Copyright

by

Marcelo Alonso Alvarez

2006

The Dissertation Committee for Marcelo Alonso Alvarezcertifies that this is the approved version of the following dissertation:

Structure Formation and the End of the Cosmic Dark Ages

Committee:

Paul R. Shapiro, Supervisor

Eiichiro Komatsu

Hugo Martel

John Scalo

Gregory A. Shields

Structure Formation and the End of the Cosmic Dark Ages

by

Marcelo Alonso Alvarez, B.S.

DISSERTATION

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN

December 2006

To my family

Acknowledgments

I want to express my sincere thanks to family, friends and colleagues who

have given me support over the years. First, I would like to thank my advisor Paul

Shapiro. I remember when I was just an undergrad, going to talk to Paul, and being

treated with kindness and respect. Those early discussions got me interested in

structure formation and gave me confidence, and there was no turning back after

that. In the years that followed, Paul taught me how to get to the bottom of

things, not to give up before achieving a clear understanding, and not to glaze

over crucial details out of laziness. Imitation is the sincerest form of flattery, and

I havel learned much from Paul through the example he sets. I will always look

back fondly on our many long discussions, and look forward to many more.

Of course, the UT Astronomy department is a wonderful place with many

great people. When I first started, Hugo Martel helped me get oriented, sharing

vital knowledge, from the Crown and Anchor to friends–of–friends. Hugo is simply

a great guy. I also feel lucky to have interacted with my collaborator and good

friend, Eiichiro Komatsu. Eiichiro’s intellect can be intimidating, but his jovial

nature makes it easy to get over, and his laughter is contagious. When Volker

Bromm arrived here, I started showing up at his office, asking questions and eager

to collaborate. I began to make lots of progress on our work together, which re-

sulted in a big confidence boost right when I needed it. I enjoyed our collaboration

very much and benefitted greatly from his sage advice. I also thank my committee

members John Scalo and Greg Shields for their comments and suggestions. My

officemate and collaborator for many years, Kyungjin Ahn, deserves special men-

v

tion. Having him as an officemate helped maintain my sanity and sense of humor

when things were tough. It would have been much harder without him there.

I am grateful for many discussions with my colleagues in the cosmology group:

Leonid Chuzhoy, Martin Landriau Beth Fernandez, Jun Koda, Jarrett Johnson,

Yuki Watanabe, and Donghui Jeong. Our cosmology meetings were lots of fun,

and I’ll miss them. Finally, I’d like to thank all my friends in the astronomy de-

partment for the good times, especially Eva, Claudia, Jeong-Eun, Robert, David,

Andrea, Niv, Mike, Martin. As I look back, I get nostalgic thinking of all those

parties, road trips, movie nights, dinners and happy hours. It’s moments like these

when you realize how important your friends really are. I know I’ll see you guys

in the future for sure.

Where would I be without my family. My mother Consuelo and her husband

Rex have always believed in me, and showed great patience with me in my often

foolish youth. Their support has been a daily source of strength throughout my

graduate career. My older siblings, Claudia, Carlos, and Monica, helped raise me

and I call upon them time and again to listen to my problems and offer their advice

when I ask. I look up to all of you, and each one of you is a wonderful parent to

your children, something I hope to emulate when I have kids. Daniel, you’re all

grown up now, but you’ll always be my little brother and have my support as you

make your way through life. Finally, my wife Shizuka. You are my best friend.

Words cannot describe how happy I feel to know we’ll always be together. As if

that weren’t enough, I’ve benefitted immeasurably from our scientific discussions

and your thoughtful comments on this thesis. I could not have done any of this

without you at my side.

vi

Structure Formation and the End of the Cosmic Dark Ages

Publication No.

Marcelo Alonso Alvarez, Ph.D.

The University of Texas at Austin, 2006

Supervisor: Paul R. Shapiro

We present results on the evolution of dark matter halos and reionization.

Dark matter halos enshroud galaxies, quasars and stars. As such, they are funda-

mentally important to structure formation. In studying reionization, we focus on

photoionization by the first stars, the 21-cm and cosmic microwave backgrounds,

and its large-scale structure. Several new and important results are presented.

First, we analyze the evolution of dark matter haloes that result from col-

lapse within cosmological pancakes. Their mass accretion history and concentra-

tion are very similar to those reported simulations of CDM. Thus, fundamental

properties of virialized halo formation and evolution are generic and not limited to

hierarchical clustering or Gaussian-random-noise initial conditions. We also find

that a simple one dimensional fluid model can explain this universal behaviour,

implying that the evolving structure of CDM halos can be well understood as the

effect of a universal, time-varying rate of smooth and continuous mass infall on an

isotropic, collisionless fluid.

We discuss cosmological reionization, from small scales and early times, to

large scales and late times. We have simulated ionization fronts (I-fronts) created

vii

by the first stars forming in “minihalos”. We find that nearby minihalos trap the

I-front, so their centers remain neutral, contrary to the suggestion that these stars

would trigger a second generation by ionizing neighboring minihalos cores. We

then turn to the cross-correlation of cosmic microwave background (CMB) and

21–cm maps. We find that its measurement can be used to reconstruct the reion-

ization history of the universe. Afterwards, we discuss the three versus first-year

data from the Wilkinson Microwave Anisotropy Probe (WMAP). Surprisingly, the

delay of reionization from three-year data is matched by a similar delay in struc-

ture formation. These effects cancel to leave the source halo efficiency constraints

unchanged. We conclude by analyzing the results of simulations of reionization,

and find that suppression and clumping reduce the size of H II regions. In addi-

tion, the analytical model of Furlanetto et al. overestimates the size distribution

of our simulated H II regions. We also study reionization topology through the

Euler characteristic.

Selected additional results and background are presented in the appendices.

viii

Table of Contents

Acknowledgments v

Abstract vii

List of Figures xiv

Chapter 1. Introduction 1

1.1 Dark Matter Halos . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Cosmic Dark Ages . . . . . . . . . . . . . . . . . . . . . . . . . 3

Chapter 2. A Model for the Formation and Evolution of Cosmolog-ical Halos 6

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Halo Formation via Pancake Instability . . . . . . . . . . . . . . . . 9

2.2.1 Unperturbed Pancake . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.3 N-body and Hydrodynamical Simulations . . . . . . . . . . . 13

2.3 Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Velocity Dispersion and Thermal Energy . . . . . . . . . . . . 16

2.3.3 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Virial Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 Jeans equation . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.1.1 Singular Isothermal Sphere . . . . . . . . . . . . . . . 21

2.4.1.2 Nonsingular Truncated Isothermal Sphere . . . . . . 22

2.4.1.3 Simulated Pancake Haloes . . . . . . . . . . . . . . . 24

2.4.2 Virial Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.2.1 Singular Isothermal Sphere . . . . . . . . . . . . . . . 26

2.4.2.2 Truncated Isothermal Sphere . . . . . . . . . . . . . 27

2.4.2.3 NFW Haloes . . . . . . . . . . . . . . . . . . . . . . 28

2.4.2.4 Simulated Haloes . . . . . . . . . . . . . . . . . . . . 30

2.5 Halo Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

ix

2.5.1 Accretion Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.2 Density Profile . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5.3 Virial Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5.4 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 3. The Universal Density Profile of CDM Halos from theirUniversal Mass Accretion History 44

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 The Universal Halo Profile of CDM N-body Simulations . . . . . . . 49

3.2.1 Density Profile . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2.2 Evolution of Halo Mass . . . . . . . . . . . . . . . . . . . . . 49

3.2.3 Evolution of Halo Concentration Parameter . . . . . . . . . . 50

3.3 Halo Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.1 Instantaneous Equilibration Model . . . . . . . . . . . . . . . 51

3.3.2 Radial Orbits Model . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.3 Fluid Approximation . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 4. The H II Region of the First Star 61

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Physical Model for Time-dependent H IIRegion . . . . . . . . . . . . 65

4.2.1 Early Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.2 Model for Breakout . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.1 Cosmological SPH Simulation . . . . . . . . . . . . . . . . . . 73

4.3.2 Ray Casting . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.3 Mass-conserving SPH Interpolation onto a Mesh . . . . . . . 75

4.3.4 Ionization Front Propagation . . . . . . . . . . . . . . . . . . 76

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4.1 Escape Fraction . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4.2 Ionization History . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4.3 IMF dependence . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4.4 Structure of H IIregion . . . . . . . . . . . . . . . . . . . . . . 83

4.4.5 I-front trapping by neighboring halos . . . . . . . . . . . . . . 85

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

x

Chapter 5. The cosmic reionization history as revealed by the CMBDoppler–21-cm correlation 95

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 21-cm Fluctuations and CMB Doppler Anisotropy . . . . . . . . . . 99

5.2.1 21-cm Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.2 Doppler Signal . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3 Doppler–21-cm Correlation . . . . . . . . . . . . . . . . . . . . . . . 101

5.3.1 Generic Formula . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3.2 Ionized Fraction–Density Correlation . . . . . . . . . . . . . . 103

5.3.3 Illustration: Homogeneous Reionization Limit . . . . . . . . . 107

5.3.4 Reionization History . . . . . . . . . . . . . . . . . . . . . . . 109

5.4 Prospects for Detection . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.4.1 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.4.2 Square Kilometer Array . . . . . . . . . . . . . . . . . . . . . 116

5.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . 118

5.6 Appendix 1: Density-ionization Cross-correlation . . . . . . . . . . . 120

5.6.1 Stromgren Limit . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.6.2 Photon Counting Limit . . . . . . . . . . . . . . . . . . . . . 122

5.6.3 Dependence of Collapsed Fraction on δ . . . . . . . . . . . . . 124

5.6.4 Final Expression . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.6.5 Bias in a Recombining Universe . . . . . . . . . . . . . . . . . 126

5.7 Appendix 2: Exact Expression for Cross-correlation . . . . . . . . . 128

5.7.1 Numerical integration . . . . . . . . . . . . . . . . . . . . . . 131

Chapter 6. Implications of WMAP 3 Year Data for the Sources ofReionization 132

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2 Structure formation at high redshift . . . . . . . . . . . . . . . . . . 137

6.3 Reionization History . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.3.1 Effect of recombinations . . . . . . . . . . . . . . . . . . . . . 141

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

xi

Chapter 7. The Characteristic Scales of Patchy Reionization 145

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.2.1 N-body simulations . . . . . . . . . . . . . . . . . . . . . . . 148

7.2.2 Radiative transfer runs . . . . . . . . . . . . . . . . . . . . . 150

7.3 Size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.3.1 Friends-of-friends method . . . . . . . . . . . . . . . . . . . . 154

7.3.2 Spherical Average method . . . . . . . . . . . . . . . . . . . . 162

7.3.2.1 Simplified toy model . . . . . . . . . . . . . . . . . . 165

7.3.2.2 Simulation results . . . . . . . . . . . . . . . . . . . . 167

7.3.2.3 Comparison to analytical model . . . . . . . . . . . . 168

7.4 Power spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.5 Topology: Euler Characteristic . . . . . . . . . . . . . . . . . . . . . 173

7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.7 Appendix: Distribution of H II region size for constant mass to lightratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Chapter 8. Discussion 186

Appendices 190

Appendix A. 21–cm Observations of the High Redshift Universe 191

A.1 Basics of 21-cm radiation . . . . . . . . . . . . . . . . . . . . . . . . 192

A.1.1 Spin temperature . . . . . . . . . . . . . . . . . . . . . . . . . 192

A.1.2 Radiative transfer of 21–cm radiation . . . . . . . . . . . . . 197

A.1.3 Observational considerations . . . . . . . . . . . . . . . . . . 199

A.1.3.1 Application to the first sources of 21–cm radiation . . 201

A.2 Minihalos and the Intergalactic Medium before Reionization . . . . 205

A.2.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 207

A.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

A.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Appendix B. Relativistic Ionization Fronts 215

B.1 Uniform Static Medium . . . . . . . . . . . . . . . . . . . . . . . . . 216

B.2 Static medium with a power-law profile . . . . . . . . . . . . . . . . 218

B.3 Cosmologically expanding medium . . . . . . . . . . . . . . . . . . . 223

B.4 Plane-stratified Medium . . . . . . . . . . . . . . . . . . . . . . . . 225

xii

Bibliography 229

Vita 245

xiii

List of Figures

2.1 Dark matter particles at a/ac = 3. . . . . . . . . . . . . . . . . . . . 10

2.2 Density profile of dark matter at four different scale factors . . . . . 11

2.3 Same as previous figure, but for simulation with gas included. . . . 12

2.4 Dimensionless specific thermal energy profiles . . . . . . . . . . . . 14

2.5 Anisotropy profile, defined two different ways . . . . . . . . . . . . . 15

2.6 Particle trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7 Departure from equilibrium in the Jeans equation . . . . . . . . . . 17

2.8 Virial ratio for TIS solution . . . . . . . . . . . . . . . . . . . . . . 23

2.9 Virial ratio calculated two different ways . . . . . . . . . . . . . . . 27

2.10 Virial ratio versus concentration parameter . . . . . . . . . . . . . . 29

2.11 Mass accretion history . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.12 Evolution of halo in self-similar regime . . . . . . . . . . . . . . . . 32

2.13 Radial velocity profile in dimensionless units . . . . . . . . . . . . . 34

2.14 Evolution of concentration parameter . . . . . . . . . . . . . . . . . 36

2.15 Virial ratio vs. scale factor. . . . . . . . . . . . . . . . . . . . . . . 41

2.16 Anisotropy parameter vs. scale factor. . . . . . . . . . . . . . . . . 43

3.1 Density profile from equilibrum model . . . . . . . . . . . . . . . . 48

3.2 Mass accretion history . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 Density profile at the end of the radial orbit simulation. . . . . . . 52

3.4 Density and circular velocity profile at end of fluid calculation . . . 55

3.5 Evolution of NFW concentration parameter with scale factor in thefluid approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.6 Phase-space density profiles . . . . . . . . . . . . . . . . . . . . . . 57

4.1 Density profile in star forming halo . . . . . . . . . . . . . . . . . . 63

4.2 Density and velocity profile in Shu solution . . . . . . . . . . . . . . 65

4.3 Timescales for breakout . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Instantaneous and time-averaged escape fraction . . . . . . . . . . . 71

4.5 Mean escape fraction vs. stellar mass . . . . . . . . . . . . . . . . . 73

4.6 Ratio of ionized gas mass to stellar mass vs. stellar mass . . . . . . 78

xiv

4.7 Volume visualization of H II region . . . . . . . . . . . . . . . . . . 80

4.8 Mass ionized vs. time . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.9 Position of selected SPH particles . . . . . . . . . . . . . . . . . . . 85

4.10 Recombination time and clumping factor vs. stellar mass . . . . . . 88

5.1 Schematic diagram of CMB-21–cm Correlation . . . . . . . . . . . . 104

5.2 Power spectrum of the cross-correlation . . . . . . . . . . . . . . . . 108

5.3 Peak correlation amplitutde vs. redshift . . . . . . . . . . . . . . . 111

5.4 Double reionization model . . . . . . . . . . . . . . . . . . . . . . . 113

5.5 Effect of bias on cross-correlation . . . . . . . . . . . . . . . . . . . 123

5.6 Power spectrum for power law fluctuation spectrum . . . . . . . . . 128

5.7 Power spectrum for CDM fluctuation spectrum . . . . . . . . . . . 129

6.1 Fluctuations vs. mass for differnt tilts and normalizations . . . . . . 135

6.2 Halo abundance vs. mass . . . . . . . . . . . . . . . . . . . . . . . . 137

6.3 Collapsed fraction vs. redshift . . . . . . . . . . . . . . . . . . . . . 139

6.4 Evolution with redshift . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.1 H II and H I size distributions, by number . . . . . . . . . . . . . . 149

7.2 H II and H I size distributions, by volume . . . . . . . . . . . . . . 152

7.3 Effect of varying the ionization threshold in the FOF method . . . . 153

7.4 Evolution of FOF region sizes . . . . . . . . . . . . . . . . . . . . . 155

7.5 Comparison of FOF size distributions . . . . . . . . . . . . . . . . . 156

7.6 Simple toy model for spherical average method . . . . . . . . . . . . 160

7.7 Spherical average size distributions . . . . . . . . . . . . . . . . . . 161

7.8 Comparison of size distributions using the spherical average method 163

7.9 Comparison to analytical model . . . . . . . . . . . . . . . . . . . . 164

7.10 Power spectra of ionized fraction at the half-ionized epoch. . . . . 170

7.11 Cross-correlation coefficient at the half-ionized epoch. . . . . . . . 171

7.12 Euler characteristic and number of FOF regions for each run . . . . 174

7.13 Comparison of Euler characteristic for selected runs . . . . . . . . . 181

7.14 Linear barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.15 Comparison of analytical models for size distribution . . . . . . . . 184

A.1 Fit to data in Zygelman (2005) . . . . . . . . . . . . . . . . . . . . 195

A.2 Lymanα pumping by cluster of stars . . . . . . . . . . . . . . . . . 203

A.3 Power spectrum of 21 cm signal at z = 20 . . . . . . . . . . . . . . 204

xv

A.4 Map of differential brightness temperature . . . . . . . . . . . . . . 209

A.5 Evolution of mean differential brightness temperature . . . . . . . . 210

B.1 I-front evolution for static gas . . . . . . . . . . . . . . . . . . . . . 218

B.2 I-front evolution for a power-law density profile . . . . . . . . . . . 220

B.3 I-front evolution for cosmologically-expanding gas . . . . . . . . . . 222

B.4 I-front peculiar velocity . . . . . . . . . . . . . . . . . . . . . . . . . 223

B.5 Two-dimensional I-front evolution in a plane-stratified medium . . . 226

B.6 I-front evolution along the symmetry axis . . . . . . . . . . . . . . . 227

xvi

Chapter 1

Introduction

Just after recombination, the universe was a relatively simple place. There-

after, small amplitude density fluctuations grew, eventually collapsing to form

virialized structures – halos. In some of these halos, stars and perhaps even black

holes formed, emitting a copious amount of energy, which affected subsequent

structure formation profoundly. The universe was eventually reionized, populated

by the myriad galaxies and globular clusters that we see today. This begs the

question: how did this transition from relative simplicity to nearly unimaginable

complexity occur, and what can observations tell us about it? The answers will

allow us to relate fundamental global properties of our universe, such as the shape

and amplitude of the initial power spectrum of density fluctuations, to its observed

properties.

1.1 Dark Matter Halos

Dark matter halos are the building blocks of structure formation. In order

to understand how astronomical observations constrain particle physics models of

dark matter, it is vital that we develop a complete theory of their structure and

evolution. Insofar as dark matter interacts only gravitationally and the statistics

of its initial linear fluctuations are Gaussian, the problem of dark matter halo

formation is very well posed. In spite of this, our understanding is still incomplete.

This is illustrated most vividly by the core/cusp discrepancy: halos forming in

1

N-body simulations of cosmic structure formation in a cold dark matter (CDM)

(Blumenthal et al. 1982) universe have radial density profiles ρ ∝ r−α with α ∼ 1

(e.g., Navarro, Frenk, & White 1997; Moore et al. 1999; Ghigna et al. 2000; Power

et al. 2002; Diemand et. al. 2005), whereas rotation curves of galaxies imply

α ∼ 0 (e.g. Flores & Primack 1994; Marchesini et al. 2002; de Blok & Bosma

2003). Similar evidence for α ∼ 0 is also present in observations of galaxy clusters

(e.g., Tyson, Kochanski, & dell’Antonio 1998; Sand et al. 2004; Broadhurst et al.

2005). Several explanations have been proposed to resolve this discrepancy, such as

self-interacting dark matter (Spergel & Steinhardt 2000), gas dynamical processes

(El-Zant et al. 2001; Weinberg & Katz 2002), or the shape of the primordial power

spectrum (Zentner & Bullock 2002; Ricotti 2003). The solution to the discrepancy,

however, still remains elusive.

The discussion above illustrates the need for a firm theoretical of the un-

derstanding of how halos form, merge, evolve, and grow in mass. The nonlinear

outcome of the collapse of even collisionless dark matter, without including the

baryonic component or exotic collisional interactions, is itself a difficult problem

which has not yet been solved. Numerical simulations are ultimately necessary,

but without an intuitive qualitative understanding, the simulation results are a

“black box”, and we are left unprepared for how to solve the descrepancies which

inevitably arise. In this dissertation, we attempt to contribute to this qualitative

understanding, by investigating certain universal properties of halo formation and

evolution, such as the mass accretion history and density profile. N-body simula-

tions of halo formation from cosmological pancake instability and fragmentation

are presented in Chapter 2, while one-dimensional fluid approximation simulations

are presented in Chapter 3.

2

1.2 The Cosmic Dark Ages

New observations are revealing a complex picture of the high redshift uni-

verse, in the epoch that marks the aftermath of the end of the “cosmic dark ages”,

a phrase which was first coined as the “dark age” by Rees (1997). The dark ages

ended when the first stars and quasars formed, beginning and eventually complet-

ing the reionization process. As these observations peer further and further back

into cosmic history, we approach the promise of finally seeing that epoch when

the very first structures were forming, providing the initial conditions for all that

would follow. The theory of reionization provides a crucial missing link, connecting

the relatively simple universe in the dark ages to the complex universe that is now

observed.

The appearance of a Gunn-Peterson trough (Gunn & Peterson 1965) in the

spectra of distant quasars shows that reionization was ending at a redshift z ∼ 6

(Becker et al. 2001; Fan et al. 2002). The large-angle polarization anisotropy of

the cosmic microwave background (CMB) observed by the Wilkinson Microwave

Anisotropy Probe (WMAP) indicates a large optical depth to Thomson scattering,

implying the universe was substantially reionized by a redshift z ∼ 11 (Spergel et

al. 2006). The Hubble Ultra Deep Field has revealed galaxies in which stars

comprise as much as a few times 1011M at redshifts z ∼ 6 − 7 (Mobasher et al.

2005; Panagia et al. 2005), perhaps the main sources responsible for reionization.

Spectroscopic observations of Lyman-α emitting galaxies at redshifts z ∼ 5 − 7

are posing puzzles with regard to the ionized state of the intergalactic medium

(IGM) at those redshifts (e.g., Haiman 2002; Hu et al. 2002; Malhotra & Rhoads

2004). Observations of the near infrared background excess (e.g. Matsumoto

et al. 2005) can be interpreted as originating in a very abundant population of

3

massive stars forming at redshifts z > 7 (e.g. Santos, Bromm, & Kamionkowski

2002; Salvaterra & Ferrara 2003; Fernandez & Komatsu 2005). Taken in their

totality, these observations pose several challenges to our current understanding of

the theory of reionization. The next generation of telescopes, such as the James

Webb Space Telescope (JWST) and the Square Kilometer Array (SKA) will probe

even more deeply into the dark ages, no doubt answering some of these questions

while confronting us with new ones.

At the same time that observations are revealing the universe in its earliest

stages of structure formation, modern computational power and numerical tech-

niques are revolutionizing our understanding of this epoch. At small scales and

early times, where the theories of star formation and structure formation meet,

simulations are converging on the formation of the first generation of stars (e.g.

Nakamura & Umemura 2001; Abel, Bryan, & Norman 2002; Bromm, Coppi, &

Larson 2002). The problem of their formation is well-posed, since ambiguities hav-

ing to do with stellar feedback and metal pollution are not present in the initial

conditions. These stars likely began reionization, creating highly asymmetric H II

regions and substantially altering the conditions for subsequent star formation. On

larger scales, simulations have begun to model the global process of reionization,

including physical processes occuring over a wide range of length and time scales

(e.g. Ricotti, Gnedin, & Shull 2002; Ciardi, Ferrara, & White 2003; Sokasian

et al 2003; Iliev et al. 2006a). On these larger scales, however, the problem is

very complex, with many free parameters. Because of resolution limitations, much

of the physics on sub-grid scales is uncertain, making it difficult to interpret the

simulation results. Analytical models (e.g., Haiman & Holder 2003; Furlanetto,

Zaldarriaga, & Hernquist 2004a; Iliev, Scannapieco, & Shapiro 2005) will therefore

4

continue to play a complementary role to the numerical simulations.

In this dissertation, we study several different aspects of the end of the dark

ages and cosmological reionization. We begin at early times and on small scales

in Chapter 4, where we use detailed three-dimensional smoothed particle hydro-

dynamics (SPH) of early structure formation to study the structure of the highly

asymmetric H II regions that were likely to have formed around the first generation

of population III stars. These stars likely began the reionization process, and it

is vital to understand how their radiative feedback affected their environment and

subsequent structure formation. In Chapter 5, we examine the cross-correlation

between cosmic microwave background (CMB) and 21–cm maps of the early uni-

verse. In particular, we address the question of what the large scale correlation,

on degree angular scales, can tell us about the global reionization history of the

universe. In Chapter 6, we use the latest state-of-the-art large-scale simulations of

reionization to examine its characteristic scales. We discuss several different quan-

titative descriptions of the geometry and topology of the reionization process, and

investigate how different assumptions, such as the degree of clumping and source

suppression, affect the results. Several supplemental results are presented in the

appendices, including a brief introduction to the basics of 21–cm radiation from

the early universe, and a discussion of relativistic ionization fronts.

5

Chapter 2

A Model for the Formation and Evolution of

Cosmological Halos

We study the collapse and evolution of dark matter haloes that result from

the gravitational instability and fragmentation of cosmological pancakes. Such

haloes resemble those formed by hierarchical clustering from realistic initial con-

ditions in a CDM universe and, therefore, serve as a convenient test-bed model

for studying halo dynamics. Our halos are in approximate virial equilibrium and

roughly isothermal, as in CDM simulations. The halo density profile agrees quite

well with the fit to N -body results for CDM haloes by Navarro, Frenk, & White

(NFW).

This test-bed model enables us to study the evolution of individual haloes

as they grow. The masses of our haloes evolve in three stages: an initial collapse

involving rapid mass assembly, an intermediate stage of continuous infall, and

a final stage in which infall tapers off as a result of finite mass supply. In the

intermediate stage, halo mass grows at the rate expected for self-similar spherical

infall, with M(a) ∝ a. After the initial collapse and virialisation at epoch (a =

a0), the concentration parameter grows linearly with the cosmic scale factor a,

c(a) ∼ 4(a/a0). The virial ratio 2T/|W | just after virialisation is about 1.35, a

value close to that of the N -body results for CDM haloes, as predicted by the

truncated isothermal sphere model (TIS) and consistent with the value expected

for a virialized halo in which mass infall contributes an effective surface pressure.

6

Thereafter, the virial ratio evolves towards the value expected for an isolated halo,

2T/|W | ' 1, as the mass infall rate declines. This mass accretion history and

evolution of concentration parameter are very similar to those reported in N -

body simulations of CDM analyzed by following the evolution of individual haloes.

We therefore conclude that many of the fundamental properties of virialized halo

formation and evolution are generic to their formation by gravitational instability

and are not limited to hierarchical clustering scenarios or even to Gaussian-random-

noise initial conditions1.

2.1 Introduction

Dark matter haloes are the fundamental structures within which galaxies

and clusters form. Halo formation, internal structure, and evolution are therefore

key elements in the theory of galaxy formation. In the current structure forma-

tion paradigm, small-amplitude random Gaussian density fluctuations present at

high redshift are amplified over time by gravity, leading to the formation of self-

gravitating virialized dark matter haloes. Due to the complex three-dimensional

nature of this problem, a fully analytical treatment of halo formation and evolution

is not possible. N-body simulations with three-dimensional Gaussian random noise

initial conditions are ultimately necessary, even with gas dynamics neglected.

Realistic halo models should share important characteristics with those

formed in the realistic N-body simulations from Gaussian-random-noise, such as

the mass accretion rate and density profile shape. Recent analytical and numerical

studies have revealed several “universal” characteristics of halo formation. Using

the extended Press-Shechter formalism (EPS; e.g. Lacey & Cole 1993), van den

1This work appeared in part in Alvarez, Shapiro, & Martel 2003, RevMexSC, 17, 39

7

Bosch (2002) has discovered that the mass accretion histories of individual haloes

built up by simulated merger trees in a CDM universe have a universal shape,

which can be fit by two parameters. Wechsler et al. (2002) have also found a uni-

versal mass accretion history, with only one free parameter, by direct examination

of a large sample of haloes in an N-body CDM simulation. These two accretion

histories are similar, although van den Bosch (2002) suggests that the Wechsler

et al. (2002) fit is a better description of the mass accretion late in the evolution

of a halo, while the van den Bosch (2002) fit is a better description early in its

evolution. Wechsler et al. (2002) also found that the concentration parameter

of the simulated haloes is strongly correlated with the mass accretion history, in-

creasing linearly with the scale factor a as the halo evolves. A natural question

which emerges is whether the fundamental properties of dark matter haloes, like

their density profiles and mass accretion histories, are a consequence of hierarchical

clustering from Gaussian-random-noise density fluctuations or are in fact a more

general result.

Several studies have attempted to answer this question with regard to the

density profile, by truncating the power spectrum of initial fluctuations for N-body

simulations of halo formation, leaving only the large-scale modes. The smallest

haloes in these simulations form by a “top down” process, yet are still well-fit by

a cuspy density profile (e.g. Titley & Couchman 1999; Avila-Reese et al. 2000;

Colin, Avila-Reese & Valenzuela 2000; Knebe et al. 2001). Huss, Jain, & Stein-

metz (1998) studied halo formation using N-body simulations of the collapse of

a spherical overdensity, varying the amount of substructure present. They con-

cluded that the haloes formed in this way are very similar to those formed in

CDM, irrespective of the amount of merging and substructure present. These re-

8

sults show that singular, NFW-like density profiles are a more general outcome

of halo collapse, not limited to hierarchical clustering scenarios. In the current

work, we extend this result to simulations using initial conditions that are much

simpler than Gaussian-random-noise initial conditions, while retaining the realistic

feature of continuous infall. Because we focus on one halo, we are able to follow

its formation and evolution, and find that many of the same trends reported for

halo evolution in the CDM simulations are also present here.

The chapter is organized as follows: In §2.2 we describe our test-bed model

for halo formation, based upon the pancake instability investigated previously by

Valinia et al. (1997). We give the initial conditions which lead to halo formation,

our simulation method, and numerical parameters. In §2.3, we summarize our

simulation results for the density, temperature, velocity dispersion, and anisotropy

profiles of the halo at various times, both with and without gas. In §2.4 we examine

the virial equilibrium of our pancake haloes, including an analysis based upon

the Jeans equation. The evolution of halo global properties is described in §2.5,

where we describe the similarities between haloes formed by pancake instability

and those in CDM, in particular the evolution of concentration parameter and the

mass accretion rate. The discussion is in §2.6.

2.2 Halo Formation via Pancake Instability

2.2.1 Unperturbed Pancake

Consider the growing mode of a single sinusoidal plane-wave density fluc-

tuation of comoving wavelength λp and dimensionless wavevector kp = x (length

unit = λp) in an Einstein-de Sitter universe dominated by cold, collisionless dark

matter. Let the initial amplitude δi at scale factor ai be chosen so that a density

9

Figure 2.1 Dark matter particles at a/ac = 3.

caustic forms in the collisionless component at scale factor a = ac = ai/δi.

2.2.2 Perturbations

Pancakes modeled in this way have been shown to be gravitationally un-

stable, leading to filamentation and fragmentation during the collapse (Valinia et

al. 1997). As an example, we shall perturb the 1D fluctuation described above

by adding to the initial primary pancake mode of amplitude δi two transverse,

plane-wave density fluctuations with equal wavelength λs = λp, wavevectors ks

pointing along the orthogonal vectors y and z, and smaller initial amplitudes, εyδi

and εzδi, respectively, where εy 1 and εz 1. A pancake perturbed by two

such density modes will be referred to as S1,εy ,εz . All results presented here refer

to the case S1,0.2,0.2 unless otherwise noted. The initial position, velocity, density,

and gravitational potential are given by

x = qx +δi

2πkpsin 2πkpqx, (2.1)

10

Figure 2.2 Density profile of the dark matter halo as simulated without gas at fourdifferent scale factors, a/ac =3, 5, 7, and 10, with spherically-averaged simulationresults in radial bins (filled circles) and the best-fitting NFW profiles (solid curves)for several epochs, as labeled. Shown above each panel are fractional deviations(ρNFW −ρ)/ρNFW from the best-fitting NFW profiles for each epoch. Vertical linesindicate the location of rsoft, the numerical softening-length, and r200, the radiuswithin which 〈ρ〉 = 200ρb, where ρb is the cosmic mean density.

y = qy + εyδi

2πkpsin 2πkpqy, (2.2)

z = qz + εzδi

2πkpsin 2πkpqz, (2.3)

vx =1

2πkp

(

dδ

dt

)

i

sin 2πkpqx, (2.4)

vy =εy

2πkp

(

dδ

dt

)

i

sin 2πkpqy, (2.5)

vz =εz

2πkp

(

dδ

dt

)

i

sin 2πkpqz, (2.6)

11

Figure 2.3 Same as previous figure, but for simulation with gas included.

ρ =ρ

1 + δi(cos 2πkpqx + εy cos 2πkpqy + εz cos 2πkpqz), (2.7)

and

φ = 〈φ〉 (cos 2πkpx+ εy cos 2πkpy + εz cos 2πkpz) , (2.8)

where qx, qy, and qz are the unperturbed particle positions.

Such a perturbation leads to the formation of a quasi-spherical mass con-

centration in the pancake plane at the intersection of two filaments (Fig. 2.1). As

we shall see, haloes formed from pancake collapse as modeled above have a density

profile similar in shape to those found in N-body simulations of hierarchical struc-

ture formation in a CDM universe, with realistic initial fluctuation spectra. As

12

such, pancake collapse and fragmentation can be used as a test-bed model for halo

formation which retains the realistic features of anisotropic collapse, continuous

infall, and cosmological boundary conditions.

2.2.3 N-body and Hydrodynamical Simulations

The code we use to simulate the formation of the halo couples the Adaptive

SPH (ASPH) algorithm, first described in Shapiro et al. (1996) and Owen et al.

(1998), to a P3M gravity solver (Martel & Shapiro 2002). The ASPH method

improves on standard SPH by introducing nonspherical, ellipsoidal smoothing ker-

nels to better track the anisotropic flow that generally arises during cosmological

structure formation. Another innovation of ASPH involves using the smoothing

kernel to predict the location of shocks in a manner which minimizes the spuri-

ous preheating which accompanies the use of artificial viscosity. Thus, ASPH is

well-suited for a problem like pancake collapse and fragmentation.

Two simulations were carried out, one with gas and one without. In both

cases, there were 643 particles of dark matter, while there were also 643 gas particles

when gas was included. The P3M grid was 1283 cells in a periodic cube size λp on

a side, with a comoving softening length of rsoft = 0.3∆x = 0.3λp/128, where ∆x

is the cell size. The initial conditions were those described in §2.2.2.

The adiabatic pancake problem (i.e. without radiative cooling) is self-

similar and scale-free, once distance is expressed in units of the pancake wave-

length λp and time is expressed in terms of the cosmic scale factor a in units of

the scale factor ac at which caustics form in the dark matter and shocks in the

gas (Shapiro & Struck-Marcell 1985). In the currently-favored flat, cosmological-

constant-dominated universe, however, this self-similarity is broken because ΩM/ΩΛ

13

Figure 2.4 Dimensionless specific thermal energy profiles at a/ac = 7 in the(gas+DM)-simulation, for the dark matter (top panel) and gas (middle panel),with dotted lines indicating the rms scatter within each logarithmic bin, and theirratio (bottom panel).

decreases with time, where ΩM and ΩΛ are the matter and vacuum energy density

parameters, respectively. For objects which collapse at high redshift in such a uni-

verse (e.g. dwarf galaxies), the Einstein-de Sitter results are still applicable as long

as we take (ΩB/ΩDM )EdS = (ΩB/ΩDM )Λ, where ΩB and ΩDM are the baryon and

dark matter density parameters. If ΩB = 0.045, ΩDM = 0.255, and ΩΛ = 0.7 at

present, then the EdS results are applicable if we take ΩB = 0.15 and ΩDM = 0.85,

instead.

14

Figure 2.5 Anisotropy profile, defined two different ways, at a/ac = 7. The curvesare labeled according to the definitions given in the text.

2.3 Profiles

All profiles are spherical averages computed using logarithmically-spaced

radial bins between the softening length rsoft and r200, the radius within which the

mean density is 200 times the mean cosmic density at that epoch.

2.3.1 Density

The density profiles at different epochs for the simulations both with and

without gas are shown in Figures 2.2 and 2.3, along with the best-fitting NFW

profile for each epoch, which has the form

ρ

ρ=

δc

(r/rs)(1 + r/rs)2, (2.9)

with δc given by

δc =200

3

c3

ln(1 + c) − c/(1 + c), (2.10)

15

Figure 2.6 Trajectories for a subset of particles in the simulation with dark matteronly at a/ac ∼ 6. The initial and final positions are shown by solid red spheres.Particles are colored according to the local dark matter density, the redder thedenser.

where c = r200/rs. The NFW profile has only one free parameter, the concentration

parameter c. Our pancake halo density profiles have concentrations which range

from 3 to 15, increasing with time, and are usually within 20% of the best-fit NFW

profile at all radii. This NFW profile is a fit to N-body results for CDM haloes,

but it is also consistent with the haloes which form in simulations using Gaussian-

random-noise initial fluctuations with small-scale fluctuations suppressed.

2.3.2 Velocity Dispersion and Thermal Energy

The profiles of the dimensionless specific thermal kinetic energy εDM =

〈|v−〈v〉|2〉/2 = 3σ2DM/2 of the dark matter and the dimensionless specific thermal

energy εgas = 3kBT/2m of the gas are shown in Figure 2.4, for the simulation with

gas and dark matter at a/ac = 7. Although the dark matter velocity dispersion

rises towards the centre, the rise is shallow and the kinetic energy distribution is

16

Figure 2.7 Departure from equilibrium in the Jeans equation. Solid and dashedlines show local and global departures from equilibrium, respectively.

approximately isothermal inside the radius r200. As seen from Figure 2.4, the gas is

even more isothermal than the dark matter, with εgas(rsoft)/εgas(r200) ' 2.5, while

the density varies by more than three orders of magnitude over the same region.

We also show the ratio of the specific thermal energy of the dark matter to that

of the gas. This ranges from εDM/εgas ∼ 1.6 in the centre to εDM/εgas ∼ 1 at r200.

