk )
D2 L
(3.23)
where µk is the cosine between the wave vector and the line of sight.
We have for the angular power spectrum for separate values of j that
(including redshift space distortions)
p
2
2
2 2 P ( (l/D) + (j 2π/L) )
Cl,j ≡ (1 + µk )
,
(3.24)
D2 L
where P now represents the spherically averaged power spectrum. We have
also introduced the notation Cl,j , denoting the power in a mode with angular
component l and radial component kj = j 2π/L.
62
Lensing Reconstruction using redshifted 21 cm fluctuations1
We show in the Appendix that because modes with different j can be
considered independent, the best estimator can be obtained by combining
the individual estimators for separate j’s without mixing them (in the sense
of making quadratic combinations of them). As long as the IGM is not substantially ionized, the assumption of Gaussianity is justified at the redshifts
of interest, z ' 6 − 12, where non linearites in the gravitational clustering
are small on observed scales of several Mpc/h. We find the three dimensional
lensing reconstruction noise defined by
hD(L)D∗ (L0 )i = (2π)2 δ(L − L0 )(CLDD + NLD )
(3.25)
to be (Equation 3.56 of the Appendix, where e.g. Cl → Pl )
NLD = P
1
k L2
1
R
1
−1
k NL,k
= P
[Cl,k L·l+CL−l,k L·(L−l)]2
tot
(2π)2
2C̃tot
l,k C̃L−l,k
d2 l
(3.26)
(3.27)
where in the last line we have just substituted the standard expression, Equation 3.11. Note that analogous to the CMB case, C̃tot
= C̃l + ClN , where ClN
l
is the noise power spectrum, and C̃l ' Cl in the case of 21 cm fluctuations.
If there is a connected four point function contribution during the epoch
of extended HII regions, this will add a term to the variance of the estimator.
This will change the weightings AL as well, and make the estimator suboptimal. We will not try to come up with an estimator that is fully compatible
with the non Gaussian signal due to patchy reionization. Instead, in the next
section we will conservatively estimate that sensitivities get worse by a factor
√1 , where xH is the ionized fraction. In other words, we will effectively be
xH
treating the patchy regions as part of the source field which can be masked.
Let us examine the contribution of the components Cl,j to the total noise
in the estimation of the deflection field. From Equation 3.24 we see that
with higher values of the component kk, higher values of the three dimensional power spectrum P (k) will translate into each Cl value. But P (k) is
monotonically falling on all scales of interest (foregrounds can be expected
to contaminate constraints below k < 10−2 h/Mpc, the scale of the horizon
at matter radiation equality and the turnover of the power spectrum), so
that effectively the angular power spectrum amplitude will drop with j towards the noise level. We show this in Figure 3.3 for our bandwidth choice
of 5 MHz. Notice that on the smallest resolved scales, l ' 2000 − 5000 the
signal increases slightly as we go from the fundamental kk = 0 to the next
higher modes in our line-of-sight decomposition. This is because the radial
3.2 Weak Lensing Reconstruction
63
10
Cl21cm l (l+1) /2π, Clnoise l (l+1) /2π (mK2)
cored, r-2
flat, r0
1
j=0
1
2
0.1
5
10
15
0.01
20
25
40
60
0.001
100
1000
l
10000
Figure 3.3 The dotted lines show the angular power spectrum for different
kk,j = j 2 π/L, labeled by j. Going to higher values of j, the signal decays
quickly below S/N=1 and only the first 15 or so modes contribute to the
final lensing estimator. This number depends of course on the thickness
of the redshift interval probed (in this case B = 5 MHz, corresponding to
∆z = 0.286 at z = 8). The thick and thin solid curves are for a flat and a
cored antenna configuration of SKA (see the text).
component of the power spectrum is increased due to redshift space distortions. Because the smallest resolved angular scales contribute most to the
lensing reconstruction, in case of the first few modes in the kk decomposition
this increase overcompensates for the general decay of Cl,j on the smallest
scales. However overall for each rectangular data field of a given bandwidth
we will only be able to sample a limited number of modes kk with signalto-noise greater than one, this number being proportional to the frequency
depth/bandwidth of the field.
The 21 cm power spectrum and, as we will see in the next section, the
noise of the experiment depend on redshift, so we calculate both for volume
elements corresponding to the above choice a 5 MHz bandwidth. Beginning
at the end of reionization, each volume element contributes to reducing the
reconstruction noise while we go along, until the signal-to-noise of the ex-
Lensing Reconstruction using redshifted 21 cm fluctuations1
64
periment at high redshift becomes negligible. We should wonder whether
long wavelength modes overlapping neighbouring volumes lead to an underestimation of the final lensing noise, as we are treating each volume element
as uncorrelated. We made a simple test by comparing the NL of a pair of
neighbouring volumes to the sum of NL ’s of each element. With our choise of
a 5 MHz bandwidth we found no excess information > 1% in the sum of the
seperated field’s NL ’s, meaning that we can safely neglect those correlations.
A multiplicity of those volumes can thus be used to reconstruct each lens
just by summing over them.
3.3
Antenna Configuration, Sensitivity Calculation
We have found out in the previous section that in order to measure the lensing signal with redshifted 21 cm fluctuations, we need small angular scale
resolution as well as wide redshift coverage. Sensitivity calculations for 21
cm experiments were used in Chapter 2 to constrain cosmological parameters. We give here more details about how this sensitivity is calculated in
practice. The sensitivity diminishes quickly with rising foreground temperature at longer wavelengths. The foreground temperature is dominated by
galactic synchrotron. To observe at high redshift one needs to compensate
for the increased foreground by increasing the collecting area or number of
antennas. The angular resolution is improved by increasing the maximal
baseline. To have good angular Fourier mode coverage we want all baselines
to be represented though. Hence an increase in angular resolution has to
be compensated for by increasing the collecting area in order to keep the
covering fraction the same. In this section we will discuss various aspects of
the first two generations of 21 cm experiments in the context of using them
for lensing reconstruction.
A standard approach in radio astronomical measurements (see e.g. the
Very Large Array (VLA) 5 , the Low Frequency Array LOFAR 6 , and the
Atacama large millimeter array (ALMA) 7 configurations) is to have a power
law decay in the number density of antenna P ∝ rα with radius. This translates into a drop in the number of baselines with separation. The distribution
flattens out towards the center to n(r) = r0 simply because antennas cannot
be stacked closer to each other than their individual physical size. We found
5
http://www.vla.nrao.edu/
http://www.lofar.org/
7
http://www.alma.nrao.edu/
6
3.3 Antenna Configuration, Sensitivity Calculation
65
in the previous section that the contribution to the weak lensing estimator of
Cl,j becomes smaller quickly with higher line of sight modes j used. This is
just an expression of the fact that what we are interested in is displacements
of 21 cm photons perpendicular the line of sight. Lensing reconstruction
works through a large number of individual sources being aligned around a
big deflector, hence the imaging of small scales is important. For probing
small angular scales a flat array profile (that is without a power law decay)
may offer an advantage, especially if the observable angular power spectrum
falls slightly beyond l ' 1000 and we want to probe it on those scales.
We can calculate the number of baselines as a function of visibility u from
performing the autocorrelation
Z
2
n(u) = λ
d2 r Pground (r + x) Pground (r)
(3.28)
where λ is the observed wavelength and P(r) denotes the radial profile of the
circularly symetric antenna distribution. x is the vector of seperation of an
antenna pair, x = λu.
The time a particular visibility u is observed, tu is given by
tu =
Ae t0
n(u)
λ2
(3.29)
where Ae is the effective antenna area and t0 is the total observing time. We
will assume 2000 hours for our calculations, which might be achieveable with
planned observatories within a single seasons.
The sensitivity for a given array distribution and specifications can then
be calculated as follows. The RMS fluctuation of the thermal noise per pixel
of an antenna pair is [99, 22]
∆T N (ν) =
λ2 BTsys
√
Ae Bt
(3.30)
where B is the bandwidth of the observation. For a single baseline the
thermal noise covariance matrix becomes
2
2
λ BTsys
δij
N
Cij =
(3.31)
Ae
Btu
where Tsys is the system temperature, dominated by galactic synchrotron
radiation (roughly, Tsys ∝ ν −2.55 with Tsys = 440K at z = 6) and B is
the bandwidth of this frequency bin of the total observation. From this we
get the noise versus angular multipole number because of 2πu = l through
d2 l N
C = d2 uCuN .
2π l
Lensing Reconstruction using redshifted 21 cm fluctuations1
66
We are now going to asses the potential of planned 21 cm experiments
to measure the lensing imprint using this quadratic estimator. There are
currently four major experiments under way, the Mileura Wide Field array
(MWA) [100], the primeaval structure telescope [101], the Low Frequency
Array (LOFAR) 8 and in the second generation planning stage, the Square
Kilometer Array (SKA) 9 . While these projects differ qualitatively in the
hardware used, the most crucial difference is their total collecting area. For
SKA, current proposals call for 5000 antennas with an individual collecting
area of 120m2 at z = 8 (the effective antenna area depends on wavelength
and hence redshift through the square). The collecting areas for MWA and
LOFAR/PAST are significantly smaller, 1% and 10% of that of SKA respectively. The optimal array can be designed by distributing the total collecting
2
area over a large number of antennas. The number of baselines goes as Nant
which enters into Equation 3.29. In other words, since an array with say ten
times the number of dishes (each ten times smaller) as SKA has a ten times
larger survey speed, it is better suited for EOR observations. However the
computational requirements for the associated correlator unfortunately also
2
scale as Nant
.
We show the result of our imaging sensitivity calculation for a flat (r0 )
to rmax = 1500 m and a cored (r0 to r = 80 m, then r−2 to rmax = 1500 m)
array of SKA antenna in Figure 3.3, together with the hierachy of decreasing
angular power spectra as the line of sight component of the 3D measurement
kk is decreased. We find that a constant radial density of antennas turns
out to offer the best compromise between the number of (k⊥ , kk) modes that
can be probed and overall angular resolution. For a cored r−2 distribution
for example, the improved sensitivity on those intermediate scales does not
compensate for the loss in angular resolution that comes from the lack of
large baselines relative to a flat distribution.
For statistical detections of the convergence on the skyp
a large field of
view is desired. The angular power spectrum errors go as 1/ fsky . Hence it
is advantageous to distribute the same total collecting area in many smaller
antennas, thus increasing the speed of the survey. On the down-side, it is
more difficult to scale small dipole like antenna to high freqencies (below the
redshift of reionization), where other interesting science with 21 cm emitters
lies (see e.g. [102]).
8
9
http://www.lofar.org
http://www.skatelescope.org/
3.4 Results and a Comparison with the CMB
3.4
67
Results and a Comparison with the CMB
In this section we will discuss a specific configuration that seems likely within
reach of the second generations of EOR observatories and calculate our three
dimensional lensing estimator for a particular reionization scenario. This way
we will gauge the prospects of lensing reconstruction using redshifted 21 cm
fluctuations and how they relate to using other backgrounds.
The prospects of lensing reconstruction with the quadratic estimator have
been explored in depth by [95, 96, 103, 104] in the context of the CMB. A
systematic problem with temperature is that it is contaminated by Doppler
related anisotropies (the kinetic Sunyaev-Zel’dovich effect [105]) which share
the same frequency dependence as the primordial CMB. [104] showed that
this contamination is significant, and might eventually have to be taken care
of by masking thermal SZ detected areas from the map. It is found that using
the CMB polarization an order of magnitude improvement over temperature
reconstruction might be possible [96]. On scales where the B component
of polarization can be resolved (above l ' 100 this component is almost
entirely produced by gravitational lensing rotation of the E modes), with
iterative likelihood techniques astonishingly there may be no limit at all to
delensing [106], on angular scales where the B modes can be resolved.
As discussed in the previous section (Equation 3.12), the effectiveness of
lensing reconstruction depends on the slope of the power spectrum, and the
21 cm background suffers a disadvantage compared to the CMB in that there
is no damping tail and the traces of baryonic oscillations are comparatively
small. On the other hand the lack of an exponential decay in the 21 cm
fluctuations suggests that these can be used for reconstruction out to smaller
scales. Figure 3.2 shows CMB strengths and limitations in comparison to
the shape of the 21 cm power spectrum (which has been rescaled to fit in the
same plot).
The damping tail of the CMB at scales l ≥ 3000 leads to a limit for the
deflection scales l ' 1200 that can be probed using quadratic techniques [96].
The 21 cm power spectrum does not decrease substantially at small scales,
the only theoretical limit being set by the Jeans scale.
Figure 3.4 shows a rather constant shape of L2 NL versus the angular
multipole of the lensing field L. This is essentially a consequence of the
rather flat shape of the 21 cm power spectrum, as we discussed following
Equation 3.12: in this case we would expect NL to not change much up to
scales where the signal becomes comparable to the noise of the experiment.
This property will allow us to image the lensing field down to small scales.
We also show in the same figure a plot of the temperature based CMB lensing
reconstruction based on the Planck satellite experimental specifications.
Lensing Reconstruction using redshifted 21 cm fluctuations1
68
j=0
j=1
j=2
j=5
j=10
j=15
j=20
CLδθδθL(L+1)/2π, NL L(L+1)/2π
1e-06
Planck
1e-07
z=1089
z=8
jmin=10
jmin=3
21cm, ∆ν=5.0 MHz, z=7.86-8.14
1e-08
10
100
1000
L
Figure 3.4 Lensing reconstruction noise NL for one redshift interval centered
at z = 8 corresponding to a bandwidth of 5 MHz. The curves labeled ‘z=8’
and ‘z=1089’ are the displacement field power spectra for 21 cm and CMB
as source respectively. The thick solid curves labeled ‘Planck’ and ‘21 cm’
are the lensing reconstruction noises we find. The 21 cm noise is based on
combining all kk modes. The thin lines labeled ‘j=0-20’ on the other side
are the results for individual kk -modes. We see that from this redshift range
alone the combined temperature and polarization information of Planck can
be beaten. The necessity to substract foregrounds lessens the constraint from
21 cm somewhat. They effectively render the first few kk modes useless for
the reconstruction, see the text. The resulting noise levels are shown in the
thick dashed curves labeled jmin = 3 and jmin = 10, for a less and more
conservative assumption about the complexity of foreground contamination
respectively.
3.4 Results and a Comparison with the CMB
69
The CMB power spectrum sensitivity is given in terms of detector noise
w and beam σFWHM [107]
−1
2
Clnoise = w−1 el(l+1)σFWHM /8ln2
(3.32)
We assume two types of experiments, Planck, with w−1/2 = 27µK − arcmin
at 5 arcmin angular resolution, and a futuristic experiment with noise level
w−1/2 = 3µK − arcmin at an angular
of 3 arcmin. For the polariza√ resolution
−1/2
−1/2
if all detectors are polarized. We
tion we use as usual that wP
= 2wT
include the noise power spectra for Planck temperature, and for polarization
and temperature measurements of our reference CMB experiment in Figure
3.2.
We calculate the minimum variance lensing noise level following [94]
1
−1
α,β (N (L))αβ
Nmv (L) = P
(3.33)
where α and β run over T, E, and B temperature and polarization fluctuations. The noise levels N (L)αβ take on different forms depending on whether
α and β are equal (the BB term is generally considered to have vanishingly
small signal to noise), or different, αβ = θE, θB, and EB.
We compare Planck constraints to 21 cm observation sensitivities for the
deflection angle power spectrum that we get from observing a 5 MHz slice at
redshift 8 in Figure 3.4. The corresponding redshift depth is ∆z = 0.5. If we
want to gather all the information accessible to us from fluctuations in the 21
cm brightness temperature, we can add up a number of redshift intervals with
the constant bandwidth, employing the whole redshift range covered by the
observation. We find that even when using only a fraction of the entire data
volume available in 21 cm, ∆z ' 0.3, Planck’s combined temperature and
polarization potential for doing lensing reconstruction can be beaten with the
three dimensional generalization of the quadratic estimator. If we want to
use a larger redshift range for the 21 cm reconstruction, we need to take into
account the redshift evolution in the power spectrum of matter fluctuations,
and that of the ionization fraction. For an SKA type experiment in the
optimized configuration we use, this range is z = 6 − 12, at higher redshifts
the sensitivity becomes too small for imaging, the limitation being set by the
increased temperature of the foreground. Note that if reionization completes
earlier than z = 12 lensing reconstruction using 21 cm will be impossible
with SKA. In our particular reionization scenario we achieve a constraint
at L ≈ 1000 about ten times as good (i.e. an NL ten times lower) for the
lensing reconstruction, in comparison to our reference CMB experiment with
3 arcminute resolution and a noise level of w−1/2 = 3µK − arcmin. We show
70
Lensing Reconstruction using redshifted 21 cm fluctuations1
this result in Figure 3.5, together with nonlinear/linear lensing field power
spectra. The Figure also shows (in the thick dashed curve) the increase in
noise when our patchy model is used for the range 6-8 in place of an extension
of the neutral phase. This increased noise relative to the case of probing the
pre-reionization IGM has two reasons: first, as shown in Figure 3.1, the
fluctuation level of the 21cm signal is decreased on the smallest resolved
scales, making this source less valuable for lensing reconstruction. Secondly,
the connected four-point function contribution adds a non Gaussian term
to the noise covariance matrix of the power spectrum, acting as a sample
variance term in correlating different band-powers. We treated this term in
a simplified manner by assuming that the majority of bubble features arises
on scales well above the resolution scale of SKA, about 1’. Then the bubbles
can be resolved and masked when establishing the final estimator. The mask
increases the sample variance simply as δCl∗ = √x1H δCl , where xH is the
neutral fraction at this redshift. Our estimate of the lensing reconstruction
noise for the patchy epoch is conservative, in that power in the regions outside
the bubble mask will not be suppressed, however we use the global average
power spectrum which is suppressed by x2H . For completeness we also show
(in the thick dotted curve) the total lensing reconstruction noise if we only
use the neutral regime above z = 8 in the analysis.