2.3.3 Anisotropy

In Figure 2.5, we plot the profile of the anisotropy parameter β, defined two

different ways, according to the frame of reference in which the velocity dispersion

is calculated. In the Eulerian case, where the bulk motion of the shell contributes to

17

the anisotropy, it is defined as β = βE ≡ 1− 〈v2t 〉/(2〈v2

r 〉). In the Lagrangian case,

however, the bulk motion of the shell is subtracted out, β = βL ≡ 1 − σ2t /(2σ

2r),

where σ2i = 〈(vi − 〈vi〉)2〉. The values β = 1, 0, and −∞ correspond to motion

which is purely radial, fully isotropic, and tangential, respectively.

Simulations of CDM typically find values of β near 0 at the centre, slowly

rising to a value of β ∼ 0.5− 0.7 at r200. (Eke, Navarro, & Frenk 1998; Thomas et

al. 1998; Huss, Jain & Steinmetz 1999; Colin, Klypin, & Kravtsov 2000; Fukushige

& Makino 2001). As seen from Figure 2.5, the halo formed by pancake instability

is somewhat more anisotropic, with values of β rising from ∼ 0.2 near the centre

to ∼ 0.8 at r200. This reflects the strongly filamentary substructure of the pancake

within which the halo forms, and perhaps the absence of strong tidal fields or

mergers as well, which might otherwise help convert radial motions into tangential

ones.

While the two definitions of anisotropy give nearly indistinguishable profiles

at r ≤ r200, the two profiles depart significantly at r > r200. This is expected, since

the bulk tangential motion is always zero because of the symmetry of the pancake,

and the bulk radial motion is nearly zero inside the halo where equilibrium is a

reasonable expectation. The two definitions of β are identical when the system is

in equilibrium. Outside the halo, however, equilibrium is violated because the bulk

radial motion is not zero, and the difference between the profiles arises due to the

detailed nature of the region outside the halo. Explaining the difference between

the β profiles at r > r200 will therefore lead to a more thorough understanding of

the geometry and velocity structure of the pancake-halo system.

In the Lagrangian case, where bulk radial motion does not contribute to the

anisotropy, the profile can best be understood by considering particle trajectories

18

as they fall into the halo from the filaments. Particles fall first into the pancake,

then the filament, and finally into the halo itself. Most of the particles outside

the halo are in the filaments, within which all of the particles are moving towards

the halo, leaving a relatively small radial velocity dispersion (see Figure 2.6). As

the particles spiral around the filaments as they move along them into the halo,

however, their paths cross in the tangential direction, leaving the velocity ellipsoid

elongated in the tangential direction. The large negative value in βL at r > r200

reflects this tangential bias.

In the Eulerian case, there is also a region of β << 0, but which occurs

only within a narrow range of radii. This is best understood by considering the

bulk radial motion. Just outside the halo, there is a region of infall (vr < 0)

which is surrounded by a region where matter is just turning around and falling

back in (vr ' 0), outside of which matter is still expanding (vr > 0). In the infall

region, the velocity is radially biased, as it is within the halo. Near the turn-around

radius, however, the bulk radial motion becomes approximately zero, giving the

large negative value of β. Outside the turn-around radius, radial motion once

again dominates over tangential motion, this time because of the global Hubble

expansion.

2.4 Virial Equilibrium

A state of equilibrium is commonly assumed in analytical modeling of dark

matter haloes (Lokas & Mamon 2001; Taylor & Navarro 2001). Such modeling

is important because it allows us to gain an understanding of the physical pro-

cesses at work in the simulations and extrapolate beyond them to include physics

that cannot yet be simulated directly. In realistic cosmological collapse, however,

19

equilibrium is not always achieved. Mergers, continuous infall, and tidal effects

are all processes which can affect equilibrium. Using N -body simulations, Tor-

men, Bouchet, & White (1997) found that haloes which formed from CDM initial

conditions roughly obey the Jeans equation for dynamical equilibrium in spheri-

cal symmetry, within a radius of order r200, suggesting that CDM haloes are in

approximate virial equilibrium. Let us test this for our pancake haloes.

In what follows, we shall show that our pancake haloes are in virial equi-

librium by determining how well the simulated haloes satisfy the Jeans equation

and the virial theorem. We will also interpret the numerical halo results further

by comparing them with some simple analytical equilibrium distributions.

2.4.1 Jeans equation

If the assumption is made of spherical symmetry and a stationary state

(vr = vt = 0 everywhere), the collisionless Boltzmann equation,

∂f

∂t+ v · ∇f −∇Φ · ∂f

∂v= 0, (2.11)

together with the Poisson equation

∇2Φ = 4πGρ, (2.12)

gives the Jeans equation

d

dr(ρv2

r) +2ρβv2

r

r= −ρdΦ

dr, (2.13)

where the anisotropy parameter is now the “Eulerian” quantity, β = βE, as defined

previously in §2.3.3 (Binney & Tremaine 1987). To the extent that a numerical

simulation conserves phase space density as it should according to the collisionless

Boltzmann equation, a disagreement between the simulation results and the Jeans

20

equation indicates either a departure from equilibrium or from spherical symmetry,

or both.

If β = 0 at all radii, then the Jeans equation takes the form

1

ρ

d

dr(ρσ2) = −dΦ

dr, (2.14)

where σ is the one-dimensional velocity dispersion of the dark matter (i.e. σ2 =

v2/3 = v2r for isotropic orbits). If we make the substitution

σ2 =kBT

m, (2.15)

wherem is the mass per particle and T is temperature for an ideal gas, and combine

this with the ideal gas law

P =ρkBT

m= ρσ2, (2.16)

we obtain

1

ρ

dP

dr= −dΦ

dr, (2.17)

the well-known equation of hydrostatic equilibrium for an ideal gas. The equation

of hydrostatic equilibrium is thus a special case of the Jeans equation with isotropic

orbits. Next, we will describe some simple models that follow from the assumption

of Jeans equilibrium, for comparison with our simulated haloes.

2.4.1.1 Singular Isothermal Sphere

The singular isothermal sphere (SIS) is the power-law solution of the equa-

tion of hydrostatic equilibrium with uniform temperature (i.e. the isothermal

Lane-Emden equation). The density is given by

ρ(r) =σ2

0

2πGr2=

kbT

2πGmr2, (2.18)

21

where σ0 is the 1D velocity dispersion, and T is the gas temperature. A more gen-

eral class of solutions can be found, however, by allowing the anisotropy parameter

β to be nonzero but remain independent of radius. The equation to be used then

becomes the Jeans equation with constant β. We can solve the Jeans equation in

this case for the velocity dispersion σβ as a function of β and σ0 for the same mass

distribution in the isotropic case, to show

σ2β =

3 − 2β

3(1 − β)σ2

0. (2.19)

Combining equations (2.18) and (2.19) gives

ρ(r) =σ2

t

4πGr2, (2.20)

independent of β, which shows that the singular isothermal sphere is supported

against collapse entirely by the tangential component of the velocity. According

to equation (2.19), purely radial orbits (β = 1) are not allowed.

2.4.1.2 Nonsingular Truncated Isothermal Sphere

Shapiro, Iliev & Raga (1999) derived a nonsingular equilibrium model for

cosmological haloes, the truncated isothermal sphere (TIS). The TIS is a particular

solution of the isothermal Lane-Emden equation

d

dζ

(

ζ2d ln ρ

dζ

)

= −ρζ2, (2.21)

with inner boundary conditions given by

ρ(0) = 1 (2.22)

and

dρ

dζ(0) = 0, (2.23)

22

Figure 2.8 Virial ratio versus dimensionless radius ζ for the TIS solution. Thevertical dotted line on the right corresponds to the truncation radius, ζt, at whichthe total energy is a minimum at fixed mass and boundary pressure, while theone on the left corresponds to the radius within which the mean density is 200times the background density. The horizontal dotted line is the virial ratio of thesingular isothermal sphere.

where r0 is related to σ and ρ0 by

r20 ≡ σ2

4πGρ0, (2.24)

ρ ≡ ρ/ρ0, ζ ≡ r/r0, and ρ0 and r0 are the core density and radius, respectively.

The TIS solution is truncated at a radius ζt = rt/r0, outside of which there is a

boundary pressure pt. Solutions to the Lane-Emden equation with the above inner

boundary conditions form a one-parameter family of ζt, one of which minimizes

the total energy (ζt = 29.4) for a given mass and boundary pressure. If the halo is

assumed to form from a uniform density top hat perturbation, then the minimum-

energy condition fixes all the properties of the TIS.

23

2.4.1.3 Simulated Pancake Haloes

Each term in the differential and integral forms of the Jeans equation were

evaluated using our numerical simulation results for the pancake haloes to test the

extent to which the haloes are in dynamical equilibrium. This was accomplished

by smoothing the numerical data using overlapping logarithmic radial bins with

widths that are four times the logarithmic spacing between bin centres to obtain

smoothed values of ρ, v2r , dφ/dr, and β. As measures of departure from equilibrium

at a given radius and globally within that radius, we define two parameters,

D(r) ≡ DL(r) −DR(r)

DR(r), I(r) ≡ IL(r) − IR(r)

IR(r)(2.25)

where

DL(r) ≡ d

dr(ρv2

r) +2ρβv2

r

r, DR(r) ≡ −ρdΦ

dr(2.26)

and

IL(r) ≡∫ r

εDL(r)dV, IR(r) ≡

∫ r

εDR(r)dV, (2.27)

and ε is the softening length. All derivatives were calculated by differencing the

smoothed values. The value D(r) = 0 indicates that the Jeans equation is satisfied

at r, while I(r) = 0 indicates a global agreement with the Jeans equation for the

region inside of r. As seen from Figire 2.7, the haloes are in global agreement

for a/ac > 3, while the departure from equilibrium at each radius decreases with

increasing scale factor. The haloes are therefore approximately in equilibrium for

all a/ac > 3.

24

2.4.2 Virial Ratio

The virial theorem offers a simple and powerful method for diagnosing

global equilibrium and is more straightforward than an analysis involving the Jeans

equation. Care must be taken, however, to take proper account of the surface pres-

sure at the boundary of the virialized halo. Specifically, the scalar virial theorem

states that for a self-gravitating system in static equilibrium (〈v〉 = 0) with no

magnetic fields, 2T + W + Sp = 0, where W is the potential energy, T is the

thermal and kinetic energy, and Sp is a surface pressure term,

Sp = −∫

pr · dS, (2.28)

where dS is the surface area element. If the system is isolated, there can be no

material outside to create a boundary pressure, and we have Sp = 0, implying

2T/|W | = 1. Cosmological haloes are not isolated systems, however, so we cannot

expect Sp = 0 and 2T/|W | = 1. In fact, the presence of infalling matter can act as

a surface pressure in the virial theorem. With infall present, we therefore expect

Sp/|W | < 0, implying 2T/|W | > 1. Just what this value should be depends on

the shape of the density distribution and the rate of accretion at the surface. In

realistic collapse, however, the boundary is usually ill-defined and one cannot hope

to determine precise virial parameters for the halo. The idealized models allow for

a simpler context in which to analyze the global properties of the halo and how

they to evolve.

In what follows, we will discuss and compare the expected virial values for

the SIS, TIS, NFW, and simulated halo mass and velocity profiles.

25

2.4.2.1 Singular Isothermal Sphere

Consider the singular isothermal sphere with a density given by equation

(2.18) with an anisotropy β that does not vary with radius. The potential energy

at r is

W (r) =∫

(ρφ)dV = −2M(r)σ20. (2.29)

The kinetic (or thermal) energy T is given by

T =1

2

∫

ρ〈v2〉dV =3

2

∫

ρσ2dV =3

2M(r)σ2. (2.30)

Using equation 2.19 to relate the actual σ2 to the one in the isotropic case σ20, gives

T =3 − 2β

2(1 − β)Mσ2

0. (2.31)

The virial ratio is therefore

2T

|W | =3 − 2β

2(1 − β). (2.32)

Since an object in Jeans equilibrium is also in virial equilibrium, we can use the

virial equation to find the value of the surface pressure implied by the kinetic and

potential energies just found, giving

Sp = −(2T +W ) = − 1

1 − βMσ2

0. (2.33)

Let us define an effective pressure as given by the equation for the surface pressure

term in spherical symmetry,

peff = − Sp

4πr3. (2.34)

After some manipulation, we find that

peff = ρσ2r , (2.35)

26

Figure 2.9 Virial ratio vs. radius calculated two different ways: (1) direct summa-tion and (2) assuming spherical symmetry.

consistent with the expectation that the radial velocity is the only component

which contributes to an effective surface pressure term in a spherically symmetric

collisionless system.2

2.4.2.2 Truncated Isothermal Sphere

Since the TIS is a unique solution given by the minimum energy at fixed

boundary pressure, the virial ratio is always the same value, namely 2T/|W | ' 1.37

at ζt = 29.4. This value is smaller than that for the isotropic singular isothermal

sphere, 2T/|W | = 1.5, and is near the global minimum for all values of ζ, which

is 2T/|W | ' 1.36 and occurs at ζ ' 22.6. At the intermediate radius ζ200 ' 24.2,

defined to be the radius within which the mean density is 200 times the background

density, the TIS virial ratio has a value 2T/|W | ' 1.36. As seen in Figure 2.8, the

2This can be shown for any spherically symmetric system in equilibrium by using the tensorvirial theorem.

27

inner core region of the TIS is dominated by kinetic (or thermal, in the gas case)

energy, whereas the value approaches that of the SIS at large ζ, where the core

region becomes small compared with the size of the TIS and the density profile

asymptotically approaches that for the SIS (ρ ∝ r−2). It is interesting to note

that the minimum virial ratio occurs at approximately the same location as the

minimum energy truncation radius, with a value which is nearly the same as at

minimum.

2.4.2.3 NFW Haloes

Lokas & Mamon (2001) investigated the equilibrium structure of haloes

with an NFW density profile. Using different values of β(r), they found several

analytical solutions to the velocity dispersion of the halo by integrating the Jeans

equation for a given ρ(r) and β(r). In order to integrate the Jeans equation to

find the velocity dispersion, it is necessary to set the velocity dispersion at some

r. In the absence of a physical value for σr at the boundary of the halo, as could

be inferred from some infall solution, the only other reasonable choice is to have

σr → 0 as r → ∞. The disadvantage of making this choice is that there is no

physical basis for using a boundary condition at infinity for an object which is of a

finite extent. The velocity dispersion at the boundary in this case is determined by

an imaginary mass distribution outside of the halo which is simply an extension of

the NFW profile to radii at which it is not valid. The advantage is that it gives a

qualitatively correct view: for an object with the same internal velocity anisotropy

profile, the virial ratio 2T/|W | → 1 as c → ∞. This is expected because the

more concentrated a halo, the closer it is to being completely isolated, implying a

virial ratio consistent with isolation, 2T/|W | = 1. We can therefore expect that

28

Figure 2.10 Virial ratio versus concentration parameter. Solid, dashed, and long-dashed lines represent expected values for β = 0, 0.5, 1, repectively. Shown also arethe values corresponding to the simulated haloes, the isotropic singular isothermalsphere (SIS), and the isotropic truncated isothermal sphere (TIS).

the results given by fixing σr = 0 at infinity should reflect the general trend of

decreasing T/|W | for increasing concentration and isotropy, while not necessarily

giving the correct values.

Shown in Figure 2.10 is the virial ratio as a function of the concentration

parameter for various values of β(r) = β0, as expected from a Jeans analysis, as well

as values found for the SIS, TIS, and simulated haloes. As seen in the figure, the

more isotropic the NFW halo, the lower the virial ratio. This is consistent with

the fact that the surface pressure term is directly related to the radial velocity

dispersion. A larger value of β implies a larger value of σ2r , which in turn implies a

higher surface pressure at fixed boundary density, giving a larger virial ratio. Also

evident is the previously mentioned trend of decreasing virial ratio with increasing

concentration.

29

2.4.2.4 Simulated Haloes

Virial ratios 2T/|W | were calculated for both simulation runs, with and

without gas. For the case with no gas,

T (r) =∑

i

1

2miv

2i , (2.36)

where the sum is over all particles within r. For the simulation with gas included,

T (r) =∑

i

1

2miv

2i +

∑

i

3

2kBTi, (2.37)

where the first sum is over all particles within r, and the second sum is over all

gas particles within r. The potential energy was found using the assumption of

spherical symmetry,

W (r) =∑

i

GMimi

rifi, (2.38)

where Mi is the mass interior to ri and fi ≡ f(ri) is a function which

represents the particular form of the softening used in the P3M algorithm (Martel

& Shapiro 2002), and the sum is over all particles withinR. As shown in Figure 2.9,

the assumption of spherical symmetry yields results which are very close at R =

r200 to those arrived at from the more rigorous definition,

W (R) ≡ 1

2

∑

i

∑

j

Gmimj

rijfij, (2.39)

where rij = |ri − rj| and fij ≡ f(rij).

Shown in Figure 2.10 are the virial ratios of both haloes at different con-

centration parameters. While the simulated haloes lie below the expected curve

for NFW haloes of comparable anisotropy, the trend of smaller virial ratios for

more concentrated haloes is clearly evident, though there is significant scatter,

particularly in the dark matter only simulation. Since the simulated haloes were

30

Figure 2.11 Mass accretion history of simulation with dark matter only (solid)along with the best-fit functional form from CDM simulations (dotted) with S = 2and a0 = 2.5ac, where M0 ≡ M(a = 12ac). Curve labeled “Gadget” was for thesame initial conditions, run with the publicly available code of the same name(Springel et al. 2000).

determined to be in approximate Jeans equilibrium, this supports the hypothesis

that an analysis which assumes some value of σr at infinity to find velocity disper-

sion profiles of a finite object is justified if the goal is to show the general trend in

the variation of anisotropy and concentration.

2.5 Halo Evolution

2.5.1 Accretion Flow

The mass growth of the halo proceeds in three stages. The first two stages

are shown in Figure 2.12. Before a/ac ∼ 3, the mass within r200 grows very quickly,

indicating initial collapse of the central overdensity. After a/ac ∼ 3, the infall rate

drops, and can be well-described by M ∝ a. Such an infall rate is reminiscent of

self-similar spherical infall. This infall rate cannot persist indefinitely, since there is

31

Figure 2.12 Evolution of the dark matter only halo in the intermediate (self-similar)regime. (left) Virial radius in units of expected caustic radius as defined in thetext (solid), and the average in the range 3 < a/ac < 7.5 (dashed). (right) Halomass (solid) and the linear best-fit (dashed), where M0 ≡ M(a/ac = 7.5). Thevertical dotted lines indicate the virialisation epoch at a = 3ac.

only a finite mass supply to accrete onto the halo because of the periodic boundary

conditions. As a consequence, the accretion rate slows after a/ac = 7. This is also

expected to occur in haloes forming from more realistic initial conditions, given

that neighboring density peaks of a similar mass scale prevent any one halo from

having the infinite mass supply necessary to sustain this mass accretion rate. In

fact, the halo can be fit at nearly all times, especially later, by the more general

fitting function

32

M(a) = M0 exp[

−Sa0

(

1

a− 1

)]

, (2.40)

where S ≡ [d(lnM)/d(ln a)]a0 is the logarithmic slope at the collapse scale factor

a = a0. For S = 2, we find a best-fit value of a0 = 2.5ac. This form was first used to

fit the evolution of haloes formed in CDM simulations (Wechsler et al. 2002). We

have identified three distinct phases in the halo evolution: initial collapse, steady

infall, and infall truncation due to finite mass supply. In realistic collapse, we see

that the halo evolves continuously from one stage to the next as evidenced by the

continous change in logarithmic slope of the fitting function, given by

d(lnM)

d(ln a)= S

a0

a. (2.41)

In the intermediate stage of collapse, the similarity to self-similar spherical

infall is also evident in the radial velocity profile of the dark matter. Plotted in

Figure 2.13 is the dark matter radial velocity profile at various epochs in dimen-

sionless units as simulated with and without gas. The dimensionless velocity is

given by

V ≡ t

rta(t)vr =

2

3H0

(

a

a0

)1/6 vr

rta,0, (2.42)

where we have used the relations rta ∝ t8/9 and a ∝ t2/3, rta is the turnaround

radius, and rta,0 ≡ rta(a = 7ac) is set by finding the radius at which vr = 0. The

dimensionless radius is given by

λ =r

rta=

r

rta,0

(

a

7ac

)−4/3

. (2.43)

In the exact case, this profile does not change with time. As seen from the figure,

the simulated halo follows this profile closely, with λ200/λc ' 0.78, where λc is the

33

Figure 2.13 Radial velocity profiles in dimensionless units as in self-similar sphericalcollapse. The thick solid line is the radial velocity profile for an ideal γ = 5/3 gasas in Bertschinger (1985).

radius at the outermost caustic and is approximately where the shock occurs in

the collisional solution, and λ200 ≡ r200/rta.

At first, the similarity of halo formation by pancake instability to self-

similar spherical infall for an intermediate epoch in the halo’s evolution may seem

surprising, given that the accretion geometry is far from spherical and the halo is

not accreting in an infinite medium. There are reasons, however, to expect that

such anisotropic collapse will look like spherical infall when quantities are spheri-

cally averaged. Most of the mass within the turn-around radius in the collisionless

self-similar spherical infall solution is located within the outermost caustic, which

corresponds in our case to approximately the virial radius, within which the mass

distribution is quasi-spherical. Within the filaments that are feeding the halo at

radii where the matter has already turned around and is falling back in, most of

the interior mass should be located within the quasi-spherical halo. As long as the

34

motion of the matter which is presently accreting onto the halo has been influenced

mostly by the halo itself, and not the neighboring halo which is present because of

the periodic boundary conditions, then we can expect the above argument to be

valid. As more and more shells turn around and fall back in, the matter currently

turning around gets progressively closer to the boundary (rta/rboundary ∝ a1/3),

and the part of the potential due to the central halo is comparable to that due

to the halo in the neighboring simulation box. Thus, it is at least self-consistent

for the simulated haloes to have a mass accretion rate and velocity structure that

resemble the self-similar solution over a range of scale factors a.

The mass evolution can be the crucial missing link for analyses which at-

tempt to model cosmological haloes as spheres in hydrostatic equilibrium under-

going continous merging and infall. As previous studies have shown, it is often

convenient to treat the collisionless matter as a fluid. From this perspective, the

boundary pressure arises from the thermal energy present in the post-accretion

shock gas, and can be related to the preshock infall velocity and density by the

shock jump conditions. For example, if the boundary of the halo is taken to en-

compass a constant overdensity and the shape of the mass distribution does not

change with time, then the typical postshock boundary pressure scales like

ρV 2 ∼ ρr2virt

−2 ∼M2/3a−4, (2.44)

where we have used M ∼ ρr3vir, ρ ∼ a−3, and the Einstein-de Sitter relation

a ∝ t2/3. Using such scalings, it is possible to provide a boundary pressure or

velocity dispersion for integration of the Jeans equation to determine the dynamical

state of the equilibrium halo. Also, in arguments such as those of Taylor & Navarro

(2002), in which the infall is assumed to follow self-similar spherical infall, a more

35

Figure 2.14 Evolution of concentration parameter of the best-fitting NFW profilefor the dark matter halo in simulations with (right) and without (left) gas. Dottedline is the actual evolution, with the filled squares showing the mean concentrationbinned in scale factor. Errorbars indicate RMS fluctuations within each bin. Thesolid lines are the best-fitting linear evolution given by cNFW = c0(a/a0), witha0 = 3ac

realistic infall rate M(a) such as that of equation (2.40) can be substituted, giving

insight not only into the form of the density profile, but also its evolution.

2.5.2 Density Profile

Although the halo generally grows by self-similar accretion for 3 < a/ac < 7,

the mass density is better fit by an NFW profile with only one free parameter, c.

Shown in Figure 2.14 is the concentration parameter versus scale factor. We find

it can be well-fit by

c = c0a

a0, (2.45)

where a0 is the scale factor at which the accretion rate becomes proportional to a,

marking the end of the collapse phase. The value a0 = 3ac is used here for both

36

simulation cases and corresponds to the vertical dotted line in Figure 2.12. The

solid lines in Figure 2.14 correspond to the best-fit values c0 = 4.3 and c0 = 3.8 for

the cases with and without gas included, respectively, where each data point was

weighted by the goodness of the corresponding NFW profile fit.

This functional form was also found by Bullock et al. (2001) and Wechsler

et al. (2002). In their analysis, they followed the mass accretion and merger

histories of individual haloes in a high-resolution CDM simulation of haloes in

the mass range ≈ 1011 − 1012M. The mass accretion histories allowed them to

determine a collapse epoch for each halo, which they correlated with the halo’s

conentration. They found that at a given redshift, the concentration was higher

for earlier collapse epoch, implying a linear evolution of the concentration of the

form given earlier as a good fit to the haloes analyzed here. They found significant

scatter in the relation, but attributed a large fraction of it to atypical haloes not

likely to be in equilibrium, reporting a best fit value of ccoll = 4.1. The closeness

of this value to our values is likely a coincidence and should not be overinterpreted

because of uncertainties present in both analyses. In our case, for example, we

define a0 ≡ 3ac, when much of the analysis points to atransition epoch for the

haloes, and we can identify an abrupt “knee” in M(a). Wechsler et al. (2002), on

the other hand, use a value of S = 2 in equation (2.40), which when we fit to our

halo evolution implies a0 = 2.5ac. Using this value instead changes our value of c0

by roughly a factor 2.5/3 ' 0.83, implying not such good agreement for the slope.

We should note that at early times, our data is not as well-fit by equation (2.40)

(see Figure 2.11), calling into question the usefulness of using this functional form

to determine a collapse epoch for our haloes. The scatter in the data which led

Wechsler et al. (2002) to their value is also a source of uncertainty. Nevertheless,

37

our values and those found by Wechsler et al. (2002) are consistent with each other

and a linear evolution of concentration versus cosmic scale factor.

2.5.3 Virial Ratio

The virial ratio was computed for the evolution of the simulated haloes,

as shown in Figure 2.15. As expected the virial ratio for both cases decreases

with time, consistent with a rising concentration parameter. It is interesting to

note that the haloes begin their evolution with a value close to that of the TIS,

and evolve toward a value close to that consistent with isolation. It seems from

this evolution that the surface pressure is dynamically strongest at the moment

of collapse and becomes continuously weaker as the halo evolves. This suggests

that the mass infall M(a) is intimately related to the conentration of the halo. If

the kinetic energy and anisotropy profiles evolve in a self-similar way, as implied

by the velocity moment profiles presented earlier, and the mass infall and hence

boundary pressure are becoming weaker dynamically, the only way for the halo to

stay in virial equilibrium is to become more concentrated.

2.5.4 Anisotropy

Shown in Figure 2.16 is the evolution of the anisotropy averaged over the

halo. Before a/ac ' 3, β ' 0.7−0.8, while after a/ac ' 3 the value drops abruptly

to and stays at β ' 0.6 − 0.7. Even though this is a modest change in anisotropy

and the halo continues with a strong radial velocity bias, the timing of the change

is consistent with some degree of relaxation occurring at a/ac = 3, marking the

end of the initial collapse of the fluctuation. The radial bias may be attributed to

the highly anisotropic collapse geometry as well as the absence of any power on

38

smaller scales to further perturb particle orbits from radial.

2.6 Discussion and Summary

We have analyzed the results of simulations of halo formation by the grav-

itational instability and accompanying fragmentation of cosmological pancakes.

These haloes are in equilibrium after their initial collapse and relaxation epoch,

and have cuspy density profiles that become more concentrated with time. During

this time the haloes grow by steady accretion of matter, mostly from the filaments,

in a manner reminiscent of self-similar spherical infall. At later times, the finite

mass supply in the box breaks the self-similar nature of the accretion rate, and it

begins to decline.

In both cases, with and without gas included, the dark matter density can

be well-fit by the NFW profile. This implies that hierarchical collapse or Gaussian-

random-noise initial conditions are not prerequisites for the persistence of a cuspy

inner density profile. As mentioned in §2.4.1, simulations using Gaussian-random-

noise initial conditions which have a cutoff in the fluctuation spectrum have suc-

ceeded in producing cuspy NFW-like density profiles, consistent with our result.

There are also theoretical grounds on which to believe that collapse of collisionless

matter by gravitational instability can lead to cuspy density profiles, as shown by

Lokas & Hoffman (2000). In their study, they used a simple spherical spherical

model similar to those of Fillmore & Goldreich (1984) and Bertschinger (1985) but

abandoned the assumption of self-similarity in favor of a more realistic collapse

in order to study the formation of the inner density cusp. Based on their work

and simulations by others, they concluded that cuspy density profiles are a generic

feature of collisionless gravitational collapse, not limited to hierarchical clustering

39

or realistic fluctuation spectra. The persistence of a cusp in our simulations of halo

formation, in cases both with and without gas included, should be interpreted as

strong evidence in support of that conclusion.

Haloes formed in the pancake instability model also reach a state of quasi-

equilibrium similar to that found in N -body simulations with more realistic fluctu-

ation spectra. Although the assumptions in assessing equilibrium are not strictly

expected to be valid during merging and infall in realistic cosmological collapse

scenarios, haloes formed in N -body simulations in such scenarios are often found

to be in approximate equilibrium. This supports a picture in which haloes can be

considered to be objects in equilibrium undergoing infall of matter, with the infall

driving an overall evolution of the halo but not disturbing the global persistence

of an equilibrium state within the halo. This is easiest to understand in the case

of continuous infall, but hierarchical collapse can also be interpreted in this pic-

ture with the mergers having the same dynamical effect as continuous infall when

averaged over periods of time longer than the typical merger timescale.

Virial ratios found for these haloes are consistent with the haloes being in an

equilibrium state, once the surface pressure term is taken into account properly.

Our results agree qualitatively with the analysis of Lokas & Mamon (2000), in

which haloes with an NFW density profile were assumed to be in equilibrium with

zero velocity dispersion at infinity and various anisotropy profiles. While our values

are typically lower than those expected using the above assumptions, the trend of

lower virial ratio with higher concentration agrees well with our simulations. This

has led us to conclude that the use of equilibrium in modeling of cosmological

haloes is an important first step, but these studies can be made more realistic by

proper modeling of the accretion of matter onto the halo in a self-consistent way.

40

Figure 2.15 Virial ratio vs. scale factor.

We have found that the mass evolution of haloes formed by pancake instability is

strikingly similar to that reported inN -body simulations of CDM. Further, we have

proposed that this mass accretion can be used to provide a boundary condition

at the virial radius rather than at infinity. This sort of modeling can be used to

predict observed properties of equilibrium haloes, as in Lokas & Mamon (2000),

and also to gain a more complete understanding of the form and evolution of the

density profile found in N -body simulations, as in Taylor & Navarro (2001).

The evolution of the concentration parameter is an important discovery re-

ported by Bullock et al. (2001) and Wechsler et al. (2002), as well as here for haloes

formed by pancake instability and fragmentation. This has important implications

for galaxy formation and clustering, given that the shape of the halo density profile

affects the dynamical interaction of the baryonic and dark matter components. In

Bullock et al. (2001), a first attempt was made to understand this evolution with

a toy model, which did not include dynamics but instead found scalings of other

41

virial parameters which would be consistent with a linear evolution of concentra-

tion with scale factor. While we do not attempt to find a self-consistent dynamical

model for the evolution of the concentration parameter, we have found several

hints which point to a connection between the infall rate and the concentration of

the halo. First, consider the limit of late times when the infall has ceased and the

halo has collapsed out of the background universe. In this case, it is reasonable

to expect that the density profile is not changing, and thus the scale radius rs is

constant with time. The universe is still expanding while the mass of the halo is

not changing, however, so r200 ∝ a, implying r200/rs = c ∝ a. At earlier times,

when infall is dynamically important but constantly decreasing, it is reasonable

to expect that the concentration will be lower at earlier times and increase as the

effective boundary pressure decreases. Based on these simple arguments, we can

hypothesize that the evolution at these two regimes together give a concentration

which increases linearly with scale factor. Exactly why the conentration is consis-

tent with a linear evolution at early times as well as late times and what the role

of anisotropy and angular momentum are will be the subject of further analysis.

We have found several similarities between haloes formed in N -body sim-

ulations of CDM with Gaussian-random-noise initial conditions and those formed

by pancake instability and fragmentation as presented here. These similarities sug-

gest the pancake instability model can be used as a “stand-in” for haloes formed in

more realistic gravitational collapse simulations without all the complications aris-

ing from Gaussian-random-noise initial conditions. Haloes formed in the pancake

instability model are desirable because of their simplicity, similar to the spherical

infall models of Fillmore & Goldreich (1984) and Bertschinger (1985). On the

other hand, the pancake instability model is more realistic than the spherical infall

42

Figure 2.16 Anisotropy parameter vs. scale factor.

model because of the finite mass supply and the anisotropic nature of the collapse,

two features which typify gravitational collapse in realistic cosmological collapse

scenarios. The pancake instability model is thus a promising tool for studying

the dynamics of halo formation and evolution which can be used as a test-bed to

guide more realistic and expensive studies involving Gaussian-random-noise initial

conditions. The haloes presented here formed from a particular set of orthogonal

fluctuation modes, S1.0,0.2,0.2 (§2.2.2). Whether the trends reported here persist for

other perturbation modes will be the subject of future work.

43

Chapter 3

The Universal Density Profile of CDM Halos

from their Universal Mass Accretion History

We use the universal mass accretion history recently reported for N-body

simulations of halo formation in the cold dark matter (CDM) model to analyze

the formation and growth of individual halos. We derive the time-dependent den-

sity profile for the virialized objects which result from this mass accretion history

by three different approximations of successively greater realism: instantaneous

equilibration, radial orbits, and a fluid approximation. For the equilibrium model,

the density profile is well-fit by the formulae suggested by Navarro, Frenk, and

White (1997; NFW) and by Moore et al. (1998) for the halos in CDM simulations,

only over a limited range of radii and cosmic scale factors. For the radial orbit

model, we find profiles which are generally steeper than either the NFW or Moore

profiles, with an inner logarithmic slope approaching -2, consistent with the limit

for a purely radial, collisionless system. In the fluid approximation, however, we

find good agreement with the NFW and Moore profiles over the full range of radii

resolved by N-body simulations (r/r200 ≥ 0.01), with a concentration parameter

which increases in time exactly as reported for CDM N-body simulations. From

this, we conclude that the evolving structure of CDM halos can be well understood

as the effect of a universal, time-varying rate of smooth and continuous mass infall

44

on an isotropic, collisionless fluid1.

3.1 Introduction

In a cold dark matter (CDM) universe, the quasi-spherical dark matter-

dominated, virialized objects – halos – which result from the nonlinear growth

of Gaussian-random-noise density perturbations, are the scaffolding within which

galaxies and their clusters are built. The complex process by which small-mass

halos form first and merge to make larger-mass halos later, in a continuous se-

quence of hierarchical clustering, and the randomness and lack of symmetry of

the initial conditions, make it necessary to solve this structure formation prob-

lem by large-scale, three-dimensional N-body simulation. As the power of these

numerical techniques and the hardware used to perform simulations have grown

in recent years, some simple, universal properties of halo structure and evolution

have emerged from this complexity. It is now well-established that these halos

have spherically-averaged density profiles which vary with radius r near the center

as ρ ∝ r−α, 1<∼α<∼ 1.5, and steepen at large radii towards an asymptotic shape,

ρ ∝ r−3 (Navarro, Frenk, & White 1997, hereafter NFW; Moore et al. 1998, her-

after Moore). While different simulation and data analysis techniques still yield

somewhat different answers with respect to the value of α at very small radii,

near the resolution limit of the N-body simulations (e.g Ricotti 2003; Power et

al. 2003; Klypin et al. 2001), there is general agreement at larger radii. Wechsler

et al. (2002) have studied the evolution of individual halos over time. They find

that individual halo profiles grow more concentrated over time, with a concen-

tration parameter which is proportional to cosmic scale factor, while the mass of

1This work appeared in part in Alvarez, Ahn, & Shapiro 2003, RevMexSC, 18, 4

45

each halo increases with a universal time-dependence to approach a finite mass at

late times. CDM N-body halos also show universal phase-space density profiles.

N-body results find ρ/σ3V ∝ r−αps , where αps = 1.875 (Taylor & Navarro 2001),

αps = 1.95 (Rasia, Tormen and Moscardini 2004), and αps = 1.9± 0.05 (Ascasibar

et al. 2004). (Also related: P (f) ∝ f−2.5±0.05; Arad, Dekel, & Klypin 2004).

An analytical understanding of these results would be of great interest.

Early, pioneering work involving self-similar, continuous mass infall in spherical

symmetry predicted density profiles that asymptotically approach power-laws in

radius (e.g. Gunn & Gott 1972; Fillmore & Goldreich 1984; Bertschinger 1985).

Hoffman & Shaham (1985) applied power-law density profile models to local den-

sity maxima of the primordial fluctuation field, predicting a correlation between

the logarithmic slopes of the density profile and primordial power spectrum. Such

models, however, are valid over a limited range of time and length scales when

applied to realistic CDM initial conditions. In the model of Hoffman & Shaham

(1985), for example, mass infall continues at the same rate, whereas in realistic

infall this rate decreases because of the finite mass supply available to any given

halo, as shown by Wechsler et al. (2002). More recent work has concentrated on

modelling halo formation and growth with a more realistic mass infall, hierarchical

growth driven by mergers. A popular way of proceeding has become to start with

the expected merger history of a given halo and to make some assumptions about

how the merging halos interact and come into some quasi-equilibrium state in the

halo, driving an overall evolution of the system. The merger history of the halo is

often derived using the extended Press-Schechter (EPS) formalism (e.g. Lacey &

Cole 1993). Avila-Reese et al. (1999) for example, combined the EPS formalism

with a spherical infall model for the dynamics of the halo as it accretes matter.

46

Although their spherically-averaged model neglected effects associated with an

inhomogeneous distribution such as dynamical friction and tidal stripping, they

found good agreement with the simulation results. Using the EPS formalism, van

den Bosch (2002) found the derived merger histories were well-fitted by a univer-

sal mass accretion history very similar to that found by Wechsler et al. (2002).

Alvarez, Shapiro, & Martel (2002) found that a halo which forms by pancake in-

stability has a mass accretion history and density profile shape and evolution which

are very close to that found by Wechsler et al. (2002). In the pancake instability

model, there is no structure on smaller scales than the halo itself. This implies

that the universal mass accretion history and density profile structure are generic

features of cosmological collapse of a halo with a finite mass supply, not limited to

hierarchical clustering scenarios. Thus, it seems that it is the global rate of mass

infall which drives the overall structure and evolution of the halo. It is useful to

interpret this mass infall as a surface pressure term in the virial theorem, as in

the truncated isothermal sphere (TIS) model (Shapiro, Iliev, & Raga 1999; Iliev

& Shapiro 2001). The TIS model is an accurate predictor of the global properties

of halos at any mass and epoch for a given background universe. A halo in the

TIS model is described as an isothermal sphere of gas in hydrostatic equilibrium

with a finite central density and a truncation radius outside of which there is a

confining boundary pressure, assumed to be created by the presence of infalling

matter. While the TIS model predicts global properties of halos very well, the

density profile does not evolve and is not a good fit to the N-body results at small

radii. The TIS model therefore illustrates the importance of the surface pressure

on driving global halo dynamics, but fails to explain the internal halo structure

because there is no connection made to a realistic mass infall rate. A dynamical

47

model of the formation and evolution of dark matter halos will therefore need to

explain the dynamical effect of realistic mass infall.