Because of the multiple background information, in reconstructing the
deflection field we can approach the limit imposed by the finite angular resolution of an experiment (determined by its maximum baseline). In Figure 3.6
we show contours for the maximum lensing deflection field multipole L that
can be probed as a function of the maximum baseline and the total collecting
area. For the r0 array there exists an optimal rmax which depends on the
amplitude of the power spectrum and the total collecting area. If one uses
a wider radius while keeping the number of dishes constant, the noise level
increases on all angular scales, so although one might be able to use slightly
larger multipoles (smaller scales) for the reconstruction, many modes of our
hierarchy do not enter the estimator. On the other hand, if one decreases
the array size, thus making the antenna distribution denser, this leads to a
lower noise level on relatively large scales, while no baselines exist anymore
to measure the small scales that are crucial for the reconstruction.
We find that after gathering 2000 hours of data, a very ambitious experiment with four times the collecting area of SKA (for which Acoll.,z=8 =
0.6km2 ) could detect lensing at scales beyond L = 6000, which is comparable
to the scale of galaxy clusters.
We will extend considerations to statistical detections of the deflection
l Dls
is larger for the CMB
power spectrum. Because the geometrical ratio DDs
3.4 Results and a Comparison with the CMB
71
1e-07
CLδθδθL(L+1)/2π, NL L(L+1)/2π
z=11-12
z=10-11
z=9-10
1e-08
CMB, 3 µK arcmin
z=8-9
21cm, SKA-type, neutral z=8-12
z=7-8
21cm, SKA-type, patchy z=6-8, neutral z=8-12
z=6-7
21cm, SKA-type, neutral z=6-12
1e-09
10
n.l.
100
1000
L
Figure 3.5 Lensing reconstruction noise with an experiment that has the total
collecting area of SKA, if the IGM is not ionized during the regime z=6-12,
shown in the thick line at bottom. The individual contributions from redshift
intervalls are shown in the dashed lines. The noise levels are compared to
the deflection angle power spectrum, where the solid (dotted) lines are for
the nonlinear (linear) density fluctuations at z=8 and z=1089. On scales
of L = 1000 and above, our method might be able to achieve an order
of magnitude lower total noise levels than what is possible with the CMB
quadratic estimator technique (shown in the other thick solid line). Here
the smallest angular scale reconstructed using EOR fluctuations is Lmax =
2250. The thick dashed line shows the increased noise level if reionization
is inhomogeneous during z = 6 − 8, calculated in the way detailed in the
text. Finally the thick dotted curve gives the total noise level if the regime
z = 6 − 8 is not used at all in the analysis.
Lensing Reconstruction using redshifted 21 cm fluctuations1
72
3500
3000
2500
Lmax=1000
2000
3000
4000
5000
2000
6000
rmax (m)
1500
1000
500
0
0.5
1
1.5
2
2.5
Acoll., z=8 = Nant Ae (km2)
Figure 3.6 Contour plot of Lmax , the largest displacement field multipole
probed. From this Figure one can infer the optimal array radius (assuming
here an r0 distribution of antennas) that one should choose given a total
collecting area Acoll . It also shows that if one would have four times the
collecting area of SKA (for which Acoll.,z=8 = 0.6km2 ), or if one were to
observe four times longer on the same patch of the sky, very high multipoles
could be probed.
3.4 Results and a Comparison with the CMB
73
(qualitatively the bulk of lensing happens at angular diameter distances that
are closer to the middle between the observer and the CMB), the respective
deflection angle power spectrum has a higher amplitude. On the other hand
the 21 cm experiment will have the advantage of measuring multiple planes
so that at comparable angular resolution smaller errors in the angular power
spectrum at high L can in principle be achieved.
In [92] it was proposed that 21 cm reconstruction of the lensing power
spectrum might be helpful in getting at B mode polarization from primordial gravity waves. Since lensing partially converts E (gradient) polarization
into a curl component, this secondary signal swamps the B modes from a
possible inflationary tensorial fluctuation background. The problem is that
at significantly lower redshifts than the last scattering surface, only part of
the lensing structure encountered by the CMB photons is traced, leading to
a delensing bias if 21 cm fluctuations are used. One would have to observe
at high enough redshifts if one were to compete with CMB polarization experiments. Indeed the authors find that in principle with an ultra sensitive
experiment (for instance space based) observing at redshift 30 one might be
able to beat CMB limits beyond the iterative likelihood approach of [106, 90].
To achieve this one would need 1-2 orders of magnitude more collecting area
than what is planned for the second generation of observatories, hence this
application is beyond the scope of this thesis.
A characteristic of 21 cm fluctuations is that the distance between observer and lens is a larger fraction of the whole distance to the source, so one
would imagine that if the same number of multipoles are probed with 21 cm
as with the CMB (by having a lower noise level), one would obtain different
constraints on dark energy. We show this in Figure 3.7, where the ratio
ΓΩΛ (L) =
ΩΛ ∂CLDD
CLDD ∂ΩΛ
(3.34)
is plotted against L. The quantity measured shows how well the dark energy
density can be constrained by using information from a given angular scale.
Its value gives the constraint that can be put on the parameter from this
scale if there were no degeneracies with other unknowns. We see that when
the same range of angular scales are resolved, the 21 cm fluctuations fair
somewhat better in constraining the dark energy density ΩΛ .
The noise for estimation of bandpowers is reduced by averaging over L
directions in a band of width ∆L
s
2
∆CLDD =
[C DD + Nmv (L)] .
(3.35)
(2L + 1)∆Lfsky L
Lensing Reconstruction using redshifted 21 cm fluctuations1
74
3
2.5
21cm
2
ΓΩ
Λ
CMB
1.5
1
0.5
0
0
500
1000
1500
angular L
2000
2500
3000
Figure 3.7 Plot of the parameter Γ defined in the text. This suggests that
21 cm fluctuations are better suited in principle to measure the value of the
cosmological constant than the CMB, assuming that the same number of
angular modes can be probed.
A comparison between the polarized reference CMB experiment and an
experiment with SKA’s sensitivity is shown in Figure 3.8 where we assumed
a sky coverage of 0.8 for the CMB experiment and a smaller field of view
0.08 for the 21 cm experiment. We plot the sample variance error (‘S’)
and sample variance plus noise (‘S+N’). Polarized CMB experiments suffer
from foreground contamination [7], so our reference experiment should be an
idealized limit. The sky coverage of the 21cm experiment might be achievable
within one year of observation with an MWA type experiment that has the
same collecting area as SKA but a ten times higher survey speed. Using
fluctuations in the 21 cm background, we should be able to measure the
deflection power spectrum to much higher multipoles, l > 10000 for this
type of experiment, than what will ultimately be possible using the CMB
quadratic estimator technique, even if the latter observes on a much larger
part of the sky (notice also that 21 cm experiments should be able to observe
separate fields in consecutive seasons). The requirement of a combination of
large collecting area and high survey speed makes this an ambitious project.
Finally we would like to estimate the effect foregrounds will have on the
estimation of the lensing field suggested here. As pointed out by [21, 22],
3.4 Results and a Comparison with the CMB
CLδθδθ, ∆CLδθδθ L (L+1)/2pi
1e-07
75
CMB S+N
CMB S
1e-08
1e-09
1e-10
1e-11
10
100
1000
10000
L
CLδθδθ, ∆CLδθδθ L (L+1)/2pi
1e-07
21cm S+N
21cm S
1e-08
1e-09
1e-10
1e-11
10
100
1000
10000
L
Figure 3.8 Displacement field power spectra and sample variance (‘S’) and
noise (‘N’) errors. An SKA like experiment might be suited to probe a large
dynamical range in the displacement field. The displacement field angular
power spectrum with errorbars on the left is for our reference CMB experiment at fsky = 0.8 and the curves on the right are for 21 cm redshift 6-12
lensing reconstruction using the ”MWA50k” (same collecting area but 10
times higher survey speed than SKA) with fsky = 0.08.
76
Lensing Reconstruction using redshifted 21 cm fluctuations1
fluctuations in the gas at high redshift can be seperated against the much
brighter foregrounds (the main source of confusion being galactic and extragalactic synchrotron), because the former vary rapidly, the latter slowly in
frequency. This can be done by fitting various smooth functions to the signal.
[108], [47], and [80] suggest quadratic or cubic polynomials, but also more
complicated functions such as Chebychew polynomials have been proposed.
[80] show that this will practically make the first few wavevector modes in
the line of sight direction unusable. The exact number of modes that can be
used depends of course on the nature of foregrounds, the bandwidth, and the
technique of fitting.
On the positive side once a model for foregrounds is given, we can implement this pretty straightforwardly within our formalism by discarding the
first few modes kk (depending on the order of fitting polynomial) in the hierarchy of Cl ’s, compare Figure 3.3. It is to be expected that a large number
of modes still will contribute to thePfinal noise level. This is quantified in
Figure 3.4 where the sum NL−1 = 1/ kj>j NL,kj L/(2π) is plotted for the two
min
cases jmin = 3, 10 meaning that the first 3/10 modes have been discarded in
the analysis. The resulting noise levels (short dashed lines) are somewhat
higher than the solid line that was obtained from assuming no foregrounds.
3.5
Conclusions and Outlook
In this chapter we have extended the quadratic estimator formalism to use
a three dimensional signal as lensing background. 21 cm fluctuations from
neutral hydrogen prior and during the epoch of reionization contain an enormous amount of data points. The correlations induced by lensing into this
signal can be used to probe the intervening matter fluctuations either on
a individual object basis, or statistically to for example probe dark energy
models. To describe fluctuations in the neutral fraction, we used an analytic
model for the morphology of HII regions to demonstrate the applicability of
our method to this regime.
Our estimator should be complete as long as non linearities in the signal
(e.g. due to the bubbles) are small. The bulk of the information we use is
coming from the neutral phase.
The first generation of experiments is likely not going to be able to image the angular fluctuations in 21 cm needed to measure the lensing effect.
However we arrive at good constraints by employing current specifications of
the SKA with a flat antenna distribution.
In comparison with the CMB, 21 cm fluctuations have a rather featureless
scale invariant unlensed spectrum which leads to a smaller lensing effect. This
3.5 Conclusions and Outlook
77
can be compensated for by using multiple redshift information when probing
each individual lens surface. The CMB quadratic estimator sensitivity can
actually be beaten this way, for example with an SKA type experiment.
Another possibility would be to combine 21 cm lensing reconstruction
with that from other observables, such as the CMB. Similar to combining
the latter with galaxy shear surveys, this will improve the constraints on the
mass/energy budget or geometry of the universe significantly.
If ambitions in the community of observational and theoretical cosmologists increase in the years to come, and third generation experiments will
be scheduled, a new prospect would be the measurement of polarized 21 cm
emission. [109] find that Thomson scattering of the quadrupole produced
by the reionized universe produces the largest effect and similar to the gains
by using CMB polarization, this could make the method suggested in this
chapter, the three dimensional form of the quadratic estimator, even more
promising.
Larger collecting areas and longer observation times also promise to allow 21 cm lensing reconstruction to map out the gravitational potential of
individual galaxy clusters. We will attempt to address this topic in a future
work.
(3.38)
(3.37)
d2 l
(2π)2
Z
dk1
2π
Z
dk2
F (l, k1 , k2 , L)I(l, k1 )I(L − l, k2 )
2π
F (l, k1 , k2 , L) = F ∗ (−l, −k1 , −k2 , −L)
(notice that F (l, k1 , k2 , L) = F (l, k2 , k1 , L)). Because δΦ(L) = δΦ∗ (−L) it can also be shown that
Φ(L) =
Z
(3.40)
(3.39)
where we have used that δθ(L) = iLφ(L). We are looking for a quadratic estimator Φ(L) for φ(L), i.e. of the form
Ĩ(l, k) = I +
Z
d2 l0
(δθ(l − l0 ) · il)I(l0 , k)
(2π)2
Z
= I(l, k) − d2 l0 I(l0 , k)φ(l − l0 )(l − l0 ) · l0
where Ĩ(θ, z) is the lensed, I(θ, z) the unlensed field. The Fourier transform of this expression is
Ĩ(θ, z) = I(θ, z) + δθ · ∇θ I(θ, z) + ...
(3.36)
Appendix A: Quadratic estimator applied to a three dimensional
observable
We want to observe the three dimensional field I(θ, z). We assume weak lensing, i.e. that
3.6
78
Lensing Reconstruction using redshifted 21 cm fluctuations1
79
"
Z
0
dk2
d2 l
F (l, k1 , k2 , L) −hI(l, k1 )
I(l0 , k2 )φ(L − l − l0 )×
2π
(2π)2
#
Z
2l 0
d
I(l0 , k1 )φ(l − l0 )(l − l0 ) · l0 i
×(L − l − l0 ) · l0 i − hI(L − l, k2 )
(2π)2
"Z
Z
Z
Z
0
d2 l
d2 l
dk1
dk2
−
F
(l,
k
,
k
,
L)
(2π)2 δ(l + l0 )(2π)δ(k1 + k2 )Pl,k ×
1 2
(2π)2
2π
2π
(2π)2
#
Z
0
d2 l
2
0
0
0
0
0
0
0
(2π) δ(L − l + l )(2π)δ(k1 + k2 )PL−l,k φ(l − l )(l − l ) · l
×φ(L − l − l )(L − l − l ) · l +
(2π)2
Z
Z
Z
d2 l
dk1
dk2
−
F (l, k1 , k2 , L)(2π)δ(k1 + k2 ) [Pl,k φ(L)L · (−l) + PL−l,k φ(L)L · (−(L − l))]
2
(2π)
2π
2π
Z
Z
Z
d2 l
dk1
dk2
F (l, k1 , k2 , L)(2π)δ(k1 + k2 ) [Pl,k φ(L)L · l + PL−l,k φ(L)L · (L − l)] ,
(3.41)
(2π)2
2π
2π
d2 l
(2π)2
d2 l
(2π)2
Z
dk2
2π
Z
Z
Z
dk1
2π
Z
dk2
F (l, k1 , k2 , L)(2π)δ D (k1 + k2 ) [Pl,k L · l + PL−l,k L · (L − l)] = 1
2π
(3.42)
where e.g. Pl,k is the power in a mode with angular component l and radial component k. With the requirement
that hΦ(L)iI = φ(L) this leads to the normalization condition
=
=
=
hΦ(L)iI =
Z
We want to find F such that it minimizes the variance of Φ(L) under the condition that its ensemble average recovers
the lensing field, hΦ(L)iI = φ(L). This becomes (to first order in φ)
3.6 Appendix A: Quadratic estimator applied to a three
dimensional observable
×(2π)2 δ(L − l − l0 )(2π)δ(k2 − k10 )P̃tot
L−l,k2
F (l, k1 , k2 , L)F ∗ (l0 , k10 , k20 , L0 )(2π)2 δ(l − L + l0 )(2π)δ(k1 − k20 )P̃tot
l,k1 ×
×(2π)2 δ(l0 − l)(2π)δ(k2 − k20 )P̃tot
L−l,k2
(3.44)
(3.43)
∂( Eq. 3.42)
d2 l dk1 dk2
= AR
(2π)δ D (k1 + k2 )[Pl,k L · l + PL−l,k L · (L − l)]
∂F (l, k1 , k2 , L)
(2π)2 2π 2π
with respect to F , where AR is a Lagrangian multiplier. In steps,
h||Φ(L)||2 i − AR × ( Equation 3.42)
(3.46)
(3.45)
Both real and imaginary part of ||F ||2 = FR2 + FI2 contribute to this variance, however the condition for the
minimization will only pick out the real part. The solution is found by minimizing the function
but from 3.39 we see with the substitution L − l → l that F ∗ (L − l, k2 , k1 , L) = F ∗ (l, k1 , k2 , L) hence
Z
Z
Z
dk1
dk2
d2 l
tot
2
2
F (l, k1 , k2 , L)F ∗ (l, k1 , k2 , L)P̃tot
h||Φ(L)|| i = 2(2π) δ(0)
l,k1 P̃l,k2 .
2
(2π)
2π
2π
Z
Z
Z
dk2
dk1
d2 l
tot
(2π)2 δ(0)F (l, k1 , k2 , L)F ∗ (l0 , k10 , k20 , L0 )P̃tot
=
l,k1 P̃L−l,k2
2
(2π)
2π
2π
Z
Z
Z
dk2
dk1
d2 l
tot
(2π)2 δ(0)F (l, k1 , k2 , L)F ∗ (L − l0 , k20 , k10 , L0 )P̃tot
+
l,k1 P̃L−l,k2
2
(2π)
2π
2π
+
Z Z Z Z Z Z
The next step is to minimize the variance
Z
Z
Z
Z
Z
0Z
d2 l
dk1 0 dk2 0
dk1
dk2
d2 l
2
h||Φ(L)|| iĨ =
F (l, k1 , k2 , L)F ∗ (l0 , k10 , k20 , L0 ) ×
(2π)2
2π
2π
(2π)2
2π
2π
×hĨ(l, k1 )Ĩ(L − l, k2 )Ĩ(l0 , k10 )Ĩ(L − l0 , k20 )i
Z Z Z Z Z Z
=
F (l, k1 , k2 , L)F ∗ (l0 , k10 , k20 , L0 )(2π)2 δ(l − l0 )(2π)δ(k1 − k10 )P̃tot
l,k1 ×
80
Lensing Reconstruction using redshifted 21 cm fluctuations1
FR (l, k1 , k2 , L) = AR (2π)δ D (k1 + k2 )
1
1
2
d2 l [Pl,k L·l+PL−l,k L·(L−l)]2
tot
(2π)2
P̃tot
l,k P̃L−l,k
Using (2π)δ D (0) =
2π
dk
2
dk 2π
2π dk
→
k,
P
2
k
XZ
k
= (2π)2 δ(0) P R
1
d2 l [Pl,k L · l + PL−l,k L · (L − l)]2
tot
(2π)2
P̃tot
l,k P̃L−l,k
[Pl,k L·l+PL−l,k L·(L−l)]2
tot
2 P̃tot
l,k P̃L−l,k
2
= 2(2π)2 δ(0)A2R
AR
δ(0)A2R
this becomes
h||Φ(L)|| i = 2(2π)
and
R
By using Equation 3.48 we find that the variance becomes
Z
Z
Z
d2 l
dk1
dk2 tot tot
2
2
h||Φ(L)|| i = 2(2π) δ(0)
P̃ P̃
F2
2
(2π)
2π
2π l,k1 L−l,k2 R
Z
Z
Z
2
dk1
dk2
d2 l
2 [Pl,k L · l + PL−l,k L · (L − l)]
2
2
[(2π)δ(k
+
k
)]
= 2(2π) δ(0)AR
1
2
tot
(2π)2
2π
2π
P̃tot
l,k P̃L−l,k
k
AR = P R
.