In what follows, we shall attempt to explain both the dynamical origin of the

universal equilibrium structure of CDM halos and its evolution as the effect entirely

of the smooth and continuous accretion of mass onto individual halos, completely

ignoring such complicated details as subhalo mergers, tidal stripping, dynamical

friction, and angular momentum transport. We seek the simplest model which will

self-consistently reproduce the universal density profile and the evolution of both

the total mass and concentration parameter reported by Wechsler et al. (2002).

We describe the universal structure and evolution of CDM halos found in N-body

simulations in §3.2. In §3.3, we describe the three different ways we model halo

growth and evolution. In §3.4 we report the results of each of the models, with a

discussion in §3.5.

3.2 The Universal Halo Profile of CDM N-body Simula-

tions

3.2.1 Density Profile

The NFW profile can be written as

ρ(x)

ρ=

δvirg(c)

3x(1 + cx)2, (3.1)

where

g(c) =c2

ln(1 + c) − c/(1 + c), (3.2)

and x ≡ r/rvir and c is the NFW concentration parameter. The Moore profile can

be written as

ρ(x)

ρ=

δvirh(c)

2x3/2(1 + (cx)3/2), (3.3)

48

Figure 3.1 (Left) Density profile from equilibrium model along with best-fittingNFW profile for this profile at present. Inset in upper-right shows same overmuch larger range. (Right) Evolution of NFW concentration parameter in theequilibrium model. Different line types indicate different ranges xin < x < 1,within which halo was fit to an NFW profile, where x ≡ r/rvir,rvir ≡ r200.

where

h(c) =c3/2

log(1 + c3/2), (3.4)

x is defined as for the NFW profile, and c is now the Moore concentration param-

eter.

3.2.2 Evolution of Halo Mass

Wechsler et al. (2002) found that the mass

Mvir ≡4π

3δvirρr

3vir (3.5)

of an individual halo in their CDM simulations grows over time according to the

universal relation

Mvir(a) = M∞ exp [−Sac/a] , (3.6)

49

-0.050

0.05

-0.050

0.05

Figure 3.2 Evolution of mass for the radial orbits (top) and fluid approxima-tion (bottom) simulations. Shown above each are the fractional deviations∆ ≡ (Mexact −M)/M .

where ρ is the mean background density, ac is the scale factor at collapse, M∞

is the asymptotic virial mass as a → ∞, and S is the logarithmic mass accretion

rate dlnMvir/dlna when a = ac. Such a relation is claimed to be a good fit to the

evolution of halos of different masses and formation epochs, with the value of Sac

chosen appropriately.

3.2.3 Evolution of Halo Concentration Parameter

A universal evolution was also found for the NFW concentration param-

eter, c. At any given epoch a0, Wechsler et al. (2002) report that the best-fit

concentration parameter is

c =c1a0

ac, (3.7)

with Sac determined from the universal mass accretion history of equation (3.6),

and a best-fit value of c1 = 8.2/S.

50

The universal mass and concentration parameter evolution described above

are good fits at any collapse epoch ac and observation epoch a0 in a ΛCDM uni-

verse. This implies that these results are independent of whether the background

universe is Λ-dominated or matter-dominated. As such, any model which finds the

same halo evolution in an E-dS universe will also serve to explain it in the more

realistic ΛCDM universe. In what follows, therefore, we assume an E-dS universe,

which results in a convenient scale-free model once time is expressed in units of

the scale factor at halo collapse ac and halo mass is expressed in units of its value

at a = ac. We use δvir = 200, so that the halo is defined with a mass Mvir = M200

and a boundary at radius rvir = r200. To be consistent with Wechsler et al. (2002),

we set S = 2 in determining the collapse epoch ac.

3.3 Halo Models

3.3.1 Instantaneous Equilibration Model

In the simplest model, we have made the assumption of equilibrium inside

the halo, so that the velocity is zero for r < rvir. Using our assumption of equilib-

rium everywhere inside of rvir, we can use mass continuity to give the density ρvir

just inside the virial radius,

dMvir

da= 4πρvirr

2vir

drvir

da. (3.8)

By differentiating equation (3.5) and combining with equations (3.6) and (3.8) to

solve for ρvir, we obtain

ρvir

ρ0= δvira

−3[

1 +3a

Sac

]−1

, (3.9)

51

where ρ0 is the mean density of the universe at a = 1. The virial radius is given

by

rvir

r0= aexp

[−Sac

3

(

1

a− 1

)]

, (3.10)

where r0 is the virial radius at a = 1. Equations (3.9) and (3.10) are parametric

in a, implying a radial density profile ρ(r) = ρvir(rvir) which is frozen in place as

matter falling in from outside passes through rvir and comes to an abrupt halt. By

taking the limit in which a→ ∞, we see that the outer density profile approaches

ρ ∝ r−4 at late times, consistent with a finite mass, while the inner density profile

asymptotically approaches flatness. Although the asymptotic logarithmic slopes of

this profile do not agree with the NFW proflie, we can still ask the question of what

concentration parameter will give the NFW profile which matches the density at

the virial radius in this profile, as a function of a. By setting x = 1 in equation

(3.1) and combining with equation (3.9), we obtain an equation for the evolution

of concentration with scale factor,

a

ac= S

[

(1 + c)2

g(c)− 1

3

]

. (3.11)

3.3.2 Radial Orbits Model

If the perturbation from which the halo forms is initially cold with no ran-

dom motion, and is expanding radially with no departure from spherical symme-

try, it can be expected that all motion will remain radial, even after collapse and

virialization. This highly-contrived model is unlikely to occur in nature, but rep-

resents a limiting case and is useful for comparison to earlier studies of collapse

and secondary infall like that of Bertschinger (1985). Furthermore, radial motion

is expected to dominate outside the shell crossing region in realistic scenarios, elu-

52

Figure 3.3 Density profile at the end of the radial orbit simulation.

cidating the effects of only allowing radial motion within the halo while treating

the infalling matter in a realistic way.

We used a finite-difference spherical mass shell code to follow the evolution

of a small amplitude perturbation which we chose so that the resulting virial mass

will evolve according to the fitting formula of equation (3.6). This is accomplished

by making an initial guess, and then “tuning” the amplitude of the perturbation

until the simulated object follows the fitting formula to reasonable accuracy over

a large range of scale factors. The shell code has an inner reflecting core, which is

necessary to treat the singular nature of the coordinate system.

In choosing the initial conditions, we assume the initial perturbation has a

velocity which follows the unperterbed Hubble flow. Assuming that a given shell

encloses a constant mass M , we may then use equation (3.6) to find the initial

perturbation with which such a mass growth is consistent, as follows. The intial

53

perturbation at some initial scale factor ai is defined to be

δi(M) ≡ M −M

M, (3.12)

where M is the unperturbed mass. If the velocity is perturbed in such a way that

only the growing mode is present, then the following analysis is equivalent with δi

replaced by 3/5δi (e.g. Bertschinger 1985). For a perturbation within which the

velocity follows the unperturbed Hubble flow and δi << 1, the turn around time

tta for any shell is given by (Fillmore & Goldreich 1984)

tta = ti

(

3π

4

)

δ−3/2i , (3.13)

where ti is the time at ai. For the Einstein-de Sitter universe adopted here, we

obtain

δi =(

3πτ

4

)2/3 ai

avir, (3.14)

where τ ≡ tvir/tta and tvir is the time at which the shell has an overdensity δvir.

Combining equations (3.6) and (3.14), we obtain

δi(M) = −(

3πτ

4

)2/3 ai

Sacln[

M

M∞

]

. (3.15)

Since M∞, ai, and ac, are determined by the mass accretion history and the initial

time, the perturbation is completely described once the parameter τ is given. To

find this value, we make use of the parametric solution of the perturbation evolution

(e.g. Padmanabhan 1993),

δvir =9 [θvir − sin(θvir)]

2

2 [1 − cos(θvir)]3 (3.16)

and

τ =[θvir − sinθvir]

π. (3.17)

For δvir = 200, we find τ ' 1.8454, implying

δi(M) ' −2.664ai

Sac

ln[

M

M∞

]

. (3.18)

54

Figure 3.4 (Top) Density profile at the end of the isotropic fluid calculation. (Bot-tom) Circular velocity profile.

3.3.3 Fluid Approximation

The collisionless Boltzmann equation in spherical symmetry yields fluid

conservation equations (γ = 5/3) when random motions are isotropic. Although

halos in N-body simulations have radially biased random motion, the bias is small,

especially in the center where it is nearly isotropic. This model is therefore a better

approximation to halo formation in the more realistic N-body simulations than one

which assumes purely radial motion. To solve for the evolution, we have used a

1-D spherical hydrodynamics code which uses the finite difference scheme given by

Thoul & Weinberg (1995). The initial conditions were chosen in the same way as

those for the radial orbit model, with an initial temperature zero.

55

Figure 3.5 Evolution of NFW concentration parameter with scale factor in the fluidapproximation.

3.4 Results

Here we will concentrate on the density profile for each of the three models.

Shown in Figure 3.1 is the radial density profile from equations (3.9) and (3.10) for

the equilibrium model. It is important to emphasize that this profile changes with

time only in the sense that the outer radius rvir moves outward as matter is added.

The density at a given radius within rvir remains constant with time. As seen in

the figure, although the asymptotic inner and outer slopes differ from those of the

NFW profile, the NFW profile is still a good fit over the range of scale factors and

radii that are typically resolved in an N-body simulation.

The evolution of the best-fit NFW concentration parameter is also shown

in Figure 3.1. Since the best-fit value depends on the radius over which the fit is

performed, we have shown curves for several relevant ranges in radius, as well al

the limiting case of equation (3.11). Shown as well is the linear relation reported by

56

Figure 3.6 Phase-space density profiles for the same non-self-similar infall solutionplotted in Figure 3.2, for a/ac = 1, 2.1, and 6.1, with arrows showing locations ofr200 at each epoch, along with best-fitting power-law r−1.93.

Wechsler et al. (2002), cNFW = 4.1a/ac. As seen from the figure, the equilibrium

model is unable to reproduce the same evolution in concentration parameter as

reported for N-body results. The evolution is in the right sense, with concentration

increasing with time, but it seems the inner slope is too flat at early times. The

higher concentration at early times can then be understood if the system’s inner

density profile was built up at a time when the system was not in equilibrium,

allowing it to be steeper than it would be if the system was stationary.

The mass accretion histories of the radial orbit and fluid approximation cal-

culations are shown in Figure 3.2. As seen from the figure, both calculations have

a mass evoloution which closely follows the analytical formula, which shows that

our formula for the initial condition is correct. Shown in Figure 3.3 is the density

profile for the last time step in the radial orbit calculation, calculated by taking

57

counting shells in logarithmically-spaced radial bins. The inner density profile is

much steeper than would be expected from the equilibrium model, approaching

what seems to be and asymptotic inner shape ρ ∝ r−2. This is consistent with the

argument of Richstone & Tremaine (1984), in which it was shown that any station-

ary collisionless system consisting of only radial motion could not have a density

profile shallower than ρ−2. Since the system is not expected to be in equilibrium

within rvir at early times and is always radial, there is no implication that it should

be as flat as given by the equilibrium model. Equilibrium is a good assumption

in the center after many windings in phase space, and the results reflect this with

the inner density profile becoming the shallowest possible one for a purely radial

system in equilibrium, while becoming steeper with increasing radius.

The density profile of the radial orbit calculation can be fit by an NFW

profile outside of the radius within which the logarithmic slope is shallower than

−2. Within that radius, the difference between the model and the NFW profile

becomes more pronounced as their slopes asymptotically approach −2 and −1,

respectively. This is likely a manifestation of the statement made in §3.2.3 that

the motion is not found to be purely radial in the centers of halos in N-body

simulations, and is in fact much closer to being completely isotropic. As such, we

may expect that a dynamical model which assumes isotropic motion will do much

better in reproducing the density profile in the N-body simulations.

Shown in Figure 3.4 are the density profile and rotation curve at the end of

the hydrodynamic simulation of the fluid approximation model. As can be seen,

the profiles are well-fit by either an NFW or Moore profile over the region typically

resolved in N-body simulations, r/rvir > 0.01. Figure 3.5 shows the evolution of

the NFW concentration parameter vs. a/ac. We find the evolution is quite close to

58

linear, with a best-fit linear relation of cNFW = 4.25a/ac. This is remarkably close

to the relation reported by Wechsler et al. (2002), cNFW = 4.1a/ac. In addition,

the fluid approximation solution yields a halo phase-space density profile, ρ/σ3v,

in remarkable agreement at all times with the universal profile reported for CDM

N-body halos, as seen in Figure 3.6 (see also Shapiro et al. 2006c). It seems that all

the complicated processes which lead to the shape and evolution of the density and

phase space density profiles can be captured by a generic spherically-symmetric

model in which the logarithmic infall rate is inversely proportional to scale factor

and the dark matter has isotropic random motion.

3.5 Discussion

In summary, we have used three seperate models of successively greater

realism to study the evolution of the density profile in models with a realistic mass

accretion history. We have found that assuming equilibrium yields density profiles

with an inner slope that is too shallow, whereas assuming radial orbits gives an

inner slope that is too steep. The model which assumes isotropic motion, however,

has a density profile whose shape and evolution are in good agreement with the

N-body simulations.

That the density profile of a halo made of a smoothly distributed isotropic

collisionless fluid has an evolution which is so close to that of one which is formed

hierarchically immediately calls into question the importance of processes which are

present only in the merging scenario, such as tidal stripping and dynamical friction.

While these effects are obviously important and can affect other characteristics of

the system such as the mass and space distribution of subhalos, the success of the

fluid approximation indicates they are only of secondary importance in establishing

59

the shape and evolution of the spherically-averaged density profile of the system.

Beyond adding to our understanding of the evolution of collisionless dark

halos, the fluid approximation with a realistic mass accretion history can allow

for the study of effects which can not yet be simulated with the same resolution

as in the collisionless case. Dynamical effects of baryonic matter on the dark

halo, such as might accompany explosions, can be studied in this model with

a proper treatment of continuous infall and a good approximation to the dark

matter dynamics. Another important application of this model is to the study of

self-interacting dark matter (SIDM). As has been shown by Ahn & Shapiro (2006),

infall of matter can delay the onset of core collapse, affecting previous estimates of

the what dark matter particle cross sections are ruled out by observations. Ahn &

Shapiro (2006) have studied self-similar models, where the mass grows as a power

of scale factor, whereas realistic collapse has an evolution like that of equation

(3.6). Future work will focus on the evolution of SIDM halos with a realistic mass

accretion history.

60

Chapter 4

The H II Region of the First Star

Numerical simulations predict that the first stars in a ΛCDM universe

formed at redshifts z > 20 in minihalos with masses of about 106M. These first

stars produced ionizing radiation that heated the surrounding gas and affected

subsequent star formation, while at the same time contributing to the reionization

of the universe. To understand these radiative feedback effects in detail, we have

simulated the three-dimensional propagation of the ionization fronts (I-fronts) cre-

ated by the first massive Population III stars (M∗ = 15 − 500M) that formed

at the centers of cosmological minihalos at a redshift z = 20, outward thru the

minihalo and beyond into the surrounding intergalactic gas and neighboring mini-

halos. We follow the evolution of the H II region created by the star within the

inhomogeneous gas density field which resulted from a cosmological gas and N-

body dynamics simulation of primordial star formation. The H II region evolves

a “champagne flow,” once the early D-type I-front which leads the expansion of

the ionized gas, preceded by a shock, moves outward down the steep density gra-

dient inside the minihalo until it detaches from the shock and runs ahead as a

weak, R-type I-front. A high resolution, 3D, ray-tracing calculation tracks the

I-front throughout this “champagne phase,” taking account of the hydrodynami-

cal back-reaction of the gas by an approximate model of the ionized wind at the

center. Our simulations determine the fraction of the ionizing radiation emitted

by the first generation of Pop III stars which escapes from their parent miniha-

61

los as a function of stellar mass. The escape fraction increases with stellar mass,

with 0.7<∼ fesc<∼ 0.9 for stellar masses in the range 80<∼M∗/M

<∼ 500. Since we

follow the evolution of the H II region into the surrounding universe, we are also

able to quantify the ionizing efficiency of these stars – the ratio of total gas mass

ionized to the stellar mass – as they begin the reionization of the universe. For

M∗>∼ 80M, this ratio is about 60,000, roughly half the number of ionizing pho-

tons released per stellar baryon during the lifetime of these stars, independent of

stellar mass. In addition, we find that nearby minihalos trap the I-front, so their

centers remain self-shielded and neutral. This is contrary to the recent suggestion

that these first stars would trigger the formation of a second generation by fully

ionizing their neighbor minihalos so as to stimulate molecular hydrogen formation

in their cores. Finally, we discuss the effect of evacuating the gas from the host

halo on the growth and luminosity of the “miniquasars” that may form from black

holes that are remnants of the first stars1.

4.1 Introduction

The formation of the first stars marks the crucial transition from an initially

simple, homogeneous universe to a highly structured one at the end of the cosmic

“dark ages” (e.g., Barkana & Loeb 2001; Bromm & Larson 2004; Ciardi & Ferrara

2005). These so-called Population III (Pop III) stars are predicted to have formed

in minihalos with virial temperatures T <∼ 104 K at redshifts z >∼ 15 (e.g., Couchman

& Rees 1986; Haiman, Thoul, & Loeb 1996; Gnedin & Ostriker 1997; Tegmark

et al. 1997; Yoshida et al. 2003). Numerical simulations are indicating that

the first stars, forming in primordial minihalos, were predominantly very massive

1This work appeared previously in Alvarez, Bromm, & Shapiro 2006, ApJ, 639, 621

62

Figure 4.1 Hydrogen number density profile in a halo of mass M = 106M atz = 20. Solid line: Spherically-averaged density profile of minihalo in the SPHsimulation. Dashed line: Density profile for a SIS with TSIS = 300 K.

stars with typical masses M∗>∼ 100M (e.g., Bromm, Coppi, & Larson 1999, 2002;

Nakamura & Umemura 2001; Abel, Bryan, & Norman 2002). In this chapter,

we investigate the question: How did the radiation from the first stars ionize the

surrounding medium, modifying the conditions for subsequent structure formation?

This radiative feedback from the first stars could have played an important role

in the reionization of the universe (e.g., Cen 2003; Ciardi, Ferrara, & White 2003;

Wyithe & Loeb 2003; Sokasian et al. 2004).

Observations of the large-angle polarization anisotropy of the cosmic mi-

crowave background (CMB) with the first-year data from the Wilkinson Microwave

Anisotropy Probe (WMAP; Spergel et al. 2003) implied a free electron Thomson

scattering optical depth of τ = 0.17, suggesting that the universe was substan-

tially ionized by a redshift z = 17 (Kogut et al. 2003). Such an early episode of

reionization may require a contribution from massive Pop III stars (e.g., Cen 2003;

63

Wyithe & Loeb 2003; Furlanetto & Loeb 2005). More recent, three-year results

from WMAP, however, indicate a lower optical depth, τ = 0.09, indicating a much

later reionization epoch, around z = 11. However, the constraints on the ionizing

efficiency of early sources remains largely unchanged because WMAP also found

a much lower fluctuation amplitude on the scale of the sources that were likely

responsible for reionization (Alvarez et al. 2006c; see Chapter 6).

Analytical and numerical studies of reionization typically parametrize the

efficiency with which these stars reionize the universe in terms of quantities such

as the escape fraction and fraction of baryons able to form stars (e.g., Haiman &

Holder 2003). In order to understand the role of such massive stars in reionization,

it is therefore crucial to understand in detail the fate of the ionizing photons they

produce, taking proper account of the structure within the host halo.

Until now, studies of the propagation of the ionization front (I-front) within

the host halo have been limited to analytical or one-dimensional numerical calcu-

lations (Kitayama et al. 2004; Whalen et al. 2004). These studies suggest that

the escape fraction is nearly unity for small halos, and as shown by Kitayama et

al. (2004), is likely to be much smaller for larger halos. These conclusions should

not be taken too literally, however, since the structure of the halos in which the

stars form is inherently three-dimensional. Rather, that work should be viewed

as laying the foundation for more detailed study in three dimensions. An effort

along these lines was recently reported by O’Shea et al. (2005), focused on the

dynamical consequences of the relic H II region left by the death of the first stars.

Here, we present three-dimensional calculations of the propagation of an

I-front through the host halo (M ' 106M) and into the intergalactic medium

(IGM). We model the hydrodynamic feedback that results from photoionization

64

Figure 4.2 Top: Density profile given by the Shu solution at different timesaftersource turn-on, t = 5×104 ,105, 2.5×105, 5×105, 106, and 2×106 yr, from left toright. Bottom: Same as top but for velocity. The peak velocity, in the post-shockgas just behind the shock is constant in time, and is about 25% lower than thevelocity of the shock itself.

heating through the use of the similarity solutions developed by Shu et al. (2002;

“Shu solution”), and calculate the propagation of the I-front by following its

progress along individual rays that emanate from the star.

4.2 Physical Model for Time-dependent H II Region

At early times, as the I-front begins to propagate away from the star, its

evolution is coupled to the hydrodynamics of the gas. The effect of this hydrody-

namic response is to lower the density of gas as it expands in a wind, and eventually

65

Figure 4.3 Timescales for breakout tB and stellar lifetimes t∗ versus stellar massM∗.Our estimate for tB becomes increasingly uncertain toward smaller stellar mass.Within these uncertainties, we estimate that no ionizing radiation will escape intothe IGM for M∗

<∼ 15M.

the I-front breaks away from the expanding hydrodynamic flow, racing ahead of

it. In what follows, we describe an approximate, spherically-symmetric model for

the relation between the I-front and the hydrodynamic flow in the center of the

halo. This allows us to account for the consumption of ionizing photons within the

halo while at the same time tracking the three-dimensional evolution of the I-front

after it breaks out from the center of the halo and propagates into the surrounding

IGM.

4.2.1 Early Evolution

In a static density field, it takes of order a recombination time for an I-

front propagating away from a source that turns on instantaneously to slow to

its “Stromgren radius” rS, at which point recombinations within balance photons

being emitted by the source. Generally, the I-front moves supersonically until it

66

approaches the Stromgren radius, at which point it must become subsonic before it

slows to zero velocity. The supersonic evolution of the I-front is generally referred

to as “R-type” (rarefied), whereas the subsonic phase is referred to as “D-type”

(dense). In the R-type phase, the I-front races ahead of the hydrodynamic response

of the photoheated gas. In the D-type phase, however, the gas is able to respond

hydrodynamically, and a shock forms ahead of the I-front (see, e.g., Spitzer 1978).

For the case considered here of a single, massive Pop III star forming in the

center of a minihalo, this initial R-type phase when the star first begins to shine is

likely to be very short lived, of order the recombination time in the star-forming

cloud, trec < 1 yr for n ' 106 cm−3. This time is even shorter than the time it takes

for the star to reach the main sequence, t ' 105 yr, given by the Kelvin-Helmholtz

time. Thus, hydrodynamic effects are likely to be dominant at early times when

the I-front is very near to the star.

The study of the formation of these stars at very small scales defines the

current frontier of our understanding, where the initial gas distribution and its in-

teraction with the radiation emitted by the star is highly uncertain (e.g., Omukai

& Palla 2001, 2003; Bromm & Loeb 2004).For example, it is still not yet known

whether a centrifugally supported disk will form (e.g., Tan & McKee 2004), or

whether hydrodynamic processes can efficiently transport angular momentum out-

ward, leaving a sub-Keplerian core (e.g., Abel et al. 2002). Omukai & Inutsuka

(2002) studied the problem in spherical symmetry, making certain assumptions

about the accretion flow and the size of a spherical H II region. Since the density

and ionization structure in the immediate vicinity of accreting Pop III protostars is

only poorly known, we must also make some assumptions here about the progress

of the I-front at distances unresolved in the simulation we use (<∼ 1 pc).

67

We will therefore assume in all that follows that the hydrodynamic response

of the gas in the subsequent D-type phase will be to create a nearly uniform den-

sity, spherically-symmetric wind, bounded by a D-type I-front that is led by a

shock. The degree to which the gas within this spherical shock is itself spherically-

symmetric depends on the effectiveness with which the high interior pressure can

reduce density inhomogeneities within. The timescale for this effect is the sound-

crossing timescale, comparable to the expansion timescale of the weakly-supersonic

D-type shock that moves at only a few times the sound speed. Since these two

timescales are comparable, it is plausible that pressure will be able to homoge-

nize the density structure behind the shock. Furthermore, the one-dimensional

spherically-symmetric calculations of Whalen et al. (2004) and Kitayama et al.

(2004) indicate that pressure is capable of homogenizing the density in radius, as

shown by the nearly flat density profiles found behind the shock in the D-type

phase. Although these are reasonable assumptions for these first calculations, we

caution the reader that the detailed evolution of the I-front, especially very close

to the star itself, can only be thoroughly understood by fully-coupled radiative

transfer and hydrodynamic simulations that resolve the accretion flow around the

star, which we defer to future study.

4.2.2 Model for Breakout

Primordial stars are expected to form enshrouded in a highly concentrated

distribution of gas. For a star forming within a halo with mass ' 106M, the

spherically-averaged density profile of the gas, just prior to the onset of protostellar

collapse, is well-approximated by that of a singular isothermal sphere (SIS) with

a temperature ∼ 300 K (Figure 1). For values relevant to a star-forming region in

68

a minihalo with mass M ∼ 106M at z ∼ 20, the hydrogen atom number density

profile of the SIS is given by

n(r) ' 2.3 × 103(

TSIS

300K

)

(

r

1pc

)−2

cm−3, (4.1)

where we have assumed a hydrogen mass fraction X = 0.75. We will take the SIS

as a fiducial density profile for the calculations presented here.

As discussed in §4.2.1, a D-type shock initially propagates outward just

ahead of the I-front, leading to an outflow and corresponding drop in central den-

sity. After some time t = tB, however, the central density is sufficiently lowered so

that recombinations can no longer trap the I-front behind the shock, and it quickly

races ahead.

This “breakout” time tB marks the moment at which ionizing radiation is no

longer bottled up within and can escape. If the lifetime of the star t∗ < tB, then the

front never escapes and the escape fraction is zero. This is essentially the reasoning

used by Kitayama et al. (2004) to explain their result that ionizing radiation does

not escape from halos with mass M > Mcrit, where Mcrit is determined by setting

t∗ = tB.

After breakout, the gas left behind is close to isothermal with the high tem-

perature of a photoionized gas (a few times 104 K). The strong density gradient

results in a pressure imbalance that drives a wind outward, bounded by an isother-

mal shock. This “champagne” flow (e.g., Franco et al. 1990) has been analyzed

through similarity methods by Shu et al. (2002), who found self-similar solutions

for different power-law density stratifications ρ ∝ r−n, 3/2 < n < 3, and is also

evident in the one-dimensional calculations of Whalen et al. (2004) and Kitayama

et al. (2004). The family of solutions obtained by Shu et al. (2002) for the n = 2

69

case are described in terms of the similarity variables (eqs. (12) and (13) of Shu

et al. 2002)

x =r

cst(4.2)

and

ρ(r, t) =mpn(r)

X=

α(x)

4πGt2, (4.3)

where cs is the sound speed of the ionized gas and α(x) characterizes the shape

of the density profile in the champagne flow. If gas within the initial SIS has a

sound speed cs,1, then different solutions are obtained for α(x), depending on the

ratio ε ≡ (cs,1/cs)2, where cs,1 and cs are the initial SIS sound speed and ionized

gas sound speed, respectively. For T1 ∼ 300K and T ∼ 2 × 104K, ε ∼ 0.007 and

the shock moves at vs = xscs ' 40 km s−1, where xs = 2.55. In Figure 4.2, we

have plotted the density and velocity profiles in the Shu solution for the above

parameters. As seen in the figure, the density drops steadily in the center and is

nearly uniform, while the velocity profile is unchanged as it moves outward. The

peak velocity, corresponding to post-shock gas, is constant in time, ∼ 30 km s−1,

and is less than the velocity of the shock itself.

In what follows, we describe a model for when and where breakout occurs

by finding the moment in the post-breakout Shu solution where the recombination

rate inside of the shock is equal to the ionizing luminosity of the star. The condition

for breakout is

Q∗ = 4παB

∫ rsh(tB)

0r2n2(r, tB)dr, (4.4)

where αB = 1.8 × 10−13 cm3 s−1 is the recombination rate coefficient to excited

states of hydrogen at T ∼ 2×104K, n(r, t) is the number density in the Shu solution,

Q∗ is the ionizing photon luminosity of the star, and rB ≡ rsh(tB) = csxstB is the

70

Figure 4.4 Top: Instantaneous escape fraction for different masses, as labeled.Bottom: Time-averaged escape fraction 〈fesc〉 as defined in the text. Although theinstantaneous escape fraction rises quickly just after breakout, the time-averagedvalue retains memory of the breakout time and therefore lags behind.

position of the shock at breakout. As shown by Bromm, Kudritzki, & Loeb (2001),

the ionizing luminosity of primordial stars with masses M > 100M is roughly

proportional to the mass of the star, Q∗ ' 1.5×1050 s−1(M/100M) (see Schaerer

2002 for more detailed calculations). Combining equations (4.2), (4.3), and (4.4),

we can solve for the breakout radius

rB =αBc

4sxS

4π(µmpG)2Q∗

∫ xS

0α2(x)x2dx. (4.5)

For fiducial values, we obtain

rB ' 2.3pc(

TSIS

300K

)2 ( Q∗

3 × 1050s−1

)−1

. (4.6)

71

Parameterizing the speed of the shock as vs, we can use the formula tB = rB/vs

to derive the time after turn on at which breakout occurs,

tB ' 5.6 × 104yr

(

vs

40 km/s

)−1 (T

300K

)2 ( Q∗

3 × 1050s−1

)−1

. (4.7)

Here and in the calculations we will present, we assume that the speed of the

shock front in the D-type and champagne phases is the same, vs ' 40 km s−1. The

lifetimes of massive stars with masses 100 < M/M < 500 are within the range

2<∼ t∗<∼ 3 Myr (e.g., Bond, Arnett, & Carr 1984), much longer than our estimate

of tB ∼ 5.6 × 104 yr for our fiducial values. Thus, we expect the time-dependent

fraction of ionizing photons that escape from the halo to rapidly approach unity

over this mass range.

For lower stellar masses, and therefore lower values of Q∗, the breakout

time tB becomes comparable to the stellar lifetime t∗, which itself increases with

decreasing mass (see Figure 4.3). The precise value of the stellar mass at which

tB = t∗ is therefore quite sensitive to the speed of the D-type shock, the density

profile used in equation (4.4), and, of course, the main sequence lifetime of the

star. In particular, the early hydrodynamic behavior of the gas in the D-type

phase depends on an extrapolation to small scales where the mass distribution is

not well understood. While our model for breakout is consistent with the one-

dimensional calculations of Whalen et al. (2004) and Kitayama et al. (2004) in

predicting that the escape fraction for massive stars M∗ > 100M approaches

unity because breakout occurs early in their lifetimes, the threshold stellar mass

at which tB = t∗ and the escape fraction goes to zero is not well determined and

deserves further attention.

72

Figure 4.5 Mean escape fraction at the end of the star’s lifetime t∗, versus stellarmass. Each symbol corresponds to 〈fesc〉 as defined by Eq. (4.22) for a differ-ent stellar mass calculation. Note that the escape fraction approaches zero forM <∼ 15M, for which tB >∼ t∗.

4.3 Numerical Methodology

4.3.1 Cosmological SPH Simulation

The basis for our radiative transfer calculations is a cosmological simulation

of high-z structure formation that evolves both the dark matter and baryonic

components, initialized according to the ΛCDM model at z = 100, to z = 20.

We use the GADGET code that combines a tree, hierarchical gravity solver with

the smoothed particle hydrodynamics (SPH) method (Springel, Yoshida, & White

2001). In carrying out the cosmological simulation used in this study, we adopt

the same parameters as in earlier work (Bromm, Yoshida, & Hernquist 2003). Our

periodic box size, however, is now L = 200h−1 kpc comoving. Employing the same

number of particles, NDM = NSPH = 1283, as in Bromm et al. (2003), the SPH

particle mass here is ∼ 70M.

73

We place the point source, representing the already fully formed Pop III

star, at the location of the highest density SPH particle in the simulation at z = 20,

n ∼ 104 cm−3, located within a halo of mass Mvir ∼ 106M and virial radius

rvir ∼ 150 pc. We assume that the ionizing photon luminosity is constant over the

lifetime of the star, with values given in Table 4 of Schaerer (2002). The final time

in each run is set to the corresponding stellar lifetime for each mass, also given in

Table 4 of Schaerer (2002).

4.3.2 Ray Casting

In order to calculate the evolution of the I-front, we must know the density

of hydrogen along the ray. We do this by means of interpolation from a mesh upon

which the density is precalculated. The density within each segment is assigned

from the mesh at the midpoint of the ray segment,

r3i+1/2 ≡

1

2

(

r3i + r3

i+1

)

. (4.8)

This discretization ensures that the midpoint of the ray is located at the point

at which half the mass within the volume element is at r < ri+1/2 and the other

half is at r > ri+1/2. The density value at the midpoint is determined by tri-linear

interpolaton from the eight nearest nodes on the mesh,

n(xm, ym, zm) =8∑

g=1

ngf(xg)f(yg)f(zg), (4.9)

where f(xg) ≡ 1 − |xm − xg|/∆c, ∆c is the mesh cell size, (xg, yg, zg) are the

coordinates of the eight nearest grid points, ng is the density at grid point g,

xm = ri+1/2 sin θ cosφ, (4.10)

ym = ri+1/2 sin θ sinφ, (4.11)

zm = ri+1/2 cos θ, (4.12)

74

and (θ, φ) are the angular coordinates of the ray.

Given the substantial dynamic range necessary to resolve the H II region

around a Pop III star, the use of only one uniform mesh to interpolate between the

SPH density field and the rays is not possible. Here we make use of the fact that

the system is highly centrally concentrated, which allows for the use of a set of

concentric equal-resolution uniform meshes, each one half the linear size of the last,

centered on the star forming region in the center of the halo. Since the segments are

spaced logarithmically in radius, the segment size at any point is smaller than the

mesh cell of the highest resolution mesh that overlaps that point. We interpolate

the SPH density to each of the hierarchical meshes (see next section). For each

ray segment midpoint, we find the highest resolution mesh overlapping that point

and use tri-linear interpolation from that mesh, as described above.

4.3.3 Mass-conserving SPH Interpolation onto a Mesh

Rather than interpolate directly from the SPH particles to our spherical

grid of rays, we first interpolate the density to a uniform rectilinear mesh. The

assignment of density to the mesh should conserve mass, which we accomplish as

follows. We use a Gaussian kernel

W (r, h) =1

π3/2h3e−r2/h2

, (4.13)

where h is the smoothing length and r is distance. This kernel is very similar to

the commonly-used spline kernel,

W (r, h) =8

πh3

1 − 6(

rh

)2+ 6

(

rh

)3, 0 ≤ r

h≤ 1

2,

2(

1 − rh

)3, 1

2≤ r

h≤ 1,

0 rh> 1.

(4.14)

75

In our case, where a spline kernel has been used in the SPH calculation, we find that

a Gaussian kernel is sufficient for interpolation purposes, provided we transform

the smoothing lengths according to hGauss = π−1/6hspline/2.

For interpolation to a uniform rectilinear mesh with cell size ∆c, we wish to

find the mean density within a cell centered at (x, y, z) contributed by a particle

with smoothing length h located at the origin,

W (r, h) ≡ 1

∆3c

∫

VW (r, h), (4.15)

where the integral is over the volume of the cell. Since the kernel is a Gaussian,

the spatial integral separates into three separate ones, so that

W (r, h) =1

8∆3c

Ξ(x)Ξ(y)Ξ(z), (4.16)

where

Ξ(s) ≡ erf

[

s+ ∆c/2

h

]

− erf

[

s−∆c/2

h

]

. (4.17)

The value of the density averaged over a cell centered at rc is thus

ρ(rc) =∑

i

miW (rc − ri, hi), (4.18)

where the sum is over all particles i such that W (rc − ri, hi)/W (0, hi) > ε, so as

not to needlessly sum over particles with a negligible contribution. We find that a

value ε = 10−5 is sufficient for our purposes.

4.3.4 Ionization Front Propagation

In deriving the I-front evolution, we make the approximation that the front

is sharp – i.e. gas is completely ionized inside and completely neutral outside.

Because the equilibration time is short on the ionized side, every recombination is

76

balanced by an absorption. Under this assumption, the I-front “jump condition”

(Shapiro & Giroux 1987) implies a differential equation for the evolution of the

I-front radius (Shapiro et al. 2005; Yu 2005),

dR

dt=

cQ(R, t)

Q(R, t) + 4πR2cn(R), (4.19)

where Q(R, t) is the ionizing photon luminosity at the surface of the front. This

equation correctly takes into account the finite travel time of ionizing photons (e.g.,

R→ c as Q(R, t) → ∞). In general, this equation can be solved numerically, once

Q(R, t) and n(R) are known.

To approximate the hydrodynamic response due to photoheating, we com-

bine the Shu solution with our ray tracing method to follow the I-front after it

breaks out into the rest of the halo. We assume that the front makes an initial

transition from R to D-type at radii r 1pc, and creates a spherical D-type front

that propagates outward to the breakout radius rB, after which the Shu solution is

expected to be valid. For each ray, we assume that the density profile of the gas at

breakout is given by the self-similar solution inside of the shock, and is undisturbed

outside, given by our cosmological SPH simulation (see §4.3.1).

The initial I-front radius is independent of angle and is initialized to the

breakout radius, so that equation (4.19) is solved with the initial value R(tB) = rB.

Q(R, t) in equation (4.19) depends on the density profile along a ray, and is given

by

Q(R, t) = Q∗ − 4παB

∫ R

0n2(r, t)r2dr, (4.20)

where n(r, t) is given by the Shu solution at t for all r < rsh(t), while for r > rsh(t)

the density is the initial unperturbed angle-dependent density distribution along

77

Figure 4.6 Ratio of ionized gas mass to stellar mass, ηHII, versus stellar mass.Notice that for M∗

>∼ 80M, this ratio is almost independent of stellar mass.

each ray. More details on the ionization front tracking method can be found in

Iliev et al. (2006c) and Mellema et al. (2006).

4.4 Results

We have carried out several ray-tracing runs, each for a different stellar

mass forming within the same host minihalo.

4.4.1 Escape Fraction

The fraction of the ionizing photons emitted by the central star which es-

cape into the IGM beyond the virial radius of the host minihalo is a fundamental

ingredient in the theory of cosmic reionization and of the feedback of Pop III star

formation on subsequent star and galaxy formation. We use our H II region cal-

culations to derive this escape fraction fesc and its dependence on time during

78

the lifetime of the star. Since our H II region density field and radiative transfer

are three-dimensional, the escape fraction is angle-dependent. Along each ray, the

escape fraction is given by

fesc(t) =

1 − 4παB

Q∗

∫ rvir0 n2(r, t)r2dr, R(t) > rvir,

0, R(t) ≤ rvir,(4.21)

where rvir = 150 pc and n(r, t) is given by the Shu solution for r < rsh(t), and by

the SPH density field in that direction for r > rsh(t). The instantaneous escape

fraction versus time, fesc, determined by taking the average over all angles of the

angle-dependent escape fraction is shown in the top panel of Figure 4.4. The

average escape fraction between turn-on and time t < t∗ is given by

〈fesc〉 ≡1

t

∫ t

0fesc(t

′)dt′, (4.22)

and is shown in the bottom panel of Figure 4.4. Figure 4.5 shows the average

escape fraction at the end of the star’s lifetime versus mass. For the very high mass

M∗ = 500M case, the mean escape fraction is 〈fesc〉 ∼ 0.9, while for M∗ = 80M

it is about 0.7. We can understand the zero lifetime-averaged escape fraction at the

smallest masses, as evident in Figure 4.5, by comparing the lifetime of the star and

the breakout time. For tB < t∗ breakout occurs before the star dies and the escape

fraction is expected to be greater than zero. For tB > t∗, little or no radiation

should escape. As can be seen in Figure 4.3, the threshold mass for which tB = t∗

is about 15 M. However, as we discussed in §4.2.2, the value of this threshold

mass is very sensitive to the parameters of our model. The escape fractions at

masses M∗<∼ 50M are not robust predictions of our calculations, but are shown

here for completeness.

79

Figure 4.7 Volume visualization at z = 20 of neutral density field (blue – lowdensity, red – high density) and I-front (translucent white surface). Top row panelsshow a cubic volume ∼ 13.6 kpc (proper) across, middle row ∼ 6.8 kpc, and bottomrow ∼ 3.4 kpc. Left column is at the initial time, middle column shows simulationat t∗ = 3 Myr for the run with stellar mass M∗ = 80M, and the right columnshows simulation at t∗ = 2.2 Myr for the run with stellar mass M∗ = 200M.The empty black region in the lower panels of middle and right columns indicatesfully ionized gas around the source, and is fully revealed as the volume visualizedshrinks to exclude the I-front that obscures this region in the larger volumes above.

80

4.4.2 Ionization History

Shown in Figure 4.8 is the evolution of the ionized gas mass outside the

halo, MHII(t), for different stellar masses. As expected, the more massive the star,

the more gas is ionized. When expressed in units of the mass of the star, however,

the quantity ηHII ≡ MHII(t∗)/M∗ is approximately constant with stellar mass for

M∗>∼ 80, ηHII ' 50, 000 − 60, 000 (Figure 4.6). In the absence of recombinations,

so that every ionizing photon results in one ionized atom at the end of the star’s

lifetime, ηHII = ηph, where

ηph ≡ Q∗t∗mp

XM∗

(4.23)

is the number of ionizing photons produced per stellar H atom over the star’s

lifetime. This efficiency is roughly independent of mass for massive primordial

stars M∗>∼ 50M, ηph ' 90, 000 − 100, 000 for X = 0.75 (e.g., Bromm et al.

2001; Venkatesan, Tumlinson, & Shull 2003; Yoshida, Bromm, & Hernquist 2004).

Recombinations cause the value of ηHII to be lower than ηph by about a factor of

two for large masses.

4.4.3 IMF dependence

Given the strong negative radiative feedback from H2 dissociating radia-

tion that is expected once a star forms within a minihalo (e.g., Haiman, Rees, &

Loeb 1997; Haiman, Abel & Rees 2000), it is unlikely that more than one star

will exist there at any given time. This negative feedback may extend to nearby

halos, though there is some uncertainty as to how strong this negative feedback

is (e.g., Ferrara 1998; Riccotti, Gnedin & Shull 2002). Thus, the first generation

of stars forming within minihalos likely formed in isolation, and the initial mass

function (IMF) of these stars was probably determined by various properties of

81

Figure 4.8 Mass ionized MHII versus time for different stellar masses, as labeled.More massive stars produce more ionizing photons in their lifetime and, therefore,ionize more of the surrounding gas.

the host halos, such as their angular momentum and accretion rate. We make the

reasonable assumption that the density structure of halos with a mass M ∼ 106M

is universal, so that our determination of the escape fraction for this halo is close

to what would be expected for other halos of comparable mass. For host halos of

this mass, therefore, variations in the escape fraction come only from variations in

stellar mass. Under these assumptions, we can convolve our results for one halo

with different IMFs to see how the average escape fraction depends on the IMF.

Usually, when applied to present-day star formation, the IMF describes the actual

distribution of stellar masses in

a cluster consisting of many members. In the primordial minihalo case,

however, where stars are predicted to form in isolation, as single stars or at most

as small multiples, the “IMF” would be more appropriately interpreted as a ‘single-

draw’ probability distribution (Bromm & Larson 2004). Our analysis here is carried

82

out in this latter sense.

For definiteness, we use a Salpeter-like functional form, given by

Φ(M) =

KM−1.35, Mmin < M < Mmax

0, otherwise,(4.24)

and normalized so that∫ ∞

0Φ(M)d lnM = 1. (4.25)

The total escape fraction, assuming one star forms per halo of mass M ∼ 106M,

is given by

f IMFesc ≡

∫∞0 Φ(M)Q∗(M)t∗(M)fesc(M)d lnM∫∞0 Φ(M)Q∗(M)t∗(M)d lnM

, (4.26)

where the total number of photons released over a star’s lifetime, Q∗t∗ appears in

the integrand because the escape fraction is being averaged over a period of time

which is long compared to the lifetime of a star. For Mmin = 0.5 and Mmax = 500,

which is a conservative estimate for the maximum Pop III stellar mass (Bromm &

Loeb 2004), for example, the escape fraction is ∼ 0.5, whereas for Mmin = 0.5 and

Mmax = 80, it is ∼ 0.35.

4.4.4 Structure of H II region

As can be seen from the visualization in Figure 4.72, the structure of the

H II region is highly asymmetric, with deep shadows created by overdense gas. In

particular, nearby halos are not ionized, but rather are able to shield themselves

and all that is behind them from the ionizing radiation of the star. This can

clearly be seen in the bottom panels of Figure 4.7, where overdense gas near to the

central star remains neutral, despite being so close. Figure 4.9 shows the location

2This visualization was produced using ray-tracing software written by the author. Morevisualizations can be found at http://galileo.as.utexas.edu

83

of neutral and ionized SPH particles close to the star, showing that the highest

density gas nearby the star remains neutral. For example, the highest density of

hydrogen that is ionized within 500 pc of the 120M star is ∼ 2 cm−3 at a radius of

' 200 pc, which corresponds to an overdensity of δ ∼ 4× 103, whereas the highest

density of neutral hydrogen is ∼ 400 cm−3 at a radius of ' 220 pc, corresponding

to δ ∼ 2.5×105. Similarly overdense gas that is further from the star is even more

likely to shield itself and remain neutral, since the flux there is weaker because of

spherical dilution. We find that ∼ 4.9% of the sky at the end of the life of the

80M star is covered by high density gas that traps the I-front, whereas ∼ 2.6%

of the sky is covered at the end of the 200M star’s life. Such shielded regions

are likely to be the sites of photoevaporation (Shapiro, Iliev & Raga 2004). The

photoevaporation time of a 2 × 105M halo which is at a distance of 250 pc from

a 120M star with luminosity 1.4 × 1050s−1 is ' 16 Myr (Iliev, Shapiro, & Raga

2005), longer than the lifetime of the star, so that most minihalo gas is likely to

retain its original density structure.

An important quantity associated with the “relic H II region” is the clump-

ing factor. Shown in Figure 4.10 is the clumping factor of the relic H II region,

cl ≡ 〈n2〉/n2, where the average is over the volume of the H II region and n is the

cosmic mean density. Thus, the recombination time in the H II region is given by

trec = trec,0/cl, where trec,0 ' 100 Myr is the recombination time of gas at the cos-

mic mean density. As the mass and luminosity of the star increase, the clumping

factor of the relic H II region decreases. Apparently, clustering of matter around

the host halo causes the clumping factor to increase near the halo. Lower lumi-

nosity sources leave behind smaller, and therefore more clumpy, relic H II regions.

The mean recombination time in the regions, however, is always less than the Hub-

84

Figure 4.9 Position of selected SPH particles within 500 pc of the 120 M star.Red particles are ionized and have a density above 1 cm−3, all the neutral particlesare shown in cyan, while only neutral particles with a density above 4 cm−3 arecolored black. The radius of the shock in the Shu solution at the end of the star’slifetime, ∼ 100 pc, is shown as the circle in the center. No SPH particles are shownwithin that radius.

ble time ∼ 175 Myr for even the largest stellar masses, with recombination times

trec < 60 Myr. These H II regions are thus likely to recombine unless other sources

are able to keep them ionized. The timing of this recombination and the associated

cooling of the recombining gas is crucial to understanding the effect of photoheat-

ing on suppressing subsequent halo formation, the so-called “entropy-floor” (Oh &

Haiman 2003).

4.4.5 I-front trapping by neighboring halos

Whether or not nearby halos trap the I-front should determine whether

ionization stimulates star formation in their centers. We can estimate the radius

85

and density at which trapping occurs, as follows. The condition for trapping is

F =∫ rvir

rt

αBn2H(r)dr, (4.27)

where F is the external flux of ionizing photons, rt is the radius at which the I-front

is trapped, and rvir is the virial radius of the halo. If we assume that the halo has

a singular isothermal sphere density profile nH(r) ∝ r−2 and an overdensity δvir,

then solving for rt we obtain

rt

rvir=

[

1 +9(36π)1/3Fm2

HΩ2m

αBX2M1/3vir (ρ(z)δvir)

5/3 Ω2b

]−1/3

, (4.28)

where ρ(z) is the mean matter density of the universe at redshift z. For r3t /r

3vir 1,

we can neglect the first term in the brackets, to see how this trapping condition

depends upon the source and halo parameters and the redshift,

rt

rvir≈

4παBρ5/30 X2Ω2

b

9(36π)1/3m2HΩ2

m

1/3

M1/9vir δ

5/9vir Q

−1/2∗ r2/3(1 + z)5/3, (4.29)

where ρ0 is the mean matter density at present. The density at the point where

the I-front is trapped is

nt ≡ nH(rt) =Xρ(z)Ωbδvir

3mHΩm

r2vir

r2t

. (4.30)

If we use the parameters of the minihalo nearest to our source halo for fiducial

values,

rt

rvir≈ 0.18

(

Mvir

2 × 105M

)1/9(r

220 pc

)2/3(δvir

200

)5/9

×(

Q∗

1.4 × 1050 s−1

)−1/3 (1 + z

21

)5/3

(4.31)

and

nt ≈ 3.6 cm−3

(

Mvir

2 × 105M

)−2/9 (r

220 pc

)−4/3

(4.32)

×(

δvir

200

)−1/9 (Q∗

1.4 × 1050 s−1

)2/3(1 + z

21

)−1/3

,

86

corresponding to a halo with total mass Mvir = 2 × 105M that is exposed to

a flux F = Q∗/(4πr2) from a source with luminosity Q∗ = 1.4 × 1050s−1 (for a

stellar mass M∗ = 120M) at a distance r = 220 pc. Since M(< R) ∝ R in a

singular isothermal sphere, MHI/Mvir = rt/rvir, and thus the neutral gas mass for

the fiducial case above is ' 5.5×103M. The density at which the I-front is trapped

is much smaller than the central density expected for a truncated isothermal sphere

(Shapiro, Iliev, & Raga 1999), nH,0 ' 30 cm−3 at z = 20. Thus, nearby halos trap

the I-front well before it reaches the central core. The weak dependence of rt/rvir

on halo mass and luminosity implies that trapping is a generic occurrence for halos

surrounding single primordial stars.

4.5 Discussion

We have studied the evolution of the H II region created by a massive Pop

III star which forms in the current, standard ΛCDM universe in a minihalo of

total mass M ∼ 106M at a redshift of z = 20. We have performed a three-

dimensional ray-tracing calculation which tracks the position of the expanding I-

front in every direction around the source in the pre-computed density field which

results from a cosmological gas and N-body dynamics simulation based on the

GADGET tree-SPH code. During the short lifetime (<∼ few Myr) of such a star,

the hydrodynamical back-reaction of the gas is relatively small as long as the front

is a supersonic R-type, and, to first approximation, we are justified in treating

the gas in this “static limit.” At early times, however, when the I-front is still

deep inside the minihalo which formed the star, the I-front is expected to make a

transition from supersonic R-type to subsonic D-type, preceded by a shock, before

it eventually accelerates to R-type again and detaches from the shock, racing ahead

87

Figure 4.10 Top: Mean recombination time of H II region at end of star’s life vs.stellar mass. Bottom: Clumping factor of H II regions at end of star’s life vs.stellar mass. Less massive stars ionize a smaller volume, which implies a higherclumping factor because of clustering around the host halo.

of it.

To account for the impact of the expansion of the gas which results from this

dynamical phase on the propagation of the I-front after it “breaks out,” we have

used the similarity solution of Shu et al. (2002) for champagne flow. This solution

allows us to determine when the transition from D-type to R-type and “break-out”

occurs and, thereafter, to account for the consumption of ionizing photons in the

expanding wind left behind in the central part of the minihalo. In this way, we are

able to track the progress of the I-front inside the host minihalo and beyond, as

88

it sweeps outward through the surrounding IGM and encounters other minihalos.

This has allowed us to investigate the link, for the first time, between the formation

of the first stars and the beginning of cosmic reionization on scales close to the

stellar source that could not be resolved in previous three-dimensional studies of

cosmic reionization. Among the results of this calculation are the following.

Our simulations allow us to quantify the ionizing efficiency of the first-

generation of Pop III stars in the ΛCDM universe as a function of stellar mass.

The fraction of their ionizing radiation which escapes from their parent minihalo

increases with stellar mass. For stars in the mass range 80<∼M∗/M<∼ 500, we find

0.7<∼ fesc<∼ 0.9. This high escape fraction for high-mass stars is roughly consistent

with the high escape fraction found for such high-mass stars by one-dimensional,

spherical, hydrodynamical calculations (Whalen et al. 2004; Kitayama et al. 2004).

For lower-mass stars, the escape fraction drops more rapidly with decreasing mass,

as it takes a longer and longer fraction of the stellar lifetime for the I-front to end

the D-type phase by reaching the “break-out” point, detaching from the shock and

running ahead as a weak, R-type front to exit the halo. For M∗<∼ 15 − 20M, in

fact, we find that the escape fraction should be zero and the I-front is D-type for the

whole life of the star. More importantly, we find that this threshold mass is very

sensitive to the hydrodynamic evolution of the I-front in the D-type phase. Given

the great uncertainty regarding the interaction of the stellar radiation and the gas

immediately surrounding the star, a definitive answer to this question can only

be obtained through three-dimensional gas dynamical simulations with radiative

transfer that properly resolve the accretion flow onto the star.

Once the H II region escapes the confines of the parent minihalo, the reion-

ization of the universe begins. Our simulations yield the ratio of the final total

89

mass ionized by each of these first-generation Pop III stars to their stellar mass.

We find that, for M∗>∼ 80M, this ratio is about 60,000, roughly half the number

of H ionizing photons released per stellar atom during the lifetime of these stars,

independent of stellar mass.

We can obtain a very rough estimate of how effective Pop III stars are at

reionizing the universe by assuming that all the stars have the same mass and form

at the same redshift, each in its own minihalo of mass ∼ 106M. In the limit where

the H II regions of individual stars are not overlapping, the ionized mass fraction

fraction is given in terms of the halo mass function by fHII = ρHII/ρH where ρHII ≡

ηHIIM∗dn/d lnM is the mean density of ionized gas, assuming each halo hosts a star

of mass M∗, and ionized a mass ηHII times the star’s own mass, and ρH is the mean

mass density of hydrogen (For M∗ ' 80M, for example, ηHII ' 50, 000). Using

the mass function of Sheth & Tormen (2002), dn/d lnM ' 130 Mpc−3 in comoving

units at z = 20 for this mass range, we obtain fHII ' 0.1[ηHII/5 × 104][M∗/80M].

If, instead, we use the ionized volume VHII obtained directly from our calculations

to derive the volume filling factor fV,HII ≡ VHIIdn/d lnM for an 80M star, the

final ionized volume is 7×10−4 comoving Mpc3, corresponding to a filling factor of

fV,HII(M∗ = 80M) ' 0.1. The similarity of the volume and mass ionized fraction

indicates that the mean density in the ionized region is approximately equal to the

mean density of the universe.

The above estimate is only the instantaneous ionized fraction, since the

recombination times of each relic H II region are small fractions of the age of the

universe at z = 20. This implies that many new generations of similar stars would

have to form to continuously maintain this ionized fraction. A more conservative

estimate of the effect of Pop III stars on reionization would also have to take

90

account of the back-reaction of the starlight from earlier generations of stars on

the star formation rate in the minihalos that follow (e.g., Mackey, Bromm, &

Hernquist 2003; Yoshida et al. 2003; Furlanetto & Loeb 2005). Since Pop III star

formation depends upon the efficiency of H2 formation and cooling inside minihalos,

a background of UV starlight between 11.2 eV and 13.6 eV is capable of suppressing

this star formation by photodissociation the H2 following absorption in the Lyman-

Werner bands. It is quite possible that the photodissociating background builds

up fast enough that minihalos are “sterilized” against further star formation before

such a large fraction of the universe can be reionized in this way (Haiman et al.

1997).

Hydrodynamic feedback due to photoionization heating of the host halo will

have a dramatic impact on its ultimate fate. Massive Pop III stars are expected to

end their lives either by collapsing to black holes or exploding as pair-instability

supernovae (e.g., Madau & Rees 2001; Heger et al. 2003). In both cases, it is

important to know the properties of the host halo gas. Our model for the hydro-

dynamic feedback, in which an ionized, nearly-uniform density bubble bounded by

an isothermal shock propagates outward at a few times the sound speed allows for

an estimate of the density and size of the bubble at the end of the star’s lifetime.

For a lifetime of 2.5 Myr, the final size and density of the bubble are rbubble ' 100

pc and nbubble<∼ 1 cm−3. If the star ended its life by exploding, rather than collaps-

ing to a black hole, then the SN remnant evolution inside this low-density bubble

and beyond will differ from its evolution in the original undisturbed minihalo gas.

This should be taken into account in models of the impact of first-generation SN

remnant blast-waves on their surroundings (e.g., Bromm, Yoshida, & Hernquist

2003).

91

This density can be used to estimate the accretion rate onto a possible

remnant black hole, MB ' 4πG2M2BHρ/c

3s (e.g., Bondi & Hoyle 1944; Springel, Di

Matteo, & Hernquist 2005). Thus, e.g., for a black hole mass of 100M and a sound

speed of 15 km s−1, we obtain MB ∼ 2×10−5M Myr−1. If we make the optimistic

assumption that after a recombination time (∼ 1.2 × 105 yr) the gas can form H2

molecules and cool back to ∼ 300 K without escaping from the halo, the accretion

rate increases by a factor of 103, to ∼ 2 × 10−2MMyr−1. These accretion rates

are small compared to the Eddington accretion rate MEdd = 4πGMBHmp/(εσTc) '

2M Myr−1, where the efficiency factor ε = 0.1 (see also O’Shea et al. 2005). Such

low accretion rates imply that remnant black holes from the first generation of

stars are unlikely to be strong sources of radiation. Calculations which do not

explicitly take into account this reduction in gas density near the remnant (e.g.,

Kuhlen & Madau 2005) risk substantially overestimating their radiative feedback as

miniquasars. These remnant black holes could begin to emit a substantial amount

of radiation only after they encounter some other environment containing cold,

dense gas. It is not clear when or whether these black holes would ever encounter

such an environment. At the very least, therefore, there should be some delay

between the formation of the first generation of stars and the X-ray emission from

their remnants, if such emission ever occurs.

Determining the fate of recombining gas in the relic H II regions left be-

hind by the first stars is crucial. The contribution of these relic H II regions

from the first-generation Pop III stars to the increasing ionized fraction of the

universe during cosmic reionization depends upon their recombination time. Be-

cause of clustering around the host halo, the clumping factor and recombination

time within the relic H II region depends on the mass of the star; higher stellar

92

masses correspond to lower clumping factors and longer recombination times. The

recombination time and clumping factor for a 40M star are about 10 Myr and

12, respectively, whereas for a 500M star they are about 35 Myr and 3 (see Fig.

10).

When ionized gas within the relic H II region cools radiatively and re-

combines, the nonequilibrium recombination lags the cooling, which enhances the

residual ionized fraction at 104 K, promoting the formation of H2 molecules which

can cool the gas to T ∼ 100 K and enhance gravitational instability (Shapiro &

Kang 1987). This corresponds to “positive feedback” if further star formation

results (e.g., Ferrara 1998; Riccotti, Gnedin, & Shull 2001).

Recently, O’Shea et al. (2005) have addressed this issue of possible second

generation star formation in the relic H II region of the first Pop III stars. They

report that the I-front of the first star will fully ionize the neighboring minihalos

and that, when the initial star dies, the dense cores of these ionized neighbor mini-

halos will be stimulated to form H2 molecules, leading to the second generation of

Pop III stars. This assumes the initial star collapses to form a black hole without

exploding as a supernova. We find, however, that the neighboring minihalos are

not fully ionized before the initial star dies, since the I-front is trapped by these

minihalos and converted to D-type outside the core region, and the photoevapo-

ration time for the minihalo exceeds the lifetime of the ionizing star. Subsequent

calculations have confirmed our result that this is not the case, but the issue of

whether the first stars stimulate or delay further star formation remains a contro-

versial one (Susa & Umemura 2006; Abel, Wise, & Bryan 2006; Ahn & Shapiro

2006). More work will be required to resolve this question.

Whether H2 molecules form in abundance or not depends on the density

93

of the recombining gas. As we discussed in §4.4.4, the highest density of gas in

the relic H II region which we found to be fully ionized and, hence, capable of

recombining to form H2 molecules, is a few cm−3. A rough estimate of the H2

molecule formation time is given by the recombination time of the gas, trec ' 105

yr. Even if H2 molecules form, however, it is not certain whether this will promote

star formation in neighboring halos. As we have shown, the densest gas located

in the center of nearby halos is not ionized. The gas that is ionized is not in the

center, and is thus likely to recombine while being ejected from the halo as part of

a supersonic, photoevaporative outflow (e.g., Shapiro, Iliev, & Raga 2004). Such

gas is less likely to be gravitationally unstable. In future work, we will investigate

whether this gas is able to cool and form stars or simply gets evaporated into the

diffuse IGM.

94

Chapter 5

The cosmic reionization history as revealed by

the CMB Doppler–21-cm correlation

We show that the epoch(s) of reionization when the average ionization frac-

tion of the universe is about half can be determined by correlating Cosmic Mi-

crowave Background (CMB) temperature maps with 21-cm line maps at degree

scales (l ∼ 100). During reionization peculiar motion of free electrons induces the

Doppler anisotropy of the CMB, while density fluctuations of neutral hydrogen

induce the 21-cm line anisotropy. In our simplified model of inhomogeneous reion-

ization, a positive correlation arises as the universe reionizes whereas a negative

correlation arises as the universe recombines; thus, the sign of the correlation pro-

vides information on the reionization history which cannot be obtained by present

means. The signal comes mainly from large scales (k ∼ 10−2 Mpc−1) where lin-

ear perturbation theory is still valid and complexity due to patchy reionization is

averaged out. Since the Doppler signal comes from ionized regions and the 21-cm

comes from neutral ones, the correlation has a well defined peak(s) in redshift

when the average ionization fraction of the universe is about half. Furthermore,

the cross-correlation is much less sensitive to systematic errors, especially fore-

ground emission, than the auto-correlation of 21-cm lines: this is analogous to

the temperature-polarization correlation of the CMB being more immune to sys-

tematic errors than the polarization-polarization. Therefore, we argue that the

Doppler-21cm correlation provides a robust measurement of the 21-cm anisotropy,

95

which can also be used as a diagnostic tool for detected signals in the 21-cm data

— detection of the cross-correlation provides the strongest confirmation that the

detected signal is of cosmological origin. We show that the Square Kilometer Array

can easily measure the predicted correlation signal for 1 year of survey observation1.

5.1 Introduction

When and how was the universe reionized? This question is deeply con-

nected to the physics of formation and evolution of the first generations of ionizing

sources (stars or quasars or both) and the physical conditions in the interstellar

and the intergalactic media in a high redshift universe. This field has been devel-

oped mostly theoretically (Barkana & Loeb 2001; Bromm & Larson 2004; Ciardi

& Ferrara 2005; Iliev et al. 2005a; Alvarez, Bromm, & Shapiro 2006a) because

there are a limited number of observational probes of the epoch of reionization:

the Gunn–Peterson test (Gunn & Peterson 1965; Becker et al. 2001), polarization

of the Cosmic Microwave Background (CMB) on large angular scales (Zaldarriaga

1997; Kaplinghat et al. 2003; Kogut et al. 2003), mean intensity (Santos, Bromm, &

Kamionkowski 2002; Salvaterra & Ferrara 2003; Cooray & Yoshida 2004; Madau &

Silk 2005; Fernandez & Komatsu 2005) and fluctuations (Magliocchetti, Salvaterra,

& Ferrara 2003; Kashlinsky et al. 2004; Cooray et al. 2004; Kashlinsky et al. 2005)

of the near infrared background from redshifted UV photons, Lyα-emitters at high

redshift (Malhotra & Rhoads 2004; Santos 2004; Furlanetto, Hernquist, & Zaldar-

riaga 2004; Haiman & Cen 2005; Wyithe & Loeb 2005) and fluctuations of the

21-cm line background from neutral hydrogen atoms during reionization (Ciardi &

Madau 2003; Furlanetto, Sokasian, & Hernquist 2004; Zaldarriaga, Furlanetto, &

1This work appeared previously in Alvarez, Komatsu, Dore, & Shapiro 2006, ApJ, 647, 840

96

Hernquist 2004) or even prior to reionization (Scott & Rees 1990; Madau, Meiksin,

& Rees 1997; Tozzi et al. 2000; Iliev et al. 2002; Shapiro et al. 2006b).

Each one of these methods probes different epochs and aspects of cosmic

reionization: the Gunn–Peterson test is sensitive to a very small amount of residual

neutral hydrogen present at the late stages of reionization (z ∼ 6), Lyα-emitting

galaxies and the wavelengths of the near infrared background probe the interme-

diate stages of reionization (7<∼ z <∼ 15), the 21-cm background probes the earlier

stages where the majority of the intergalactic medium is still neutral (10<∼ z <∼ 30),

and the CMB polarization measures the column density of free electrons integrated

over a broader redshift range (z <∼ 20, say). Since different datasets are comple-

mentary, one expects that cross-correlations between them add more information

than can be obtained by each dataset alone. For example, the information content

in the CMB and the 21-cm background cannot be exploited fully until the cross-

correlation is studied: if we just extract the power spectrum from each dataset,

we do not exhaust the information content in the whole dataset because we are ig-

noring the cross-correlation between the two. The cross-correlation always reveals

more information than can be obtained from the datasets individually unless the

two are perfectly correlated (and Gaussian) or totally uncorrelated.

Motivated by these considerations, we study the cross-correlation between

the CMB temperature anisotropy and the 21-cm background on large scales. We

show that the CMB anisotropy from the Doppler effect and the 21-cm line back-

ground can be anti-correlated or correlated at degree scales (l ∼ 100), and both the

amplitude and the sign of the correlation tell us how rapidly the universe reionized

or recombined, and locations of the correlation (or the anti-correlation) peak(s) in

redshift space tell us when reionization or recombination happened. This informa-

97

tion is difficult to extract from either the CMB or the 21-cm data alone. Our work

is different from recent work on a similar subject by Salvaterra et al. (2005). While

they studied a similar cross-correlation on very small scales (∼ arc-minutes), we

focus on much larger scales (∼ degrees) where matter fluctuations are still linear

and complexity due to patchy reionization is averaged out. Cooray (2004) stud-

ied higher-order correlations such as the bispectrum on arc-minute scales. For

our case, however, fluctuations are expected to follow nearly Gaussian statistics

on large scales, and thus one cannot obtain more information from higher-order

statistics. He also studied the cross-correlation power spectrum of the CMB and

projected 21-cm maps, and concluded that the signal would be too small to be de-

tectable owing to the line-of-sight cancellation of the Doppler signal in the CMB.

However, we show that cancellation can be partially avoided by cross-correlating

the CMB map with 21-cm maps at different redshifts (tomography). Prospects

for the Square Kilometer Array (SKA) to measure the cross-correlation signal on

degree scales are shown to be promising.

Throughout the chapter, we use c = 1 and the following convention for the

Fourier transformation:

f(n, η) =∫

d3k

(2π)3fke

−ik·n(η0−η), (5.1)

where n is the directional cosine along the line of sight pointing toward the celestial

sphere, η is the conformal time, η(z) =∫ t0 dt

′/a(t′) =∫∞z dz′/H(z′), and η0 is the

conformal time at present. Note that

η0 − η(z) =∫ z

0

dz′

H(z′), (5.2)

which equals the comoving distance, r(z) = η0−η(z), in flat geometry (with c = 1).

98

Also, using Rayleigh’s formula one obtains

f(n, η) = 4π∑

lm

(−i)l∫ d3k

(2π)3fkjl[k(η0 − η)]Ylm(n)Y ∗

lm(k). (5.3)

The cosmological parameters are fixed at Ωm = 0.3, Ωb = 0.046, ΩΛ = 0.7, h = 0.7,

and σ8 = 0.85, and we assume a scale invariant initial power spectrum for matter

perturbations.

This chapter is organized as follows. In §5.2 and 5.3 we derive the analytic

formula for the Doppler–21-cm correlation power spectrum. Equations (5.16) or

(5.24) are the main result. We then present a physical picture of the correlation

and describe properties of the correlation in detail. We also discuss the validity

of our assumptions and possible effects of more realistic reionization scenarios. In

§5.4 we discuss detectability of the correlation signal with SKA before concluding

in §5.5.

5.2 21-cm Fluctuations and CMB Doppler Anisotropy

5.2.1 21-cm Signal

Following the notation of Zaldarriaga, Furlanetto, & Hernquist (2004), we

write the observed differential brightness temperature of the 21-cm emission line

at λ = 21 cm(1 + z) in the direction of n as

T21(n, z) = T0(z)∫ η0

0dη′W [η(z)− η′]ψ21(n, η

′), (5.4)

where W (η(z) − η′) is a normalized (∫∞−∞ dxW (x) = 1) spectral response function

of an instrument which is centered at η(z)−η′ = 0, T0(z) is a normalization factor

given by

T0(z) ' 23 mK

(

Ωbh2

0.02

)

[(

0.15

Ωmh2

)(

1 + z

10

)]1/2

, (5.5)

99

and

ψ21(n, η) ≡ xH(n, η)[1 + δb(n, η)]

[

1 − Tcmb(η)

Ts(n, η)

]

→ 1 − xe(η)[1 + δx(n, η)]1 + δb(n, η) , (5.6)

where δb is the baryon density contrast,

δx ≡ xe − xe

xe, (5.7)

is the ionized fraction contrast, xH is the neutral fraction, and xe ≡ 1 − xH is

the ionized fraction. Here, we have assumed that the spin temperature of neutral

hydrogen, Ts, is much larger than the CMB temperature, Tcmb. This assumption

is valid soon after reionization begins (Ciardi & Madau 2003).

To simplify the calculation, we assume that the spectral resolution of the

instrument is much smaller than the features of the target signal in redshift space.

This is always a very good approximation. (For the effect of a relatively large

bandwidth, see Zaldarriaga, Furlanetto, & Hernquist (2004).) Therefore, we set

W (x) = δD(x) to obtain T21(n, z) = T0(z)ψ21[n, η(z)]. To leading order in δx and

δb, the spherical harmonic transform of T21(n, z) is given by

a21lm(z) = 4π(−i)l

∫

d3k

(2π)3[xH(z)(1 + fµ2)δbk − xe(z)δxk]α

21l (k, z)Y ∗

lm(k), (5.8)

where α21l (k, z) is a transfer function for the 21-cm line,

α21l (k, z) ≡ T0(z)D(z)jl[k(η0 − η)], (5.9)

D(z) is the growth factor of linear perturbations, µ ≡ k · n, and f ≡ d lnD/d ln a.

The factor (1+fµ2) takes account of the enhancement of the fluctuation amplitude

due to the redshift-space distortion, the so-called “Kaiser effect” (Kaiser 1987; see

also Bharadwaj & Ali 2004 and Barkana & Loeb 2005).

100

5.2.2 Doppler Signal

The CMB temperature anisotropy from the Doppler effect is given by

TD(n) = −Tcmb

∫ η0

0ητe−τ n · vb(n, η)

= −Tcmb

∫ η0

0dητe−τ n ·

∫

d3k

(2π)3vbk(η)e

−ik·n(η0−η), (5.10)

where Tcmb = 2.725 K is the present-day CMB temperature, x ≡ ∂x/∂η, τ (η) ≡

σT

∫ η0 dη

′ne(η′) is the Thomson scattering optical depth, and vbk is the peculiar

velocity of baryons. In deriving the above formula, we have neglected the fluctua-

tion of ionized fraction, δx(n, η), and electron number density, δe(n, η), since their

contributions to the cross-correlation would be higher order corrections (the effect

due to δevb is called the Ostriker–Vishniac effect; e.g., Ostriker & Vishniac 1986),

and such a correction is negligible for linear fluctuations on the large scales we con-

sider here. Note that the negative sign ensures that we see a blueshift, TD(n) > 0,

when baryons are moving toward us, n ·vb < 0. The peculiar velocity is related to

the density contrast via the continuity equation for baryons, vbk = −i(k/k2)δbkD.

One obtains

TD(n) = Tcmb

∫ η0

0dηDτe−τ

∫

d3k

(2π)3

δbk

k2

∂

∂ηe−ik·n(η0−η). (5.11)

The spherical harmonic transform of TD(n, z) is then given by

aDlm = 4π(−i)l

∫

d3k

(2π)3δbkα

Dl (k)Y ∗

lm(k), (5.12)

where αDl (k) is a transfer function for the Doppler effect,

αDl (k) ≡ Tcmb

k2

∫ η0

0dηDτe−τ ∂

∂ηjl[k(η0 − η)]. (5.13)

101

5.3 Doppler–21-cm Correlation

5.3.1 Generic Formula

Given the spherical harmonic coefficients just derived for the 21-cm line

(Eq. [5.8]) and the Doppler anisotropy (Eq. [5.12]), one can calculate the cross-

correlation power spectrum, C21−Dl , exactly as

C21−Dl (z)

= 〈a21lm(z)aD∗

lm 〉 2

π

∫ ∞

0k2dk

[

xH(z)(1 + f〈µ2〉)Pδδ(k) − xe(z)Pxδ(k)]

α21l (k, z)αD

l (k)

= TcmbT0(z)D(z)2

3π

∫ ∞

0dk [4xH(z)Pδδ(k) − 3xe(z)Pxδ(k)] jl[k(η0 − η)]

×∫ η0

0dη′Dτ e−τ ∂

∂η′jl[k(η0 − η′)], (5.14)

where we have defined the matter power spectrum, Pδδ(k), as 〈δkδ∗k′〉 ≡ (2π)3δ(k−

k′)Pδδ(k), and the cross-correlation power spectrum between ionized fraction and

density Pxδ(k), as 〈δxkδ∗k′〉 ≡ (2π)3δ(k − k′)Pxδ(k). In the last line of equation

(5.14), we have used f〈µ2〉 = 1/3 for a matter-dominated universe. Note that δ

used in these power spectra is the density contrast of total matter, δ, as baryons

trace total matter perturbations, δb = δ, on the scales of our interest (scales much

larger than the Jeans length of baryons). Equation (5.14) can be simplified by

integrating it by parts:

C21−Dl (z) = −TcmbT0(z)D(z)

2

3π

∫ η0

0dη′

∂

∂η′Dτ e−τ (5.15)

×∫ ∞

0dk [4xH(z)Pδδ(k) − 3xe(z)Pxδ(k)] jl[k(η0 − η)]jl[k(η0 − η′)].

A further simplification can be made by using an approximation to the integral of

the product of spherical Bessel functions for l 1:

2

π

∫ ∞

0dkP (k)jl(kr)jl(kr

′) ≈ P

(

k =l

r

)

δ(r − r′)

l2, (5.16)

102

where r(z) = η0 − η(z) is the comoving distance out to an object at a given z. We

obtain

l2C21−Dl (z) ≈ −TcmbT0(z)D(z)

[

4

3xH(z)Pδδ

(

l

r(z)

)

− xe(z)Pxδ

(

l

r(z)

)]

× ∂

∂η(Dτe−τ ). (5.17)

In what follows we will use the exact expression given by equation (5.16) in our

main quantitative results, while we will retain the approximate expression given

by equation (5.17) to develop a more intuitive understanding of the origin of the

cross-correlation. We have found that the exact expression gives results which

are about 10% lower (at l ∼ 100) than the approximate expression of equation

(5.17) for the single reionization history we will use in §5.3.4, while for the double

reionization history the exact result is smaller by about 40%. This is because the

line-of-sight integral in equation (5.14) acts to smooth out features in redshift,

an effect which dissappears when the delta function is used in the approximation.

Since the double reionization model fluctuates much more strongly in redshift than

the single reionization model, the effect is more apparent for double reionization.