[Pl,k L · l + PL−l,k L · (L − l)]
tot
P̃tot
l,k1 P̃L−l,k2
d2 l dk1 dk2
∂h||Φ(L)||2 i
tot
= 2(2π)2 δ(0)
2FR (l, k1 , k2 , L)P̃tot
l,k1 P̃L−l,k2
2
∂F (l, k1 , k2 , L)
(2π) 2π 2π
and by inserting this into the normalization condition 3.42 we get that
so
and
(3.52)
(3.51)
(3.50)
(3.49)
(3.48)
(3.47)
3.6 Appendix A: Quadratic estimator applied to a three
dimensional observable
81
k
NLΦ = P R
1
[Pl,k L·l+PL−l,k L·(L−l)]2
tot
(2π)2
2P̃tot
l,k P̃L−l,k
d2 l
(3.55)
(3.53)
(3.54)
R
1
[Pl,k L·l+PL−l,k L·(L−l)]2
tot
(2π)2
2P̃tot
l,k P̃L−l,k
d2 l
so we have justified that we can simply sum over seperate k modes to arrive at our final lensing noise.
= P
1
−1
k NL,k
1
k L2
NLD = P
(3.56)
and since the variances of estimator D of the displacement hDiens. = δθ and Φ are just related by h||D(L)||2 i =
L2 h||Φ(L)||2 i we obtain
it follows that
hΦ(L)Φ∗ (L0 )i = (2π)2 δ D (L − L0 )NLΦ
or h||Φ(L)||2 i = (2π)2 δ D (0)NLΦ
With the definition of the noise power spectrum NLΦ ,
82
Lensing Reconstruction using redshifted 21 cm fluctuations1
3.7 Appendix B: Quadratic estimator lensing reconstruction in
practice
83
3.7
Appendix B: Quadratic estimator lensing
reconstruction in practice
In this section we would like to test how well the quadratic estimator performs
in practice. We remind us of the form of this estimator
Z 2
dl
ψ̂(L) = N (L)
θ̃(L − l)θ̃∗ (l)g(L, l) .
(3.57)
2π
By requiring the estimator to be unbiased and having maximal signal-tonoise, we had found that
g(L, l) =
with the normalization
"Z
N (L) =
θ
(L − l) · LC|L−l|
+ l · LClθ
(3.58)
tot
2C̃ltot C̃|L−l|
θ
θ 2
d2 l [(L − l) · LC|L−l| + l · LCl ]
tot
(2π)2
2C̃ltot C̃|L−l|
#−1
.
Let us reshape the result for the estimator a little:
Z 2
θ
θ
d l (L − l)C|L−l| + lCl
θ̃(l)θ̃∗ (L − l)
ψ̂(L) = N (L) L ·
tot tot
2π
2C̃l C̃|L−l|
(3.59)
(3.60)
For computational convenience, this convolution in harmonic space can be
expressed as a product in real space. The advantage of this is that all integrals
can be evaluated using FFT’s. Let us look at the first term of Equation 3.60
(the second term will be the equivalent). We have
!
!
Z 2
˜
1
d l −ilClθ θ(l)
θ̃(L − l)
ψ̂(L) = N (L)L
·
(3.61)
tot
−i
2π
2C̃ltot
C̃|L−l|
Define F1 (l) =
˜
−ilClθ θ(l)
C̃ltot
and F2 (L − l) =
θ̃(L−l)
.
tot
C̃|L−l|
Omit factor
1
2
in F1 (because
of second term in equation 3.60), so that the estimator of ψ becomes
Z 2
d x −iL·x
ψ̂(L) = iN (L)L ·
e
F1 (x)F2 (x)
(3.62)
2π
The convergence can then simply be obtained through
1
κ̂(L) = − L2 ψ̂(L)
2
(3.63)
84
Lensing Reconstruction using redshifted 21 cm fluctuations1
This algorithm can be applied to a three dimensional observable such as
21 cm radiation from both before and during the epoch of reionization10 .
Here we show the results of implementing the 3D algorithm to reconstruct a
Gaussian convergence field.
The result of the lensing reconstruction for both Planck and for our 5
MHz 21cm data slice (the representative lensing reconstruction noise levels
are shown in Figure 3.4) are shown in Figure 3.9. These maps have an angular extent of 10◦ . In the case of Planck (the left panel), no resemblance
can be seen between input and reconstructed convergence. The 21 cm reconstruction is better (many degree scale features seen in the input are located
in the reconstruction) in this case because of a slightly better angular resolution assumed in this experiment compared to Planck (3’ versus 5’), but also
because we are probing multiple planes in redshift. Finally, in Figure 3.10,
we show two profiles of input and reconstructed convergence from the maps
shown in Figure 3.9.
10
although our quadratic estimator will be sub-optimal during the patchy regime
conv_in.fts_0
conv_reco_5Mhz.fts_0
Figure 3.9 Middle panel: Input convergence, a Gaussian random field on 10◦ . The convergence power spectrum
used was obtained from integrating over the matter power spectrum (from CMBFAST) to z = 8. The field looks
qualitatively similar when integrating to z = 1089, only the temperature gradient has been adjusted. Left panel: the
result of the lensing reconstruction using the specifications of Planck (polarization reconstruction is not very effective
in this case). Right panel: result of lensing reconstruction using the quadratic estimator introduced in this chapter
for a 5 MHz wide band centered around z=8, corresponding to ∆z = 0.26. The corresponding lensing reconstruction
noise level is shown in Figure 3.4. Notice that for the case of 21 cm we have only used a fraction of the redshift
regime potentially accessible for the reconstruction. All three maps have been smoothed after the reconstruction
with a 5’ beam.
conv_reco_planck.fts_0
3.7 Appendix B: Quadratic estimator lensing reconstruction in
practice
85
Lensing Reconstruction using redshifted 21 cm fluctuations1
86
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
Input κ
Reconstructed κ
-0.08
0
100
200
300
400
500
600
Figure 3.10 A random profile through input and 21 cm reconstructed convergence maps. The abscissa units are number of pixels. Even for our small
redshift window probing the lensed EoR fluctuations, there is a clear match
between original and reconstructed convergence profiles.
Chapter 4
The influence of inhomogeneous
reionization on the CMB1
In this chapter we investigate the impact of spatial variations in the ionized
fraction during reionization on temperature anisotropies in the CMB. To do
so we combine simulations of large scale structure to describe the underlying
density field with the analytic model based on extended Press-Schechter theory described in Chapter 2 to track the reionization process. We find that
the power spectrum of the induced CMB anisotropies depends sensitively on
the character of the reionization epoch. Models that differ in the extent of
the “patchy phase” could be distinguished by future experiments such as the
Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT).
In our models, the patchy signal peaks at l ' 2000, where it can be four times
larger than the kinetic Sunyaev-Zel’dovich (kSZ)/Ostriker-Vishniac (OV) signal (∆Ttot ' 2.6µK). On scales beyond l ' 4000 the total Doppler signal is
dominated by kSZ/OV, but the patchy signal can contribute up to 30% to the
power spectrum. The effect of patchy reionization is largest on scales where
the primordial CMB anisotropies dominate. Ignoring this contribution could
lead to significant biases in the determination of cosmological parameters
derived from CMB temperature measurements. Improvements in the theoretical modeling of the reionization epoch will become increasingly important
to interpret the results of upcoming experiments.
1
Based in part on O. Zahn, M. Zaldarriaga, L. Hernquist, M. McQuinn, Astrophysical
Journal, 630, 657 (2005)
87
88
4.1
The influence of inhomogeneous reionization on the CMB1
Introduction
Cosmic Microwave Background (CMB) anisotropy experiments have now
constrained temperature fluctuations down to scales of 40 [110, 111] resulting in a much improved understanding of inhomogeneities during the time
of decoupling at z ' 1100 and of the global properties of the Universe. In
combination with data from Supernovae Ia, measurements of the local expansion rate, galaxy clustering studies, and observations of the Lyman α
forest, the CMB data has lead to the establishment of a “standard” cosmological model. We live in a Universe dominated by dark energy and with
significantly more dark matter than baryons. Structure grew from a scale independent spectrum of primordial Gaussian fluctuations, in good agreement
with predictions of inflationary models.
In the CMB community, theoretical and experimental interest is shifting
to the study of secondary anisotropies on smaller angular scales, discussed in
the preliminary chapter. We will overlook the effects, caused by fluctuations
in the distribution of baryons and dark matter in the redshift regime z '
0 − 30, here again briefly. Future data on the different secondary effects will
constrain the way structure formation proceeded from the linear into the nonlinear regimes. Secondary anisotropies can be divided into three categories:
gravitational lensing effects, inverse Compton scattering of CMB photons by
hot plasma, and Doppler related anisotropies.
Mass concentrations along the line of sight such as clusters, sheets, and
filaments lead to weak deflection of the CMB photons, or Gravitational Lensing (for a review see chapter 9 of [112]). Gravitational lensing of the CMB
can be used as a cosmological probe in various ways. The lensing effect on the
CMB power spectrum could help break degeneracies between cosmological
parameters [113, 114]. Lensing induces departures from Gaussianity on CMB
maps that can be used to reconstruct the spatial distribution of the lensing
mass [115, 94]. The CMB lensing signal can be correlated with galaxy lensing
shear providing additional information. Thus, lensing can be used to probe
the evolution of gravitational clustering, constrain the properties of the dark
energy, the shape of the matter power spectrum, and neutrino physics (see
e.g. [116, 94, 117]).
Inverse Compton scattering in the hot intracluster medium, also called
the thermal Sunyaev-Zel’dovich effect [118], changes the spectrum of the
CMB photons, leading to cold spots (decrements) in the microwave background at frequencies below 217 GHz and to hot spots (increments) at
frequencies above. The effect is proportional to the line-of-sight integrated
pressure. Because it is independent of redshift, it is a unique probe of collapsing structures out to z ' 3 (see Figure 4.6). The thermal SZ effect has
4.1 Introduction
89
been measured in follow-up observations of X-ray clusters (see for instance
[119, 120]). Comparison of the cluster SZ temperatures with the X-ray measured temperatures leads to constraints on the angular diameter distance.
The SZ effect also leaves a signature in the CMB power spectrum on small
scales where the primordial CMB vanishes. BIMA (the Berkeley Illinois
Maryland Association) and CBI (the Cosmic Background Imager) claim detections of this effect at ≥ 2σ significance levels [110, 111]. CBI infers a LSS
clustering amplitude on 8 Mpc/h scales of σ8 = 0.9, which is at the upper
level of constraints from cluster observations and weak lensing (for a recent
compilation of experimental results see table 5 of [121]). Future large angular
scale SZ surveys such as SPT promise to measure the cluster abundance as a
function of mass and redshift, which will offer the possibility of constraining
the matter density and the equation of state of the dark energy.
Finally, Doppler related effects are produced by the scattering of CMB
photons off electrons moving as a result of the structure formation process.
This process, also known as the kinetic Sunyaev-Zel’dovich effect when applied to clusters of galaxies [105], leads to hot or cold spots, depending
on whether the ionized baryons move toward or away from the observer.
The frequency dependence of the photons is left unchanged, except for tiny
relativistic corrections. These “Doppler” induced anisotropies are the only
known way to measure the high redshift large scale velocity field.
Several analytical models for the kinetic SZ effect due to patchy reionization have been presented in the literature. They vary significantly in the
assumed size and time evolution of spherically shaped bubbles, and whether
the ionized regions are correlated [122, 123] or not [124]. Some models employ a prescription for correlating the ionizing sources by using the bias that
can be calculated for the dark matter halos in which they presumably reside [123]. All authors agree that on scales below 40 , where the primordial
CMB signal falls rapidly owing to photon diffusion, the Doppler effect induced by patchiness could contribute enough to use it as a tool to study the
reionization epoch.
In this chapter, we use a hybrid approach between the analytic model for
the formation of HII regions described in Chapter 2, and smoothed particle
hydrodynamics (SPH) simulations of large scale structure. The advantage
over purely analytic models of patchy reionization kinetic SZ is that our
sources follow the complex clustering behavior of dark matter and baryons.
An advantage over currently feasible full radiative transfer calculations of the
reionization epoch is that we can investigate the morphological properties of
reionization on scales an order of magnitude larger with rather small memory
and CPU requirements. We re-iterate that our prescription does not directly
address many of the (uncertain) physical details of reionization related to
The influence of inhomogeneous reionization on the CMB1
90
star formation, feedback processes, clumpiness, recombinations, and radiative
transfer. However, by combining these properties into a single parameter, we
are able to explore the basic parameter dependencies, and we believe it is
a good starting point to help us understand the morphological properties of
cosmological reionization and its influence on the CMB.
In Section 4.3, we explore different parametrizations of reionization. In
one model we assume that the source properties are constant so that reionization evolves over a redshift interval of ∆z ' 4. In a second model try to
do justice on one hand to the rapid decline of the neutral fraction around
z = 6 − 7, seen in measurements of the Gunn-Peterson optical depth by
measuring spectra of distant quasars, while at the same time allowing for
earlier onset of reionization, at the high end of the range of values allowed by
WMAP [14]. We model this by assuming two succeeding reionization epochs,
in which HII regions first are produced by low metallicity (Pop. III) stars,
and later expand further during an epoch dominated by Pop. II stars, so
that on the whole patchiness lasts longer.
In Section 4.4 we investigate whether the next generation of ground based
bolometric arrays will be able to measure reionization and distinguish between different scenarios. The specifications of the Atacama Cosmology
Telescope (ACT) 2 [88] and the South Pole Telescope (SPT)3 [89] will be
used to predict how well these experiments can measure CMB temperature
power spectra in the region where secondary effects dominate over primordial CMB anisotropies. We find that ACT/SPT should be able to distinguish
between sudden homogeneous reionization and patchy reionization at a high
level of significance, even if the total optical depth in both scenarios were the
same.
Since the patchy signal peaks on scales where the primordial anisotropy
dominates, cosmological parameter estimation may become biased. We will
show in Section 4.4.2 that precision experiments such as Planck should take
this bias into account either by focusing on their polarization data or by
adding a reionization parameter to their analysis.
Our numerical simulations of large scale structure and the power spectra
we used to generate realizations of the primordial CMB assume a cosmology
in agreement with WMAP constraints: Ωdm = 0.26, ΩΛ = 0.7, Ωb = 0.04,
fluctuations on 8Mpc/h scales of σ8 = 0.9, and no tilt (nS = 1) or running
(αS = 0) of the spectral index.
2
3
see http://www.hep.upenn.edu/ angelica/act/act.html
see http://astro.uchicago.edu/spt/
4.2 Simulation of secondary anisotropy and patchy reionization 91
4.2
Simulation of secondary anisotropy and
patchy reionization
Our approach is to use large scale simulations of the cosmic web as a basis for
applying the model described in Chapter 2 as a post-processing step to generate HII regions during the reionization phase. The underlying large scale
structure was simulated using the parallel Tree-PM/SPH solver GADGET
[125], based on the fully conservative implementation of SPH by [126]. Here,
we used the results of runs with a boxsize of 100Mpc/h and 2163 particles,
with parameters corresponding to the G-series runs of [127]. In what follows,
we employ a simulation that included only “adiabatic” gas physics; i.e. the
gas can heat or cool adiabatically and be shocked, but we do not include
radiative effects or the consequences of star formation and associated feedback processes. Snapshots of the simulation are produced every light crossing
time interval. This leads to 77 outputs between z = 0 and z = 20. These
simulations are described in more detail in [128].
We implemented the model for the HII morphology as described in Chapter 2. As underlying density field we used the high redshift snapshots from
our simulation. Although the analytic model by definition calls for a Gaussian random density field, the small non-linearities present in our box should
not constitute a grave violation. We expect the ionized regions to be large
in comparison to the non-linear scale, the former being 10 Mpc/h on average, the latter being below 1 Mpc/h. An advantage of using the non-linear
field for the density is that the mode coupling effects described in Section
?? should be properly modeled. Through a detailed comparison we verified
however that these effects are small on the level of the kinetic SZ effect.
To obtain the underlying density field, we discretize the matter fluctuations in our simulation boxes into 2563 cells. The overdensity, δ, is then
smoothed with a top hat window function.
To sample a wide range of bubble sizes, and obtain a smoothly varying
ionization fraction, we varied the barrier in radial direction throughout each
box by adjusting the redshift dependent minimal ionization radius. The
reionization information for each of the 2563 cells is stored. A 8 Mpc/h deep
cut through the stacked outputs between z ' 12 and z ' 17 is shown in
Figure 4.1, for one of the models we will describe in the next section, in
which the ionization efficiency is constant, ζ = 60. The periodic boundary
conditions of our simulation boxes are apparent in this plot4 .
4
note that to simulate maps of the CMB, we will randomly translate the simulation
volumes before projecting them, see below
z=15.53
z=14.19
z=13.16
z=12.24
Figure 4.1 Time development between redshifts z = 12 and z = 17 of an 8Mpc/h deep slice through an inhomogeneously ionized ζ = 60 (constant) box, with increasing ionization fraction. The outputs with periodic boundary
conditions are simply stacked behind each other here. In order to obtain realistic maps, the individual outputs where
randomly rotated and translated in our simulations. At the low redshift end, the bubbles become comparable in size
to the box.
z=16.61
92
The influence of inhomogeneous reionization on the CMB1
4.2 Simulation of secondary anisotropy and patchy reionization 93
To compute the electron scattering and gravitational lensing effects from
the content of the simulation boxes, we only need to store two-dimensional
maps of the product of pressure and volume (Compton scattering), the velocity weighted free electron density (Doppler effect), and the matter density
(lensing). We simulate regions on the sky with an angular extent of 1◦ each,
at a resolution of 2562 pixels. In the small angle approximation we can simply project the content of each simulation box onto a plane in its center. At
high redshifts, z > 5.7, the field of view exceeds the boxsize, so we use the
periodic boundary conditions to cover it. During the patchy regime, only
those gas particles that are located in an ionized cell according to the information we stored earlier contribute to the signal. At later times all particles
are assumed ionized. We translate and rotate the positions and velocities of
the particles randomly before doing the projection onto the plane. The gas
particle properties are distributed over surrounding pixels weighted by the
SPH kernel. For dark matter particle masses we use the cloud-in-cell algorithm. Photons are traced through the planes from a regular grid close to
the observer towards the last scattering surface. We produced Gaussian random fields of the primordial CMB using spherical harmonic decomposition
coefficients generated with the publicly available code CMBFAST5 [56].