Equation (5.17) implies one important fact: the cross-correlation vanishes

if Dτ e−τ is constant. In other words, the amplitude of the signal directly de-

pends on how rapidly structure grows and reionization proceeds, and the sign of

the correlation depends on the direction of reionization (whether the universe re-

combines or reionizes). Moreover, the shape of l2Cl(z) directly traces the shape

of the matter power spectrum at k = l/r(z). It is well known that P (k) has a

broad peak at the scale of the horizon size at the epoch of matter-radiation equal-

ity, keq ' 0.011 Mpc−1 (Ωmh2/0.15). Since the conformal distance (which is the

same as the comoving angular diameter distance in flat geometry) is on the order

103

of 104 Mpc at high redshifts, the correlation power spectrum will have a peak at

degree scales, l ∼ 102.

5.3.2 Ionized Fraction–Density Correlation

While Pδδ(k) is a known function on the scales of interest here, the cross-

correlation between ionized fraction and density, Pxδ(k), is not. In order to under-

stand its importance in determining the observable signal, we have estimated its

value on large scales. Since we give the full details of derivations in Appendix, we

quote only the result here:

xe(z)Pxδ(k) = −xH(z) lnxH(z)[

bh(z)− 1 − f]

Pδδ(k), (5.18)

where bh(z) is the average bias of dark matter halos more massive than mmin,

bh(z) = 1 +

√

2

π

e−δ2c(z)/2σ2

min

fcoll(z)D(z)σmin, (5.19)

mmin is the minimum halo mass capable of hosting ionizing sources, fcoll(z) is

the fraction of matter in the universe collapsed into halos with m > mmin, and

σmin ≡ σ(mmin) is the r.m.s. of density fluctuations at the scale of mmin at z = 0.

We take mmin to be the mass of a halo with a virial temperature Tmin, which we

will treat as a free parameter. Here, f is a parameter characterizing the physics

of reionization: f = 0 is the “photon counting limit”, in which recombinations are

not important in determining the extent of ionized regions. On the other hand,

f = 1 is the “Stromgren limit”, in which the ionization rate is balanced by the

recombination rate, as would occur in a Stromgren sphere. While our choice for the

range of f is resonable, larger values are possible if recombinations limit the size of

H II regions and the clumping factor increases with increasing density. Equation

(5.19) is general in the sense that it can accomodate such a scenario. It is easy

104

Figure 5.1 Simplified schematic diagram illustrating the nature of the correlationbetween the Doppler and 21-cm anisotropies. Red arrows pointing away fromthe observer indicate ionized gas falling into the positive density perturbation(represented by the black oval) from the near side, whereas blue arrows representionized gas falling in from the far side. During reionization, there is more ionizedgas on the near side of the perturbation (at lower redshift) than on the far side. Thisimplies that the net effect from this perturbation is a redshift of the CMB in thatdirection (labeled as δDOP < 0). Because the sources responsible for reionization arelocated in halos which are very biased relative to the underlying linear density field,the overdense region shown here is actually underdense in neutral hydrogen, so thatthis overdensity represents a negative fluctuation in the 21-cm signal (labeled asδ21−cm < 0). Because both the 21-cm and the Doppler fluctuations from a regionthat is undergoing reionization are both the same sign, the signature of reionizationis a positive correlation, while recombination (in which the situation is reversedfor the Doppler signal) results in an anti-correlation. In reality, the growth offluctuations and the dependence of the density on redshift complicate the picture,so that the sign of the signal is determined not by the derivative of the ionizedfraction d[xe]/dz, but rather d[xe(z)(1 + z)3/2]/dz (see equation 5.23).

105

to check that Pxδ naturally satisfies the physical constraints: it vanishes when the

universe is either fully neutral, xH = 1, or fully ionized, xH = 0. Although we

have derived an explicit relationship between Pxδ and Pδδ (which is based on several

simplifying assumptions – see Appendix), we note that the formulae presented here

are sufficiently flexible so that any model for the large-scale bias of reionization

with Pxδ = bxδPδδ can be substituted for the one we use here.

We simplify equations (5.14) and (5.17) by making the approximations that

e−τ ≈ 1 (justified by observations of the CMB polarization; see Kogut et al. 2003)

and

D(z) =1 + zN

1 + z, (5.20)

which is a very good approximation at z 1 when the universe is still matter-

dominated. The linear growth factor has been normalized such that D(zN ) = 1;

thus,

D = −H(z)d

dz

1 + zN

1 + z=

(ΩmH20 )1/2(1 + zN)

(1 + z)1/2. (5.21)

We also use the relation

τ (z) = σTρb0

mp(1 − Yp)(1 + z)2xe(z)

= 0.0525H0Ωbh(1 + z)2xe(z), (5.22)

where ρb0 is the baryon density at present, Yp = 0.24 is the helium mass abundance

(hydrogen ionization only is assumed), and xe(z) is the ionized fraction. One finds

∂

∂η(Dτ e−τ) = −0.0525H3

0 ΩmΩbh(1 + zN)(1 + z)3/2 d

dz

[

xe(z)(1 + z)3/2]

. (5.23)

Combining equations (5.14) and (5.23), we obtain

l2C21−Dl (z)

2π= 0.37 µK2

(

Ωbh2

0.02

)2 (Ωmh

2

0.15

)1/2

xH(z)[

4/3 + lnxH(z)(

bh − f − 1)]

×(

1 + z

10

)−1/2 ∫ ∞

0dz′

Fl(z, z′)

H(z′)(1 + z′)3/2 d

dz′

[

xe(z′)(1 + z′)3/2

]

, (5.24)

106

where

Fl(z, z′) = l2

∫ ∞

0dkP (k)(1 + zN)2

105Mpc3 jl [k(η0 − η(z))] jl [k(η0 − η(z′))] . (5.25)

The approximation of equation (5.17) and equation (5.23) imply

l2C21−Dl (z)

2π' 18.4 µK2

(

Ωbh2

0.02

)2 (Ωmh

2

0.15

)1/2

xH(z)[

4/3 + lnxH(z)(

bh − f − 1)]

× Pδδ[l/r(z), zN ](1 + zN)2

105 Mpc3

d

dz

[

xe(z)(1 + z)3/2]

(

1 + z

10

)

. (5.26)

Note that P (k, zN)(1 + zN)2 is independent of the normalization epoch, zN , for

zN 1; the result is independent of the choice of zN , as expected. These equations

are the main result of this chapter, and we shall use these results to investigate the

properties of the correlation in more detail. Since we have a product of dxe/dz and

xH, we expect that the largest contribution comes from the “epoch of reionization”

when xe(z) changes most rapidly. Therefore, by detecting the Doppler–21-cm

correlation peak(s), one can determine the epoch(s) of reionization. The sign of

the cross-correlation is also very important. The sign of the cross-correlation is

determined by the sign of the derivative term and the difference between xHPδδ

and xePxδ. For the case in which xHPδδ > xePxδ, the Doppler effect and the 21-cm

emission are anti-correlated when the ionized fraction, xe(z), increases toward low

z faster than (1 + z)−3/2. For our simplified model of inhomogeneous reionization

(see Appendix), we find that xHPδδ < xePxδ, however, and in this case we find a

positive correlation as the universe is being reionized. This is unique information

that cannot be obtained by present means. See Fig. 5.1 for a schematic diagram

which describes the nature of the cross-correlation.

107

5.3.3 Illustration: Homogeneous Reionization Limit

It may be instructive to study the nature of the signal by taking the “ho-

mogeneous reionization limit”, in which the ionized fraction is uniform, δx ≡ 0.

Such a situation may be more relevant than our model for biased reionization if,

for example, the photons responsible for reionization have a very long mean free

path or clumping in denser regions cancels the effect of bias. In the homogeneous

limit, the approximate formula (eq. 5.26) implies

l2C21−Dl (z)

2π' 24.5 µK2

(

Ωbh2

0.02

)2 (Ωmh

2

0.15

)1/2Pδδ[l/r(z), zN ](1 + zN)2

105 Mpc3

× xH(z)d

dz

[

xe(z)(1 + z)3/2]

(

1 + z

10

)

. (5.27)

One may estimate the amplitude of the signal at the epoch of reionization, z = zr,

using a duration of reionization at zr, ∆z as follows (omitting factors of order

unity):

l2C21−Dl (zr)

2π' −195 µK2

∆z

xH(zr)xe(zr)

0.25

(

1 + zr

10

)5/2

. (5.28)

The remarkable feature is that the predicted signal is rather large. For zr = 15

(which is consistent with early reionization suggested by Kogut et al. (2003)) and

∆z = 1, we predict l2C21−Dl /(2π) ∼ −600 µK2 at l ∼ 102. Under these assump-

tions, therefore, detection of the anti-correlation peak should not be too difficult,

given that the Wilkinson Microwave Anisotropy Probe (WMAP) has already ob-

tained an accurate CMB temperature map at l ∼ 102 (Bennett et al. 2003). When

any experiment for measuring the 21-cm background at degree scales becomes

on-line, one should correlate the 21-cm data on degree scales with the WMAP

temperature map to search for this peak. Note also that in the homogeneous reion-

ization limit the sign is reversed, so that reionization results in an anti-correlation.

108

Figure 5.2 (Left) Power spectrum of the cross-correlation between the cosmic mi-crowave background anisotropy and the 21-cm line fluctuations, l2C21−D

l /(2π). Weassume z = 15 for a reionization history given by equation (5.29) with a reioniza-tion redshift of zr = 15 and duration of ∆z = 0.5. Note that its shape followsthat of the linear matter power spectrum, Pδδ(k) with k ' l/r(z), where r(z) isthe comoving angular diameter distance. (Right) Evolution of the peak amplitudeof l2C21−D

l /(2π) at l ∼ 100 from a homogeneous reionization history described byequation (5.29), for ∆z = 0.5 and different reionization redshifts, zr = 7, 11, 15and 19, from left to right.

109

The sign of the correlation therefore depends sensitively on the degree to which

reionization is biased on large scales.

5.3.4 Reionization History

To calculate the actual cross-correlation power spectrum, we need to specify

the evolution of the ionized fraction, xe(z). Computing xe(z) from first principles is

admittedly very difficult, and this is one of the most challenging tasks in cosmology

today. To illustrate how the cross-correlation power spectrum changes for different

reionization scenarios, therefore, we explore two simple parameterizations of the

reionization history.

In one case, we assume that the ionized fraction increases monotonically

toward low z. We use the simple parameterization adopted by Zaldarriaga, Furlan-

etto, & Hernquist (2004):

xH(z) =1

1 + exp [−(z − zr)/∆z], (5.29)

where zr is the “epoch of reionization” when xH(zr) = 1/2 and ∆z corresponds to

its duration. In this case, one obtains a fully analytic formula for the correlation

power spectrum:

l2C21−Dl (z)

2π' 58 µK2

[

4/3 + lnxH(z)(

bh − f − 1)] P [l/r(z), zN ](1 + zN)2

105 Mpc3

×(

Ωbh2

0.02

)2 (Ωmh

2

0.15

)1/2

xHxe

[

3

2− xH(z)(1 + z)

∆z

]

(

1 + z

10

)3/2

.(5.30)

In the homogeneous reionization limit, Pxδ ≡ 0, one gets l2Cl/(2π) ' −165 µK2

for z = 9 = zr and ∆z = 1/2, and the amplitude of the signal scales as (1 + zr)5/2,

as expected (see Eq. [5.28]). For an early reionization at zr = 15, the homogeneous

reionization model predicts l2Cl/(2π) ' −570 µK2.

110

Figure 5.3 (top and left panels) Peak correlation amplitude vs. redshift. Eachpanel is labeled with a different value of f which parameterizes the uncertaintyin the physics of reionization (see Appendix for the definition of f and detaileddiscussion). The most likely value of f is somewhere between 0 and 1. The dottedline corresponds to the homogeneous reionization limit in which fluctuations inthe ionized fraction are totally ignored (Eq. [5.27]), while the thick line takes intoaccount fluctuations in the ionized fraction (Eq. [5.26]). The dashed line is thedifference between the homogeneous reionization and the total signal. (bottomright) Evolution of xe with redshift. Note that in all cases the reionization of theuniverse results in a positive correlation.

111

The left panel of Figure 5.2 shows the absolute value of the predicted cor-

relation power spectrum, l2C21−Dl /(2π), for the homogeneous reionization model

with z = 15 = zr and ∆z = 0.5. As we have explained previously, the shape

of l2|C21−Dl | exactly traces that of the underlying linear matter power spectrum,

Pδδ. The right panel of Figure 5.2 shows the the redshift evolution of the peak

value of the power spectrum at l ∼ 100, for different values of zr. As discussed

at the end of §5.3.2, the reionization of the universe leads to an anti-correlation

between the Doppler and 21-cm fluctuations. The magnitude of the signal in-

creases with redshift when the duration of reionization in redshift, ∆z, is fixed

(see equation (5.29)). We could instead fix the duration of reionization in time,

∆t, in which case ∆z increases with redshift as ∆z ∝ (1 + z)5/2∆t; according to

equation (5.28), therefore, the peak height in this case would be approximately

independent of redshift.

To gain more insight into how the prediction changes with the details of

the reionization process, let us use a somewhat more physically motivated model

for the ionized fraction,

ln[1 − xe(z)] = −ζ0(z)fcoll(z). (5.31)

The ionized fraction increases monotonically toward low z when ζ0 does not de-

pend on z. Using this model with ζ0 = 200 and Tmin = 104 K, we calculate the

cross-correlation power spectrum. Figure 5.3 plots the peak value of l2C21−Dl as

a function of z, showing the contribution from Pδδ, Pxδ, and the sum of the two

(Eq. [5.26]). The bottom-right panel shows the evolution of the ionized fraction

predicted by equation (5.31). In this figure we explore the dependence of the signal

on the details of reionization by varying the parameter f . (See Appendix for the

112

Figure 5.4 Same as in Fig. 2, but for a “double reionization” model in which theuniverse undergoes a brief period of recombination. Note that in all cases therecombination epoch results in a negative correlation.

precise meaning of f .) In all cases, the contribution from Pδδ is negative, whereas

that from Pxδ is positive; because the halo bias is relatively large for our fiducial

case of Tmin = 104 K, with 4 < bh < 17 for 10 < z < 30, the Pxδ term dominates

over the Pδδ term, and the correlation is positive (see also Iliev et al. 2005). In-

creasing the value of f towards a more recombination dominated scenario decreases

the importance of the dominant Pxδ term, reducing the total amplitude of the sig-

nal further. What happens when the universe was reionized twice (Cen 2003; see

113

however Furlanetto & Loeb 2005)? In Figure 5.4 we showed the case where the

ionized fraction is a monotonic function of redshift. As seen in the figure, there

is a prominent correlation peak, regardless of the details of reionization process,

encoded in f . The situation changes completely when the universe was reionized

twice. We parameterize such a double reionization scenario using a z-dependence

for Tmin(z) and ζ0(z):

ζ0(z) = ζi + (ζf − ζi)g(z) (5.32)

and

Tmin(z) = Ti + (Tf − Ti)g(z), (5.33)

where

g(z) =exp [−(zcrit − z)/∆ztran]

1 + exp [−(zcrit − z)/∆ztran](5.34)

is a function that approaches zero for z > zcrit and unity for z < zcrit, with

a transition of duration ∆ztran. We take zcrit = 15, ∆ztran = 0.25, ζi = 100,

ζf = 40, Ti = 103 K, and Tf = 104 K. In this case, the minimum source halo

virial temperature makes a smooth transition from 103 K at high redshift to 104 K

at low redshift, as might occur if dissociating radiation suppresses star formation

in “minihalos” with virial temperatures < 104 K. The drop in ζ0(z), which for

convenience coincides with the transition in Tmin, could be due to, for example,

metal pollution from Pop III stars creating a transition to Pop II, accompanied

by a transition from a very top heavy IMF to a less top heavy one (e.g. Haiman

& Holder 2003). In this scenario, the universe may recombine until enough Pop

II stars and halos with virial temperatures > 104K form to finish reionization.

We emphasize that this is a simple parameterization for illustration, and is not

meant to represent a realistic double reionization model. However, this model is

sufficient to show that the signature of a recombination epoch during reionization

114

is a reversal in the sign of the correlation. Because of a rapid change in the ionized

fraction during recombination, the negative correlation peak is very prominent,

reaching l2C21−Dl ∼ 700 − 900 µK2 for f = 0 − 1.

5.4 Prospects for Detection

5.4.1 Error Estimation

Assuming that CMB and instrumental noise for 21-cm lines are Gaussian,

one can estimate the error of the correlation power spectrum by

(∆Cl)2 =

1

(2l + 1)fsky∆l

[

Ccmbl C21

l + (C21−Dl )2

]

, (5.35)

where ∆l is the size of bins within which the power spectrum data are averaged

over l−∆l/2 < l < l+∆l/2, and fsky is a fraction of sky covered by observations,

fsky ≡Ω

4π= 2.424 × 10−3

(

Ω

100 deg2

)

. (5.36)

In the l range we are considering (l ∼ 102), CMB is totally dominated by signal

(i.e., noise is negligible), which gives l2Ccmbl /(2π) ∼ (50 µK)2 at l ∼ 102. On

the other hand, the 21-cm lines will most likely be totally dominated by noise

and/or foreground and the intrinsic signal contribution to the error may be ignored.

We also assume that the foreground cleaning reduces it to below the noise level.

We calculate the noise power spectrum based upon equation (59) of Zaldarriaga,

Furlanetto, & Hernquist (2004):

l2C21l

2π=

1

∆νtobs

(

llmax

2π

λ2

A/T

)2

, (5.37)

where ∆ν is the bandwidth, tobs is the total integration time, and A/T is “sen-

sitivity” (an effective area divided by system temperature) measured in units of

115

m2 K−1. The maximum multipole, lmax, for a given baseline length, D, is given by

lmax = 2πD

λ= 2994

(

D

1 km

)(

10

1 + z

)

. (5.38)

Here, we have used λ = 21 cm(1 + z). Note that we have implicitly assumed

uniform coverage for the interferometer in deriving equation (5.37), which may

not be realistic. Making the baseline distribution more compact would enhance

the detectability.

5.4.2 Square Kilometer Array

The current design of the Square Kilometer Array (SKA) aims at a sen-

sitivity of A/T ∼ 5000 m2 K−1 at 200 MHz. 2 Of which, 20% of the area

forms a compact array configuration within a 1 km diameter, whereas 50% is

within 5 km, and 75% is within 150 km. Since we are interested in a relatively

low-l part of the spectrum, we use the compact configuration, D = 1 km, and

A/T = 5000 × 0.2 = 1000 m2 K−1. We obtain

l2C21l

2π=

(130 µK)2

Nmonth∆νMHz

[(

l

100

)

(

1 + z

10

)(

D

1 km

)

(

103 m2 K−1

A/T

)]2

, (5.39)

where Nmonth is the number of months of observations and ∆νMHz is the bandwidth

in units of MHz. Note that we have assumed here that all the time spent during

the observation is on-source integration time. However, a more realistic assesment

would be that a smaller (e.g., ∼ 1/3) fraction of the total time is spent integrating.

In this case, one should compensate by increasing the total time of the observation

accordingly. Since the noise power spectrum is much larger than the amplitude of

the predicted correlation signal, we safely ignore the contribution of C21−Dl to the

error. (We ignore the second term on the right hand side of Eq. [5.35]).

2Information on SKA is available at http://www.skatelescope.org/pages/concept.htm.

116

The planned contiguous imaging field of view of SKA is currently 1 deg2

at λ = 21 cm and it scales as λ2. Using the number of independent survey fields,

Nfield, the total solid angle covered by observations is given by Ω ' 100 deg2[(1 +

z)/10]2Nfield. This estimate is, however, based on the current specification for the

high frequency observations, and may not be relavant to low frequency observations

that we discuss here. It is likely that there will be different telescopes with much

larger field of view at low frequencies, and thus we shall adopt a field of view which

is three times larger:

Ω ' 300 deg2(

1 + z

10

)2

Nfield. (5.40)

Using equation (5.35) and the parameters of SKA, we find the expected error per√NmonthNfield∆νMHz to be on the order of

Err

(

l2Cl

2π

)

' 938 µK2

√

√

√

√

l/∆l

NmonthNfield∆νMHz

l2Ccmbl /(2π)

2500 µK2. (5.41)

Therefore, for the nominal survey parameters, Nmonth = 12 and Nfield = 4, the SKA

sensitivity to the cross-correlation power spectrum reaches Err[l2C21−Dl /(2π)] '

135 µK2, which gives ∼ 3-σ detection of the correlation peak for the normal reion-

ization model, and ∼ 6-σ detection of the anti-correlation peak for the double

reionization model. Increasing the integration time or the survey fields will obvi-

ously increase the signal-to-noise ratio as√NmonthNfield. One would obtain more

signal-to-noise by choosing a larger value for ∆ν, which is equivalent to stacking

different frequencies. (However, ∆ν must not exceed the width of the signal in

frequency space.) Therefore, we conclude that the cross-correlation between the

Doppler and 21-cm fluctuations is fairly easy with the current SKA design. For

more accurate measurements of the shape of the spectrum, however, a larger con-

tiguous imaging field of view may be required. The more promising way to reduce

117

errors may be to increase sensitivity (i.e., larger A/T ) by having more area, A, for

the compact configuration. This is probably the most economical way to improve

the signal-to-noise ratio, as the error is linearly proportional to (A/T )−1, rather

than the square-root.

5.5 Discussion and Conclusion

We have studied the cross-correlation between the CMB temperature anisotropy

and the 21-cm background. The cross-correlation occurs via the peculiar veloc-

ity field of ionized baryons, which gives the Doppler anisotropy in CMB, coupled

to density fluctuations of neutral hydrogen, which cause 21-cm line fluctuations.

Since we are concerned with anisotropies in the cross-correlation on degree angu-

lar scales (l ∼ 100), which correspond to hundreds of comoving Mpc at z ∼ 10,

we are able to treat density and velocity fluctuations in the linear regime. This

greatly simplifies the analysis, and distinguishes our work from previous work on

similar subjects that dealt only with the cross-correlation on very small scales

(Cooray 2004; Salvaterra et al. 2005). Furthermore, because the 21-cm signal con-

tains redshift information, the cross-correlation is not susceptible to the line of

sight cancellation that is typically associated with the Doppler effect. Finally, be-

cause the systematic errors of the 21-cm and CMB observations are uncorrelated,

the cross-correlation will be immune to many of the pitfalls associated with ob-

serving the high redshift universe in 21-cm emission, such as contamination by

foregrounds3. We argue that detection of the predicted cross-correlation signal

3A potential source of foreground contamination is the Galactic synchrotron emission affectingboth the CMB and 21-cm fluctuation maps; however, the amplitude of synchrotron emission inthe CMB map at degree scales is much smaller than the Doppler anisotropy from reionization,and thus it is not likely to be a significant source of contamination.

118

provides the strongest confirmation that the signal detected in the 21-cm data is

of cosmological origin. Without using the cross-correlation, it would be quite chal-

lenging to convincingly show that the detected signal does not come from other

contaminating sources.

We find that the evolution of the cross-correlation with redshift can con-

strain the history of reionization in a distinctive way. In particular, we predict

that a universe undergoing reionization results in a positive cross-correlation at

those redshifts, whereas a recombining universe results in a negative correlation

(this dependends on our simplified model of biased reionization – a model in which

reionization is homogeneous would imply a reversal of the sign of the correlation).

Thus, the correlation promises to reveal whether the universe underwent a period

of recombination during the reionization process (e.g., Cen 2003), and to reveal

the nature of the sources of ionizing radiation responsible for reionization. The

signal we predict, on the order of l2Cl/(2π) ∼ 500 − 1000 µK2, should be eas-

ily detectable by correlating existing CMB maps, such as those produced by the

WMAP experiment, with maps produced by upcoming observations of the 21-cm

background with the Square Kilometer Array (SKA).

Our derivation of the cross-correlation rests upon linear perturbation theory

and the reasonable assumption that the sizes of ionized regions are much smaller

than scales corresponding to l ∼ 100. However, assuming that the sizes of ionized

regions are much smaller than the fluctuations responsible for the signal we predict

is not equivalent to assuming that the ionized fraction is uniform. Because our

prediction depends on the correlation between ionized fraction and density, Pxδ,

we have derived a simple approximate model for it (see Appendix). In future

work, we will use large-scale simulations of reionization to verify the accuracy of

119

the relation we derive, and perhaps to refine our analytical predictions. Whatever

the result of more detailed future calculations, we are confident that the CMB

Doppler-21–cm correlation will open a new window into the high redshift universe

and shed light on the end of the cosmic dark ages.

5.6 Appendix 1: Density-ionization Cross-correlation

The size of H II regions during reionization is a function of the neutral

fraction: as the neutral fraction decreases, the typical size of H II regions increases,

quickly approaching infinity as the neutral fraction approaches zero and the H II

regions percolate. As predicted by analytical and numerical studies of the large

scale topology of reionization (Furlanetto, Zaldarriaga, & Hernquist 2005; Iliev et

al. 2005), the typical H II region size approaches only up to a few tens of comoving

Mpc at even near the end of reionization. Since we are interested in epochs during

which the ionized fraction is about a half, we can safely assume for our purposes

that the typical H II region size is smaller than the length scales of the fluctuations

relevant here (∼ 100 Mpc). In this case, the ionized fraction within a given volume

can be determined by considering only sources located inside that volume.

Let us suppose that we take a region in the universe which has an overden-

sity of δ, where δ 1. If we assume that each baryon within a collapsed object can

ionize ζ(δ) baryons, then the ionized fraction within some volume can be written

as a function of its overdensity δ,

ln[1 − xe(δ)] = −ζ(δ)fcoll(δ), (5.42)

where fcoll(δ) is a local fraction of the collapsed mass to mean mass density, which

would be different from the average collapsed fraction in the universe, fcoll(0). Note

120

that this functional form correctly captures the behavior at low and high ionizing

photon to atom ratio. For ζfcoll 1, xe ' ζfcoll 1, which corresponds to all the

ionizing photons emitted within the volume ionizing atoms within that volume, as

expected before H II regions have percolated. For ζfcoll 1, which corresponds

to many more ionizing photons than atoms, xe ' 1, as expected after percolation.

Given that we are only considering sources located within the region, however, this

expression is only an approximation during percolation, when sources from outside

of the volume become visible. We emphasize that in any case equation (5.42) is

based on a simplifying assumption and does not capture many of the subtleties

included in more sophisticated models of reionization.

Here, we present two functional forms for ζ(δ) which are meant to bracket

two important physical limits. The first limit we will refer to as the “Stromgren

limit”, while the second we will refer to as the “photon counting limit”. In both

limits, we will assume that each hydrogen atom in a collapsed object will produce

εγ(z) ionizing photons.

5.6.1 Stromgren Limit

If we assume that a fraction η∗(z) of collapsed gas is undergoing a burst of

star formation of duration ∆t∗(z), then the ionizing photon luminosity per unit

volume, Nγ, is given by

Nγ =εγη∗n(1 + δ)

∆t∗fcoll(δ), (5.43)

where n is the mean density of the universe. The “Stromgren limit” is defined such

that every recombination is balanced by an emitted photon; the following equation

121

therefore applies:

ln[1 − xe(δ)] = − Nγ

Nrec

= − εγη∗fcoll(δ)

αcl∆t∗n(1 + δ), (5.44)

where α is the recombination coefficient, cl is the clumping factor, and Nrec is the

recombination rate per unit volume in a fully-ionized IGM. The last two terms in

equation (5.44) are the ratio of photon luminosity within a given volume to the

number of recombinations per unit time which would occur in that volume were it

to be fully ionized. Note again that this expression ensures the proper behavior of

xe in the low and high photon luminosity limits. Combining equations (5.42) and

(5.44), we find

ζ(δ) = ζ0(1 + δ)−1 ≈ ζ0(1 − δ), (5.45)

where ζ0 ≡ εγη∗/(αcl∆t∗n) and the approximation is valid in the limit δ 1. In

deriving this relation, we have assumed that the clumping factor, cl, is independent

of δ. While it is unlikely that clumping decreases with increasing δ, it is plausible

that it could increase. This would decrease the correlation between density and

ionized fraction, Pxδ. Because this term typically dominates the cross-correlation

(see §5.3.4), this would have the effect of reducing the predicted signal.

5.6.2 Photon Counting Limit

In the “photon counting limit”, recombinations are not important in de-

termining the the extent of ionized regions. Instead, it is the ratio of all ionizing

photons ever emitted to hydrogen atoms which determines the ionized fraction.

The number density of photons that have been emitted within a volume with

overdensity δ is given by

nγ(δ) ≡ εγn(1 + δ)fcoll(δ). (5.46)

122

In the photon counting limit, we assume that the ionized fraction is given by

ln [1 − xe(δ)] = − nγ(δ)

n(1 + δ)= −εγfcoll(δ). (5.47)

Combining equations (5.42) and (5.46), we find that ζ is independent of overdensity,

ζ(δ) = ζ0 ≡ εγ. (5.48)

5.6.3 Dependence of Collapsed Fraction on δ

Motivated by these two limits, we parameterize the δ-dependence of ζ as

ζ(δ) = ζ0(1 − fδ). (5.49)

In the photon counting limit, f = 0, while in the recombination dominated limit

f = 1. Now that we have specified the form of ζ(δ), we turn to the collapsed

fraction, fcoll(δ). The average collapsed fraction in the universe, i.e., fcoll(δ) with

δ = 0, is given by

fcoll(0) = erfc

[

δc(z)√2σmin

]

. (5.50)

According to the extended Press-Schechter theory, the local collapsed fraction is

(Lacey & Cole 1993)

fcoll(δ,m) = erfc

δc(z) − δ/D(z)√

2 [σ2min − σ2(m)]

, (5.51)

where m is the mass of the region. On large scales, σ(m) σmin and δ 1, so

that equation (5.51) can be expanded in a Taylor series around δ = 0,

fcoll(δ) ' fcoll(0) +

√

2

π

e−δ2c(z)/2σ2

min

σminD(z)δ. (5.52)

An alternative expression for the local collapsed fraction can be written in terms

of the mean halo bias bh,

fcoll(δ) = fcoll(0)1 + bhδ

1 + δ, (5.53)

123

Figure 5.5 The effect of bias on the relative importance of the Pxδ and Pδδ termsof the cross-correlation signal. The dotted, solid, and dashed curves in each panelcorrespond to models in which the sources responsible for reionization have a mini-mum virial temperature of 103, 104, and 105 K, respectively. (bottom-right) Shownare the reionization histories with a value of ζ0 chosen such that the Thomsonscattering optical depth τes = 0.15. The model with Tmin = 104 K is a reioniza-tion history obtained with ζ0 = 200, and is the same as the single-reionizationmodel presented in the main part of the chapter. (bottom-left) Mean halo bias vs.redshift. (top-left) Ratio Pxδ/Pδδ as obtained using equation (5.61). (top-right)The ratio [xePxδ]/[xHPδδ]. Note that when the ratio equals one, the correlationvanishes, while larger values indicate a dominant contribution from the Pxδ term.The top panels assume the photon counting limit (f = 0).

124

which, for δ 1, is well-approximated by

fcoll(δ) ' fcoll(0)[

1 + (bh − 1)δ]

. (5.54)

Equations (5.52) and (5.54) are consistent only if

bh ≡ 1 +

√

2

π

e−δ2c(z)/2σ2

min

fcoll(0)σminD(z). (5.55)

The reader can easily verify that this expression is the same as that found by

averaging over the halo bias derived by Mo & White (1996),

b(ν) = 1 +ν2 − 1

δc, (5.56)

which gives

bh =

∫∞νmin

dνf(ν)b(ν)∫∞νmin

dνf(ν), (5.57)

where

f(ν) ∝ exp[

−ν2/2]

. (5.58)

The Taylor expansion of the collapsed fraction used in deriving equation (5.52) is

therefore consistent with the standard linear bias formalism of Mo & White (1996).

5.6.4 Final Expression

Equations (5.42), (5.49), and (5.54) imply that

ln[1 − xe(δ)] = ln[1 − xe][

1 + (bh − 1 − f)δ]

(5.59)

where we have used the fact that ln[1−xe] = −ζ0fcoll(0). In the limit where f = 0

and we are simply counting photons, the ionized fraction does not depend on δ if

the mean halo bias bh = 1, since the additional photons emitted within a given

region are exactly canceled by the additional atoms contained within that region.

125

If recombinations are important, however, the condition for xe to remain constant

is given by bh = 2. In this case, the bias must compensate for the additional

photons necessary to balance the enhanced recombination rate per atom within

the volume. As was noted in §5.6.1, this relies on assuming a clumping factor

which is independent of density. If the clumping factor is an increasing function

of density, then the condition would be bh > 2. The cross-correlation of ionized

fraction and density fluctuations is given by (for δ 1 and δx 1)

〈δxδ〉 ' −1 − xe

xe

ln(1 − xe)(bh − 1 − f)〈δδ〉, (5.60)

so that

xePxδ(k) ' −(1 − xe) ln(1 − xe)(bh − 1 − f)Pδδ(k)

= −xH lnxH(bh − 1 − f)Pδδ(k). (5.61)

A comparison of the different terms implied by equation (5.61) is shown in Figure

5.5.

If the universe begins to recombine, then equation (5.61) would be correct

in the limit where the recombination time is short compared to the time it takes

for sources to dininish in intensity. For simplicity, we will assume this is the case

and use the relation of equation (5.61) exclusively in the main body of the chapter.

In the following section, we will investigate the departure from that relation for

the case where the sources decay faster than the recombination time.

5.6.5 Bias in a Recombining Universe

Equation (5.61) was derived under the assumption that the ionized fraction

is determined by the abundance of reionization sources and the density of their

126

environment. However, in the limit where the intensity of ionizing radiation due

to these sources drops precipitously, as may be expected from metal enrichment or

some other form of negative feedback, the ionized fraction will be determined by

the rate of recombination. In order to understand the effect of a “recombination

epoch” on the cross-correlation, we will derive a simple relation for Pxδ for the

extreme case in which a region of the universe recombines with no sources present.

In the absence of ionizing radiation, recombination is expected to proceed

according to

dxe

dy= −(1 + δ)x2

e, (5.62)

where y ≡ t/trec is time in units of the mean recombination of the universe, trec. We

will take the initial ionized fraction to be a deterministic function of the overdensity,

so that the initial fluctuation of ionized fraction (when recombination begins occur)

is δx,i = bx,iδ, where the subscript i refers to the initial value. If the bias in ionized

fraction just before recombination begins is described by equation (5.60), then we

have

bx,i = −1 − xe,i

xe,iln(1 − xe,i)(bh − 1 − f). (5.63)

For the sake of generality, however, we will report our results in terms of bx,i.

Solving for equation (5.62), we obtain the time evolution of xe,

xe(y) =xe,i

1 + xe,i(1 + δ)y, (5.64)

from which it follows that

δx(xe) =

[

xe

xe,i(bx,i + 1) − 1

]

δ, (5.65)

where we have assumed δ 1 and have used the relation

xe(y) =xe,i

1 + xe,iy. (5.66)

127

Figure 5.6 (left) Fl(z, z′) (with P (k) = Ak) for l = 10, 100, 1000 from widest to

narrowest, where the solid is from the analytical expression (eq. 5.71) and the pointsare from numerical integration. (right) Cross-correlation coefficient vs. l from theexact analytical expression (solid black), the approximate analytical expression(eq. 5.68; dashed), and the numerical integration (red). The top panel is as afraction of the approximate analytical expression. The red and black curves arenearly indistinguishable, which shows the numerical integration does a good job.

When xe = xe,i, δx = bx,iδ, as expected. As the universe recombines to become fully

neutral, xe → 0 and δx → −δ. Since we expect bx,i > 1 because overdense regions

have an overabundance of ionizing dources, a period of recombination is expected

to weaken the importance of Pxδ term. When xe/xe,i = 1/2 and bx,i 1, for

example, the bias determined from (5.61) is too large by a factor of two. Because

we have assumed that sources turn off instantaneously in equation (5.62), this is

an upper limit to the effect of a recombination epoch on Pxδ.

128

Figure 5.7 Same as previous figure, but for the actual power spectrum P (k). Inthis case, there is no analytical solution to compare to. At the peak, the numericalresult is 65% of the analytical approximation.

5.7 Appendix 2: Exact Expression for Cross-correlation

In this section, we compare the exact expression for the CMB-21–cm corre-

lation to an approximate relation which is much easier to deal with and compute.

The exact expression for the cross-correlation, when the fluctuations in density

and ionized fraction are linear, is given by equation (5.16):

C21−Dl (z) = −TcmbT0(z)D(z)

2

π

∫ η0

0dη′

∂

∂η′Dτ e−τ (5.67)

×∫ ∞

0dk [xH(z)Pδδ(k) − xe(z)Pxδ(k)] jl[k(η0 − η)]jl[k(η0 − η′)],

To simplify the comparison, we will assume Pxδ = 0, which does not affect our

results, since the shape Pxδ(k) is the same as that of Pδδ(k). In this case, equation

(5.68) reduces to

l2C21−Dl (z)

2π' 11.7 µK2

(

Ωbh2

0.02

)2 (Ωmh

2

0.15

)1/2

xH(z)(

1 + z

10

)−1/2

(5.68)

129

×∫ ∞

0dz′H−1(z)

(

1 + z′

10

)3/2d

dz

[

xe(z′)(1 + z′)3/2

]

Fl(z, z′),

where

Fl(z, z′) ≡ l2

∫ ∞

0dkPδδ[k, zN ](1 + zN)2

105 Mpc3 jl [k(η0 − η(z)] jl [k(η0 − η(z′)] . (5.69)

This exact equation contains an integral over redshift because the Doppler

effect is the result of peculiar motions over the entire line of sight. The 21–cm

maps, on the other hand, come from a particular redshift at a single point along

each line of sight. When we cross correlate CMB and 21–cm maps, we expect that

the only part of the Doppler signal which contributes to the cross-correlation is

that which originates from the same distance as the 21–cm map. Implicit in this

expectation is that the 21–cm angular fluctuations, which are perpendicular to the

line of sight, are related by the continuity equation to the velocity fluctuations

along the line of sight, which are the source of the Doppler effect. This “flat sky

approximation” is valid on small scales, and thus large values of l. On larger

scales, however, the angular fluctuations do not exactly correspond to those along

the line of sight, and we expect a reduction in the amplitude of the correlation.