In models of patchy reionization, CMB anisotropies are caused by two
types of contributions induced by Doppler scatterings: perturbations in the
baryon density ωb , given by δωb = ρb /ρtot − 1, and local changes in the
ionization fraction δxe . These produce a change in temperature of the CMB
blackbody. The total contribution to the temperature anisotropy is given by
the integral over conformal time 6
Z
σT
∆TkSZ
(n̂) = − np (η0 ) dη[a−2 e−τri (η) x̄e (η)]n̂ · q
(4.1)
TCM B
c
with
δxe
)(1 + δωb )v .
(4.2)
x̄e (η)
The Thomson scattering optical depth is σT , τri is the optical depth from
the observer to conformal time η, and n̂ is the line of sight unit vector.
To implement patchy reionization, we use the semi-analytic model described in Chapter 2. We implement this model into the large scale structure
realization given by the high redshift outputs of our SPH simulation. Besides
the simplifications involved in the analytic modeling, there are two potential
q = (1 +
5
http://www.cmbfast.org
During homogeneous reionization and at lower redshifts, where galaxy clusters are
present, this is traditionally called the kinetic Sunyaev-Zel’dovich effect, at higher redshifts
the Ostriker-Vishniac [129] effect
6
94
The influence of inhomogeneous reionization on the CMB1
dynamical range limitations in our approach that we should address. First,
because the mean overdensity on scales larger than the simulation box is
artificially set to zero in numerical simulations, in order to achieve periodic
boundary conditions, we expect a systematic bias in the overall ionization
fraction. [45] estimated this effect and showed that it should be less than 1%
in simulations of 100 Mpc/h size. A different bias arises from our finite mass
resolution. Our lack of structure on scales smaller than Lbox /(Npart. )1/3 leads
to a slight delay in the onset of reionization. Very small overdensities on
scales below 0.5 Mpc/h, that could harbor ionizing sources with HII regions
around them, are not captured by our analysis.
4.3
Results for various time dependence of ζ
In our analysis we will compare three different reionization scenarios. These
are tuned such that they all lead to a comparable integrated optical depth,
τri ' 0.125 − 0.1337 The different histories of the fraction of ionized volume
elements Q(z) are shown in Figure 4.2.
Model A describes a universe that undergoes homogeneous and instantaneous reionization at redshift 14. This is the standard scenario [56, 130]
assumed in likelihood analyses of large scale CMB polarization anisotropy
(in its correlation with temperature, see [3, 14]). This model has τri = 0.133.
Model B is the patchy model described above, with the assumption of
constant source properties. An ionization efficiency ζ = 60 leads to reionization beginning at z = 19 and concluding by z = 12. This model has
τri = 0.130.
Model C exhibits extended patchiness. We describe this by assuming a
first generation of metal free sources starting at redshift 20 with an ionization
efficiency of ζ = 200 (it has been suggested that their photon output could
be 10-20 times higher than that of normal stars, see [131, 132, 133]). In our
model, these sources sustain themselves only to redshifts around 15, because
their hard photons dissociate fragile H2 they need for cooling (e.g. [134, 135]).
However, the first sources are assumed to leave pockets of ionized medium
which begin to harbor Pop II stars. The initial mass function becomes less
7
This value is at the high (1-σ) end of the currently preferred range of the WMAP
analysis [14]. We chose our reionization optical depth before the release of this updated
analysis, when the WMAP data pointed to an even higher value of τri = 0.17 ± 0.05 [3].
Our choice of τri was a compromise between the CMB data and quasar spectra results at
the time (pointing to a lower value of τri ' 0.06. We repeated our analysis for a lower
redshift scenario and find that the results are very similar. This is because the patchy
kSZ signal we are simulating is determined by the extend of reionization more than its
absolute redshift.
4.3 Results for various time dependence of ζ
95
top heavy with time, and we describe the resulting Population II stars with
an efficiency ζ = 12. This leads to a total optical depth of τri = 0.125. Our
model has a monotonically evolving ionization fraction. We note that another
way of reconciling CMB with quasar spectra observations is by resorting to
recombinations that leave 0.1% of the universe neutral until z = 6 − 7 [32].
In Figure 4.2, we plot for all three models the fraction of ionized volume
elements as a function of redshift, Q(z). In Model A, the neutral fraction
drops instantaneously at z = 14, in Model B sources of a constant ionization
efficiency ζ = 60 lead to a brief partly ionized phase, and in Model C the
ionization fraction “freezes in” for some time, while metal free Pop. III stars
cease to exist and leave residual ionized bubbles for “normal” Pop. II stars
to be created and to ionize the medium fully at z = 7 − 8.
We divide the total Doppler signal into three different redshift regimes
in Figure 4.3. The maps have a side length of 1◦ and are smoothed with
a Gaussian beam of width θ ' 10 , corresponding to a multipole number of
lmax ' 10000, comparable to the angular resolution of ACT, and slightly
below that of SPT. It is expected that during the interval from z = 0 to
z = 3, more strongly clustered regions make the Doppler effect maps highly
non-Gaussian. We show this by the solid line (PDF) in Figure 4.4. In the
picture in the center (z = 3 − 11) structure formation is less advanced. The
corresponding histogram can be well-approximated by a Gaussian, as shown
by the long-dashed curve in Figure 4.4. Signals created by radially moving
inhomogeneities during this epoch are referred to as the Ostriker-Vishniac
effect [129]. The plot on the right of Figure 4.3 is the patchy epoch of
reionization Model B. Again, this epoch has a largely Gaussian morphology
(short-dashed line of Figure 4.4), because structures on scales of the ionized
bubbles are just beginning to collapse. According to this figure, large bubbles
form on scales of tens of Mpc. The redshift regimes represented by the left
and middle picture have the same properties in all three of our reionization
scenarios.
96
The influence of inhomogeneous reionization on the CMB1
Figure 4.2 Evolution of the fraction of ionized volume elements Q(z) in our
calculations of reionization. Solid curve corresponds to instantaneous reionization at redshift z = 14, yielding an optical depth of τri ' 0.125 for standard cosmological parameters. The dashed curve represents our patchy model
in the case of a constant ionizing efficiency of sources. The dotted curve is for
a model where a first generation of metal-free stars can survive the negative
feedback of H2 photodissociation for a limited period of time, then formation
of these stars comes to a halt, leaving a network of HII regions. After a while,
Population II stars are born inside these regions and their photons gradually
lead to a homogeneously ionized universe (Model C). Model C spends the
longest time in the partly ionized regime and will therefore lead to the largest
signal of the three models which have comparable total optical depth.
Figure 4.3 Based on the implementation of Model B with a constant value of ζ, we divide the Doppler signal into
three redshift epochs. Plot on the left shows the kinetic Sunyaev-Zel’dovich effect out to redshift 3, in the middle the
Ostriker-Vishniac regime from z = 3 out to z = 11 is shown, and on the right the regime of non-uniform (patchy)
reionization. Each picture shows the same angular extent, 1◦ . The left panel exhibits highly clustered structures.
With increasing redshift, the clustering becomes more linear and patchy reionization leads to CMB signals on large
scales. We smoothed the maps obtained from our simulations by a Gaussian beam corresponding to l = 10000 ' 10 ,
comparable to the angular resolutions of ACT and SPT.
4.3 Results for various time dependence of ζ
97
98
The influence of inhomogeneous reionization on the CMB1
The upper panel of Figure 4.5 shows that in the homogeneously reionizing
universe, the largest portion of the Doppler signal comes from low redshifts,
the kSZ regime. The upper panel represents the case of homogeneous reionization, Model A. Contrary to the thermal SZ effect, the signal still carries
information about higher redshifts, z ≤ 10. We can use this principle to gain
knowledge about the details of reionization taking place at high redshifts.
The additional contributions from larger redshifts (z > 10) in Model C are
shown in the lower panel of the figure.
For comparison, we show in Figure 4.6 how different redshift regimes
contribute to the thermal SZ effect (where ∆T /T = −2y for low frequencies).
The plot shows that this signal is basically saturated at z = 3; the thermal
SZ effect receives most of its contribution from galaxy clusters in the more
nearby universe.
4.4
Observability of patchy reionization with
future CMB experiments
In this section, we assess the observability of patchy reionization with future
experiments. ACT and SPT should be able to distinguish different reionization scenarios with high significance by measuring temperature power spectra. In their cosmological parameter analyses, all-sky experiments such as
Planck will have to account for patchy reionization as a possible source of
bias and need to rely on their polarization data in order to obtain proper
constraints.
4.4.1
Power spectral constraints from ACT and SPT
The effect of patchy reionization on the CMB power spectrum is of similar
magnitude to that of the Doppler effect induced by variations in the mean
density. The first source population is heavily biased so the patchy reionization signal peaks on larger angular scales. Figure 4.1 shows that for the
regime in which the ionization fraction is roughly 50% (which is where most
of the signal comes from) the bubbles reach sizes of tens of comoving Mpc.
In Figure 4.7 we plot the different contributions to the Doppler power spectrum. Patchy reionization in Model C with ζ = 12 − 200 lasts from redshifts
20 to 7. The signal imprinted in the CMB from this epoch is shown in solid.
At later times, the universe is homogeneously reionized and the dashed line
shows the kinetic SZ effect for this period.
Inverse Compton scattering in galaxy clusters leads to a larger signal
than that caused by Doppler scattering. Because only a small fraction of the
4.4 Observability of patchy reionization with future CMB
experiments
99
Figure 4.4 The probability distribution functions for the three maps in Figure
4.3 are shown. The solid line is for the kinetic SZ regime z=0-3, the longdashed line for the Ostriker-Vishniac era z=3-11, the short-dashed line for
the patchy epoch of model B with constant ionization efficiency, z=11-19.
The two histograms representing the earlier cosmological epochs are wellapproximated by Gaussians.
100
The influence of inhomogeneous reionization on the CMB1
Figure 4.5 Contribution to the kinetic SZ signal for Models A (upper panel)
and C (lower panel) out to different redshifts. Although a large fraction of the
total Doppler effect comes from low redshift, high overdensity regions, there
is a significant contribution out to z ' 10. In the patchy model (bottom)
the regime z = 11 − 19 leads to further enhancement of the signal.
4.4 Observability of patchy reionization with future CMB
experiments
101
Figure 4.6 Redshift dependence of the thermal Sunyaev-Zel’dovich ∆T /T =
−2y effect as generated with our simulations. The major contribution comes
from galaxy clusters in the regime z < 3. Hence, the thermal SZ effect is less
suited for studying the reionization epoch than is the kinetic SZ effect.
102
The influence of inhomogeneous reionization on the CMB1
CMB photons present are likely to scatter inside the cluster medium (' 1%
for a massive 15 keV cluster), the distribution will not thermalize to that
of the hot gas. The effect has a characteristic frequency dependence, and
it can be distinguished from Doppler scatterings which leave the frequency
distribution of the photons unchanged. The dot-dashed line in Figure 4.7 is
the combination of the secondary signal with the primordial CMB, once the
thermal SZ with its characteristic frequency distribution has been removed.
On scales where patchiness during reionization contributes more to the
total signal than the Doppler effect owing to modulations in the density,
(l < 4000), the primordial anisotropies dominate. To study the secondary
anisotropies created during reionization one has to observe angular scales
where photon diffusion smoothes out the primary anisotropies.
Upcoming experiments such as ACT and SPT will observe the sky in a
number of frequency bands with comparable sensitivity and angular resolution. ACT [88] will observe in three frequency bands: 145, 225, and 265
GHz, with FWHM beam-widths θF W HM of 1.7, 1.1, and 0.9 arcminutes, respectively. We use the specifications of the 225 GHz channel for our power
spectrum analysis because the thermal SZ effect almost vanishes in this frequency regime, its zero being at ν ' 218 GHz. The sensitivity per resolution
pixel for this channel is σ = 2 µK. The South Pole Telescope [89] will observe
in five bands at 95, 150, 219, 274, and 345 GHz. The 219 GHz channel will
have an angular resolution of θF W HM = 0.690 and a sensitivity of 10µK 8 .
We assume sky coverages of 0.5% and 10% for ACT and SPT, respectively.
From these specifications and a template for the power spectrum of primary and secondary anisotropies at arcminute scales (which we obtain from
our simulations), we can calculate the errors in the Cl determination, including noise as additional random field on the sky [136]):
s
2
fsky
Cl +
,
(4.3)
∆Cl =
fsky (2l + 1)
wBl2
where we assume errors bars corresponding to a Gaussian map. Here, w =
(θf whm σ)−2 , and Bl is a beam
√ profile (assumed Gaussian) given by Bl =
−θb2 l(l+1)/2
e
(with θb = θf whm / 8ln2).
We assume that experiments such as ALMA 9 will lead to a good understanding of the point source frequency spectrum and angular clustering. In
more pessimistic scenarios where only the frequency dependence or nothing
is known about point sources, they can strongly degrade our ability to detect the kinetic SZ [137] (we included their estimate for the IR source power
8
9
J.E. Ruhl, private communication.
http://www.eso.org/projects/alma/science/alma-science.pdf
4.4 Observability of patchy reionization with future CMB
experiments
103
Figure 4.7 Different contributions to the total Doppler signal (kinetic SZ) in
our extended patchy reionization Model C. The solid curve is the contribution
to temperature anisotropies from the patchy regime alone at z = 7 − 20. The
dashed curve gives the Doppler effect from density modulations at homogeneous ionization out to z = 11. The dotted curve sums up those contributions
to the total Doppler signal. The total temperature fluctuations, cleaned of
the thermal SZ and IR sources (shown at 145 GHz in the thin dashed lines),
is given by the dot-dashed curve. These curves are smoothed versions of the
power spectra generated from 50 maps.
104
The influence of inhomogeneous reionization on the CMB1
spectrum at 145 GHz in Figure 4.7). On the other hand the assumption of
perfect cleaning of the thermal SZ effect is safer because we understand its
frequency dependence well.
To compare the experimental constraints with the power spectra extracted from our simulations on degree patches on the sky, we bin the errors
into bands of width ∆l = 360. The predicted measurements errors are plotted in Figure 4.8, together with power spectra generated from the “cleaned”
maps of primordial CMB and kinetic SZ combined. Since the kinetic SZ
appears almost featureless on the scales accessible to the next generation of
experiments, reionization models can be distinguished by their average amplitude on these scales. ThePerror bars in the relevant band-powers of ACT
2
= ( i σi−2 )−1 . It follows that the overall amplitude
are combined using σtot
can be measured with an accuracy of σ∆T = 0.011µK (we assumed a 1% calibration uncertainty). Given that the plateau of the extended patchy model
(Model C) lies at ∆T ' 2.407µK, while Model A has an average amplitude
on these scales of ∆T ' 2.076µK, the two will be distinguishable at the
30σ level with ACT, if we ignore contamination by point sources and by the
thermal SZ, which is not a realistic assumption. This is the same for the
South Pole Telescope, which has better angular resolution, but will look less
deep (it has a much larger survey area).
It is also of interest to ask whether additional constraints on reionization
scenarios could be achieved by using non-Gaussian statistics, in particular
the four point function (the skewness arising from a line-of-sight velocity
effect should vanish). It may be expected, that because the HII regions are
created inside large overdensities at high redshift, patchy models are more
Gaussian on the whole. The main problem with this notion is that in any
patchy model with a comoving bubble size comparable to ours this effect will
suffer from “washing out” by the primordial CMB. On scales where Doppler
induced fluctuations become larger than the primordial CMB, the impact of
patchiness amounts only to a fraction of the total signal (15% in Model B,
30% in Model C), the rest being attributed to scatterings owing to density
modulations alone. Also, the majority of non Gaussian contributions comes
from clusters and filaments at z < 3, compare Figure 4.4. We computed the
kurtosis
h(∆T /T )4 i
−3
(4.4)
Θ4 ≡
σT4
of thermal SZ “cleaned” maps (i.e. primordial CMB+Doppler) that were
filtered with a Gaussian in Fourier space, approximately cutting out “contamination” by the primordial CMB anisotropies below l ' 3500, and beam
smearing at scales of l ' 9000 comparable to the angular resolution of ACT
4.4 Observability of patchy reionization with future CMB
experiments
105
Figure 4.8 With future experiments like ACT, the various reionization models should be easily distinguishable by measuring the temperature power
spectrum. Since the kinetic SZ signal is almost featureless on the scales of
experimental relevance, the band-powers can be combined to a simple amplitude when distinguishing the different models. We find that the double
reionization scenario (given by the upper power spectrum) could thus be distinguished at high significance from a uniform, instantaneous model where
the universe reionizes at z = 14 (bottom curve). The change of the ionizing
efficiency with time ζ̇ affects the slope of the secondary anisotropy power
spectra.
106
The influence of inhomogeneous reionization on the CMB1
(similar to the window function proposed by [137]). Using this method, we
do unfortunately not find a statistically significant difference between patchy
and homogeneous models of reionization.
4.4.2
Expected bias in cosmological parameter determination from Planck
Cosmic microwave background measurements have been and will likely remain the most precise tools for the measurement of cosmological parameters.
The best constraints will come from a combination of CMB temperature and
polarization power spectra which encapsulate all the relevant information in
the sky maps. The aim is to make the data cosmic variance limited to as
high as possible multipole numbers. WMAP is cosmic variance limited up
to l ' 500.