Mathematically, this is manifested by the fact that the integrand of the line of

sight integral, Fl(z, z′), approaches a δ-function for large l, since

2

π

∫ ∞

0dkP (k)jl(kr)jl(kr

′) ≈ P

(

k =l

r

)

δ(r − r′)

l2. (5.70)

This relation is an expression of what is commonly referred to as Limber’s ap-

proximation (e.g., Limber 1954; Kaiser 1992). Inserting this approximation into

equation (5.69), we obtain

l2C21−Dl (z)

2π' 24.5 µK2

(

Ωbh2

0.02

)2 (Ωmh

2

0.15

)1/2

xH(z)Pδδ[l/r(z), zN ](1 + zN)2

105 Mpc3

× d

dz

[

xe(z)(1 + z)3/2]

(

1 + z

10

)

. (5.71)

130

Note that this equation is much simpler than equation (5.69) because there is no

integral over redshift and the cross-correlation is proportional the matter power

spectrum, l2C21−Dl (z) ∝ Pδδ(k = l/r(z)). The question thus arises: is l ∼ 100,

where the correlation peaks, a sufficiently large l such that we can use equation

(5.71) instead of (5.68)? We will answer this question by comparing the numerical

evaluation of equation (5.68) to the analytical approximation of equation (5.71).

5.7.1 Numerical integration

As a test of our numerical integration of equation (5.69), we will assume

Pδδ(k, zN)(1 + zN)2/(105Mpc3) = Ak, in which case

Fl(z, z′) =

A√π

2

l2Γ(l + 1)

Γ(l + 32)

[

η0 − η(z)

η0 − η(z′)

]l

[η0 − η(z′)]−2

× 2F1

1

2, l+ 1, l +

3

2;

[

η0 − η(z)

η0 − η(z′)

]2

, (5.72)

where 2F1 is a hypergeometric function. The shape of Fl(z, z′) is plotted in the left

panel of Figure 5.6, for the case P (k) = Ak. As l increases, Fl(z, z′) approaches

a delta function, as expected. The right panel shows the cross-correlation as a

function of l. This shows that the numerical integration is accurate. All curves are

for the analytical reionization history given by equation (5.29) with z = zr = 15

and ∆z = 0.5. For this power-law power spectrum, the exact numerical integration

leads to a result which is about 80% of the approximate Limber approximation.

Figure (5.7) is the same as Figure (5.6), except that the actual P (k) is used. In

this case, there is no exact analytical result to compare to. Near the peak, the

numerical integration leads to a value which is about 65% of the approximate one.

In our quantitative results, we therefore use the exact numerical integration.

131

Chapter 6

Implications of WMAP 3 Year Data for the

Sources of Reionization

New results on the anisotropy of the cosmic microwave background (CMB)

and its polarization based upon the first three years of data from the Wilkinson

Microwave Anisotropy Probe (WMAP) have revised the electron scattering optical

depth downward from τes = 0.17+0.08−0.07 to τes = 0.09 ± 0.03. This implies a shift

of the effective reionization redshift from zr ' 17 to zr ' 11. Previous attempts

to explain the high redshift of reionization inferred from the WMAP 1-year data

have led to widespread speculation that the sources of reionization must have

been much more efficient than those associated with the star formation observed

at low redshift. This is consistent, for example, with the suggestion that early

star formation involved massive, Pop III stars which early-on produced most of

the ionizing radiation escaping from halos. It is, therefore, tempting to interpret

the new WMAP results as implying that we can now relax those previous high

demands on the efficiency of the sources of reionization and perhaps even turn the

argument around as evidence against such high efficiency. We show that this is not

the case, however. The new WMAP results also find that the primordial density

fluctuation power spectrum has a lower amplitude, σ8, and departs substantially

from the scale-invariant spectrum. We show that these effects combine to cancel

the impact of the later reionization implied by the new value of τes on the required

ionizing efficiency per collapsed baryon. The delay of reionization is surprisingly

132

well-matched by a comparable delay (by a factor of ∼ 1.4 in scale factor) in the

formation of the halos responsible for reionization1.

6.1 Introduction

One of the most important outstanding problems in cosmological structure

formation is how and when the universe was reionized. Observational constraints

such as the Thomson scattering optical depth to the last scattering surface (Kogut

et al. 2003; Page et al. 2006) from the large-angle polarization anisotropy in the

CMB detected by WMAP and the intergalactic, hydrogen Lyα absorption spectra

of high-redshift quasars (e.g., Becker et al. 2001) provide crucial constraints on the

theory of cosmic reionization and the structure formation which caused it during

the early epochs that have thus far escaped direct observation. The WMAP first-

year data implied an electron scattering optical depth, τes = 0.17, which seemed

surprisingly large at the time, since it was well in excess of the value, τes ' 0.04, for

an intergalactic medium (IGM) abrubtly ionized at zr ' 6.5, the reionization epoch

which had been suggested by quasar measurements of the Gunn-Peterson (Gunn

& Peterson 1965; “GP”) effect. In order for such an abrupt reionization to explain

the high value of 0.17 observed by WMAP for τes, in fact, zr ' 17 is required.

This presented a puzzle for the theory of reionization: How was reionization so

advanced, so early in our observed ΛCDM universe, and yet so extended in time as

to accumulate the high τes observed by WMAP, while ending as late as z ' 6.5 to

satisfy the quasar spectral constraints?

This stimulated widespread speculation regarding the efficiency for the for-

mation of the early stars and/or miniquasars which were the sources of reionization,

1This work appeared previously in Alvarez, Shapiro, Ahn, & Iliev 2006, ApJ, 644, L101

133

as well as for the escape of their ionizing photons into the IGM (e.g., Haiman &

Holder 2003; Cen 2003; Wyithe & Loeb 2003; Kaplinghat et al. 2003; Sokasian et

al. 2004; Ciardi, Ferrara, & White 2003; Ricotti & Ostriker 2004). A general con-

sensus emerged that the efficiencies for photon production and escape associated

with present-day star formation were not adequate to explain the early reionization

implied by the high τes value, given the rate of early structure formation expected

in the ΛCDM universe. Common to most attempts to explain the high τes was

the assumption that early star formation favored massive Population III stars,

either in “minihalos,” with virial temperatures Tvir < 104 K, requiring that H2

molecules cool the gas to enable star formation (Abel, Bryan, & Norman 2002;

Bromm, Coppi, & Larson 2002), or else in larger halos with Tvir > 104 K, for

which atomic hydrogen cooling is possible, instead. A high efficiency for turning

halo baryons into stars and a high escape fraction for the ionizing radiation into

the IGM were generally required as well. Several effects were suggested that could

extend the reionization epoch, too, including the rising impact of small-scale struc-

ture as a sink of ionizing photons (e.g., Shapiro, Iliev, & Raga 2004; Iliev, Shapiro,

& Raga 2005; Iliev, Scannapieco, & Shapiro 2005), the suppression of low-mass

source-halo formation inside the growing intergalactic H II regions (e.g., Haiman &

Holder 2003), and a general decline of the efficiency for releasing ionizing radiation

over time (e.g., Cen 2003; Choudhury & Ferrara 2005).

With three years of polarization data, WMAP (henceforth, “WMAP3”),

has now produced a more accurate determination of τes, which revises the optical

depth downward to τes = 0.09 ± 0.03 (Page et al. 2006). This value is consistent

with an abrupt reionization at zr = 11, significantly later than that implied by

the WMAP first-year data (henceforth, “WMAP1”’). It is natural to wonder if

134

Figure 6.1 R.m.s. fluctuation as a function of mass for different values of the“tilt”, ns = 1, 0.98, 0.96, 0.94, 0.92, 0.9, from top to bottom. Top: Ratio of thethree year WMAP to the one year variance. The most likely value for the threeyear data, ns = 0.95 is shown as the dashed line. Note that all curves intersectat the normalization mass scale corresponding to 8 h−1 Mpc. Bottom: Variancederived from the three year WMAP data in units of the variance for a scale-freepower spectrum with ns = 1.

this implies that the high efficiency demanded of ionizing photon production by

WMAP1, described above, can now be reduced, accordingly, to accommodate the

later epoch of reionization determined by WMAP3. In what follows, we will show

that this is not the case.

Structure formation in the ΛCDM universe with the primordial density

fluctuation power spectrum measured by WMAP3 is delayed relative to that in

the WMAP1 universe, especially on the small-scales responsible for the sources of

135

reionization. This, by itself, is not surprising, since there was always a degeneracy

inherent in measuring the amplitude of the primordial density fluctuations using

the CMB temperature anisotropy alone, resulting from the unknown value of τes.

Higher values of τes, that is, imply higher amplitude density fluctuations to produce

the same level of CMB anisotropy. This degeneracy is broken by the independent

measurement of τes made possible by detecting the polarization anisotropy, as well.

Hence, when WMAP3 revised the value of τes downward relative to WMAP1, so it

revised downward the amplitude of the density fluctuations. This same decrease

of τes implies a tilt away from the scale-invariant power spectrum, P (k) ∝ kns with

ns = 1, which lowers the density fluctuation amplitude on small scales more than on

large scales. As we will show, this delays the structure formation which controls

reionization by just the right amount such that, if reionization efficiencies were

large enough to make reionization early and τes = 0.17 in the WMAP1 universe,

the same efficiencies will cause reionization to be later in the WMAP3 universe

and τes ∼ 0.09, as required.

In §6.2, we compare the rate of structure formation in ΛCDM according to

WMAP1 and WMAP3, on the scales relevant to reionization. In §6.3, we relate

the history of reionization to the growth of the mass fraction collapsed into source

halos, and use this to compare the reionization histories in WMAP3 and WMAP1

universes. Our conclusions are summarized in §6.4.

We adopt cosmological parameters (Ωmh2, Ωbh

2, h, ns, σ8) =

(0.14, 0.024, 0.72, 0.99, 0.9) (Spergel et al. 2003) and (0.127, 0.022, 0.73, 0.95, 0.74)

(Spergel et al. 2006) for WMAP1 and WMAP3, respectively. The most notable

changes from old to new are: a reduction of normalization of the power spectrum

on large scales (σ8 = 0.9 → 0.74) and more “tilt” (ns = 0.99 → 0.95). Throughout

136

Figure 6.2 Halo abundance vs. mass for new (WMAP3) and old (WMAP1) pa-rameters at z = 15, as labelled. Top: Press-Schechter mass function. Bottom:

Fraction of matter fcoll(> M) in halos with mass greater than M .

this paper, we use the transfer function of Eisenstein & Hu (1999).

6.2 Structure formation at high redshift

A fundamental building block of models of reionization is the fraction of the

mass contained in virialized halos – the “collapsed fraction” – the sites of ionizing

photon production and release. Using the Press-Schechter formalism (Press &

Schechter 1974), this collapsed fraction is given by:

fcoll(z) = erfc[

νmin(z)/√

2]

, (6.1)

where νmin(z) ≡ δc/[D(z)σ(Mmin)], σ2(M) is the variance in the present-day matter

density field according to linear perturbation theory, as filtered on the mass scale

M , D(z) is the linear growth factor (D(z) ∝ 1/(1+z) and δc = 1.686 in the matter-

dominated era), and Mmin(z) is the minimum mass for collapsed objects. For

studies of reionization, the minimum mass is typically parameterized in terms of

137

the minimum virial temperature, Tmin, of halos capable of hosting ionizing sources,

Mmin ' 4 × 107M[(Tmin/104K)(10/(1 + z))(1.22/µ)]3/2,

where µ = 1.22 for fully neutral gas (Iliev & Shapiro 2001).

For ΛCDM, σ(M) for M ∼ 106 − 108M is lower for WMAP3 than for

WMAP1 by about 30 percent (Figure 6.1). During reionization, when such halos

are still rare, we expect their abundance to be exponentially suppressed by this

factor. This is clearly shown in Figure 6.2, where the new halo abundance and

collapsed fraction are lower than the old ones by 1-2 orders of magnitude. Since

the threshold for halo collapse scales at these redshifts as δc/D(z) ∝ 1 + z, struc-

ture formation on these mass scales is delayed by a factor 1/0.7 ∼ 1.4 in scale

factor. This is illustrated in Figure 6.3, where we plot fcoll(T > Tmin = 104 K)

versus redshift and show that the shift of 1.4 in scale factor provides an excellent

description of the delay in structure formation which results.

For the simplest possible reionization model, in which the universe is in-

stantly and fully ionized at some redshift zr, the optical depth τes ∝ (1 + zr)3/2.

If we assume that fcoll(zr) is a constant, so that reionization occurs when the

collapsed fraction reaches some threshold value, then our simple estimate implies

that the change in the cosmological parameters alone reduces τes by a factor of

1.43/2 = 1.65, from τes ∼ 0.17 to τes ∼ 0.1. In the next section we discuss the

motivation behind tying the reionization history to the collapsed fraction.

6.3 Reionization History

An important quantity in the theory of cosmic reionization is the num-

ber of ionizing photons per hydrogen atom in the universe required to complete

138

Figure 6.3 Collapsed fraction vs. redshift for halos with virial temperatures greaterthan Tmin = 104 K, for WMAP1 and WMAP3, as labelled. Dashed curve, nearlyon top of “WMAP3” curve, is “WMAP1” curve with 1 + z → (1 + z)/1.4.

reionization2. In the absence of recombinations, this ratio is unity. Given some

observational constraint on the epoch of reionization, such as the onset of the GP

effect at z ' 6.5, we can deduce that at least one ionizing photon per atom had

to have been released by that time. This ratio can then be used to predict other

quantities, such as the associated metal enrichment of the universe (e.g., Shapiro,

Giroux, & Babul 1994) or the intensity of the near infrared background (e.g., San-

tos, Bromm, & Kamionkowski 2002; Fernandez & Komatsu 2006). Most models of

cosmic reionization link the ionized fraction of the IGM to the fraction of matter

in collapsed objects capable of hosting stars (e.g., Shapiro, Giroux, & Babul 1994;

Chiu & Ostriker 2000; Wyithe & Loeb 2003; Haiman & Holder 2003; Furlanetto,

Zaldarriaga, & Hernquist 2004; Iliev et al. 2005; Alvarez et al. 2006). For a given

model, reionization is complete whenever the total number of ionizing photons

2For simplicity, we will neglect helium reionization. This does not effect our basic conclusionshere.

139

emitted per hydrogen atom reaches some threshold value. Along with the escape

fraction, star formation efficiency, and stellar initial mass function, the evolution

of the collapsed fraction fcoll(z) forms the basis for calculation of this ratio and

thus the reionization history.

To relate τes to the halo abundance encoded in fcoll, it is necessary to deter-

mine the relationship between the reionization history and the collapsed fraction.

If we assume every H atom which ends up in a collapsed halo releases on average

fγ(z) ionizing photons, and that εγ(z) is the number of ionizing photons consumed

per ionized H atom, then we can write a simple relation between fcoll and the mean

ionized fraction,

xe(z) =fγ(z)

εγ(z)fcoll(z) ≡ ζ(z)fcoll(z) (6.2)

(e.g., Furlanetto, Zaldarriaga, & Hernquist 2004). For simplicity, we will assume a

constant value, ζ(z) = ζ0 (this simplification does not affect our main conclusions),

and fix the value of ζ0 for a given Tmin, so that τes = 0.17 for WMAP1. Reionization

is complete when the collapsed fraction reaches a threshold given by fcoll(zr)ζ0 = 1.

In Figure 6.4, we plot the value of νmin which corresponds to Tmin = 104 K.

As mentioned in §6.2, WMAP3 implies a delay of structure formation by ∼ 1.4 in

scale factor. In the lower panel, we compare the reionization histories for WMAP1

and WMAP3 according to equation (6.2), for the same efficiency ζ0. The same

shift by a factor 1.4 in scale factor is also present in the reionization histories,

which is not surprising, since we have assumed that xe ∝ fcoll, and fcoll is a unique

function of νmin. As mentioned in §6.2, this change can account for a shift in the

implied value of τes from 0.17 to 0.1, quite close to the WMAP3 value of 0.09±0.03.

On the basis of this simple calculation, we conclude that the reduction of τes from

140

Figure 6.4 Evolution with redshift for WMAP1 and WMAP3, as labelled. Top:

Threshold for collapse, νmin, for a halo with virial temperature 104 K. Dashed curve,nearly on top of “WMAP3” curve, is “WMAP1” curve with 1 + z → (1 + z)/1.4.Bottom: Reionization histories given by xe = ζ0fcoll, labelled by the correspondingvalues of τes, for ζ0 = 170 and Tmin = 104 K (solid), and ζ0 = 35 and Tmin = 2×103

K (dotted). The two dashed curves are “WMAP1” curves with 1+z → (1+z)/1.4.

WMAP1 to WMAP3 does not, itself, significantly reduce the demand for high

efficiency of ionizing sources imposed previously by WMAP1.

6.3.1 Effect of recombinations

Recombinations undoubtedly play an important role during reionization.

To first approximation, they should determine by what amount the parameter

εγ(z) appearing in equation (6.2) exceeds unity. The quantity εγ − 1 is equal to

the average number of recombinations that all ionized atoms must undergo during

reionization, Nrec. As shown by Iliev et al. (2005), Nrec ' 0.6 at percolation for

large scale simulations of reionization that resolve all sources with masses greater

than ≈ 2 × 109M, but do not resolve clumping of the IGM on scales smaller

than ≈ 700 comoving kpc. Surely, smaller scale structure affects reionization

141

strongly (e.g., Iliev, Shapiro, & Raga 2005; Iliev, Scannapieco, & Shapiro 2005),

and therefore the number of recombinations per ionized atom is likely to be higher.

For example, Alvarez, Bromm, & Shapiro (2006) found that the recombination

time in the gas ionized by the end of the lifetime of a 100 M star embedded in a

106M halo at z = 20 is ∼ 20 Myr, roughly one tenth of the age of the universe

at that time.

At the high redshifts considered here, the ratio of the age of the universe

to the recombination time is proportional to (1 + z)3/2. Since structure formation

is later for WMAP3 than for WMAP1 by a factor of 1.4 in scale factor, photon

consumption due to recombinations is lower for WMAP3 by a factor ∼ 1.43/2 =

1.65. Even if recombinations dominate the consumption of ionizing photons during

reionization, therefore, the new WMAP data require an efficiency ζ0 which is at

most a factor of only ∼ 1.65 lower than that for the first year data. This is true

even if clumping increases toward lower redshift, since the evolution of clumping

follows structure formation and is, therefore, similarly delayed.

6.4 Discussion

We have shown that the new cosmological parameters reported for WMAP3

imply that structure formation at high redshift on the scale of the sources respon-

sible for reionization was delayed relative to that implied by WMAP1. This delay

can account for the new value in τes without substantially changing the efficiency

with which halos form stars. Recombinations are fewer when reionization is later,

but the reduction is modest. Even the IGM clumping factor on which this recom-

bination correction depends follows the delay in structure formation.

An important additional constraint on reionization is that it end at a red-

142

shift z >∼ 6.5, in order to explain the lack of a GP trough in the spectra of quasars

at z <∼ 6.5. Because the GP trough saturates at a very small neutral fraction, the

quasar data alone do not tell us when the universe became mostly ionized. Indeed,

it is possible for the ionized fraction to have been quite high already at high red-

shift z ∼ 10 while there remained a neutral fraction sufficiently high to satisfy the

GP constraint at z ∼ 6.5 (e.g., Choudhury & Ferrara 2006). While the universe

may become mostly ionized well before z ∼ 6.5, it cannot be later than this, how-

ever. Because of the shifting in time of structure formation we have described, any

model of reionization which previously satisfied the WMAP1 τes ∼ 0.17 constraint

and became mostly ionized at z <∼ 9 would now reionize too late to be compatible

with the quasar observations. Recently, Haiman & Bryan (2006) used this fact to

deduce that the formation of massive Pop III stars was suppressed in minihalos.

While σ8 = 0.744 and ns = 0.951 are the most likely values obtained from

the new WMAP data alone, there remain significant uncertainties. When combined

with other data sets such as large-scale structure (e.g., Spergel et al. 2006) and the

Lyman-α forest (e.g., Viel, Haehnelt, & Lewis 2006; Seljak, Slosar, & McDonald

2006), important differences arise. While these differences may seem small from

the point of view of the statistical error of the observational data, the implications

for reionization can be quite dramatic, as we have seen here. It is also important

to note that these measurements of the power spectrum are on scales much larger

than those relevant to the sources of reionization. As such, the theory of reioniza-

tion requires us to extrapolate the power spectrum by orders of magnitude beyond

where it is currently measured. This means that the study of reionization is cru-

cial to extending the observational constraints on the origin of primordial density

fluctuations (e.g., by inflation) over the widest range of wavenumbers accessible

143

to measurement. Direct observations of the high redshift universe such as 21-cm

tomography (e.g., Iliev et al. 2002; Zaldarriaga, Furlanetto, & Hernquist 2004;

Shapiro et al. 2006; Mellema et al. 2006) and large-aperture infrared telescopes

such as JWST promise to diminish the uncertainties which currently prevent us

from making reliable statements about the nature of the first sources of ionizing

radiation.

144

Chapter 7

The Characteristic Scales of Patchy Reionization

We use large-scale simulations of reionization to calculate the characteristic

length scales of patchy reionization. By examining the size distribution of H I/H II

regions, the power spectrum of their fluctuations, and Euler characteristic of the

ionized fraction field, we investigate how various assumptions about the sources of

reionization, such as their mass-to-light ratio, susceptibility to positive and nega-

tive feedback, and bias, affect the topology of reionization. We use two different

methods for identifying the size distribution of H I/H II regions, the friends-of-

friends (FOF) method and the spherical average method. In the FOF method,

there is typically one very large connected region even when the volume is only

half-ionized, in which most of the ionized volume is contained. For the spherical

average method, the bubble distribution typically peaks on comoving Mpc scales.

Suppression and clumping reduce the size and increase the number of H II re-

gions, although the effect is modest, reducing the typical radius for the spherical

average method by factors of a few. We find that density and ionized fraction

are almost perfectly correlated on large scales at the half-ionized epoch, and that

suppression and clumping actually increase the degree of correlation at fixed scale.

The Minkowski functional V3, also known as the Euler characteristic, proves to be

very sensitive to suppression and clumping, and is closely related to the number

of H I/H II regions found by the FOF method.

145

f2000 f250 f2000C f250C f2000 250 f2000 250S f250 250S f2000C 250S f250C 250S

mesh 2033 2033 2033 2033 2033 2033 2033 2033 2033

box size 100 100 100 100 35 35 35 35 35

(fγ)large 2000 250 2000 250 250 250 250 250 250

(fγ)small - - - - 2000 2000 250 2000 250

Csubgrid 1 1 C(z) C(z) 1 1 1 C(z) C(z)

z50% 13.6 11.7 12.6 11 16.2 14.5 12.6 13.8 11.6

zoverlap 11.3 9.3 10.2 8.2 13.5 10.4 9.9 9.1 8.4

τes 0.145 0.121 0.135 0.107 0.197 0.167 0.138 0.151 0.122

Table 7.1 Simulation parameters and global reionization history results for simula-tions with WMAP1 cosmology parameters. Comoving box sizes are in [h−1Mpc].

7.1 Introduction

Cosmic reionization is closely related to the earliest structure and galaxy for-

mation. Much progress has been made recently both observationally (e.g. Becker

et al. 2001; Spergel et al. 2003; Spergel et al. 2006) and theoretically (e.g.

Choudhury & Ferrara 2005; Furlanetto & Loeb 2005; Iliev et al. 2006a; Zahn et

al. 2006). In the near future, 21-cm surveys such as LOFAR, PAST, MWA, and

SKA, observations of the kinetic Sunyaev-Zel’dovich (kSZ) effect in the cosmic mi-

crowave background (CMB) (e.g. Iliev et al. 2006d), and very high-redshift galaxy

surveys carried out using planned and existing observational facilities (e.g. Mal-

hotra & Rhoads 2005), promise to open new windows into the reionization epoch.

The prospect of observing the topology of reionization directly and on large scale

demands a thorough understanding of the geometry of reionization, which is our

present goal.

In the simplest description of the reionization process, sources of ionizing

radiation form separate and distinct “H II regions” (Shapiro & Giroux 1987), which

146

grow and merge until space is filled with ionized gas and reionization is complete1.

Even in this simplified picture, there can arise a great amount of complexity. For

example, sources of reionization are likely to be clustered in space, implying that

individual H II regions may contain many sources (Iliev et al. 2005a; Furlanetto

et al. 2004a) Rare sources are more biased relative to more abundant ones, and

it is expected that the level of bias will largely determine the topology of the

reionization process. Accurate theoretical predictions for the morphology and size

of H II regions depend on a detailed understanding of the the abundance and

clustering of the ionizing sources themselves.

In addition to the complexity that arises in attempting to model reioniza-

tion, there is also a significant amount of ambiguity in characterizing the reion-

ization process itself. Recently, for example, much emphasis has been placed on

the size distribution of H II regions as a basis for predicting observed quantities,

such as the 21-cm background fluctuation power spectrum (e.g. Furlanetto et al.

2004b; Mellema et al. 2006b). The geometrical shapes of H II regions can be quite

complex, and near the end of reionization it may be more appropriate to refer to

the size distribution of H I regions. Even the definition of “H II region size” is

subject to ambiguities (Iliev et al. 2006a; Zahn et al. 2006).

Among the most useful ways to characterize the reionization process is

through the power spectra of spatially fluctuating quantities, such as ionized frac-

tion and neutral hydrogen density. Use of the power spectrum allows for rigorous

quantitative comparison between different reionization scenarios. Furthermore, the

power spectrum of neutral hydrogen fluctuations is expected to be the most easily

1For the present work, we will neglect sources of very long mean free path radiation such asX-rays, which would complicate the picture further. It is not known whether such sources weresignificant sources of ionizing photons during reionization (Ricotti & Ostriker 2004)

147

observed quantity via 21–cm radio observations (Zaldarriaga, Furlanetto, & Hern-

quist 2004). Obviously, the size distribution of H II regions is closely related to the

power spectrum. In fact, recent analytical models for the power spectrum during

reionization are built upon a quantitative description of the size distribution of

H II regions themselves. It is therefore important to compare the results of three

dimensional simulations to both the analytical predictions for the power spectrum

and the models for the size distribution of H II regions from which the analytical

power spectra are derived.

In this paper, we will use large-scale simulations of reionization to identify

the characteristic scales of reionization. The simulations on which our analysis is

based were described in Iliev et al. (2006a,b). We will use two different methods

to measure the size distributions of H II and H I regions (§7.2), and study how

each of these quantities relates to other measures of reionization geometry, such

as the power spectrum (§7.3) and Euler characteristic (§7.4). We will end with a

discussion in §7.5.

7.2 Simulations

Our basic methodology has been previously described in detail (Iliev et al.

2006a; Mellema et al. 2006a) Here, we will briefly describe the underlying N-body

simulations that were performed and the radiative transfer simulations that we

analyze.

7.2.1 N-body simulations

As a basis for our radiative transfer calculations, we begin with the time-

dependent density field extracted from N-body simulations of structure formation.

148

Figure 7.1 Distribution of H II (solid) and H I (dashed) region sizes, by numberat the half-ionized epoch. The corresponding redshifts are given by entries in the“z50%” row in the table.

149

We use the PMFAST code (Merz et al. 2005), with 16243 equal mass dark matter

particles, where the Poisson equation is solved on a 32483 particle mesh grid. Two

different simulations were carried out, one within a box which is 100h−1 Mpc on a

side, and the other within a 35h−1 Mpc box. The particle mass is 2.5×107M and

1.1× 106M for the 100h−1 and 35h−1 Mpc boxes, respectively. The cosmological

parameters used were for a flat ΛCDM universe with Ωm = 0.27, Ωb = 0.044,

h = 0.7, n = 1, and σ8 = 0.9, consistent with the first-year results from WMAP

(Spergel et al. 2003).

7.2.2 Radiative transfer runs

Table 1 summarizes the 11 different radiative transfer runs that we will

analyze here. All the radiative transfer simulations were performed using the C2-

Ray method (Mellema et al. 2006) on a uniform rectilinear grid containing 2033

grid cells. The density is assigned to the mesh using the dark matter particles from

the underlying N-body simulation, so there is the implicit assumption that the gas

distribution follows that of the dark matter, which is valid on the large scales

considered here, much larger than the Jeans mass of the mean IGM. Simulations

are labelled by the parameter fγ , which is an efficiency factor for halo ionizing

photon production, so that each halo of mass M is assigned a steady luminosity

for the duration of each radiative transfer step

Nγ = fγMΩb

∆tiΩ0mp, (7.1)

where Nγ is the number of ionizing photons emitted per unit time, M is the

halo total mass, and mp is the proton mass, and ∆ti of each radiative transfer

timestep. In some cases, halos were assigned different luminosities according to

whether their mass was above (“high-mass sources”) or below (“low-mass sources”)

150

109M. This mass scale corresponds roughly to the value below shich low-mass

halos are suppresssed as sources of ionizing radiation if they form inside an H II

region becaus their baryonic content and star-forming efficiency are suppressed by

Jeans-mass filtering where the IGM is photoionized.

For example, f2000 250S indicates that high-mass sources had an efficiency

fγ = 250, while low-mass sources had an efficiency fγ = 2000, and were sup-

pressed within ionized regions. Simulations f2000 was analyzed in detail in Iliev

et al. (2006a), and we refer the reader to that paper for further details on our

application of the C2-Ray method to large scale simulations of reionization.

This suite of simulations allows us to see how the morphology and character-

istic scales of reionization depend upon various important numerical and physical

effects which are not yet well understood. An important physical effect which may

be present during reionization is source suppression, in which ionizing radiation

from sources hosted by halos with a virial emperature below some threshold is

suppressed when the halos are located within ionized regions. By comparing, for

example, f2000 250 to f2000 250S, it is possible to isolate the effects due solely to

source suppression. Sub-grid clumping can also be an important effect, illustrated

for example by comparison of f2000 with f2000C. Throughout this study, we will

make comparisons like these, in order to see how these physical effects manifest

themselves and to find which quantitative measurements best descriminate among

different reionization scenarios.

7.3 Size distribution

One of the most basic measures of reionization is the size distribution of H II

regions. Under the assumption that most of the volume is either highly-ionized

151

Figure 7.2 Distribution of H II (solid) and H I (dashed) region sizes, by volumefraction.

152

Figure 7.3 Effect of varying the threshold for the f2000 simulation at z = 13.6,when the total ionized fraction is about 50 percent.

or highly-neutral, H II regions can be considered to be topologically connected

volumes of space. We have previously used a friends-of-friends (FOF) method

(Iliev et al. 2006a) to identify such regions, using the condition for a cell to be

considered ionized of x > 0.5. Recently, Zahn et al. (2006) have also considered

the size distribution of H II regions from radiative transfer simulations of cosmic

reionization. They used a different method, which we will here call the “spherical

average” method. In what follows we will use both methods on the same numerical

data, exploring the differences and similarities between the two. We will also

study how the size distribution of H II regions is affected by the parameters of

the simulation, such as box size, source suppression, and treatment of sub-grid

clumping. Because H II region sizes diverge during percolation, we will also follow

the H I region sizes, which can be defined in exactly the same manner as for H II

regions, allowing for a more precise characterisation of the end of reionization.

153

Once reionization overlap occurs, the rise of the ionizing UV background due to

the arrival of photons from distant sources is ultimately limited by the bound-free

opacity of the relic neutral atomic component. Our simulations do not attempt to

provide an accurate model for the limiting effect on ionizing photon mean free path

after overlap due to the presence of Lyman limit systems, which can be comparable

to our simulation box sizes. Here we are concerned with an earlier epoch, however,

when the photon mean free path is still determined by the size of H II regions.

7.3.1 Friends-of-friends method

Our first method for identifying the size distribution of H II/H I regions

relies on a literal definition of “H II (H I) region”: a connected region in which

hydrogen is mostly ionized (neutral). For grid data, the obvious way to identify

such a connected region is to use a “friends-of-friends” (FOF) approach, in which

two neighboring cells are considered friends if they both fulfill the same condition.

Cells are grouped into distinct regions according to whether they are linked to-

gether in an extended network of mutual friends. The algorithm we use to group

cells together is the equivalence class method, described in Press et al. (1992).

Unless otherwise specified, we use x > 0.5 for a cell to be considered ionized, and

x < 0.5 for a cell to be considered neutral, so that every point in the simulation

box is either in an H I or an H II region. Our method was first described in Iliev

et al. (2006a).

The FOF method has been used extensively for halo finding in cosmological

N-body simulations (Davis et al. 1985). Our implementation is more straightfor-

ward, since each cell always has only 26 possible neighbors, the identities of which

are known in advance, as opposed to particle data, in which it is necessary to

154

Figure 7.4 Evolution of the H II (top) and H I (bottom) region size distributionsfor the f2000 simulation.

155

Figure 7.5 Size distributions using the friends-of-friends method. Each panel com-pares two different simulations when each is half-ionized.

156

perform costly searches to identify the groups. Another significant difference be-

tween the two methods is the role played by free parameters. In the halo finding

FOF method, the free parameter is the linking length, which is the distance within

which two particles are considered to be friends. In the region finding method, the

free parameter is the threshold, xth, for a cell to be considered ionized or neutral.

As we will see, our results are not very sensitive to the choice of xth.

Shown in Figure 7.1 is the number-weigted volume distribution, dP/dV of

H I/H II region sizes for each of the 2033 resolution runs, with the normalization∫

dP/dV = 1. Each panel shows the distribution at a redshift for which the

corresponding simulation was approximately half ionized (see the entries under

“z50%” in Table 1 for the corresponding redshifts). The small “bump” at far right

in each panel corresponds to one large region, comparable in size to the simulation

box, while the peak in the leftmost bin corresponds to a significant number of

isolated, individual ionized or neutral cells. Although the total volumes of H I and

H II regions are comparable at the selected redshifts, there are always more small

H I regions than H II regions at the half-ionized epoch. Some simulations have a

sharp decrease in the number of H II regions below a certain volume. For example,

there are no H II regions (except for the single-cell regions in which there are

sources) below ∼ 100 Mpc3 in f2000. The absence of H II regions of size below this

scale is an artifact of our time discretization of the radiative transfer. Over each

radiative transfer timestep, lasting ∆t ∼ 20 Myr, all halos present at the beginning

of a timestep are assumed to emit at constant luminosity for the duration of the

timestep. At the end of the timestep, therefore, each source has emitted at least

Nmin = Nγ∆t = fγMminΩb

mpΩb(7.2)

157

photons, where Mmin is the minimum halo mass. Neglecting recombinations and

assuming the region around a given source is large enough to have a density close

to the cosmic mean, the smallest H II region size is

Vmin = Nmin/nH ' 180Mpc3(

Mmin

2.5 × 109

)

(

fγ

2000

)

, (7.3)

where nH is the mean number density of hydrogen. For the f2000 simulation,

the truncation of the size distribution occurs at a slightly smaller scale, around

50-100 Mpc3, than that implied by this simple formula, Vmin ' 180 Mpc3. For

f2000 250, Mmin ' 108M, for which equation (7.3) gives Vmin ' 7 Mpc3, in good

agreement with the location of the truncation of the bubble distribution. For the

f250 simulation, we obtain Vmin ' 20 Mpc3, which is substantially larger than the

scale at which the truncation occurs in the simulation data. We attribute this to

the departure from uniformity that occurs on these smaller scales, implying that

the density is likely to be substantially higher than that assumed in equation (7.3),

which also neglected recombinations.

Figure 7.2 shows the same distribution, but weighted by volume rather than

number. In this case, it is evident that most of the volume is contained within

either a single large H I region, or a single large H II region. This is an inherent

property of the FOF method, where regions are grouped together as soon as they

touch. Thus, two regions that might otherwise be considered distinct may be

linked together when using the FOF method. One might be tempted to vary the

single free parameter, xth, in order to find a value for which the distribution of

region sizes is smoother. In practice, however, the great majority of cells are either

fully ionized or fully neutral, and the results are not very sensitive to the precise

value of xth. This is illustrated in Figure 7.3 where the dependence of H II region

158

size distribution on xth is shown for the f2000 simulation at the half-ionized epoch.

While there are small changes in the bubble distribution as xth is varied in the

range 0.01-0.99, the qualitative picture remains unchanged, with a large number

of single-cell regions, a similar number of regions of intermediate size, and a single

large region comparable in size to the simulation box. This weak dependence on the

choice of free parameter is in contrast to the FOF method applied to halo finding

in N-body simulations, where the linking length can be tuned to give the most

“reasonable” results, which correspond, for example, to halos with some average

overdensity.

The evolution of the distributions is shown in Figure 7.4 for the f2000

simulation. Even at early times, when the ionized fraction is only about 10 percent,

there are already a large number of intermediate size H II regions, as seen in the

top-right panel of the figure. At this time, some of the H II regions have already

merged into one larger region. As reionization proceeds, the largest H II region

grows, due not only to sources within the region but also to merging with other

H II regions. As those H II regions grow and merge into the larger one, their

numbers decrease, as well as the fraction of the ionized volume that they occupy,

as seen in the top-left panel of the figure.

The situation is rather different for the H I regions. Initially there is one

very large H I region (the dashed line is obscured by the solid lines in the bottom

panels of Figure 7.4), accompanied by a small number of isolated neutral cells.

Such isolated neutral cells are most likely to occur at the edges of H II regions,

inside isolated cells which are dense enough to be mostly neutral but not dense

enough to shield lower density cells on the far side of the source, leaving those cells

mostly ionized. As the H II regions grow and overlap, isolated H I regions begin

159

Figure 7.6 Spherical averaging method applied to a distribution of non-overlappingspherical H II regions. The right curves are the actual log-normal distributionsgiven by equation (7.5) for different values of σ, while the left curves show theresult which would be obtained on the same distribution using the spherical averagemethod. As σ → 0 and dP/dR approaches a delta function, dPsm/dR approachesthe kernel function W (R, 〈R〉).

160

Figure 7.7 Size distributions ionized (solid) and neutral (dashed) region using thespherical average method for each simulation. The vertical dotted line in eachpanel corresponds to half the cell size of the radiative transfer grid.

161

to appear. Note that the volume distribution of these small, isolated H I regions

is rather flat. At late times, the large H I region continues to shrink and fragment

into smaller ones, until finally the H I regions completely dissappear.

Including clumping and suppression reduces the typical sizes of H II regions.

This can clearly be seen in Figure 7.5, where the size distribution of pairs of

simulations are compared in each panel. For example, f2000C, which included

sub-grid clumping, has many more small H II regions with sizes 1-100 Mpc3 than

does f2000, which does not include sub-grid clumping. The comparison is not as

dramatic for f250 and f250C, where the difference is hardly noticable. This may be

in part because reionization occured later in these simulations, reducing the overall

recombination rate and thereby the relative importance of clumping. Also, the

smallest H II regions in the f250 simulation are much closer to the spatial resolution

scale, leaving less room for the effect of clumping to be discerned. The trend of

smaller H II regions is also evident for suppression. For example, f2000 250S, which

includes suppression of small source inside H II regions, has more H II regions with

sizes ∼ 10−1 Mpc3 than f2000 250, which does not include source suppression.