The Planck satellite should achieve cosmic variance limitation out to l =
2500 for its temperature power spectrum measurement. Hence, Planck will
reach into the regime where the secondary anisotropies become important. To
avoid biases in parameter estimation, systematic changes that the secondaries
may produce in the power spectra need to be considered.
The contribution to the power spectrum from patchy reionization is one
to two orders of magnitude smaller than the primordial CMB anisotropies on
the relevant scales, but owing to the exquisite sensitivity of these experiments
it biases parameter estimates. For the analytic models of patchy reionization
suggested by [122] and [123] this parameter bias was estimated. In these
models the mean bubble size is smaller than in our computation, and the
signal peaks at higher multipoles.
The bias can be estimated from the Fisher matrix coefficients
Fij =
∂Cl
∂pj
(4.5)
P l D
Cl
Fij−1 l ∂C
∂p
p −1 j
Fii
(4.6)
X ∂Cl
∂pi
l
wl
in the following manner:
∆pi
Bi ≡
=
σi
P
j
where ∆pi are the systematic biases in the determination of parameters pi , wl
are the inverse squares p
of the statistical errors in the Cl estimation, given by
Equation 4.3 and σi = Fii−1 is the estimate of the error bars for parameter
i. In this expression, ClD denotes the combined Doppler power spectrum
owing to kSZ/OV and patchy reionization.
4.4 Observability of patchy reionization with future CMB
experiments
107
model
τri
ΩΛ
ωdm
ωb
ns
As
1σ
0.0035 0.010 0.0017 0.00018 0.0045 0.0050
A
0.40
1.18
-1.10
1.71
2.14
0.16
B
1.26
1.66
-1.54
2.40
3.05
0.78
C
2.25
2.89
-2.69
4.20
5.62
1.58
Table 4.1 Bias in units of the statistical error (Bi ) expected for cosmological
parameter estimation with Planck, if temperature and polarization power
spectra are used and the influences of kSZ/OV and patchy reionization are
neglected in the power spectrum analysis. The maximum multipole in our
analysis was l = 4000.
We combined the three frequency channels of Planck with the highest angular resolution and took into account the number of polarized instruments.
For a one year observation period, the three channels (217, 143 and 100 GHz)
have θfwhm = 5.0,7.1 and 9.2 arcminutes. This leads to the raw sensitivities:
wT−1 = (0.0084µK)2
wP−1 = (0.0200µK)2 .
We assumed a sky coverage of fSky = 0.8.
In Table 4.1 we show the results of our analysis, using the power spectra
for
our
p −1 different models as bias. The first line shows the statistical error alone,
Fii , the following lines show the parameter bias Bi for Models A, B, and
C. For Planck, we used the specifications of the High Frequency Instrument
10
. The power spectra derivatives for the Fisher analysis were computed for
the fiducial cosmological model given in the introduction.
It is clear from Table 4.1 that secondary anisotropies need to be taken into
account. This is the case in particular for parameters that influence the shape
of the power spectra at intermediate scales (ωb , ωdm , ns ). Constraints on the
amplitude As and the optical depth owing to reionization τri are less heavily
biased (besides the amplitude, the reionization optical depth only affects
CMB polarization on large scales). Note that even in a simple homogeneous
reionization model, as our Model A, most of the biases are of order unity. The
individual systematic shifts depend strongly on what parameters are used.
The analysis of [123] has extra parameters that make the biasing source (the
patchy power spectrum) distribute differently into each parameter offset.
The polarization anisotropies generated during the reionization epoch are
expected to be four orders of magnitude smaller than the temperature power
10
http://www.rssd.esa.int/index.php?project=PLANCK&page=perf top
108
The influence of inhomogeneous reionization on the CMB1
model
τri
ΩΛ
ωdm
ωb
ns
As
1σ
0.0041 0.013 0.0020 0.00026 0.0079 0.0056
A
0.010 -0.018 0.017
0.024
0.086 0.068
B
0.027 -0.028 0.026
0.037
0.135 0.196
C
0.050 -0.070 0.067
0.058
0.301 0.487
Table 4.2 Bias in cosmological parameter estimation with Planck is reduced
significantly, when temperature power spectra are used only until l = 1000
and polarization power spectra are used in the whole range (Planck should
be able to measure ClEE out to l ' 1800). On the other hand, the parameter
errors become only slightly larger by leaving out the high l temperature
information.
spectra [138]. This suggests that to avoid biases one could use temperature
information down to an angular scale where the Doppler contamination is still
negligible but use the polarization data for all l. We show the results of an
analysis in which we used temperature information only out to l ' 1000, but
polarization in the full range accessible to Planck (this will be out to l ' 1800)
in Table 4.2. The 1σ error bars for the parameters are only slightly larger
than in Table 4.1, indicating that the parameter estimates are more or less
“saturated” at l ' 1000. On the other hand, the expected parameter biases
owing to reionization are much smaller. The use of polarization information
in future CMB experiments thus can play an important role beyond breaking
degeneracies between traditional cosmological parameters and improving the
error bars.
The other strategy is to include an extra parameter to model the effect
of the Doppler contributions. This approach has the added advantage that
a positive constraint on reionization could be obtained with Planck alone.
Patchiness shifts the power spectrum on small scales, and we model this by
a “patchy amplitude” parameter. This parameter is included by adding the
Doppler spectra given in the last section with a variable amplitude AD to
the primordial CMB. If the parameter derivatives in the Fisher matrix center
(AD = 1) around Model B, we find that the amplitude parameter could be
constrained with σAD = 0.51. The model is only ' 0.3σ away from homogeneous reionization (Model A), so Planck will not be able to make a strong
statement. If patchy reionization turns out to be extended, similar to Model
C, Planck should observe this at higher significance σAD = 0.20, amounting
to a 5σ detection of Doppler induced secondary anisotropies, with the amplitude of Model C being 3σ away from homogeneous reionization. Finally,
if reionization proceeded homogeneously, Planck will not gain knowledge be-
4.5 Conclusions and Outlook
109
yond its large scale polarization measurements of τri by using small scale
temperature fluctuations, given that the uncertainty for our Model A lies at
σAD = 0.71.
When Planck’s power spectra are modeled with the “patchy amplitude”
parameter, constraints for the standard cosmological parameters are slightly
improved over the analysis that abandoned temperature data beyond l =
1000 (Table 4.2), despite the introduction of an extra parameter. Concretely, we find in this analysis that στri =0.037, σΩΛ =0.011, σωdm =0.0018,
σωb =0.00021, σΩns =0.0058, σAs =0.0052.
The Doppler power spectrum can bias the result of parameter analyses
with future CMB surveys. Planck may be able to improve constraints on
models with an extended epoch of patchiness. We hope to have shown in
this section that careful modeling of the epoch of first stars will become
crucial for doing precision cosmology with progressively more refined CMB
experiments.
4.5
Conclusions and Outlook
In this chapter we have presented simulations of secondary anisotropies in the
cosmic microwave background calculated using smoothed particle hydrodynamics simulations of large scale structure, focusing on the effect produced
by patchy reionization. We incorporated our semi-analytic model for the
morphology of the HII regions into our numerical treatment and investigated
whether such patchy scenarios can be distinguished from homogeneous reionization.
An important advantage of our technique over pure analytical predictions
of patchy reionization kinetic SZ is that we follow the complicated clustering
of dark matter and baryons into the slightly non-linear regime. In contrast
to full radiative transfer calculations of the reionization epoch, we combine
uncertainties in the physics of source formation, feedback processes and radiative transfer into a single parameter and explore the consequences of varying
this parameter. This simplification allows us to make predictions on scales
an order of magnitude larger than current radiative transfer schemes can
accomplish with a small expense of memory and CPU.
We extracted power spectra from sky maps produced by tracing rays
across our simulation volumes. The patchy reionization signal peaks on multipole scales of l ' 2000, and it increases the amplitude of the “cleaned”
CMB power spectrum by up to 30% on scales l ≥ 4000, so that the total
level of Doppler related anisotropies is ∆T ' 2.4µK. We found that with
the next generation of ground based CMB experiments (ACT, SPT) the dif-
110
The influence of inhomogeneous reionization on the CMB1
ferent reionization models we investigated could be distinguished with high
significance by using the power spectrum. Additional information about the
morphological properties of that epoch that could be obtained for instance
by measuring the four point function or other deviations from Gaussianity
will probably be difficult to obtain, because the reionization signal peaks on
angular scales where the primordial CMB anisotropies dominate. In the future, it may be possible to combine measurements of the CMB with other
observations such as 21 cm fluctuations from neutral hydrogen [75] or Lymanalpha emission from high redshift galaxies (e.g. [19]) to further constrain the
topology of the reionization process.
We investigated the bias in the determination of cosmological parameters
that will be produced by the additional patchy reionization signal, when extracting cosmological parameters from CMB anisotropy measurements. We
find that for Planck this bias is significant. The bias may be circumvented
by focusing completely on polarization information in the multipole regime
where patchiness peaks, with only a slight disadvantage in parameter constraints. Alternatively, a template for the Doppler spectrum could be introduced in the parameter analysis which may lead to a detection of the effects
of an extended reionization phase.
4.6
Appendix: Simulations of the thermal SunyaevZel’dovich effect and comparison to other
authors
The thermal Sunyaev-Zel’dovich effect (tSZ) is caused by the hot thermal
distribution of electrons in the intra-cluster medium (ICM) of clusters of
galaxies. For a massive cluster, the probability of a CMB photon passing
through to encounter a single interaction with an ICM electron is about
1%. Inverse Compton scattering increases the energy of the CMB photon by
roughly kB Te /me c2 , leading to a small distortion in the CMB spectrum. This
leads to a decrement in comparison to the primordial CMB at frequencies
below 217 GHz, and an increment above. This follows from the derivation in
[139], leading to the following result
∆Tth.SZ
(θ) = f (x) y(θ) where
TCMB
Z
σT kB
y(θ) =
dlne (θ, l)Te (θ, l)
m e c2
(4.7)
(4.8)
4.6 Appendix: Simulations of the thermal Sunyaev-Zel’dovich
effect and comparison to other authors
111
where the amplitude y is commonly referred to as the thermal Comptonization parameter. For an isothermal cluster it equals the optical depth, τe ,
times the fractional energy gain per scattering. The Thomson cross-section
is denoted σT , ne is the electron number density, Te the electron temperature,
kB Boltzmann’s constant, and me c2 the electron rest mass energy in the line
of sight integral. The characteristic frequency dependence of the thermal SZ
is
x
e +1
− 4 (1 + δSZE (x, Te )),
(4.9)
f (x) = x x
e −1
where δSZE (x, Te ) is a correction due to relativistic electron velocities in the
ICM. Note that f (x) → −2 in the non-relativistic and Rayleigh-Jeans (RJ)
SZE
limits so the equations simplify to ∆T
' −2y.
TCMB
From the above expression we realize that ∆TSZE /TCMB is independent
of redshift. This feature of the SZE makes it a unique tool for investigating
the high redshift universe.
To give a rough order of magnitude estimate of the thermal SZE, note
that the probability of scattering is 1%, and the spectral distortion for a 5
keV cluster is roughly 0.01, so the temperature fluctuation imprinted in the
CMB in this case would be of order 10−4 .
Our simulation of the thermal SZ effect was decribed in Section 4.2. We
show the map resulting from our simulation in Figures 4.9 (the Compton-y
parameter). It is the result of ray tracing through 76 simulation boxes, while
including the effect of weak gravitational lensing. Each data cube is randomly
rotated and translated along the line of sight before computation of its SZ
contribution. The final map has been smoothed with a beam corresponding
to a half arcminute, slightly smaller than the nominal resolutions of ACT
and SPT.
In Figure 4.10, we show the angular power spectrum, Cl l(l + 1)/2π of our
simulation in the gray curve. This is the result of running our ray tracing
algorithm ten times while using different seeds for the random orientations
of each box, to make the power spectrum smoother. We note that the effects
of gravitational lensing on the relatively featureless SZ power spectrum is
small.
As the simulation of the thermal SZ effect has a long history of analytic
and numerical modeling attempts, we wanted to compare our results to other
predictions. In particular we obtained the angular power spectra results of
[140, 141, 142, 143]. We included these results in Figure 4.10. We made
one slight modification to make this comparison more realistic: the different
authors have used different cosmologies. In particular their values for the
overall normalization of the matter power spectrum, σ8 , which denotes fluctuation amplitude on an 8 Mpc/h scale, is different. Also the baryon density
112
The influence of inhomogeneous reionization on the CMB1
2e-05
4e-05
6e-05
(counts)
8e-05
0.0001
Figure 4.9 Map of the thermal SZ effect, on a 1◦ map, extracted from our
simulations. This is the result of tracing 5123 rays through the 76 simulation
volumes between z=0-20. The color scheme shows the Compton-y parameter
for the thermal SZ (which corresponds in the Rayleigh-Jeans regime to 50%
of the temperature fluctuation).
4.6 Appendix: Simulations of the thermal Sunyaev-Zel’dovich
effect and comparison to other authors
113
which enters in the power spectra simply as ωb ≡ Ωb h, is different. Reference
[144] found a scaling of the thermal SZ with σ8 of Clth.SZ ∝ σ87 . Although
this is probably not the final answer, and other authors find scaling between
σ86−8 , we adopt the [144] scaling and multiply the curves obtained from the
authors by
0.97
0.0282
rescaled
authors
= Cl
× 7
× 2
Cl
σ8,authors ωb,authors
(σ8 = 0.9 and ωb = 0.028 being the parameters in our simulation) to achieve
a more proper comparison. We plan to investigate these scaling relation
further in future work. We find that the the re-scaled predictions for the
thermal effect differ from each other by a large factor, up to 70% on large
as well as small scales. In part this can be ascribed to modeling differences
in the gas distribution throughout the intracluster medium (ICM), in other
cases the analytic models have an advantage in that they can assemble more
massive objects, whereas the largest cluster we find in our simulation has a
mass of ' 8 × 1014 M at z=0, and there is a gap in the mass function down
to ' 2 × 1014 M . Massive clusters contribute a substantial fraction to the
signal [144], and their abundance needs to be modeled carefully.
The influence of inhomogeneous reionization on the CMB1
114
Recent thermal SZ models, re-scaled by σ87 (Ωb h)2 of the various cosmologies
1e-10
Cl l (l+1)/2π
primordial CMB
SPT errorbars
Holder&Carlstrom 1999
Zhang&Pen 2000
Seljak et al. 2000
Cooray 2000
Komatsu&Seljak 2002
our result based on SPH simulation
1e-11
1e-12
1000
10000
l
Figure 4.10 Recent estimates of the thermal Sunyaev-Zel’dovich angular
power spectrum based on various analytic and numerical modeling schemes,
as referenced in the key. The result from our simulation is shown in the
gray line. The various results differ significantly in amplitude and shape,
even after accounting for the different cosmological parametrizations used.
The green errorbars are estimated sensitivities for the Atacama Cosmology
Telescope. This shows that models have to improve substantially before the
scientific harvest from upcoming experiments can be fully realized.
Chapter 5
Simulations and Analytic
Calculations of Reionization
Morphology 1
In this chapter we present results from a large volume simulation of Hydrogen reionization. We combine 3d radiative transfer calculations and an Nbody simulation, describing structure formation in the intergalactic medium
(IGM), to detail the growth of HII regions around high redshift galaxies.
Our N-body simulation tracks 10243 dark matter particles, in a cubical box
of co-moving side length Lbox = 65.6 Mpc h−1 . This large volume allows
us to accurately characterize the size distribution of HII regions throughout
most of the reionization process. At the same time, our simulation resolves
many of the small galaxies likely responsible for reionization. It confirms a
picture anticipated by analytic models: HII regions grow collectively around
highly-clustered sources, and have a well-defined characteristic size, which
evolves from a sub-Mpc scale at the beginning of reionization to R & 10
co-moving Mpc towards the end. We show that in order to obtain this qualitative picture, source resolution must not be sacrificed at too great a level.
We present a detailed statistical description of our results, and compare them
with our numerical hybrid scheme based on semi-analytic modeling, described
in Chapter 2. We find that the analytic calculation reproduces the size distribution of HII regions, the power spectrum of the ionization field, and the
21 cm power spectrum of the full radiative transfer simulation remarkably
well. The ionization field from the radiative transfer simulation, however, has
more small scale structure than the analytic calculation, owing to Poisson
1
Based in part on O. Zahn, A. Lidz, M. McQuinn, S. Dutta, L. Hernquist, M. Zaldarriaga, S. Furlanetto, Astrophysical Journal, 654, in press (2006)
115
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Simulations and Analytic Calculations of Reionization Morphology
1
scatter in the simulated abundance of galaxies on small scales. We propose
and validate a simple scheme to incorporate this scatter into our calculations.
Our results suggest that analytic calculations are sufficiently accurate to aid
in predicting and interpreting the results of future 21 cm surveys. In particular, our fast numerical scheme is useful for forecasting constraints from
future 21 cm surveys, and in constructing mock surveys to test data analysis
procedures.
5.1
Introduction
In Chapter 2 we motivated the introduction of analytic models to describe the
morphology of reionization: modeling the largest scales while resolving the
smallest sources responsible is challenging. In this chapter, we push forward
by running a large volume radiative transfer simulation. Our work represents progress on several fronts. First, we simulate reionization in a larger
volume than most previous works (although see [34, 35]), while maintaining
high mass resolution. This allows us to reliably calculate the size distribution of HII regions as well as power spectra of ionization and 21 cm fields,
impossible with previous small volume simulations. Second, we compare our
results with the analytic calculations based on FZH04. These models are
now widely used, and while elegant and inspired by previous small volume
reionization simulations [32, 33], they remain untested. Our comparison also
gauges the level of theoretical control in our modeling of reionization – i.e.,
how robust are our conclusions to the details of our modeling? One convincing way to dissuade the above-mentioned skeptic is to demonstrate that we
can understand the gross features of our radiative transfer simulations analytically. Additionally, if analytic models are sufficiently accurate then they
are useful tools to forecast constraints from future experiments, and to construct mock surveys, providing important tests of data analysis procedures.