Apparently, clumping and suppression increase the number of small H II regions

which are present at the half-ionized epoch. This is expected, given that both

clumping and suppression reduce the ionizing efficiency of collapsed matter within

a given H II region, resulting in smaller H II regions. In order to achieve a given

ionized fraction, therefore, more, smaller H II regions are required.

7.3.2 Spherical Average method

The spherical average method was described in detail by Zhan et al. (2006).

In their technique, each cell in the computational volume is considered to be in an

162

Figure 7.8 Size distributions using the spherical average method. Each panel com-pares two different simulations when each is half-ionized.

163

Figure 7.9 Comparison of f2000 simulation (solid) and analytical model (dashed).left: H II region size probability distributions for x ∼0.2, 0.5, and 0.8, from left toright. The corresponding redshifts are z =14.5, 13.6, and 12.3 for the simulationand z =14.5, 13.6, and 13.3 for the analytical model.right: Evolution of the volumeionized fraction for the simulation and analytical model. The inset shows the samehistory but for a linear scale in the ionized fraction.

164

ionized region if a sphere centered on that cell has a mean ionized fraction greater

than some threshold, usually taken to by xth = 0.9. The size of the H II region

to which it belongs is taken to be the largest such sphere for which the condition

is met. We use essentially the same method, and our determination of H I region

size is also done here in a similar way, with a threshold of xth = 0.1. With this

method, a smoother distribution of H II region sizes is obtained than by the FOF

method, as is illustrated by the following simple toy model.

7.3.2.1 Simplified toy model

We begin by assuming that gas is either fully ionized or fully neutral, and

that all ionized bubbles are non-overlapping spheres with a volume-weighted distri-

bution dP/dR, so that P (R+dR)−P (R) is the fraction of the ionized volume that

lies within bubbles with radii between R and R + dR. What bubble distribution,

dPsm/dR, would we obtain by using the spherical average method? To simplify

further, we will take the threshold for the spherical average, xth, to be arbitrarily

close to unity, so that a point is considered to be within an ionized sphere of a

given radius only if all the matter in that sphere is ionized. In this case,

dPsm

dR= 3

∫ ∞

Rdr

(r −R)2

r3

dP

dr=∫ ∞

RdrW (r,R)

dP

dr, (7.4)

where W (r,R) = 3(r − R)2/r3 is the bubble size distribution obtained by the

spherical average method for the case of a single sphere of radius r. The lower limit

of the integral is R because only spheres which are larger than R can contribute to

the spherical average bubble distribution at R; the largest ionized sphere that can

be drawn around any given point is always smaller than the actual ionized sphere

in which it lies.

165

Shown in Figure 7.6 are dP/dR and the corresponding dPsm/dR for the

log-normal distribution

dP

d lnR=

1√2πσ2

exp

[

(ln(R) − ln(〈R〉)2

2σ2

]

, (7.5)

for a few different σ. As can be seen from the figure, the spherical average tends

to change the true bubble distribution in two ways. First, it smooths the actual

bubble distribution with the kernel W (r,R). Second, it lowers the value of the

mean bubble size, Rav =∫

RdP/dR. In our simple toy model, the mean bubble

size obtained by the spherical average method is always 1/4 of the actual mean

bubble size.

Our toy model is admittedly crude, most notably in the assumption of a

threshold xth = 1 and spherical H II regions. A lower value of xth would allow

small pockets of neutral gas to be attributed to large ionized regions. In fact, for

the case where x = 1 and x = 0 in ionized and neutral regions, respectively, lower

values of xth lead to an overestimate of the volume which is ionized, leading to a

violation of the normalization condition,

∫ ∞

0dR

dP

dR= xv. (7.6)

In most cases this overestimate is not very large. A lower value of xth would also

yield larger H II regions, but this effect has been shown to be rather modest (

Zahn et al. 2006). The assumption of spherical symmetry is a conservative one,

however, in the sense that it provides a lower limit to how much the spherical

average method underestimates the “true” H II regions sizes. This is because the

spherical average method is sensitive to the smallest dimension of the region in

which it lies, since if the radius is larger than the smallest dimension, the part of

166

the sphere lying in that direction would lie outside the region, and the condition

would no longer be satisfied.

7.3.2.2 Simulation results

Shown in Figure 7.7 is the distribution of H I/H II region sizes found by the

spherical average method, dP/dR, which is normalized such that∫

dR(dP/dR) =

1, at the half-ionized epoch for each simulation. The H I and H II region dis-

tributions have strikingly similar shapes, with peaks typically around 1 − 4 Mpc.

The most notable exception is f250 250S, where the neutral H I region distribution

peaks at a radius a few times smaller than that of the H II region peak. The

similarity of H I and H II region size distributions found by the spherical average

method is in contrast to the differences between the distributions found by the

FOF method. In the FOF method, the overwhelming majority of the volume at

the half-ionized epoch is contained in one large region, and the differences between

H I and H II region sizes occur for smaller regions, which are relevant to only a tiny

fraction of the total volume. The spherical average method, on the other hand,

attributes the volume in the largest FOF region to a continuous range of smaller

regions, and in the process the detailed differences between H I and H II region

distributions is for the most part averaged out.

In Figure 7.8, we make the same comparisons as were made for the FOF

method in Figure 7.5. The same trends are seen in the spherical average dis-

tributions, namely that clumping and suppression lead to smaller H II regions.

However, the impression can be quite different. For example, the comparison of

f2000 and f2000C in Figure 7.5 shows quite a difference between the two, while

the same comparison in Figure 7.8 shows only a very small overall leftward shift

167

of the f2000C curve relative that of f2000. As we have already noted, much of

the noticeable difference in the FOF curves comes from a very small fraction of

the volume, and a log scale is required to see such differences in the FOF plots.

The spherical average distributions are much smoother, and therefore offer a less

detailed, more global picture of the spatial structure of the ionized regions.

7.3.2.3 Comparison to analytical model

A natural question which arises in applying the spherical average method

to our simulations is how well the H II region size distribution is predicted by the

analytical model which inspired the method, first developed by Furlanetto et al.

(2004a). In that work, it was assumed that the ionized fraction follows the in-

stantaneous collapsed fraction of source halos, fcoll, with a region being considered

ionized if ζfcoll > 1, where ζ is an efficiency parameter. The more relevant con-

dition here, where source halos are assumed to have a luminosity proportional to

their mass, is

α∫ t

0fcoll(t

′)dt′ > 1, (7.7)

where α = fγ/∆ti is an efficiency parameter ( Zahn et al. 2006). In the appendix,

we describe how we generalize the model of Furlanetto et al. (2004a) to reflect

this modified condition. Because it is beyond the scope of this paper to generalize

the model to include such effects as clumping, suppression, and mass-dependent

efficiency, we will focus here exclusively on the f2000 simulation, and defer more

detailed analysis to future work.

Figure 7.9 shows the evolution of both the analytical model and f2000 sim-

ulation with redshift. We adopted the parameter α = fγ/∆ti = 100 Myr−1, which

corresponds to the timestep and efficiency for the f2000 simulation. The evolution

168

of the ionized fraction is surprisingly well-matched by the analytical model for this

choice of α. This is surprising because the analytical model is based on the Press-

Schechter formula for the collapsed fraction, which, as was shown in Iliev et al.

(2006a) and Zahn et al. (2006), underpredicts the abundance of the rare halos that

are especially important in the initial stages of reionization in the f2000 simulation.

On the other hand, the analytical model neglects recombinations. It is plausible,

therefore, that these two effects cancel each other, leading to the agreement in the

ionization history seen here.

The situation is not so fortuitous for the H II region size distributions. The

simulation consistently gives smaller H II region size than the analytical model,

although the qualitative evolution is quite similar. In either case, the peak of the

H II region size distribution moves to larger scales as reionization proceeds. At

early times and smaller scales, the distributions are symmetric in appearance, while

at the largest scales the distributions take on a more asymmetric shape, become

narrower and more strongly peaked, steeply declining beyond the peak. While the

origin of the discrepancy between the model and simulations is not certain, one

very likely source is that recombinations cause the ionized regions to be smaller

in the simulation than in the model. As we have already seen from the FOF size

distributions, the presence of clumping, and hence an enhanced recombination rate,

has the effect of reducing the size of H II regions relative to no clumping. This is

also true for spherical average size distributions, as shown in Figure 9. However,

the effect of clumping, which increases the cumulative number of recombinations

by a factor of about 5 between f2000 and f2000C (Iliev et al. 2006b), causes only a

small change in the size distribution at the half-ionized epoch, as seen in the lower

left panel of Figure 7.8. It thus remains to be seen how important recombinations

169

actually are in explaining the discrepancy between our results and the analytical

predictions.

7.4 Power spectra

We have also calculated power spectra of the density and ionized fraction

fields, Pδδ, Pxδ, and Pxx, where 〈δkδ∗k′〉 ≡ (2π)3δ3(k − k′)Pδδ(k), 〈δxkδ∗k′〉 ≡ δ3(k −

k′)(2π)3Pxδ(k), and 〈δxkδ∗xk′〉 ≡ δ3(k−k′)(2π)3Pxx(k). Here, δ is the overdensity of

matter, while δx ≡ x−xv. Note that we do not normalize δx by xv. When plotting

the actual power spectrum, we use the dimensionless power per logarithmic interval

in wavenumber, ∆2(k) ≡ k3P (k)/(2π2).

Shown in Figure 7.10 is the ionized fraction power spectrum, ∆2xx(k), for

selected simulations at the half-ionized epoch. As seen from the figure, there is a

broad peak in the power spectrum at scales of order kmax ∼ 0.5 − 2 Mpc−1. We

expect this peak to be associated with the the size of ionized or neutral bubbles,

for the following simple reason. On scales smaller than the bubbles, the correla-

tion function ξxx(r12) = 〈(x(r1) − xv)(x(r2) − xv)〉 reduces to the constant value

xv(1 − xv), while on scales much larger than the bubbles, the ionized fraction

is uncorrelated, and the correlation function should approach zero (Zaldarriaga,

Furlanetto, & Hernquist 2004). This behaviour for the correlation function implies

that the power spectrum ∆2(k) should approach zero at large and small scales, with

a peak at the characteristic size of the bubbles. Indeed, comparison of Figures 7.8

and 7.10 shows that the peaks of RdP/dR and ∆xx are related by kmax ∼ 1/Rmax.

In addition, the effects of clumping and suppression are also evident in the power

spectra, with the curves shifting towards higher k, and thus smaller scales, when

suppression or clumping are included.

170

Figure 7.10 Power spectra of ionized fraction at the half-ionized epoch.

171

Figure 7.11 Cross-correlation coefficient at the half-ionized epoch.

172

Figure 7.11 shows the cross-correlation coefficient of ionized fraction and

density field,

rxδ(k) ≡∆2

xδ(k)

[∆2xx(k)∆

2δδ(k)]

1/2. (7.8)

When rxδ = (−1)1, the ionized fraction and density field are perfectly (anti-

)correlated, while rxδ = 0 implies they are uncorrelated. As seen from the figure,

the ionized fraction and density fields at the half-ionized epoch are nearly per-

fectly correlated on large scales, k ∼ 0.1 Mpc−1. The scale above which the cross-

correlation coefficient rises coincides with the location of the peak of the ionization

power spectrum. This is consistent with the “inside-out” picture of reionization, in

which biased, overdense regions are ionized first, as our simulations indicate (Iliev

et al. 2006a). Recombinations are enhanced in ionized regions, and their effect

could plausibly reverse the sign of the correlation (Alvarez et al. 2006b; Zahn et al.

2006). Our simulations show no sign of such reversal, however. Most of the change

from the addition of clumping, which enhances the role of recombinations, appears

to be a shifting of the curves to smaller scales. Apparently, the decrease in H II re-

gion size, which decreases the scale at which the density and ionized fraction begin

to become correlated, completely overwhelms the enhanced recombination rate in

slightly overdense regions, and the net result is an increase in the cross-correlation

at a given scale. The same holds true for the effect of suppression.

7.5 Topology: Euler Characteristic

Minkowski functionals have been used extensively in cosmology to charac-

terise the topology of large scale structure (Gott, Melott, & Dickinson 1986; Mecke

et al. 1994; Schmalzing & Buchert 1997) and also the non-Gaussianity of the cosmic

microwave background (Komatsu et al. 2003). Recently, Gleser et al. (2006) have

173

Figure 7.12 Euler characteristic V3 (squares) versus mean volume averaged ionizedfraction for each of the 2033 resolution runs. Shown also are the number of H II(triangles) and H I (circles) regions found by the FOF method that are larger thanone cell.

174

used Minkowski functionals as a way to characterise the morphological structure

of reionization. In their work, they focused on the topology of the H I density field.

Here, we will take a complementary approach based upon the topology of the H I

regions themselves.

Consider a scalar function f(x) defined at each point x in three dimensional

space. The first Minkowski functional, V0(fth), is simply the fraction of the volume

in which f(x) < fth,

V0(fth) =1

Vtot

∫

Vd3xΘ[fth − f(x)], (7.9)

where Θ is the Heaviside step function. The next three Minkowski functionals are

defined as surface integrals over the boundary of the volume defined by f(x) < fth:

V1(fth) =1

6Vtot

∫

∂Fth

d2S(x) (7.10)

V2(fth) =1

6πVtot

∫

∂Fth

d2S(x)(

1

R1

+1

R2

)

(7.11)

V3(fth) =1

4πVtot

∫

∂Fth

d2S(x)1

R1R2

, (7.12)

where R1 and R2 are the principal radii of curvature along the surface ∂Fth. The

Minkowski functional V3, which is the integral of the Gaussian curvature over the

surface, is also known as the Euler characteristic, χ, and is equal to the number

of parts minus the number of tunnels of the structure. For example, a torus has

V3 = 0, since it has zero total curvature, and has equal one part and one tunnel.

A sphere, on the other hand, has V3 = 1, since it has one part but not tunnels,

and positive total curvature.

Shown in Figure 7.12 is the evolution of the Euler characterstic V3 for ionized

fraction with a threshold xth. Just as in the case of the FOF method, the value of

V3 at any given redshift is not sensitive to the threshold xth, since most of the gas

175

is either highly ionized or highly neutral. Shown also in the figure are the number

of H I and H II regions found by the FOF method that are larger than one cell. As

reionization begins, the Euler characteristic is close to the number of H II regions.

This is expected, since the topology is intially simple, consisting of many individual

H II regions, each with no tunnels. As more and more sources form, the number

of H II regions increases. At some point, typically around xv ∼ 0.1, the number

of H II regions decreases as their mergers and collective growth begins to outpace

the formation of new H II regions. At this point the Euler characteristic ceases to

track the number of H II regions and drops precipitously, becoming negative. This

indicates a transition to a more complex topology; as H II regions merge, they do so

incompletely, leaving many neutral tunnels through the H II regions. Eventually,

these tunnels are pinched off into individual H I regions, and the topology becomes

simple once again, with the Euler characteristic now matching the number of H I

regions instead of the number of H II regions, as was the case at the beginning of

reionization.

Apparently, clumping has a dramatic effect on the evolution of the Euler

characteristic, causing the minimum value, reached at an ionized fraction of x ∼

0.5, to be much lower. This can be seen most clearly in f2000 vs. f2000C and

f250 vs. f250C, where the amplitude of the minimum increases by a factor of two

when clumping is added. This is a manifestation of the greater amount of small

scale structure and larger number of H II regions that accompany the addition of

clumping. The situation is more complicated for suppression. As seen in Figure

7.13, there is one simulation, f2000 250S, which is very different from the rest. In

contrast to f2000 250, which is the same except without suppression, the minimum

is reached earlier and is relatively more shallow compared to the subsequent peak

176

associated with neutral regions. It seems that the suppression has accelerated

the evolution of the topology, leading to more neutral regions at intermediate and

late stages of reionization. One possibility is that suppression increases the role

of small sources forming within neutral tunnels and patches, thereby increasing

the number of H II regions and pinching off the tunnels sooner. The simulation

f250 250S, which also includes the effects of suppression, however, shows no such

enhancement of the Euler characteristic at late times. For f250 250S, the efficiency

of suppressible halos is fγ = 250, while for f2000 250S their efficiency is fγ = 2000.

In this case, the small halos that form in neutral regions are not so important,

and the large increase of V3 at the late stage of reionization seen in f2000 250S

dissappears.

7.6 Discussion

We have analyzed the results of several large scale, high resolution simula-

tions of cosmic reionization. We have calculated three different quantities from the

simulations results; the size distribution of H I/H II regions, the density and ionized

fraction power spectra, and the Euler characteristic. In doing so, we have inves-

tigated how parameters of the simulations, such as clumping, suppression, source

efficiency, and box size, affect these quantities. In addition, we have compared two

different methods for calculating the size distribution of H I/H II regions, the FOF

method and the spherical average method. Finally, we have explored how these

quantities relate to each other, such as Euler characteristic and number of H I/H II

regions, or the ionized fraction power spectrum and size distribution peak scales.

The nature of the size distribution of H I/H II regions is a matter of def-

inition. Applying a literal definition leads to the friends-of-friends approach, in

177

which the regions are considered to be connected regions of space. Because the

topology of reionization can be quite complex, this definition does not lend itself

easily to analytical modeling, and the connection of the size distribution obtained

in this matter to other statistical descriptions, such as the power spectrum, is by

no means trivial. For the FOF method, what is lost in its complexity is gained

in the detailed description of the reionization process that it provides. Although

most of the volume is always in one large bubble when the box is greater than

half-ionized, there is a wealth of information contained in the number and sizes of

the smaller bubbles, which only occupy a small fraction of the volume. The FOF

data is also essential to the interpretation of other measures of the topology of

reionization, such as the Euler characteristic of the inoization field.

The spherical average method, on the other hand, gives distributions which

are much smoother and better suited for comparison with analytical predictions.

When the universe is mostly neutral, the peak of the ionization power spectrum

is related to the size distribution of ionized regions. When the universe is mostly

ionized, however, it is the peak of the size distribution neutral regions which is

related to the peak in the ionization power spectrum. In either case, the peak

of RdP/dR, Rmax is related to the peak of the ionization power spectrum ∆2(k),

through kmax ∼ 1/Rmax. We have compared one of our simulations, f2000, to a

modified version of the analytical model of Furlanetto et al. (2004a). Although

the qualitative evolution of the spherical average size distributions agrees with

the analytical predictions, the simulation size distributions themselves are smaller

by as much as a factor of five. It is possible that recombinations, which are not

included in the analytical model, could cause the simulated H II regions to be

smaller, but this is unlikely to be the full explanation. We find disagreement

178

even when maximum predicted analytical bubble size is ∼ 20 Mpc, much smaller

than our box size of ∼ 140 Mpc, so it is also unlikely that finite box size is to

blame. Numerous inconsistencies and shortcomings of the analytical model have

previously been pointed out (McQuinn et al. 2005), but it remains unclear where

the explanation for the discrepancies lies.

The Euler characteristic, which is the same as the Minkowski functional

V3, offers a rich description of the evolution of the topology of reionization. In

the early stages of reionization, it tracks the number of H II regions, while at

the late stages it tracks the number of H I regions. At intermediate times, it

tracks the complexity in the transition from ionized bubbles to neutral patches.

Its evolution turns out to be very sensitive to clumping and suppression, implying

its measurement may provide a powerful way of descriminating between different

reionization scenarios in observations of the high redshift universe. Finally, since it

is a detailed description of the morphology of the ionization field, it shows promise

as a diagnostic tool for the numerical simulations themselves.

Clumping and suppression both lead to a decrease in the typical size of

H II regions and an increase of small scale structure. This is because both of

these effects tend to decrease the ionizing efficiency of sources within existing H II

regions. For the case of clumping, recombinations consume more photons. With

suppression, smaller mass sources do not form inside existing H II regions, and

thus the production of ionizing photons is decreased. This results in smaller H II

regions, so that more H II regions must form to achieve the same global ionized

fraction. In both the FOF and spherical average size distributions, the effects of

clumping and suppression show up as an increase in the number and filling fraction

of small H II regions. For the FOF distribution, this is seen as filling in of the “gap”

179

at small scales, while for the spherical average distribution it is seen as a modest

overall shift towards smaller scales. The effects of clumping and suppression are

also seen in the power spectra, ∆xx(k), with an effect very similar in nature to

that on the size distributions. The cross-correlation between ionized fraction and

density is also shifted toward smaller scales when clumping and suppression are

included, which leads to an increase in the cross-correlation coefficient. In no

case do we find that clumping or suppression leads to an anti-correlation between

ionized fraction and density. The effect of clumping is also seen in the Euler

characteristic, where the amplitude of the minimum is increased by a factor of two

when clumping is included. This is a reflection of the greater number of small H I

and H II regions and the more complex morphological structure that is present

when the H II regions begin to overlap. For suppression, the Euler characteristic

can even distinguish between more and less efficient small, suppressible sources.

When the small sources are more efficient, there can be a dramatic increase in the

Euler characteristic at late times, due to small halos forming in neutral patches.

The simulations presented here neglected the role of gas dynamics. While

gas dynamical effects, such as those that accompany the photoevaporation of mini-

halos, may not dramatically affect the overall evolution of the ionized fraction, they

may substantially affect the morphology of the reionization process itself, as we

have seen in our discussion of the effect of clumping and suppression on the Euler

characteristic. In addition to neglect of gas dynamics, another shortcoming of our

simulations is our assumption of a constant mass to light ratio for the source halos.

In reality, star formation is likely to be triggered by mergers, and the efficiency

is likely to be strongly mass dependent. As these and other important physical

effects are treated more and more realistically, it will be important to understand

180

Figure 7.13 Euler characteristic versus ionized fraction for selected 2033 resolutionruns, as labeled.

how the results presented here are affected.

7.7 Appendix: Distribution of H II region size for constant

mass to light ratio

In this paper, we compare our numerical simulations to the analytical model

for H II region size distribution first presented in Furlanetto et al. (2004a). In that

paper, the assumption was made that a region is fully ionized if its collapse fraction

times an efficiency is greater than one,

ζfcoll > 1. (7.13)

This corresponds, for example, to the assumption that ζfcoll ionizing photons are

181

Figure 7.14 Comparison of linear barrierB(m, z) (dotted) to actual barrier δx(m, z)(solid) with z =21, 18, and 15, from top to bottom, and Mmin = 108M. left:Case for the condition ζfcoll > 1, with ζ = 240. right: Case for the conditionα∫

fcolldt > 1, with α = 6.7 Myr−1.

released per atom per unit time. If recombinations are neglected, then equation

(7.13) results by ensuring that the time-integrated number of ionizing photons

released is greater than the number of the atoms.

If, on the other hand, we assume that photons are released at a rate of

αfcoll, then the criterion for a region to self-ionize becomes ( Zahn et al. 2006)

α∫ t

0fcoll(t

′)dt′ > 1. (7.14)

This is the assumption that we have made in our simulations, by assuming that

each source halo in our volume emits in proportion to its mass. Our parameter

fγ is related to α by fγ = α∆ti. For our timestep of ∆ti = 20 Myr, fγ = 250

corresponds to α = 12.5 Myr−1 and fγ = 2000 corresponds to α = 100 Myr−1.

Here, we will describe our modification to the original model of Furlanetto et al.

(2004a) to include constant mass-to-light ratio.

182

Figure 7.15 Comparison of time-integrated collapse fraction criterion (dotted) vs.instantaneous collapse fraction critersion (solid). left: Reionization history. right:H II region size distribution at x=0.1, 0.5, and 0.9, from left to right. The efficiencyparameters α and ζ are the same as those used in Figure 7.14.

The collapse fraction in a region of mass m is given by

fcoll(t) = erfc

δc(t)− δm√

2 [σ2min − σ2(m)]

, (7.15)

where σ(m) is the r.m.s. fluctuation on the scale m, σmin corresponds to the

minimum source halo mass mmin, δm is the overdensity of the region, and δc(t)

is the linear threshold for collapse in the spherical tophat model. Inserting this

expression into equation (7.14), we obtain the condition for a region to self-ionize,

δm ≥ δx(m, t) ≡ δc(t) − Σ(m)Ierfc−1

[

δc(t)D

αΣ(m)D

]

, (7.16)

where Σ(m) =√

2 [σ2min − σ2(m)], D(t) is the growth factor, and we have intro-

duced the function

Ierfc(u) ≡∫ ∞

uerfc(y)dy =

e−u2

π− uerfc(u). (7.17)

183

This “barrier” can be approximated by a linear function in the same manner as

in Furlanetto et al. (2004a), by using the value and slope evaluated at σ2(m) = 0.

We obtain

δx(m, t) ' B(m, t) ≡ B0(t) +B1(t)σ2(m), (7.18)

where

B0(t) = δc(t) −√

2σminIerfc−1ξ, (7.19)

B1(t) =1√

2σmin

ξ[

erfc(

Ierfc−1ξ)]−1

+ Ierfc−1ξ

, (7.20)

and

ξ ≡ δc(t)D√2σminαD

. (7.21)

Shown in Figure 7.14 are comparisons between the approximate, linear barrier

B(m, z), and the actual one δx(m, z). It is clear from the figure that the linear

barrier is just as good an approximation in the constant mass to light ratio case

as in the original case.

Shown in Figure 7.15 are the reionization histories for the instantaneous

and time integrated criteria, given by

x = ζerfc[

νmin/√

2]

(7.22)

for the instantaneous case, and

x =

√2

νmin

α

H(z)Ierfc

[

νmin/√

2]

, (7.23)

for the time-integrated case, where νmin ≡ δc/σmin. The evolution of the average

ionized fraction is slightly faster in the time-integrated case than in the instanta-

neous case. Also shown is the bubble mass function, given by

mdn

dm=

√

2

π

ρ

m

∣

∣

∣

∣

∣

d ln σ

d lnm

∣

∣

∣

∣

∣

B0

σ(m)exp

[

−B2(m, t)

2σ2(m)

]

. (7.24)

184

This formula for the bubble mass function is the same for both the instantaneous

and time-integrated criteria, with the only difference being the definitions of B0(z)

and B1(z). As can be seen in the figure, the time-integrated criterion gives slightly

larger H II regions, although the results are qualitatively very similar. This is

because bubbles retain memory of earlier times, when the sources were more rare

and further apart. This was first pointed out by Zahn et al. (2006).

185

Chapter 8

Discussion

We have studied the evolution of dark matter halos and the epoch of reion-

ization. In Chapter 2, we focused on dark matter halos formed in simulations of

cosmological pancake instability and fragmentation. These simulations yielded the

surprising result that the overall evolution and structure of these “pancake halos”

are very similar to those formed in CDM simulations of hierarchical clustering.

The global evolution is the same, whether halos form hierachically, as in CDM,

or by fragmentation, as in the pancake instability model. This led us to investi-

gate the connection between the mass accretion history and the evolving density

profile. We did this through one-dimensional modelling, presented in Chapter 3.

There we showed that the “fluid approximation”, which is equivalent to assum-

ing that dark matter random motions are isotropic, yields results that match the

three-dimensional CDM and pancake instability results. The isotropization that

accompanies dark matter collapse and virialization is therefore a key element in

the universal halo properties discussed in Chapters 2 and 3. Departures from this

universal behaviour will likely be accompanied by departures from isotropy.

In Chapter 4, we presented the first three-dimensional simulations of the

propagation of ionization fronts around the first stars. We accomplished this

through a novel application of the self-similar solutions of champagne flow (Shu et

al. 2003) to Pop III stars embedded in minihalos. We found that the lowest mass

stars, with masses less than ∼ 15M, have ionization fronts that never “breakout”

186

of their halos, and therefore do not contribute substantially to the reionization of

the universe. Our results are in good agreement with the one-dimensional calcula-

tions of Whalen et al. (2004) and Kitayama et al. (2004) with regard to the escape

fraction of the radiation from minihalos with mass ∼ 106M at z ∼ 20. Because we

followed the I-front into the surrounding medium, we were able to go beyond the

one-dimensional simulations, to calculate the ionizing efficiency of these first stars,

finding that the ratio of ionized mass to stellar mass is roughly constant, 5-6×104,

for stellar masses M∗>∼ 80M. At lower masses, the efficicency is much lower. We

also found that lower mass stars leave behind clumpier H II regions. Thus, while

the efficiency is constant at high mass, the survival time of the H II region after

the star dies is strongly dependent on stellar mass, even for M∗ > 80M. Larger

mass stars leave behind longer lasting H II regions. Finally, we found that nearby

minihalos trap the I-front, preventing gas in their centers from being ionized. This

calls into question the results of O’Shea et al. (2005), in which star formation is

stimulated in a nearby halo by fully ionizing its core. Subsequent calculations have

confirmed our result that this is not the case, but the issue of whether the first

stars stimulate or delay further star formation remains a controversial one (Susa

& Umemura 2006; Abel, Wise, & Bryan 2006; Ahn & Shapiro 2006).

In Chapter 5, we proposed a novel observational technique for reconstruct-

ing the history of reionization: the evolution of the CMB Doppler–21 cm corre-

lation. We focused on large angular scales of about a degree, corresponding to

about 500 comoving Mpc, where the complexity of patchy reionization is averaged

out and density fluctuations are still in the linear regime. The 21 cm fluctuations

are caused by density fluctuations, while the Doppler CMB fluctuations are due

to velocity fluctuations. The correlation arises because the density and velocity

187

fluctuations are related via the continuity equation. We found that the expected

signal can be detected by the Square Kilometer Array (SKA) at the 3-σ level. In

order to predict the signal, we devoloped a simple analytical model for the ion-

ization density cross-correlation power spectrum, Pxδ(k), which is based upon the

linear bias of halos (e.g., Mo & White 1996). Future work will utilize numerical

simulations of reionization to test and refine the analytical predictions.

The implications for reionization of the 3-year WMAP results were dis-

cussed in Chapter 6. The additional two years of polarization data allowed for

a more precise determination of the optical depth to Thomson scattering, which

constrains the duration of reionization, revising it downward from τes ∼ 0.17 to

τes ∼ 0.09. Consequently, the epoch of reionization was changed from z ∼ 17 to

z ∼ 11. The new polarization measurements also broke the degeneracy between

τes and the amplitude and “tilt” of the primordial power spectrum; the lower the

optical depth, the lower the amplitude of fluctuations necessary to reproduce the

observed temperature anisotropies. We found that these two effects cancel each

other, so that the delay of reionization is matched by a delay of structure formation,

leaving the constraints on the ionizing efficiency of collapsed matter unchanged.

Subsequent calculations by Iliev et al. (2006b) have confirmed that this is also the

case in realistic simulations of cosmological reionization.

Finally, we used large-scale radiative transfer simulations to identify the

characteristic scales of reionization in Chapter 7. These simulations confirmed

the analytical predictions presented in Chapter 5, that the ionized fraction and

density are correlated on large scales, because of halo bias. We used two differ-

ent models to determine the size distribution of H II regions: the friends-of-friend

(FOF) method, and the shperical average method. The FOF method, however,

188

is the only method with which can actually determine a catalogue of individual,

non-overlapping H II regions. The spherical average method only provides the size

distribution. The scale at which the spherical average size distribution, RdP/dR,

peaks, coincides with the scale at which the ionization power spectrum, ∆xx(k),

peaks. Both clumping and suppression reduce this characteristic scale. The Euler

characteristic turns out to be a very useful measure of the complex topology of

H I/H II regions, and is very sensitive to suppression and clumping. This implies

that observations which can measure the Euler characteristic, perhaps through

high-resolution tomographic 21 cm observations, will be very useful in descrimi-

nating between different reionization scenarios. Future work will concentrate on

the connection between the characteristic scales of reionization and the correlation

function of the dark matter halos which host ionizing sources.

In this dissertation we have considered structure formation, both in the

context of dark matter halo structure and evolution, and in the epoch of reion-

ization. While the original motivations for studying these two topics were quite

independent of one another, this does not mean that the results obtained in one

topic do not hold implications for the other. In the future, we will apply the knowl-

edge gained through the study of dark matter halo evolution to reionization, and

vice-versa.

189

Appendices

190

Appendix A

21–cm Observations of the High Redshift

Universe

One of the most promising means by which to observe the high redshift

universe in the cosmic dark ages is through the 21–cm wavelength hyperfine tran-

sition of the atomic hydrogen that is abundant prior to reionization (e.g. Scott

& Rees 1990; Subramanian & Padmanabhan 1993). Motivated by the prospect

of new radio telescopes that will be able to observe such a signal, several specific

observational techniques have been proposed (e.g. Tozzi et al. 2000). Among these

are the study of absorption features in the spectra of bright, high-redshift quasars

(Carilli, Gnedin, & Owen 2002; Furlanetto & Loeb 2002), features in the frequency

spectrum of the signal averaged over a substantial patch of the sky (Shaver et al.

1999; Gnedin & Shaver 2004), and angular fluctuations (Madau, Meiksin, & Rees

1997; Iliev et al. 2002; Ciardi & Madau 2003; Iliev et al. 2003; Zaldarriaga,

Furlanetto, & Hernquist 2004; Furlanetto, Sokasian, & Hernquist 2004). Aside

from the study of absorption along the line of sight to a quasar, all the techniques

proposed thus far depend on the spin temperature of the gas, TS, differing from the

temperature of the cosmic microwave background, TCMB. Otherwise, the intensity

of the radiation at the redshifted 21 cm wavelength will be indistinguishable from

that of the CMB.

In this appendix, we present the relevant background and some selected

results which are important for the study of 21–cm radiation from the high redshift

191

universe. Much of the discussion here serves as background to Chapter 5, which

is concerned with the cross correlation between 21–cm and CMB data on degree

angular scales.

A.1 Basics of 21-cm radiation

We begin with a brief review of the basic physics of 21–cm radiation in the

early universe.

A.1.1 Spin temperature

The n = 1 ground state of atomic hydrogen is split into the so-called hyper-

fine transition, with an excitation temperature of kT∗ = hν21, where T∗ = 0.068 K

and ν21 = 1.4 GHz. The spin temperature, Ts, is defined by the relative popula-

tions of the triplet excited state, n1, and the singlet ground state, n0, given by the

Boltzmann equation,

n1

n0= 3 exp [−T∗/Ts] , (A.1)

where the factor of three is a statistical weight for the triplet excited state. There

are only two physical mechanisms by which the spin temperature is decoupled

from the CMB temperature; Lyα pumping by radiation with a wavelength in the

Lyα transition (the so-called “Wouthuysen-Field effect” – e.g. Wouthuysen 1952;

Field 1959), and spin exchange during collisions between neutral hydrogen atoms

(e.g. Purcell & Field 1956). Both of these mechanisms tend to bring the spin

temperature into thermal equilibrium with the local kinetic temperature TK, so

that TS → TK. The efficiency of Lyα pumping depends upon the intensity of

the UV radiation field in the Lyα transition, whereas the efficiency of collisional

coupling depends upon the local density and temperature.

192

We will first discuss the effect of the radiation background in the 21–cm

transition. We will consider the CMB temperature TCMB T∗ (always a good

assumption), so that the intensity of the CMB at the 21–cm transition is given by

the Rayleigh-Jeans limit of the thermal blackbody spectrum,

I(ν21) ≡ I21 =2ν2

21

c2kTCMB. (A.2)

Given a sufficient period of time, atoms bathed in the CMB will reach equilibrium

according to

n0B01I21 = n1(A10 +B10I21), (A.3)

where A10 and B10(B01) are spontaneous and stimulated emission (absorption)

Einstein coefficients, respectively, and A10 = 2.85× 10−15 s−1. Since Ts = TCMB in

equilibrium, we obtain the Einstein relations in the Rayleigh-Jeans limit,

B01 = 3c2

2hν3A10 (A.4)

B01 = 3B10 (A.5)

(e.g., Rybicki & Lightman 1979). The timescale for equilibrium to be reached is

given by

t ∼ [n0B01I21/(n0 + n1)]−1

=2T∗

3TCMBA−1

10 (A.6)

' 8 × 103yr (TCMB/60K)−1, (A.7)

which shows that equilibration happens on timescales much shorter than the the

age of the universe for typical radiation background temperatures (i.e. CMB at

z ∼ 20).

The situation changes when collisions between neutral atoms are important

Purcell & Field (1956) showed that there is a finite probability that a hydrogen

193

atom, upon collision with another atom, will undergo spin exchange, effectively

resulting in a collisionally-induced transition in the hyperfine levels. The upward

and downward transition rates can be found by considering thermodynamic equi-

librium; in the absence of other processes, the spin temperature would come into

equilibrium with the kinetic temperature, and the following balance would apply:

C01n0 = C10n1, (A.8)

where C01 and C10 are transition probabilities per atom. From Boltzmann eqilib-

rium, assuming only collisions, we can deduce that

C01 = 3C10 exp [−T∗/Tk] , (A.9)

where Tk is the kinetic temperature of the gas. The probability, C10(K), that

an atom will undergo a downward transition can be obtained through quantum

mechanical calculations (e.g., Purcell and Field 1956; Allison & Dalgarno 1969;

Zygelman 2005). Adding these terms to the detailed balance of equation A.3, we

obtain

n0 [3C10 exp(−T∗/Tk) +B01I21] = n1 [C10 +A10 +B10I21] . (A.10)

Using the Einstein relations above, equation A.10 implies

Ts =TCMB + ycTk

1 + yc, (A.11)

where

yc =C10T∗

A10Tk. (A.12)

The most up-to-date calculations of the dependence of yc on temperature were

presented by Zygelman (2005), for which we find a good fit is given by

log[

yc

nH

]

=

−4.134 + 9.781x − 4.7755x2 + 1.075x3 − 0.091x4, T > 10 K,0.5206 − 0.419x + 1.272x2 + 0.4760x3, 1 K < T < 10 K,0.5206, T < 1 K

(A.13)

194

Figure A.1 Fit to data in Zygelman (2005). Also shown is the data presented inAllison & Dalgarno (1969). Note the disagreement at low temperature T < 10 K.

where x ≡ log T and nHI is the neutral hydrogen number density.