This is important given our ignorance of the nature of the ionizing sources:
we would like to cover a large parameter space in the source properties, prohibitive with time-consuming radiative transfer simulations. Furthermore,
future surveys will span volumes of several cubic Giga-parsecs, a challenging
task for detailed simulations.
We emphasize that our present work is only a first step towards more
realistic simulations of Hydrogen reionization. As we describe subsequently,
our radiative transfer simulations miss potentially important aspects of the
physics of reionization. Specifically, we include only a crude prescription
for the sources of ionizing photons, our coarse resolution underestimates the
importance of recombinations – especially if mini-halos are present during
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117
reionization [145, 146, 147] – and misses small galaxies that may contribute
ionizing photons, and we ignore feedback effects entirely. We intend to model
some of these effects in the near future.
Our work has overlap with the recent simulation and analysis of [35]. In
comparison to these authors, reionization finishes significantly later in our
simulation, near z ∼ 6.5, as compared to z ∼ 12, a consequence of our
more conservative prescription for the ionizing sources. Besides improving
the accuracy of simulations of the reionization epoch, an important emphasis
of our present work is in comparing our radiative transfer simulation results
with ‘hybrid simulations’ based on analytic models.
The layout of this chapter is as follows. In Section 5.2 we describe our
N-body simulation, source prescription and radiative transfer calculation. In
Section 5.3 we describe our ‘analytic model simulation’, which is more precisely an implementation of a model based on FZH04 into the cosmological
realization used for the radiative transfer simulation. We will sometimes refer
to this scheme losely as an ‘analytic calculation’ although the implementation of the model is entirely numerical. In Section 5.4 we present a detailed
statistical description of our radiative transfer and analytic results. We describe a numerical scheme that incorporates the stochasticity of the source
distribution into our analytic calculations in Section 5.5. We also show that
if extremely bright and rare sources reionize the IGM, bubble growth is less
collective than in our fiducial model.
In Section 5.6 we compare radiative transfer and analytic model predictions for the 21 cm signal. We conclude in §5.7, mentioning future research
directions and emphasizing possible improvements to our simulations.
Throughout we assume a flat, ΛCDM cosmology parameterized by: Ωm =
0.3, ΩΛ = 0.7, Ωb = 0.04, H0 = 100h km/s/Mpc with h = 0.7, and a scaleinvariant primordial power spectrum with n = 1, normalized to σ8 (z = 0) =
0.9 2
5.2
Simulations
We begin by running a large N-body simulation to locate dark matter halos,
and produce a cosmological density field. Next, we populate the dark mat2
This value for σ8 is slightly different than the value preferred by the WMAP satellite
alone of 0.76 ± 0.05 [14]. When combined with other datasets, such as the Lyman-α
forest [148, 149], and weak lensing (see e.g. section 4.1.7 and Table 6 of [14]), a higher
value of σ8 can be found. Furthermore, changes in the fluctuation amplitude within this
experimental range can be incorporated in our comparison between radiative transfer and
”analytic” schemes by changing slightly the ionization efficiency parameter. This does not
qualitatively affect our results.
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1
ter halos with ionizing sources, using a simple prescription to connect mass
and light (Section 5.2.2). In a subsequent post-processing step, we perform
a radiative transfer calculation, casting rays of ionizing photons from our
sources through the cosmological density field (Section 5.2.3). We make two
approximations with this approach. First, we assume that the gas distribution perfectly traces the dark matter distribution, as characterized by our
N-body simulation. Second, we neglect the interplay between gas dynamics
and radiation transport – i.e, in reality, structure formation responds to the
passage of ionization fronts, and gas motions in turn influence the propagation of the fronts. These effects are essential in calculating the detailed
small-scale behavior of ionization fronts, as fronts slow down upon impacting dense clumps [147], but are less important for our goal of capturing the
large-scale size distribution of HII regions.
5.2.1
N-body simulations
As noted in the introduction, we require a cosmological simulation with a
large dynamic range, in order to adequately sample the distribution of HII
regions, while simultaneously resolving small galaxies. Ideally, we would
resolve halos with virial temperatures of Tvir & 104 K – corresponding to a
dark matter halo mass of Mdm ∼ 108 M at z ∼ 6 – above which atomic line
cooling is efficient. In halos more massive than this, gas can cool, condense
to form stars, and produce ionizing photons. This ‘cooling mass’ therefore
represents a plausible guess as to the minimum host halo mass for ionizing
sources. If molecular Hydrogen cooling is efficient despite radiative feedback,
however, even smaller mass halos should host sources [150]. Presently, we
ignore this possibility. Additionally, high resolution is required to capture
the clumpiness of the IGM, and properly account for recombinations during
reionization. On the other hand, HII regions may be larger than R & 20 Mpc
h−1 at the end of reionization [50], necessitating a large volume simulation.
As described in Chapter 2, to resolve a 108 M halo with 32 particles, in
a simulation box of side-length L = 100 Mpc h−1 , for example, requires a
prohibitively large number of particles, Np ∼ 35003 !
Our present N-body simulation is meant to represent a compromise between these competing requirements of large volume, and high mass resolution. Specifically, our N-body simulation follows 10243 dark matter particles
in a box of side-length, L = 65.6 Mpc h−1 , using an enhanced version of
the TreePM code, Gadget-2 [151]. We run the simulation assuming the flat
LCDM cosmology specified in the introduction, with initial conditions generated using the [152] transfer function.
Dark matter halos are identified from simulation snapshots, using a friends-
5.2 Simulations
119
Figure 5.1 Halo mass function from our N-body simulation. The black points
with (Poisson) error bars indicate the halo mass function from our simulation as a function of redshift. The green curve is the Sheth-Tormen fitting
function for the halo mass function, while the red dashed line shows the
Press-Schechter fitting function.
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1
of-friends algorithm (e.g., [153]). Specifically, particles are grouped into halos using a linking length of b = 0.2 times the mean interparticle separation.
Linked groups of greater than 32 particles are considered resolved, and to
constitute dark matter halos. This corresponds to a minimum halo mass of
109 M , just an order of magnitude above the cooling mass.
The resulting mass function is shown in Figure 5.1, spanning a broad redshift range between z ∼ 6−20. [45] showed that if a simulation is normalized
to the cosmic mean density, the halo mass functions will be biased. According to Figure 3 of [45] the bias introduced in our calculations should only
be of order 0.1 %. The halo mass function is sampled with large dynamic
range, roughly three orders of magnitude near z ∼ 6. The simulated mass
function is always larger than predicted by the Press-Schechter formalism
[52], but generally in good agreement with the Sheth-Tormen [154] fitting
formula. At the highest redshifts sampled, however, our results fall in between the two fitting formula. This is in qualitative agreement with recent
measurements from [155], and [156], although our mass function appears systematically higher than that of [35]. The figure shows that the abundance
of our lowest mass halos is systematically below theoretical expectations,
likely a consequence of our limited mass resolution. As a conservative measure, we therefore place ionizing sources only in halos of mass larger than
Mmin = 2 × 109 M , corresponding to a 64−particle halo.
5.2.2
Ionizing Sources
Our next step is to connect mass with light – that is, we wish to populate
the dark matter halos from our N-body simulation with ionizing sources. In
this thesis, we will adopt a very crude prescription for our ionizing sources,
leaving a more sophisticated prescription to future work. This will facilitate comparison with the analytic models (see Section 5.3). Specifically, we
populate each dark matter halo with a single source whose luminosity in
Hydrogen ionizing photons is directly proportional to the host halo mass,
Ṅ = cMhalo . Clearly the parameter c encodes a good deal of complicated
physics, involving the efficiency of star formation, the efficiency of producing
ionizing photons, the fraction of ionizing photons that escape from the host
halo, etc. With this single simplifying assumption, the cumulative number
of ionizing photons released
the sources, per hydrogen atom in the IGM,
R t by
0
at time t is Nph /NH ∝ 0 dt fcoll (t0 ). Here fcoll (t0 ) is the fraction of mass
in halos with mass M ≥ Mmin = 2 × 109 M . Using the [154] mass function, which closely matches our simulation results (Figure 5.1), we find that
c = 3.1 × 1041 photons/sec/M yields one photon per hydrogen atom at
z = 6.5. (See Figure 5.2 and associated text for a discussion). This choice of
5.2 Simulations
121
c corresponds roughly, for example, to Pop II stars, forming with an efficiency
of f? = 0.1 from a Salpeter IMF, with a stellar lifetime of ∆t ∼ 5 × 107 yrs,
and a modest escape fraction of fesc ∼ 0.01 [157]. We adopt this conversion
in all subsequent calculations.
5.2.3
Radiative Transfer
We next form a coarse density field for many snapshots, spaced in equal time
intervals of ∆t = 5 × 107 years and spanning a broad redshift range from
z ∼ 6 − 16, by gridding our dark matter particles onto a uniform, Cartesian
grid with 2563 mesh points. Our sources (Section 5.2.1 and Section 5.2.2)
are tabulated at the same time-sampling, and moved close to the center of
their corresponding cell, with a slight offset of a fraction of a cell to avoid grid
artifacts. Occasionally, several sources land in a single cell and our considered
to be a sole, more luminous source. At z ∼ 6.5, near our assumed completion
of reionization, there are ' 330, 000 ionizing sources in our simulation.
With the ionizing sources and cosmological density field in hand, we trace
rays of ionizing photons through the simulation box using the adaptive raytracing scheme of [158], and the code of Sokasian et al. [159, 32]. We refer the
reader to these papers for the details of this code, but give a brief summary
here. In short, the code assumes a sharp ionization front, and tracks the
position of the front by casting rays and integrating over the ionization front
jump condition [160]. The jump condition amounts to tabulating the number
of photoionizations and recombinations along a ray, halting the ray when its
photon supply is exhausted. Each source is considered separately, although
the order in which sources are processed is randomized at each timestep to
avoid artifacts [32].
Behind the ionization front, each source hitting a given cell contributes
a photoionization rate of ΓHI,s = σ̄ Ṅ /(4πrs2 ), i.e. assuming optically thin
conditions within the front. Here rs is the distance from the cell in question
to a source, Ṅ is the number of Hydrogen photons per second from a source,
and σ̄ is a frequency-averaged cross section, computed here assuming each
source has a spectrum ∝ ν −4 [159]. Within the front, ionization fractions are
computed assuming ionization equilibrium, a uniform temperature of T =
104 K, case A recombination rates [159], and neglecting sub-grid clumping.
Helium is assumed to be at most singly-ionized by our soft sources, and we
assume that the HeII front precisely tracks the HII front. Similarly, inside
the front we assume that the ionized Helium (HeII) fraction traces the HII
fraction [32]. Note that all of these assumptions impact mainly the detailed
ionization fractions within the front, and are less important for tracking the
overall size distribution of HII regions. In contrast to [159, 32], we do not
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1
include a diffuse background radiation field, simply allowing rays to wrap
around the periodic box.
Our assumption of a sharp ionizing front is justified given the short mean
free path of Hydrogen ionizing photons in the pre-reionization IGM. [161]
present explicit comparisons between ‘ionization front tracking’ and more
detailed calculations that self consistently solve for the optical depth, ionization fraction, and temperature. At least in the case of a single source (their
Figure 16), ionization front tracking reproduces very closely the results of
more detailed calculations, further justifying our approach.
In Figure 5.2 we plot the redshift evolution of the ionization fraction in
our simulation. The black circles show the mass-weighted ionization fraction, while the blue squares show the volume-weighted ionization fraction.
The mass-weighted ionization fraction is somewhat larger than the volumeweighted ionization fraction. This is because the ionizing sources in our simulation are highly biased, and ionize their overdense environs before breakingfree to ionize neighboring voids (e.g. [32, 35]). The reionization process takes
a fairly significant stretch of cosmic time, with the mass-weighted ionization
fraction at the level of xi,m ∼ 0.1 at z ∼ 9, and attaining xi,m ∼ 1 only by
z ∼ 6.5.
The evolution of the neutral fraction in our model is consistent with, although not required by the measurements of e.g. [8], which demand only that
the IGM reionize sometime before z & 6. Our model produces an electron
scattering optical depth of τe = 0.06, on the low side of CMB constraints [7],
which suggest τe = 0.09 ± 0.03. We emphasize that our choice of c (Section
5.2.2) was calibrated so that reionization ends slightly above z & 6, so this
should be viewed as a consequence of our assumptions, rather than a theoretical prediction. Although our model is tuned to give late reionization,
analytic models find that the size distribution of HII regions depends primarily on the bias of the ionizing sources, with only an implicit dependence on
redshift [57]. The size distribution of HII regions at a given ionization fraction is therefore expected to be a robust result, independent of our detailed
assumptions about the efficiency of the ionizing sources, (although see [57],
Section 5.5 for caveats).
Note that our simulation terminates slightly before reionization completes
(xi (z) ∼ 1). We stop our calculation early because we do not include a ‘diffuse
background’ in our simulation [159], and so our calculation becomes very
expensive at the end of reionization when rays wrap around the simulation
box several times. In any event, the tail end of reionization is likely poorly
modeled in our simulation, since this stage may be regulated primarily by
Lyman limit systems [42, 55], which are missing in our analysis.
In our simulation, the mass-weighted ionization fraction closely tracks
5.2 Simulations
123
Figure 5.2 Ionization fraction as a function of redshift. The black circles
show the mass-weighted ionization fraction from the simulation, while the
blue squares show the volume-weighted ionization fraction. The red line is
the cumulative number of ionizing photons per hydrogen atom expected for
our ionizing sources. The close resemblance between the number of photons
per atom and the measured ionization fractions owes to the poor resolution
of our radiative transfer calculation, which underestimates the importance of
recombinations.
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Simulations and Analytic Calculations of Reionization Morphology
1
the cumulative number of ionizing photons per Hydrogen atom emitted by
our ionizing sources (see the solid red line in Figure 5.2), but this is partly
an artifact of the poor resolution of our radiative transfer calculation. Our
low grid resolution underestimates the amount of small scale structure in the
density field, and hence the importance of recombinations. In the future, we
intend to model recombinations as ‘subgrid physics’, accounting for enhancements in the recombination rate owing to unresolved small scale structure
(see e.g. [34]). Presently, we caution that we are under-estimating the number of ionizing photons per Hydrogen atom required to complete reionization.
Furthermore, we expect reionization to be even more extended than in our
calculation, since recombinations should slow the growth of HII regions.
5.3
Numerical scheme based on analytic considerations
As motivated in the introduction, we compare our results with the hybrid
scheme inspired by the analytic model of FZH04. In this chapter we only give
a brief review of this model and describe a modification we use to be able to
better compare it to the radiative transfer simulation. The major advantage
of our implementation over a purely analytic calculation is that the hybrid
scheme, which amounts to a Monte-Carlo realization of the analytic model,
can capture the asphericity of HII regions during reionization.
To remind ourselves briefly, the ionization criterion described in Chapter
2 compares the instantaneous luminosity of sources to the amount of neutral
hydrogen present in a region. The criterion was
√
(5.1)
δr ≥ δx (m, z) ≡ δc (z) − 2erfc−1 (ζ −1 )[σ 2 (rmin ) − σ 2 (r)]1/2 .
For constant mass to light sources, the criterion for a region to self-ionize
is modified to capture the integrated flux over collapsed objects at all time
instead of the luminosity of the present sources alone
Z t
α
dt0 fcoll. (m ≥ mmin |δm , t0 ) ≥ 1 ,
(5.2)
0
where α is an efficiency factor linking halo mass and ionizing photon yield.
A region can self-ionize if it is sufficiently overdense to satisfy the inequality
in Equation (5.2). In practice we find that the threshold criterion of Equation (5.2) gives quantitatively similar results to that of FZH04, although it
produces slightly larger HII regions. We show the new barrier in Figure 5.3,
together with a number of ‘random walks’ from our simulation. This content
5.3 Numerical scheme based on analytic considerations
6
125
constant M/L
FZH04
5
4
δx
3
2
1
0
-1
-2
1
10
R
Figure 5.3 Figure comparing the constant mass-to-light barrier with the barrier used in Chapter 4, where the instantaneous luminosity was compared
to the amount of neutral hydrogen seen. The new scheme produces slightly
larger bubbles, but the difference is not large. Both schemes are extreme
examples of what we would expect to see in reality: gas collapsing into halos
will take a finite amount of time to be converted into stars, however due
to feedback effects we expect the influence of small sources formed at the
beginning stages of reionization to cease toward the end.
of this figure is similar to Figure 2.3 in Chapter 2, but complementary in
that it plots the smoothed overdensities against the radius of the region of
influence of ionizing sources, as opposed to the variance of the density field
on that scale.
We described the numerical implementation of our semi-analytic simulations scheme in detail in Chapter 2.
In comparison to our radiative transfer simulation, the hybrid scheme is
quite fast: for our present 2563 grid calculation, with 50 logarithmic smoothing steps, the computation (at a given redshift) takes only ∼ 10 minutes on a
desktop computer with a 3 GHz processor. This is vastly more efficient than
our full radiative transfer calculation: our N-body simulation takes 38 hours
to run down to z ∼ 6 using 134 2 GHz processors, and our post-processing
calculation requires a few additional days of running time on a large memory
computer. With our rapid numerical scheme, we can produce an ionization
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Simulations and Analytic Calculations of Reionization Morphology
1
map based on the analytic model and compare with our radiative transfer
simulations. Using precisely the initial conditions from our N-body simulation in our hybrid calculation allows us to compare radiative transfer and
analytic ionization fields on a cell-by-cell basis.
Before presenting this comparison, there are a few more pertinent technical details. Ideally, we would compare the analytic and radiative transfer calculations with identical assumptions regarding the ionizing efficiency of our
sources, i.e. we should calibrate α in Equation (5.2) based on the source prescription of Section 5.2.2. In practice there are several difficulties with matching precisely the simulated source prescription. Most importantly, Equation
(2.4) is derived assuming sharp k-space filtering, while our smoothing procedure adopts a spherical top-hat in real space. This slight inconsistency in our
modeling means that our model does not conserve photons precisely, affecting the ionization fraction for a given source efficiency, α (see the Appendix).
Further, our simulated mass function is closer to the Sheth-Tormen fitting
formula [154] than the Press-Schechter [52] mass function, and we require
an analogue of Equation (2.4) for the Sheth-Tormen mass function [45, 57].
We improve on some of these shortcomings in Section 5.5. The upshot of
this is that, in order to compare with our radiative transfer simulations, we
adjust α in Equation (5.2) at each redshift to match the (volume-weighted)
ionization fraction. This readjustment is usually of order 20%.
We show examples of the resulting ionization maps in Figure 5.4. The
left column shows thin slices through the radiative transfer simulation at
three different stages in the reionization process: z = 8.16, 7.26 and 6.89
when the (volume-weighted) ionization fraction is xi,v = 0.13, 0.35, and 0.55
respectively. The right column shows corresponding slices from the hybrid
simulation scheme. Several conclusions are immediately apparent.
First, the ionized regions are quite large at the intermediate and late
stages of reionization. The ionizing sources are highly clustered, and HII
regions quickly start growing collectively around the sources, rapidly reaching much larger sizes than can be achieved by individual sources (FZH04).
Second, the hybrid simulation is in good general agreement with the radiative transfer simulation. The hybrid scheme seems to ‘locate’ the HII regions
found in the radiative transfer calculation, and additionally reproduces their
general morphology. Third, the HII regions in the analytic calculation are a
bit more ‘connected’ than those in the radiative transfer simulation. Equivalently, the ionization field in the radiative transfer simulation appears to
have more small scale structure than the ionization field from the hybrid
scheme. In the following sections, we will quantify the visual comparison of
Figure 5.4, diagnose differences found, and refine our numerical scheme. We
contrast the morphology seen here with that from [35] in Section 5.5.
5.3 Numerical scheme based on analytic considerations
radiative transfer
127
analytic constant M/L
z=8.16
z=7.26
z=6.89
Figure 5.4 Maps of the ionization field. The left column shows HII regions for a
thin slice through our radiative transfer simulations at redshifts z=8.16, z=7.26,
and z=6.89 (top to bottom). The volume-weighted ionization fraction at these
redshifts is xi,v = 0.13, 0.35 and 0.55, respectively. The slices are 0.25 Mpc/h
deep, and 65.6 Mpc/h on a side. The right panel shows the same using our hybrid
simulation scheme, as applied to the initial conditions used in our radiative transfer
simulation. The analytic modeling agrees well with the more detailed simulation,
although there is more small scale structure in the map from the radiative transfer
simulation (see text).
128
5.4
Simulations and Analytic Calculations of Reionization Morphology
1
Statistical Description
In this section we present a detailed statistical description of our results.
Throughout we will compare with our hybrid scheme rather than the purely
analytic calculations for two reasons. First, there are technical difficulties in
the analytic calculations at intermediate ionization fractions [54], and second,
we would like to be able to model non-spherical bubble shapes.
5.4.1
The Bubble PDF
The first statistic we consider is the probability distribution of bubble sizes.
That is, we calculate how large the HII regions are at different stages of
reionization. This depends somewhat on how one chooses to define contiguous ionized volumes – Figure 5.4 clearly illustrates that the ionized regions
are not spherical, particularly at the end of reionization. The ionized regions
do, however, obtain a reasonably well-defined characteristic size at each redshift. In order to quantify this, we require a convenient and well-motivated
definition of ‘bubble’ that we can apply consistently to the radiative transfer
simulation and the hybrid scheme.
Here we adopt a definition of bubble size inspired by the excursion set
formalism, upon which our analytic calculation is based (see [35] for an alternate approach). Specifically, we ‘draw’ spheres around each point in our
simulation box of varying radius, R, and average (smooth) the ionization field
within each such sphere. We start by considering large spheres, of volume
comparable to that of our simulation box, and step downward in size until we
eventually get to the size of our simulation pixels. At each smoothing radius,
R, we compare the average ionization in each sphere to a threshold ionization, xth . A pixel is marked as ‘ionized’ and belonging to a bubble of radius
R, when R is the largest smoothing radius at which the pixel’s smoothed ionization exceeds the threshold ionization, xth . If a given pixel fails to exceed
the threshold ionization at all smoothing scales, it is considered neutral (not
ionized).
The bubble pdf is then derived by tabulating the fraction of ionized pixels
that lie within bubbles with radius between R and R+dR. With this convention, the bubble pdf is normalized to unity rather than to the mean ionization
fraction. The results of this calculation are shown in Figure 5.5, for an ionization threshold of xth = 0.9. The figure illustrates quantitatively the visual
impression of Figure 5.4 : the HII regions have a well-defined characteristic
size at each stage of reionization, and this characteristic scale evolves as bubbles around neighboring sources overlap and grow collectively (FZH04, [57]).
The characteristic scale evolves from sub-Mpc scales at z = 8.16, when the
5.4 Statistical Description
129
Figure 5.5 Size distribution of HII regions as a function of redshift. The solid
curves show results from the radiative transfer simulation, while the dotted
curves are from the analytic calculation. We adopt a threshold ionization of
xth = 0.9 (see text). The volume-weighted ionization fractions at the redshifts shown are xi,v = 0.13, 0.22, 0.35, 0.55, 0.82 at z = 8.16, 7.68, 7.26, 6.89
and z = 6.56 respectively.
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Simulations and Analytic Calculations of Reionization Morphology
1
volume-weighted ionization fraction is xi,v = 0.13 to R & 10 Mpc co-moving
at z = 6.56 when the volume-weighted ionization fraction is xi,v = 0.82. The
large size of HII regions at high ionization fraction implies that large volume simulations are required to adequately sample this stage of reionization
([45], FZH04, [35]). The precise value of the characteristic bubble size depends somewhat on the number we adopt for the threshold ionization. For
instance, if we instead adopt the less stringent threshold of xth = 0.7, the
characteristic size increases by a factor of ∼ 2 near z = 8.16. Again, while
our definition of bubble-size is somewhat arbitrary, the bubbles nevertheless
have a well-defined characteristic scale [57], and our algorithm can be applied
consistently to each of the analytic model and radiative transfer ionization
maps.
The dotted lines indicate that our hybrid scheme reproduces the bubble
pdf simulated through radiative transfer quite accurately, roughly matching
the characteristic bubble size and its trend with redshift. The hybrid scheme
however leads to slightly larger HII regions at all but the final redshift. We
will discuss this difference in future sections. At the final redshift, the agreement is almost exact, however here our simulated volume is too small to
provide a representative sample.
5.4.2
Power Spectra of the ionized fraction
For further comparison, we measure the (spherically averaged) 3d ionization
power spectrum as a function of redshift. We consider the ionization field
δx = x(~r) − hxi, where x(~r) denotes the ionization at spatial position ~r, and
hxi denotes the volume-averaged ionization. Note that we do not normalize
by the mean ionization here, i.e. we consider the absolute ionization fluctuation, rather than the fractional fluctuation. The result of the power spectrum
calculation is shown in Figure 5.6, with power spectra calculated from the
radiative transfer simulation plotted in red. Throughout this chapter we plot
the dimensionless power spectrum, ∆2 (k) = k 3 P (k)/(2π 2 ), which yields the
contribution to the variance per logarithmic interval in k. On large scales
at high redshift the ionization power spectrum is proportional to the density
power spectrum, while it turns over or flattens on scales in which there are
ionized bubbles. Finally, on very small scales (k ' 10 h Mpc−1 ) the power
spectra ramp up, an artifact of discreteness noise. The bubble ‘feature’ moves
to progressively larger scales (small k) as reionization proceeds, a further illustration of the bubble growth seen in Figure 5.5. The blue dashed curves
show power spectra from our hybrid simulation, which are similar to the radiative transfer power spectra, except with slightly more large scale power,
and slightly less small scale power. One can also infer from the figure that
5.4 Statistical Description
131
0.1
2
∆ XX(k)
z=8.16, xi=0.11
0.01
z=7.26, xi=0.33
2
∆ XX(k)
0.1
0.01
Radiative Transfer
Halo-Smoothing
Analytic constant M/L
z=6.89, xi=0.52
2
∆ XX(k)
0.1
0.01
0.1
1
10
k [h/Mpc]
Figure 5.6 Power spectra of the ionized fraction, going from large redshift
(small ionization fraction) to small redshift (large ionization fraction). The
red lines are from the radiative transfer simulation, the blue dashed lines
are from the analytic hybrid calculation, while the purple dotted lines show
results from the improved scheme of the next section. The high-k behavior
(k & 10h Mpc−1 ) is an artifact from discreteness noise.
132
Simulations and Analytic Calculations of Reionization Morphology
1
1
z=6.89, xi=0.52
z=7.26, xi=0.33
z=8.16, xi=0.11
0.6
0.4
rx
i,radtrans.xi, analytic model
0.8
0.2
0
0.1
1
k [h/Mpc]
10
Figure 5.7 Cross correlation coefficient between the ionization fields from the
radiative transfer simulation and the analytic model calculations. The thin
lines show the cross correlation coefficient between the radiative transfer
and hybrid simulations at a few different redshifts. The thick lines show
corresponding results from the improved hybrid simulation described in the
next section.
an even larger volume simulation is preferable, in order to better sample the
large scale ionization power spectrum. Finally, the purple dotted lines are
from an improved numerical scheme which we discuss in the next section.
In order to further quantify the agreement between the radiative transfer
simulation and the hybrid scheme, we calculate the cross correlation coefficient between the two ionization fields. The cross correlation coefficient is
1/2
defined by r(k) = ∆2x1,x2 (k)/ [∆2x1 (k)∆2x2 (k)] . In this equation ∆2x1,x2 (k)
is the cross power spectrum between the radiative transfer simulation and
hybrid scheme ionization fields, while ∆2x1 (k), and ∆2x2 (k) are their respective
power spectra. The cross correlation coefficient is bounded between 1 and
−1, with r(k) = 1 indicating perfectly correlated modes, and r(k) = −1 designating perfectly anti-correlated modes. The results of this calculation are
shown as thin lines in Figure 5.7 (ignore, for now, the thick lines which show
results from the improved hybrid scheme introduced in the next section).
5.4 Statistical Description
133
The correlation coefficient is always larger than r ∼ 0.5 for scales larger than
k . 1h Mpc−1 , while it drops off on smaller scales. This quantifies the qualitative agreement suggested by Figure 5.4: the radiative transfer and hybrid
scheme ionization fields trace each other closely on scales larger than k . 1h
Mpc−1 . The cross correlation between the two fields becomes slightly weaker
at low redshift, as the average ionization increase. A plausible explanation
for the slightly worse agreement at low redshift is that our hybrid simulation
scheme has difficulty with ‘bubble mergers’ (see the Appendix), which are
more frequent at high ionization fraction.
Why does the cross correlation between the two fields drop off around
k & 1h Mpc−1 ? The analytic model assumes a one-to-one correspondence
between the abundance of halos and the (Lagrangian) matter overdensity on
a given smoothing scale. We know this is inexact. For one, the abundance
of our minimum mass sources is M dn/dM . 1 Mpc−3 . On ∼ 1 Mpc scales,
we therefore expect significant Poisson scatter in the abundance of ionizing
sources in our radiative transfer simulation (see also [57, 162]). To explore
this further, we compute the cross power spectrum between the halo density
field and the matter density field. The cross correlation coefficient between
the halo and matter density fields qualitatively mirrors the cross correlation
between the two ionization fields seen in Figure 5.7, dropping off at k & 1h
Mpc−1 . In other words, the halo bias is stochastic on scales of k & 1h Mpc−1
for our assumed source population. This stochasticity is not incorporated in
our analytic hybrid scheme, and likely leads to the lack of small scale structure compared to the ionization field simulated through radiative transfer.
We will return to this issue in Section 5.5. We note here, however, that this
Poisson scatter would presumably be less important if our radiative transfer
simulation resolved smaller, more abundant galaxies.
The analytic model connects ionized regions with large scale overdensities, which contain more sources and are reionized before underdense regions
(FZH04, [45]). The model therefore predicts that the ionization field is positively correlated with the matter density (e.g. McQuinn et al. 2005b), before
turning over on scales comparable to that of the ionized bubbles (FZH04).
Figure 5.8 shows the cross power spectrum between ionization and density
(bottom panel) as well as the cross-correlation coefficient between the two
fields. The radiative transfer simulation results (solid lines) nicely mirror the
analytic model predictions (dotted lines). In our radiative transfer simulation and hybrid scheme, reionization proceeds inside-out with the overdense
regions reionized before underdense regions, as emphasized by FZH04 and
Sokasian et al. (2003, 2004). Recombinations, underestimated in our present
simulations, could potentially weaken this correlation or, in an extreme case,
reverse the correlation with voids ionized first [42]. We intend to explore this
134
Simulations and Analytic Calculations of Reionization Morphology
1
in future work.
5.5
Improved numerical scheme
Although the agreement between our radiative transfer simulation and the
hybrid scheme is already quite good, we present here a modified numerical
scheme that improves upon the one presented in Section 5.3 and [163]. Specifically, we aim to fix two short-comings of the analytic calculation. First, as
mentioned previously, the analytic calculation is based on the Press-Schechter
formula for the collapse fraction. This formula is derived assuming sharp kspace filtering, while our scheme filters the initial density field with a top-hat
in real space, which is slightly inconsistent ([54], the Appendix). Second,
the mass function in our radiative transfer simulation (Figure 5.1) is closer
to the [154] mass function than the [52] mass function. Finally, the analytic
calculation assumes a one-to-one correspondence between initial over-density
and halo abundance. As we discussed in Section 5.4.2, the halo bias in our
N-body simulation is stochastic on small scales.
Each of these shortcomings can be remedied by directly using the simulated halos in our numerical scheme, rather than the Press-Schechter formula
for the collapse fraction. More specifically, we place the halo distribution
from our N-body simulation on a grid and compare, at each grid cell, the
halo mass to the total mass enclosed by a spherical top-hat. We then use
a condition analogous to Equation (5.2) to determine whether a region is
ionized by the sources within it. In other words, the calculation proceeds
exactly as in Section 5.3, except that we use the halo distribution directly
from the simulation, rather than Press-Schechter theory. Note further that
we now consider the evolved, non-linear density field rather than the initial,
linear density field to determine if a region can self-ionize. We will call this
improved numerical implementation the ‘halo-smoothing’ scheme in what follows. The CPU intensity of this scheme is again dominated by the number
of FFT’s necessary to achieve convergence in the bubble size statistic. As
with the analytic scheme, this is roughly 12 minutes on a 3GHz Intel Xeon
desktop computer.
The results of this new scheme are shown in comparison with radiative
transfer and analytic calculation in Figure 5.11, where we show 21 cm brightness temperature fluctuations (see Section 5.6) for a thin slice through the
simulation volume. This is analogous to Figure 5.4, except the ionized regions are now dark, the neutral regions now bright, and fluctuations in the
gas density are now visible in the neutral regions. The left column shows
results from our radiative transfer simulation, the right column shows the
5.5 Improved numerical scheme
135
Figure 5.8 Top panel: Cross correlation coefficient between the ionization and
density field. The solid (dotted) lines show the cross correlation coefficient
between the ionization and density fields in the radiative transfer simulation
(hybrid scheme) at several redshifts. Bottom panel: Cross-power spectrum
between the ionization and density field. Solid lines are calculations from the
radiative transfer simulation, while dotted lines are from the hybrid scheme.
136
Simulations and Analytic Calculations of Reionization Morphology
1
standard FZH04-type implementation, while the center column shows our
improved halo-smoothing scheme. The blue dots in the left and center column show the ionizing sources contained in the thin simulation slice. The
new scheme clearly resembles the full simulation more closely, with more disconnected ionized regions, owing to the presence of Poisson fluctuations in
the source distribution.
Figure 5.6 quantitatively illustrates improved agreement with the radiative transfer calculation, with our improved scheme showing more small scale
power than the hybrid simulation scheme. Figure 5.7 additionally shows the
cross correlation between the radiative transfer ionization field and the ionization field in the improved numerical scheme (thick lines). The halo-smoothing
ionization field traces the ionization field simulated through radiative transfer more closely, and down to smaller scales, than in our initial calculation.
We attribute the improved agreement largely to our incorporation, in the
improved scheme, of Poisson scatter in the halo abundance.
If the ionizing sources are even less abundant than we assume presently,
the Poisson scatter naturally becomes more important. Indeed for sufficiently
rare sources, Poisson fluctuations dominate over source clustering on the scale
of a typical bubble, and bubble growth is less ‘collective’ than in our fiducial
model. In this regime, the morphology of HII regions during reionization may
be qualitatively different. To examine this, we repeat our halo-smoothing
calculation at z = 7.26 including only halos with m ≥ 4 × 1010 M as sources.
We adjust the ionizing efficiency of these rarer sources upward to match
our usual ionized fraction at this redshift, xi,v = 0.35, in order to compare
maps at fixed ionization fraction. The result of this calculation is shown in
the left panel of Figure 5.9. One can see that the bubbles are considerably
more spherical than in our usual source prescription (middle panel), and that
the HII regions have a more sharply defined scale. The left panel further
illustrates that for this source prescription there are very few sources in each
bubble. Note that this is a thin slice, and some sources contributing to bubble
growth lie above or below it.
Figure 5.9 Dependence of reionization morphology on source density. In the left panel we show the ionization field
from our halo-smoothing procedure using only sources (white points) with mass larger than M ≥ 4 × 1010 M (note
that some sources contributing to the ionized regions lie in front or behind the thin slice shown). With this choice, the
number density of sources roughly matches that of M ≥ 2 × 109 M sources at z ∼ 14 (as in [35]). The center panel
shows the result with our usual source prescription, indicating a significantly more complex morphology. Finally the
right panel shows, for comparison, the analytic model with Mmin = 108 M . Each panel is at z = 7.26, and in each
case the source efficiencies are adjusted to match xi,v = 0.35.
5.5 Improved numerical scheme
137
138
Simulations and Analytic Calculations of Reionization Morphology
1
Furthermore, the left panel qualitatively resembles the morphology seen
in the reionization simulations of [35] (see their Figure 83 ). Their simulations
are done at higher redshift, but have a similar source number density as our
present, extreme choice of m ≥ 4 × 1010 M (with this choice our simulation
volume contains roughly 5,000 sources at z=7.26). We regard the morphology
seen in [35] as artificial and unlikely to represent the true morphology of HII
regions during reionization. Their choice of minimum source mass (Mmin =
2.5×109 M ) is driven by the low mass resolution of their simulations, and the
efficiency of their ionizing sources is boosted extremely high in order to match
first-year WMAP constraints [6]. In other words, their simulation represents
a very extreme case of reionization by rare, bright sources. Our simulation
is also missing plausible ionizing sources, given our comparable minimum
source mass. However, owing to the different assumptions about the ionizing
efficiency in our simulation, reionization occurs later and so our sources are
much more abundant (265,000 sources in the simulation volume at z=7.26).
We are hence still in the regime where HII regions grow collectively, and we
expect only small modifications to the morphology and size distribution of
HII regions when we include still smaller mass sources. This is illustrated
in the right panel of Figure 5.9 where we show predictions for our original
hybrid scheme (Section 5.3), with the minimum source mass extended down
to the cooling mass, Mmin ∼ 108 Msun . While there are some differences with
the results from our usual source prescription (center panel), the differences
are clearly far smaller than in comparison to the Poisson-dominated case (left
panel). The differences with [35] highlight the utility of our fast numerical
schemes for quickly examining many different prescriptions for the ionizing
sources and for understanding the robustness of the results.
5.6
21 cm signal and power spectra
The statistics discussed in Section 5.4 are largely diagnostic, aimed at describing the size distribution of HII regions in the simulation, and characterizing
the agreement between the radiative transfer simulation and analytic calculations. In this section we make a more observationally relevant comparison,
contrasting radiative transfer and analytic 21 cm power spectra.
We reviewed the theory of 21 cm fluctuations in Chapter 2. At the redshifts we consider presently, the 21 cm excitation temperature is likely coupled to the gas temperature, and much larger than the temperature of the
CMB (e.g. [67, 164, 73]), TS >> TCMB , implying δT ∝ (1 + δs )xH . We then
3
Note for comparison with these authors’ figure: their simulation box has a side length
of L = 100 Mpc/h, while ours has L = 65.6 Mpc/h.
5.6 21 cm signal and power spectra
139
model the 21 cm brightness temperature using the simulated density and
peculiar velocity fields, in conjunction with radiative transfer/analytic calculation simulated ionization fields. We incorporate here the effect of redshift
space distortions, taking into account the simulated peculiar velocity field.
On large scales, linear infall boosts the spherically averaged 21 cm redshift
power spectrum relative to its real space analogue, analogous to the ‘Kaiser
effect’ in galaxy surveys [165, 78, 80, 76].
The result of our power spectrum calculation is shown in Figure 5.10
for three different redshifts during reionization. The results are qualitatively
similar to those of Figure 5.6, and can be roughly understood by decomposing
the 21 cm power spectrum into three constituent pieces (FZH04):
∆221 (k) = T b [∆2xx (k) −
10
28
x̄H ∆2xδ (k) + x̄2H ∆2δδ (k)] .
3
15
(5.3)
Here ∆2xx refers to the ionization power spectrum, ∆2xδ refers to the ionizationdensity cross power spectrum, and ∆2δδ refers to the density power spectrum.
Note that, for illustrative purposes we ignore higher order terms [54, 57],
although their effects are included in our calculations. The numerical coefficients in this decomposition come from angle-averaging the redshift space
power spectrum. On scales much larger than the size of the ionized bubbles, each term in this decomposition is directly proportional to the density
power spectrum, and so the 21 cm power spectrum is directly proportional
to the density power spectrum. On the other hand, on very small scales one
would expect that the 21 cm power spectrum approaches the density power
spectrum multiplied by the neutral fraction squared (and a constant factor
' 1.87 for the spherically averaged redshift space case). The latter is shown
in the thin dashed curves in the Figure. The discrepancy see is due to the
significance of higher order terms that were neglected in Equation 5.3, that
in reality amount to corrections of order one.
These qualitative trends can be seen in Figure 5.10. For further illustration, we extrapolate our predictions to large scales using an analytic model
hybrid simulation (green long-dashed lines) which we based on a Gaussian
random field with sidelength 300 Mpc/h.
At high redshift, where the ionized regions are small, the 21 cm power
spectrum has the shape of the density power spectrum. It is evident that the
power spectrum does not trace the density power spectrum (or its rescaled
version Pδ x2H 1.87) exactly, as we would have naively expected. The reason
for this departure (in fact it is between 50% and 100% in the Figure shown)
lies in the importance of the higher order terms in Equation 2.26. The three
point terms are negative in the case of inside-out reionization, which explains
140
Simulations and Analytic Calculations of Reionization Morphology
1
z=8.16, xi=0.11
∆221(k) (mK2)
100
10
1
z=7.26, xi=0.33
∆221(k) (mK2)
100
10
1
z=6.89, xi=0.52
∆221(k) (mK2)
100
10
Radiative Transfer
Halo-Smoothing
Analytic constant M/L
1
0.1
1
10
k [h/Mpc]
Figure 5.10 The 21 cm brightness temperature power spectra in redshift
space. The solid red, short-dashed blue and dotted purple lines show the
radiative transfer, analytic, and halo-smoothing power spectra, respectively.
The green long-dashed lines show extrapolations of the analytic predictions
to large scales. Some of the differences in the predictions on large scales may
be attributable to our limited simulation volume. The redshift space 21 cm
power spectrum approaches Pδ x2H 1.87 (shown in the thin dashed curve) on
small scales. The differences seen are due to the relevance of higher order
contributions to the 21 cm power spectrum (see upcoming work).
5.7 Conclusions and Outlook
141
the discrepancy. We discuss this topic in a separate work not presented in
this thesis [77].
At lower redshifts, the 21 cm power spectrum begins to flatten on large
scales owing to the presence of ionized regions, before following the shape
of the density power spectrum again on small scales. This flattening moves
to progressively larger scales as reionization proceeds, and the bubbles grow
larger. Our first observational handle on the characteristic sizes of HII regions
at different stages of reionization will likely come from measuring the 21 cm
power spectrum, and observing this flattening. In other work, we will explore
the extent to which the size distribution of HII regions can be extracted from
future measurements of the 21 cm power spectrum (Zahn et al., in prep.).
Notice that the agreement between the analytic and radiative transfer 21
cm power spectra is even better than the agreement between the ionization
power spectra. While the ionization field in the radiative transfer simulation
has more small scale power than the analytic model ionization field, the
different approaches show similar amounts of small scale 21 cm power. This
owes to the small-scale dominance of the ∆2δδ (k) term in the 21 cm power
spectrum, which overwhelms the difference in small scale ionization power
(see Figure 5.6). The 21 cm power spectrum in each analytic scheme seems to
provide a very good approximation to the results of our full radiative transfer
simulations. Some of the difference on large scales may be attributable to
our limited simulation volume, and a convergence test with increasing boxsize
would be informative, but we leave this to future work.
5.7
Conclusions and Outlook
In this final chapter, we have presented results from a large volume radiative
transfer simulation and our fast numerical scheme based on analytic considerations, as well as an improved scheme which is based on the dark matter
halos found in the simulation. We have given a detailed comparison of the
three different reionization modeling schemes. Our basic conclusion is that
the approximate schemes agree remarkably well with the radiative transfer
simulation.
Future work should investigate the effect of recombinations which, we
anticipate, will lead to two primary modifications [55]. First, recombinations
will slow down reionization by requiring more ionizing photons to achieve a
given ionization fraction. This should mainly act to modify the redshift evolution of the ionization fraction, and not the size distribution of HII regions
at a given ionization fraction, our main focus in the present work. Second,
ionization fronts may be halted upon impacting dense clumps, where the re-
142
Simulations and Analytic Calculations of Reionization Morphology
radiative transfer
halo⌧smoothing
1
analytic constant M/L
z=8.16
z=7.68
z=6.89
Figure 5.11 21 cm brightness temperature fluctuations. We compare 21 cm maps
from the radiative transfer simulation and numerical scheme at three different
redshifts. Each map is 65.6 Mpc/h on a side, and 0.25 Mpc/h deep, comparable
to the frequency resolution of planned experiments, and shows a different cut then
Figure 5.4. The ionized fractions are xi,V = 0.13, 0.35 and 0.55 for z = 8.16, 7.26
and 6.89 respectively. Left column: Radiative transfer calculation with ionizing
sources (blue dots). Middle column: Halo-smoothing procedure (see text) with
sources/halos from the N-body simulation. Right column: Constant mass-to-light
ratio version of FZH04, based purely on the initial, linear dark matter overdensity.
5.7 Conclusions and Outlook
143
combination rate is very high (e.g. [42, 147, 55]). This latter effect might,
indeed, modify the size distribution of HII regions at a given ionization fraction. However, as long as mini-halos are destroyed by pre-heating prior to
reionization (e.g. [20]), estimates show this effect is important only at the
tail end of reionization, when xi,v & 0.85 [55], which we do not presently
simulate.
In the future we will address these issues explicitly, along with other
refinements to our radiative transfer simulations. We intend to consider a
more sophisticated prescription for the ionizing sources [166, 32], and extend
the mass range of our sources down to the cooling mass. It will be interesting
to examine how sensitive the 21 cm predictions are to the assumed properties
of the ionizing sources [57]. In particular, in Section 5.5 we found that the
morphology and size distribution of HII regions differs dramatically from our
fiducial model when extremely rare, bright sources dominate. This warrants
further quantitative investigation. Finally, we intend to examine the effect
of feedback on reionization, incorporating Jeans mass suppression (e.g. [43,
109, 167]) in reionized regions of the IGM.
In spite of these refinements, we contend that the agreement demonstrated illustrates that the analytic models are on the right track, and provide
a useful complementary tool to radiative transfer simulations. The approximate schemes described here are very fast, allowing quick coverage of a large
parameter space, convenient for forecasting constraints from upcoming 21
cm surveys (Zahn et al. in prep). Even full radiative transfer simulations
currently have a large number of free parameters related to the efficiency of
the ionizing sources, the escape fraction of ionizing photons, and sub-grid
clumping. Our numerical schemes allow one to gauge how the expected signal depends on these numerous, unconstrained parameters. It can also be
used to investigate non-Gaussianities in the 21 cm signal, as advocated by
[168], and to construct mock 21 cm survey volumes, providing a useful test of
data analysis procedures, which are presently still under development. This
is particularly relevant given that surveys like the MWA will be done in
large volumes of several co-moving cubic Gigaparsecs, prohibitive for current
radiative transfer simulations, but manageable with analytic calculations.
Finally, it might be interesting to couple the fast analytic model schemes
with a gas-dynamical calculation to investigate the impact of reionization on
galaxy formation.
144
5.8
Simulations and Analytic Calculations of Reionization Morphology
1
Appendix: Photon Conservation in our
approximate simulation schemes
The objective of this Appendix is to show that the pure FZH04 model conserves photons, but that our numerical schemes do not precisely conserve
photons. We then discuss the implications of this finding. In the pure
FZH04 model, we can prove that the global ionization fraction is given by
x̄ = ζ × fcoll. . This is just a reflection of photon conservation: as we sum
up the total ionized mass from individual HII regions, no photons are lost or
gained in our accounting of the net ionized mass.
A rigorous proof proceeds as follows. For simplicity, we outline this proof
using the pure FZH04 barrier, but the proof can be easily generalized to the
barrier of Equation (5.2). Let us consider random walks in the (δ, σ 2 ) plane
(e.g. Bond et al. 1991), generated using top-hat smoothing in k-space. We
consider the first up-crossing distributions for two types of barriers. First, we
examine the probability that a random walk crosses the ‘bubble barrier’, representing the critical density threshold for a region to self-ionize (see Figure 1
of FZH04). We denote the differential probability that a random walk crosses
this barrier, at a resolution between σ 2 and σ 2 + dσ 2 , by dPb /dσ 2 . Next, we
consider the ordinary Press-Schechter barrier, representing the critical overdensity for a region to collapse and form a halo. The differential probability
distribution for a random walk to cross the ‘collapse barrier’, at a resolution
between σ 02 and σ 02 + dσ 02 , is denoted by dPc /dσ 02 . Similarly, the probability distribution for collapse in a region with large-scale overdensity δb , on
smoothing scale σ 2 , is denoted by dPc (σ 02 |δb , σ 2 )/dσ 02 . The total ionized mass
in a region of large scale overdensity δb , at a smoothing scale σ 2 , is then given
by
Z
dσ 02 dPc (σ 02 |δb , σ 2 )
.
(5.4)
dMh ζ Mh
dMh
dσ 02
Note that the conditional probability distribution in this formula is calculated
by considering the fraction of random walks, originating at (δb , σ 2 ), that
cross the collapse barrier at higher resolution (Lacey & Cole 1993). The
mass calculated using Equation (5.4) is precisely the ionized mass in an HII
region that crosses the ‘bubble barrier’ at the point (δb , σ 2 ). In order to find
the total ionized mass in all HII regions, we merely need to integrate over
all such crossings, i.e., we integrate Equation (5.4) over σ 2 weighted by the
probability of crossing the bubble barrier. Symbolically, the total ionized
mass in the IGM is then given by
Z
2 Z
dσ 02 dPc (σ 02 |δb , σ 2 )
2 dPb (σ )
dσ
dM
ζ
M
.
(5.5)
h
h
dσ 2
dMh
dσ 02
5.8 Appendix: Photon Conservation in our approximate
simulation schemes
145
This is one expression for the total ionized mass in the IGM, obtained by
summing the ionized mass in all individual HII regions. Our proof of photon
conservation is completed by showing that this ‘local’ expression matches a
separate expression, proportional to the global collapse fraction. The total
mass in halos is simply
Z
dσ 02 dPc (σ 02 )
,
(5.6)
dMh Mh
dMh dσ 02
and the total, photon-conserving, ionized mass is just ζ times this expression.
Now, this expression, proportional to the global collapse fraction follows
by considering the crossing distribution of the collapse barrier, irrespective
of when each random walk crosses the bubble barrier. This result clearly
must match that of Equation (5.4) since for two random
variables,
x and
R
R
yR with probability distributions P (x) and P (y), dyP (y) dxxP (x|y) =
dxxP (x), i.e. in one case we are integrating (‘marginalizing’) over ‘bubble
crossings’, and in the other case we are not. This proves that the pure FZH04
model conserves photons, and our numerical implementation of the FZH04
model with a sharp k-space filter indeed conserves photons.
In practice, however the hybrid scheme of Section 5.3 smoothes the density field with a top-hat in real space, rather than a sharp k-space filter. In
this case photon conservation is not guaranteed. Specifically, the expression
in Equation (??) of Section5.3 is rigorously equal to the collapse fraction
only for sharp k-space filtering, and not for real-space smoothing (see also
[54]). One option would be to simply apply our algorithm with a sharp
k-space filter, but we find that this produces artificial features in our ionization maps (ringing in configuration space). For this reason, we prefer to
apply our algorithm using a top-hat in real space. In practice this leads to
photon non-conservation at the 20% level, with our algorithm systematically
under-shooting the expected ionization, x̄ = ζ × fcoll. . To compare with the
radiative transfer simulation, we simply boost the ionizing efficiency to make
up for this photon loss, matching the (volume-weighted) ionization fraction
in the radiative transfer simulation.
Is photon-conservation fulfilled in our improved ‘halo-smoothing’ scheme?
We consider a simple toy problem to illustrate that our improved scheme also
does not quite conserve photons. Imagine two equal luminosity sources in a
uniform density field. When the ionized regions surrounding these sources
begin to overlap, the spherical top-hat criterion can lead to somewhat unphysical features. This is sketched in the left panel of Figure 5.12. Our
algorithm does not allow for flux from one source to expand the HII region
surrounding the second source. Instead of both HII spheres (with initial radius r1 ) growing further during overlap, a new ionized region arises between
146
Simulations and Analytic Calculations of Reionization Morphology
1
them, the overlap of two spheres with radius r2 = 21/3 r1 . In Figure 5.12 we
plot the ratio of the ionized volume in our scheme, to the expected, photonconserving ionized volume. The figure clearly illustrates that our scheme
generally loses photons as two bubbles ‘merge’. The precise level of photon
loss in our ‘halo-smoothing’ scheme will depend on the ionized fraction, the
size distribution of the HII regions, the luminosity and bias of the sources
interior to merging bubbles, and the rate of merging bubbles. In practice,
the level of photon non-conservation in our halo-smoothing scheme is also
at the 20% level. Again our solution is to uniformly boost the ionizing efficiency of our sources to match the (volume-weighted) ionization fraction in
the radiative transfer simulation. Ideally, we would only boost the efficiency
in recently merged bubbles where we expect photon loss. In practice, any
error associated with this approximation appears small, although the higher
frequency of bubble mergers at late stages of reionization makes our scheme
slightly less reliable in this regime (see Figure 5.7).
5.8 Appendix: Photon Conservation in our approximate
simulation schemes
147
Vcompound/(2xVindividual)
1.05
1
0.95
0.9
0.85
0.8
0
0.5
1
1.5
D1,2 [RHII]
2
2.5
Figure 5.12 An illustration of photon non-conservation in our ‘halosmoothing’ simulations. The left panel shows the ionized region from our
numerical scheme for the toy problem of two equal-luminosity sources. Our
scheme leads to the HII region denoted by the thick solid boundary. In reality, one expects an oblong HII region, as photons from each source stream to
the edge of the HII region created by the other source, and expand its volume.
For this toy problem, our procedure initially over-estimates the ionized volume by a few percent at moderate source separations, then under-estimates
the ionized volume at smaller separations. In the limit of non-overlapping
HII regions (very large source separations), and in the limit of very small
source separations, we recover the expected ionized volume. This is illustrated in the right panel which shows the fractional photon loss/gain as a
function of source separation. The x-axis is the source separation in units of
the radius of an individual HII region.
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Simulations and Analytic Calculations of Reionization Morphology
1
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