A radiation background in the Lyα transition can also affect the spin tem-

perature via the Wouthuysen–Field mechanism, which works in two ways. First,

excitation of the Lyα transition “mixes” the hyperfine levels; the shape of the

spectrum in the Lyα transition determines the relative fraction of electrons which

are excited from the ground and excited hyperfine levels. The more steeply the

spectrum falls off, the more photons there are available to excite the electrons

from the excited hyperfine state relative to the ground state, since it takes more

energy to excite electrons in the ground state. Thus, the steeper the decline, the

lower the spin temperature. The end result is that, for a sufficiently intense back-

ground in the Lyα transition, the spin temperature Ts approaches the Lyα “color

195

temperature” Tα, defined according to

1

kTα

= −∂ logNν

∂hν(A.14)

evaluated at ν = να, where N(ν) = c2Jν/(2hν3) is the photon occumpation num-

ber (Madau, Meiksin, & Rees 1997). In this case, the Lyα transition is in the

exponential cut off of the Planck spectrum, hνα kTα. Also, note that the

color temperature depends only upon the slope, and not the intensity, of the UV

background. Having established a coupling of Tα and Ts, the second part of the

Wouthuysen–Field mechanism couples Tα to Tk. As shown by Field (1959), the

redistribution of frequencies that occurs due to recoil when Lyα photons are scat-

tered by a thermal distribution of hydrogren atoms results in just that coupling,

with Tα → Tk. The overall effect of Lyα coupling can be expressed by the simple

relation

Ts =TCMB + yαTα

1 + yα, (A.15)

where the coupling constant yα = 4PαT∗/(27A10Tα), and Pα is the total scattering

rate (Madau et al. 1997). The combined effect of collisions and Lyα pumping can

be written

Ts =TCMB + yαTα + ycTk

1 + yα + yc. (A.16)

Often times, Tα is taken to be equal to Tk.

Recently, several authors have revisited the Wouthuysen–Field effect in the

context of modern theories of structure formation, adding a great amount of detail

and accuracy to the calculations (e.g., Chen & Miralda-Escude 2004; Pritchard

& Furlanetto 2006; Chuzhoy & Shapiro 2006). Chuzhoy & Shapiro (2006), for

example, derived a simple analytical formula for the color temperature,

Tα =[1 + Tk/0.4K]Ts

1 + Ts/0.4K, (A.17)

196

resulting in an accurate formula for the spin temperature in terms of only the

kinetic temperature and Lyα background intensity,

Ts =TCMB + [yα,eff + yc]Tk

1 + yα,eff + yc, (A.18)

where

yα,eff = yα [1 + 0.4K/Tk]−1. (A.19)

This shows that for large kinetic temperatures, much greater than 1K, Tα = Tk is

an excellent approximation in equation (A.16).

A.1.2 Radiative transfer of 21–cm radiation

The radiative transfer of 21–cm radiation is greatly simplified by the fact

that Ts T∗. In this case, the local source function, S21 = S(ν21), is in the

Rayleigh-Jeans limit of the blackbody spectrum,

S21 =2ν2

21

c2kTs. (A.20)

This means that the intensity in the equation of transfer can be replaced by tem-

perature, since the constant of proportionality, 2kν221/c

2, is always the same. For

radiation which originates in the last scattering surface and traverses the uniformly-

expanding IGM before reaching the observer at the present, the formal solution is

given by

TB = TCMB,0e−τ +

∫ τ

0

Ts(τ′)

1 + z′exp [τ ′ − τ ]dτ ′, (A.21)

where TB is the brightness temperature observed at frequency νobs, TCMB,0 ≈ 2.73K

is the present-day CMB temperature, τ is the total optical depth in the 21–cm line

from the CMB to the observer, and z′ is the redshift of gas at optical depth τ ′,

when the radiation has a frequency ν = νobs(1 + z′). The optical depth is due to

197

21–cm line absorption,

dτν = κνdl = nHIσ0φ(ν)dl, (A.22)

where l is distance along the line of sight and φ(ν) is the line profile, normalized

to unity. The absorption coefficient κν is normalized by setting the net absorption

rate equal to the stimulated absorption rate minus the stimulated emission rate:

4πS21

hν21

∫ ∞

0κνdν =

1

n0 + n1nHIS21(n0B01 − n1B10), (A.23)

where we have used∫

φ(ν)dν = 1 and taken S21/(hν21) out of the integral, since it

is constant over the frequencies where κν > 0. Since equilibrium is always a good

approximation, we can use the Einstein relations (eq. A.4) and equations (A.22)

and (A.23), we obtain

σ0 =3c2A10T∗

32πν221Ts

, (A.24)

where we have used n0/(n0+n1) = 1/4, which is always an excellent approximation

in the Rayleigh-Jeans limit. If we make the substitution dl = c/[H(z)(1 + z)] in

equation (A.22), and let φ(ν) = δ(ν − ν21), then the integrated optical depth

becomes

τ =3c3nHIA10T∗

32πν321H(z)Ts

≈ 6 × 10−3(1 + δ)(

TCMB

Ts

)

(

Ωbh2

0.02

)

[(

0.3

Ωm

)(

1 + z

10

)]1/2

h−1, (A.25)

where we have used H(z) = H0Ω1/2m (1 + z)3/2 at high redshift in a flat universe.

For situations in which the gas departs from the cosmic mean expansion, it can

easily be shown that equation (A.25) is still correct, with the Hubble expansion

replaced by the local line-of-sight divergence, H(z) → dvr/dr.

For linear to mildly non-linear gas in the redshifts of interest, the 21–cm

transition is optically thin and its line width is negligible, so that the solution to

198

the equation of transfer reduces to

δTb ≡ TB − TCMB,0 =Ts − TCMB

1 + zτ (A.26)

=3c3nHIA10T∗

32πν321H(z)(1 + z)

(

1 − TCMB

Ts

)

. (A.27)

Often times, the assumption is made that Ts TCMB, which is probably a good

approximation soon after reionization begins (e.g., Ciardi & Madau 2003; Furlan-

etto, Sokasian, & Hernquist 2004; Chen & Miralda-Escude 2004). In this case,

the differential brightness temperature δTb is independent of the spin temperature,

which simplifies the analysis considerably. In §A.2, we will present results for an

epoch in which this is not the case, when emission from minihalos dominated the

brightness temperature fluctuations.

A.1.3 Observational considerations

Several radio interferometer arrays are currently being developed, one of the

main goals of which is to observe the 21–cm line of neutral hydrogen before and dur-

ing cosmological reionization. The Primeval Structure Telescope (PaST/21CMA1)

has already begun taking preliminary data, the Low Frequency Array (LOFAR2)

is currently under construction in the Netherlands, the Mileura Widefield Array

(MWA3) is planned for construction in Mileura, Australia, and the Square Kilome-

ter Array (SKA4), the most ambitious in terms of sensitivity, is still in the planning

phase.

There are several parameters which determine the ability of a radio tele-

scope to resolve structure in the high redshift universe, both in angle and in fre-

1http://web.phys.cmu.edu/~past/2http://www.lofar.org3http://www.haystack.mit.edu/ast/arrays/mwa/index.html4http://www.skatelescope.org

199

quency space. Chief among these are the total collecting area, Atot, and the base-

line, D. The baseline determines the minimum angular scale that can be resolved,

given by

∆θ ∼ λ

D=

21(1 + z) cm

D. (A.28)

One might be tempted, therefore, to place the elements of the array, each with

an individual collecting area Adish = Atot/Ndish, where Ndish is the total number

of elements, as far apart as possible, in order to increase the baseline and thus

the resolution. A compromise must be made between resolution and sensitivity,

however, which is given for these parameters as

∆Terr =TsysD

2

Atot

√∆νtint

=Tsys

fcover

√∆νtint

, (A.29)

where tint is the integration time, ∆ν is the bandwidth, and Tsys is a “system tem-

perature”. As can be seen from the equation, the error, ∆Terr, increases inversely

with the covering factor of the elements, fcover ≡ Atot/D2. As the baseline is in-

creased for a fixed number of elements, the filling factor goes down and along with

it the sensitivity. Note the similarity of the above equation to the “noise power

spectrum” given by Zaldarriaga, Furlanetto, & Hernquist (2004) (see also Eq. 5.37

in Chapter 5).√

l2CNl

2π=

Tsys

fcover

√∆νtint

2πl

lmax, (A.30)

where lmax = 2πD/λ. The factor l/lmax reflects Poisson sampling of l-modes on

the sky: l/lmax ∝√Nmodes, where Nmodes is the number of patches of angular

size 2π/l in the sky. This noise power spectrum is relevant to making maps of the

reionization epoch, which may prove to be quite difficult, especially at high-l, since

∆terr ∝ l, and the flucuations due to noise compete and sometimes overwhelm the

actual signal (Zaldarriaga et al. 2004). Statistically, however, it should be possible

200

to measure the 21–cm power spectrum, since in that case one can take the average

of as many different regions as fill the field of view at a given scale. This corresponds

to binning the data logarithmically, so that ∆l ∝ l. In that case, the l/lmax factor is

canceled, and the error in the measurement of the power spectrum is independent

of l.

The system temperature is defined as the temperature of a matched resistor

input to an ideal noise-free receiver that produces the same noise power level as

measured at the output of the actual receiver (Furlanetto 2006). While tradition-

ally Tsys is attributed to the internal workings of the instrument itself, it turns out

that the effective temperature of the sky, Tsky which is dominated by the Galactic

synchrotron background, completely dominates the system noise. Since this is the

case, it is customary to absorb the Galactic foreground noise into the definition

of the system noise, so that Tsys ≈ Tsky. The Galactic synchrotron has a strong

frequency dependence, so that the effective sky temperature at the redshifted 21–

cm line is strongly redshift dependent (Chen & Miralda-Escude 2006; Furlanetto

2006),

Tsys ≈ 2000 K(

1 + z

21

)2.5

. (A.31)

The intensity of the Galactic synchrotron varies significantly with Galactic latitude,

and this formula is valid near the Galactic poles, where it is not nearly as bright as

it is toward the Galactic center and plane. The strong redshift dependence implies

that lower redshifts will be much easier to observe than lower ones.

A.1.3.1 Application to the first sources of 21–cm radiation

Here, we briefly describe the application of these concepts to assess the pos-

sibility of observing the very first sources of 21–cm radiation (Cen 2006; Chuzhoy,

201

Alvarez, & Shapiro 2006; Chen & Miralda-Escude 2006) with the next generation

of radio telescopes. In order to spatially resolve the first sources of 21–cm radi-

ation, it is necessary to achieve sufficient angular and frequency resolution with

sufficient sensitivity. Typical radii of a few comoving Mpc for these first sources

could correspond to “Lyα spheres”, where Lyα pumping by a massive (∼ 105M)

cluster of massive Pop III stars within a single halo pump the surrounding matter,

causing it to appear in absorption with respect to the CMB with an amplitude of

about 100 mK (see Figure A.2). Angular size is related to comoving size by

∆θ ' 3 × 10−4h

(

R

2 Mpc

)

(

1 + z

10

)−0.2

, (A.32)

while frequency interval is related to comoving size by

∆ν = 0.1 MHz

(

R

1.7 Mpc

)

(

1 + z

10

)−0.5

. (A.33)

The correspondence between angular scale and baseline is therefore given by

D ∼ λ

∆θ= 26 km

(

R

2 Mpc

)−1 (1 + z

21

)1.2(

h

0.7

)−1

(A.34)

Radio interferometer sensitivity is often expressed in terms of Atot/Tsys, so

the question arises: what sensitivity is required to resolve these Lyα spheres?

202

0 2 4 6 8 10−300

−250

−200

−150

−100

−50

0

R (Mpc)

δT (

mK

)

Figure A.2 UV source with total luminosity of, respectively, L = 5×1041 (dashed),5 × 1042 (dashed-dotted) and 5 × 1043erg/s (solid line) between Lyα and Ly-limitfrequencies (for more details see Chuzhoy, Alvarez, & Shapiro 2006).

Inserting the relations above into equation (A.36), we obtain

Atot

Tsys

' 2.2 × 104m2K−1(

∆Terr

50 mK

)−1(

R

2 Mpc

)−2.5 (t

103 h

)−1/2

(A.35)

×(

1 + z

21

)2.65(

h

0.7

)−2

.

SKA, the most ambitious radio telescope proposed to date, aims for a total col-

lecting area of ∼ 106 m2 (hence the name), 50% of which is within a “compact

core”, with a baseline of about 5 km, optimized for larger scale statistical studies

of reionization. This baseline is too short to resolve the Mpc scale (∼ 100 km is

required to resolve 1 comoving Mpc at z ' 20), so it may be more appropriate

to consider a larger baseline. While 50% of the collecting area is expected to be

within a 5 km baseline, 75% is expected to be within 150 km. In this configuration,

therefore, only 25% of the collecting area, 2.5×105 m2, is available to resolve these

structures. With a system temperature Tsys ' 2000 K at z = 20, this implies a

SKA sensitivity of only ∼ 125 m2 K−1, far below the necessary value derived above,

203

10−3

10−2

10−1

100

100

101

102

k (Mpc−1)

|δT

b/yα|[P

(k)k

3 ]1/2 (

mK

)

Figure A.3 Power spectrum of the 21 cm signal at z = 20. The pumping radiation isassumed to be produced by X-ray sources in halos with Tvir > 104 K or Tvir > 5·103

K (solid and dotted lines), or by Pop III stars in halos with Tvir > 104 K orTvir > 5 · 103 K (dashed and dashed-dotted lines). We assummed yα 5. (formore details, see Chuzhoy, Alvarez, & Shapiro 2006)

Atot/Tsys ' 2.2 × 104 m2 K−1. Clearly, the steep dependence of sky temperature

on redshift makes these high redshift observations quite difficult.

While resolving the structure of these objects may be beyond currently-

proposed observations, it may be possible to statistically measure their large scale

fluctuations. Shown in Figure (A.3) is the expected spherically averaged power

spectrum P (k) of the 21–cm signal produced by a biased Lyα pumping UV back-

ground (for more details, see Chuzhoy et al. 2006). At scales of tens of Mpc, easily

accessible to SKA, the signal is of order tens of mK. According to our discussion

of the noise power spectrum above and in Zaldarriaga, Furlanetto, & Hernquist

(2004), the error in the estimate of the power spectrum is given by

∆T 21err ≡

√

l2∆C21l

2π=

2πTsys

fcover

√∆νtint

lmin

lmax, (A.36)

where lmin is given by the total field of view. For SKA, the field of view is planned

204

to be ∼ 540[(1 + z)/21]2 deg2, which corresponds to

lmin ' 16(

1 + z

21

)−1

, (A.37)

while for the compact core with D = 5 km,

lmax =2πD

λ= 7.1 × 103

(

1 + z

21

)−1

. (A.38)

For fiducial values, the error of the power spectrum measurement is

∆Terr ' 1.7 mK(

Tsys

2000 K

)(

D

5 km

)2 ( Atot

5 × 105 m

)−1

(A.39)

×(

∆νtint

0.2 MHz103 h

)−1/2(

lmin

16

)(

lmax

7.1 × 103

)−1

. (A.40)

As seen in Figure (A.3), the different power spectra can easily be detected and

distinguished at 10−2 Mpc−1 < k < 1Mpc−1, which corresponds roughly to 100 <

l < 104. Even LOFAR, which will have Atot ' 5×104 m2 and D = 2 km, will have

a power spectrum error of ∼ 7 mK for the same bandwidth and integration time.

It is therefore likely that the fluctuations of the first sources will be detected long

before the actual sources themselves.

A.2 Minihalos and the Intergalactic Medium before Reion-

ization

At very high redshifts (z >∼ 30), gas at the mean density is sufficiently dense

for collisions to couple the spin temperature to the kinetic gas temperature. At

lower redshifts, collisions become negligible for gas at or below the cosmic mean

density, and it becomes invisible until its spin temperature is again decoupled from

the CMB by Lyα pumping due to an early UV background from the first stars and

quasars. Even though gas at the mean density is no longer collisionally coupled

205

at z < 30, the gas density within a “minihalo” – a virialized halo of dark and

baryonic matter with virial temperature T < 104K and mass 104 < M < 108M

– is sufficiently high so as to couple its gas spin temperature to the halo virial

temperature, causing it to appear in emmission with respect to the CMB. Iliev et

al. (2002) used the truncated isothermal sphere model (TIS; Iliev, Shapiro, & Raga

1999) combined with the Press-Shechter approximation for the halo mass function

to predict the fluctuating 21 cm signal from minihalos at redshifts z > 6. Iliev et al.

(2003) extended these results to include non-linear biasing effects and compared

their analytical predictions to the results of N-body simulations. These authors

concluded that the fluctuations in intensity accross the sky created by minihalos

were likely to be observable by the next generation of radio telescopes. Such

observations could confirm the basic CDM paradigm and constrain the shape and

amplitude of the power spectrum at much smaller scales than previously possible.

Recently, Furlanetto & Loeb (2004) suggested that the emmission signal originating

in shocked, overdense gas that is not inside of minihalos is probably much larger

than that from minihalos alone as calculated by Iliev et al. (2002). Their conclusion

is based on an extension of the Press-Schechter approximation that is used to

determine the fraction of the intergalactic medium (IGM) that is hot and dense

enough to be produce a 21 cm emmission signal.

In this section, we present some selected results from Shapiro et. al. (2006),

in which more details can be found. We predict the 21 cm signal at z > 6 due

to collisional coupling, with the implicit assumption that the UV background is

not strong enough to make Lyα pumping important. Because the Lyα pumping

efficiency is expected to fluctuate strongly until enough sources form to make

the efficiency uniform (e.g. Barkana & Loeb 2004), these results are relevant to

206

different regions of the universe at different redshifts, depending upon location and

abundance of the first sources of UV radiation. Within these regions, we focus on

properly resolving the gasdynamics of structure formation at small scales through

the use of high resolution gasdynamic and N-body simulations. By testing the semi-

analytical prediction of the halo model of Iliev et al. (2002) for the contribution

to the mean signal from gas in minihalos, we will investigate the extent to which

IGM gas may be a non-negligible contribution to the total fluctuating signal, as

suggested by Furlanetto & Loeb (2004).

A.2.1 Numerical Simulations

We have run series of cosmological N-body and gasdynamic simulations

to derive the effect of gravitational collapse and the hydrodynamics on the pre-

dicted 21 cm signal from high redshift. Our computational box has a comoving

size of 0.5h−1 Mpc, which is optimal for adequately resolving both the minihalos

and the small-scale structure-formation shocks. We used the code described in

Ryu et al. (1993), which uses the particle-mesh (PM) scheme for calculating the

gravity evolution and an Eulerian total variation diminishing (TVD) scheme for

hydrodynamics. We generated our initial conditions for the gas and dark matter

distributions using the publicly available software COSMICS (Ma & Bertschinger

1995). The N-body/hydro code uses an N 3 grid and (N/2)3 dark matter particles.

All the results presented here were for a simulation with grid size N = 1024.

In addition to the total 21-cm signal from our simulations, δT b, IGM, we

are also interested in the relative contribution of the virialized minihalos and the

IGM to the total signal, the sum of which gives the total 21-cm signal, δT b, tot =

δT b, halo + δT b, IGM. First, we calculate the total mean signal as a simple average

207

over the simulation cells, δT b, tot ≡∑

i δTb, i/N3. The minihalo contribution is given

by δT b, halo ≡ ∑

i fiδTb, i/N3, where fi is the fraction of the DM mass in a cell i

which is part of a halo. The IGM contribution can then be obtained as

δT b, IGM = δT b, tot − δT b, halo =∑

i

(1 − fi)δTb, i/N3. (A.41)

In order to calculate the minihalo contribution to the total differential

brightness temperature, δT b, halo, one needs to first identify the halos in the simu-

lation volume. We identified the halos using a friends-of-friends (FOF) algorithm

(Davis et al. 1985) with a linking length parameter of b = 0.25. The FOF algo-

rithm applies to the dark matter N-body particles, rather than the gas in grid cells.

Once this halo catalogue is processed for each time-slice of our N-body results, the

baryonic component of each halo is identified for the grid cells of the hydrody-

namics simulation which are contained within the volume of the halos in our FOF

catalogue. We do this as follows. First, the density in each cell contributed by each

DM particle is determined by the triangular-shaped cloud assignment scheme. For

each cell in which mass is contributed by the DM particles of a given halo, the

gaseous baryonic component in that cell is assumed to contribute a fraction fi of

its mass given by the fraction of the total DM mass in that cell which is attributed

to the halo DM particles. Accordingly, each cell i contributes an amount fiδTb,i

to the signal attributed to halo gas, while (1− fi)δTb,i is assumed to be the signal

from the IGM outside of the halo, where δTb,i is calculated from the cell as a whole.

A.2.2 Results

In Figure A.4 we show (unfiltered) maps of the differential brightness tem-

perature obtained directly from our numerical simulation. We show the total

signal, as well as the separate contributions from minihalos and IGM at redshifts

208

Figure A.4 Map of the differential brightness temperature, δTb, (projected onto onesurface of the box) for the redshifted 21-cm signal obtained from the simulation.Rows, top to bottom, show redshifts z=30, 20, and 10. Columns, left to right,represent contributions from minihalos, the IGM and the total signal. Note thatthe scale is linear in δTb for the upper two rows of images, but logarithmic for thebottom row.

209

Figure A.5 Evolution of mean differential brightness temperature, δT b, of 21-cmbackground. (a)(left) Evolution of the total 21-cm signal vs. redshift. All datapoints are directly calculated from our simulation box, with the assumption thatoptical depth is negligible throughout the box. (b)(right) δT b vs. redshift belowz = 20. The contributions from minihalos (circles), the IGM (triangles), andthe total (squares) are plotted, as labelled. For comparison, the result for theunperturbed IGM is also plotted (dashed-dot curves).

210

z = 30, 20, and 10. At z = 30, the earliest redshift shown (top row), most of

the diffuse IGM gas is still in the quasi-linear regime and cold, thus largely in

absorption against the CMB. At redshift z = 20 (middle row), the diffuse gas is

still largely in absorption, while the (relatively few) halos that have already col-

lapsed are strongly in emission. The combination of the two contributions creates

a complex, patchy emission/absorption map, and absorption and emission par-

tially cancel each other in the total mean signal. Finally, at z = 10 (bottom row),

including the diffuse component, gas heated above TCMB is widespread leading to

a net emission against the CMB. The bulk of this 21-cm emission comes from the

high-density knots and filaments. Although both the halo and IGM contributions

come from roughly the same regions, the minihalo emission is significantly more

clustered, while the IGM emission is quite diffuse.

In Figure A.5, we quantify the relative contributions of the minihalos and

diffuse IGM to the total mean 21-cm signal averaged over the whole computational

box and their evolution. The total signal is deep in absorption, with δTb < −10

mK at z > 37. The 21-cm signal is completely dominated by the IGM contribu-

tion at this stage. The absorption signal follows the expected evolution for the

unperturbed universe well, since the density fluctuations are still small and the

uniform-density assumption is reasonably accurate. The absorption continually

decreases as significant nonlinear structures start forming and portions of the gas

became heated due to this structure formation. The net signal goes into emission

after redshift z ∼ 20, reaching up to ∼ 5 mK by z ≈ 8. The emission signal at

z < 18 is due to both collapsed halos and the clumpy, hot IGM gas. In terms

of their relative contributions, the minihalos dominate over the diffuse IGM at

all times when the overall signal is in emission, below z = 18. We find that the

211

relative contributions to the total signal,∣

∣

∣δT b, j

∣

∣

∣ /(∣

∣

∣δT b, halo

∣

∣

∣+∣

∣

∣δT b, IGM

∣

∣

∣

)

where

j means either “halo” or “IGM,” is nearly constant over two different redshift

regimes: for z > 20,∣

∣

∣δT b, IGM

∣

∣

∣ /(∣

∣

∣δT b,halo

∣

∣

∣+∣

∣

∣δT b, IGM

∣

∣

∣

)

≈ 1, while for z < 16,∣

∣

∣δT b,halo

∣

∣

∣ /(∣

∣

∣δT b, halo

∣

∣

∣+∣

∣

∣δT b, IGM

∣

∣

∣

)

≈ 0.7. In the transition region, 16<∼ z <∼ 20

the relative contributions exhibit more complex behavior, approximately canceling

each other out, resulting in a total signal which is close to zero.

A.2.3 Conclusions

We have run a set of cosmological N-body and hydrodynamic simulations

of the evolution of dark matter and baryonic gas at high redshift (6 < z < 100).

With the assumption that radiative feedback effects from the first light sources are

negligible, we calculated the mean differential brightness temperature of the red-

shifted 21-cm background at each redshift. The mean global signal is in absorption

against the CMB above z ∼ 20 and in overall emission below z ∼ 18. At z > 20,

the density fluctuations of the IGM gas are largely linear, and their absorption

signal is well approximated by the one that results from assuming uniform gas at

the mean adiabatically-cooled IGM temperature. At z < 20, nonlinear structures

become common, both minihalos and clumpy, hot, mildly nonlinear IGM, resulting

in an overall emission at 21-cm with differential brightness temperature of order a

few mK.

By identifying the halos in our simulations, we were able to separate and

compare the relative contributions of the halos and the IGM gas to the total

signal. We find that the emission from minihalos dominates over that from the

IGM outside minihalos, for z <∼ 20. In particular, the emission from minihalos

contributes about 70 − 75% of the total emission signal at z < 17, peaking at

212

100% at z ≈ 18, and balancing the absorption by the IGM gas at z ≈ 20. In

contrast, the absorption by cold IGM gas dominates the total signal for z > 20.

These results appear to contradict the suggestion by Furlanetto & Loeb

(2004), that the 21-cm emission signal would be dominated by the contribution

of shocked gas in the diffuse IGM. They used the Press-Schechter formalism to

estimate the fraction of the IGM outside of minihalos, which is shock-heated,

by adopting a spherical infall model for the growth of density fluctuations and

assuming that all gas inside the turn-around radius is shock-heated. This method

is apparently not accurate enough to describe the filamentary nature of structure

formation in the IGM.

On the other hand, our results are consistent with the analytical estimates

of the mean 21-cm emission signal from minihalos by Iliev et al. (2002). This indi-

cates that the statistical prediction of the collapsed and virialized regions identified

as minihalos by the Press-Schechter formalism, with virial temperatures T < 104K,

with halos characterized individually by the TIS model, is a reasonably good ap-

proximation for the mean 21-cm signal for minihalos at all redshifts and a good

estimator even for the total mean signal including both minihalos and the diffuse

IGM at z <∼ 20. This encourages us to believe that the angular and spectral fluc-

tuations in the 21-cm background predicted by Iliev et al. (2002) based on that

model will also be borne out by future simulations involving a much larger volume

than was simulated here. The current simulation volume is too small to be used to

calculate the fluctuations in the 21-cm background because current plans for radio

surveys to measure this background involve beams which will sample much larger

angular scales (> arcminutes) than are subtended by our current box. A larger

simulation volume than ours will also be necessary to sample this minihalo bias in a

213

statistically meaningful way. This bias is likely to affect the minihalo contribution

to the 21-cm background fluctuations substantially more than it does the diffuse

IGM contribution, thereby boosting the relative importance of minihalos over the

IGM even above the ratio of their contributions to the mean signal.

214

Appendix B

Relativistic Ionization Fronts

From the very first stars at z = 20 − 30 to the brightest quasars at z = 6,

ionization fronts (I-fronts) play a central role in the story of reionization. Since

the pioneering work of Shapiro & Giroux (1987), it has become customary to

approximate the growth of H II regions during reionization by treating the IGM

as a two-phase medium – ionized on the inside of I-fronts and neutral on the

outside. In calculating the time-dependent progress of these I-fronts, the “I-front

jump conditions” are used, in which the flux of neutral atoms crossing the front is

balanced by the flux of ionizing photons at the front, leading to an I-front velocity

v. In most cases, the speed of the front is much less than the speed of light, and the

finite speed of light need not be taken into account. In some situations, however,

this is not the case. Because of recent detection of a Gunn-Peterson trough in the

spectra of high-redshift quasars, much attention has been focused on the I-fronts

that may have surrounded these quasars. In this case, where the quasar is quite

bright and short-lived, the finite travel time of light becomes important.

In this appendix, we revisit the original equations of Shapiro & Giroux

(1987) and extend them to take account of the finite speed of light. In addition,

we also treat relativistic I-fronts in several other situations in which their speed

can approach the speed of light, such as a plane stratified medium and a centrally

concentrated halo. All of this work appears in Shapiro et al. (2006b), to which

the reader can refer for additional details.

215

B.1 Uniform Static Medium

In deriving the position of the I-front w.r.t. time, we should ensure that at

time t the number of photons that have been emmitted prior to the photon which

is just reaching the ionization front at position R(t) is equal to the sum of the total

number of atoms within R(t) and the total number of recombinations that have

taken place,

∫ tR

0N (t′)dt′ = 4π

∫ R(t)

0n(r)r2dr + 4πα

∫ t

0dt′∫ R(t′)

0n2(r)r2dr, (B.1)

where tR ≡ t − R(t)/c is the retarded time at R(t), N(t) is the ionizing photon

luminosity of the source, n(r) is the hydrogen number density as a function of

radius, and α is the recombination rate coefficient. Differentiating this equation

w.r.t. t, we obtain

4π

(

Rn(R)R2 + α∫ R

0n2(r)r2dr

)

= N(tR)

(

1 − R

c

)

. (B.2)

Solving for R implies

R = cN (tR) − 4πα

∫R0 n2(r)r2dr

4πR2cn(R) + N (tR). (B.3)

In the limit of c→ ∞, we recover the non-relativistic result,

R =N (t)− 4πα

∫R0 n2(r)r2dr

4πR2n(R), (B.4)

so that the velocity is given by the recombination-corrected flux divided by the

density. Let us make the simplification that the density is uniform and that the

source luminosity is constant. In this case, equation (B.3) simplifies to

R = cN − 4παn2R3/3

4πR2cn+ N. (B.5)

216

Defining the Stromgren radius RS as

N =4π

3αn2R3

S, (B.6)

we can then define the nondimensional radius y and time x by

y ≡ R/RS (B.7)

and

x ≡ t/trec = αnt, (B.8)

where trec is the recombination time. The nondimensional form of (B.5) is thus

given by (after some algebra)

dy

dx=

(1 − y3)

q + 3y2, (B.9)

where

q ≡ RS

ctrec. (B.10)

This equation reduces to the non-relativistic one when q → 0,

dy

dx=

1 − y3

3y2, (B.11)

which has the standard solution

y = (1 − e−x)1/3. (B.12)

The solution of the relativistically correct equation (B.9) is

x =∫ y

0duq + 3u2

1 − u3, (B.13)

which has the implicit solution

x =q

6

[

ln

(

y2 + y + 1

y2 − 2y + 1

)

+ 2√

3tan−1

(

2y + 1√3

)

− π√3

]

− ln(1 − y3). (B.14)

Plotted in Figure 6.2 is y(x) for different values of q.

217

Figure B.1 Relativistic I-front for a steady source in a static, uniform gas: (a)(top) radius (in units of Stromgren radius rS) and (b) (bottom) velocity (in unitsof rS/trec). Curves are labeled by values of the dimensionless light-crossing timeof the Stromgren radius, q ≡ rS/(ctrec), with q = 0 (i.e. nonrelativistic limit) andq = 0.01, 0.1, 1, 10, and 100. In these dimensionless units, the speed of light is q−1.

B.2 Static medium with a power-law profile

Consider a source that switches on in the center of a spherically-symmetric

density profile n(r). The case in which the density decreases with increasing radius

is relevant whenever the source, like a massive star, is forming by gravitational

instability, in the middle of a centrally-concentrated gas profile. We shall assume

that the source luminosity is time-independent and that the clumping factor is

constant in space and time. The nonrelativistic problem of an H II region in a

spherically-symmetric density distribution that varies as a power of radius outside

of a flat-density core was discussed by Franco et al. (1990). Let the H atom number

218

density in the undisturbed gas be defined by

nH(r) =

n0(r/r0)−γ , r > r0,

n0, r ≤ r0.(B.15)

As long as the I-front is inside the core, its propagation follows the solution for

a uniform-density gas derived above. In this phase the I-front continually slows

down from vI ≈ c at small radii rI . If the core Stromgren radius rS,0 ≤ r0, where

rS,0 ≡ [3Nγ/(4πCαBn20)]

1/3 ≤ r0, then the I-front will slow down to zero velocity

just as it fills this Stromgren sphere, thereby remaining trapped within the core.

If rS,0 > r0, instead, then the I-front continues to expand beyond the core radius.

We shall assume from now on that rS,0 > r0.

We nondimensionalize the resulting velocity equation by expressing all radii

in units of rS,0 and time in units of trec,0 ≡ (αBn0C)−1, the recombination time in

the core

Y ≡ r

rS,0, (B.16)

so YI ≡ rI/rS,0, Y0 ≡ r0/rS,0, and YS ≡ rS/rS,0, while

w ≡ t

trec,0. (B.17)

We also define the dimensionless ratio of the light crossing and the recombination

time in the core as

q ≡ rS,0

ctrec,0. (B.18)

The solutions for the I-front radius and velocity for each value of the density profile

slope γ are then fully characterized by the two dimensionless parameters q and Y0,

as follows:

dYI

dw=

1−( 2γ2γ−3)Y 3

0 +( 3

2γ−3)Y 2γ0

Y 3−2γI

3Y γ0

Y 2−γI +q[1−( 2γ

2γ−3)Y 30

+( 3

2γ−3)Y 2γ0

Y 3−2γI ]

, for γ 6= 3/2,

Y −3

0−1−3 ln(YI/Y0)

(

3

Y0

)(

YIY0

)1/2

+q[Y −3

0−1−3 ln(YI/Y0)]

, for γ = 3/2.(B.19)

219

Figure B.2 Relativistic I-front for a steady source in a static gas with a power-lawdensity profile, nH ∝ r−γ , and a constant density core for r ≤ r0 (left panels forγ = 2, right panels for γ = 2.5): (a) (top) radius (in units of core Stromgren radiusrS,0) and (b) (bottom) velocity (in units of rS,0/trec,0). Curves are labeled by valuesof the dimensionless light-crossing time of the core Stromgren radius, expressed inunits of the core recombination time, q ≡ rS,0/(ctrec,0). In these units, the speed oflight is q−1. The case q = 0 corresponds to the nonrelativistic solution. All curvesassume Y0 = 4−1/3.

For comparison, the speed of light in these dimensionless units is q−1. The I-front

leaves the core with initial velocity

dYI

dw

∣

∣

∣

∣

∣

YI=Y0

=1 − Y 3

0

3Y 20 + q(1 − Y 3

0 ), (B.20)

which is relativistic (roughly) if q >∼ (3Y 20 )/(1 − Y 3

0 ) = [vI,NR(r0)/(rS,0/trec,0)]−1,

where vI,NR(r0) is the nonrelativistic I-front solution speed when rI = r0 (i.e.

take q=0 in eq. (B.20) above). Hence, for any given r0 and n0, a sufficiently high

luminosity Nγ is required to make the I-front velocity still relativistic once the front

220

reaches rI = r0. It is possible for the I-front to accelerate afterwards, however,

depending upon the values of γ and Y0, so even if the front is not relativistic when

it leaves the core, it may become relativistic at larger radii.

Franco et al. (1990) derived some of the properties of the R-type I-front

phase for the nonrelativistic solution of this problem. We can use this nonrela-

tivistic solution directly to derive additional properties of the relativistic solution.

The nonrelativistic I-front has a velocity vI,NR which depends upon its radius rI as

follows:

vI,NR(rI) =vI,NR(r0)

Y −30 − 1

u(γ), (B.21)

where u(γ) is given by

u(γ) =

(YI/Y0)γ−2

[

Y −30 − 2γ

2γ−3+ 3

2γ−3(YI/Y0)

3−2γ]

, for γ 6= 3/2,

(YI/Y0)1/2[

Y −30 − 1 − 3 ln YI/Y0

]

, for γ = 3/2.(B.22)

Equations (B.21) and (B.22) then yield the correct relativistic velocity vI(rI) for

the I-front.

For any I-front that expands beyond the radius of the core, the relativistic

I-front will only expand until it reaches the same Stromgren sphere radius rS as in

the nonrelativistic solution, given by

rS(γ)

rS,0

=

[

3−2γ3

+ 2γ3Y 3

0

] 13−2γ Y

2γ2γ−3

0 , for γ 6= 3/2,

Y0 exp

13

[

Y −30 − 1

]

, for γ = 3/2(B.23)

(Franco et al. 1990). Finally, there is a (flux-dependent) critical value of the loga-

rithmic slope of the density profile, −γf , below which there is no finite Stromgren

radius rS at which the I-front is trapped, given by

γf =3

2

[

1 − Y 30

]−1(B.24)

221

Figure B.3 Cosmological I-front radius versus time for steady source in the meanIGM, with comoving radius in units of the time-varying, comoving Stromgrenradius, rS(t) = a(t)rS(ti), and time in units of the age of universe ti at source turn-on. Different curves in each panel correspond to different values of q = rS,i/(ctrec,i),with q = 0 (i.e. nonrelativistic limit, top line) and (from top to bottom) q = 0.1, 0.2and 0.5. Each panel is for different values of λ ≡ ti/(χefftrec,i), as labeled. We haveassumed that χeff = 1 for simplicity.

(Franco et al. 1990). For density profiles that decline more steeply than r−γf , the

relativistic I-front expands without bound, just as it does in the nonrelativistic

solution.

The relativistic and nonrelativistic I-front propagation solutions for power-

law density profiles are plotted in Figure B.2 for the illustrative cases of γ = 2

and 2.5, for the particular value of Y0 = 4−1/3 for which the critical slope γf = 2.

In that case, for γ = 2, the I-front is never trapped at a finite Stromgren radius,

but it decelerates continuously and reaches zero velocity at infinite radius. Such

222

Figure B.4 Same as Figure B.3 but for the I-front peculiar velocity, instead, wherevI,pec is in units of (rS,i/ti). The speed of light, c, corresponds in these units to(χeffλ/q). We assume χeff = 1 for simplicity.

I-fronts are therefore relativistic only at early times. For γ = 2.5 > γf , on the

other hand, the I-front reaches a minimum velocity and thereafter accelerates, so

it is relativistic both at early and late times.

B.3 Cosmologically expanding medium

In a uniform expanding universe described by a scale factor a ≡ 1 at t = ti

(when the constant-luminosity source turns on), the conservation of photons with

all recombinations counted as absorptions implies

N(

t− R

c

)

= V nia−3 + αn2

i

∫ t

0V (t′)a−6(t′)dt′, (B.25)

223

where V (t) ≡ 4πR3(t)/3 is the volume within the spherical ionization front, and N

is the ionizing photon luminosity. Let us define the comoving volume VC ≡ V/a3

and differentiate equation (B.25) w.r.t. time,

N

(

1 − R

c

)

=dVC

dtni + αn2

i VCa−3. (B.26)

As in Shapiro & Giroux (1987), we define the dimensionless volume according to

y ≡ VC/VS,i and dimensionless time x ≡ t/ti, where the initial Stromgren volume

is defined according to

N = VS,iαn2i . (B.27)

Substituting these relations in equation (B.26), we obtain

dy

dx= λ(1 − R/c − y/a3), (B.28)

where λ ≡ αniti is the ratio of recombination time to the age of the universe

when the source turned on. Since R = aRS,iy1/3, we can express R/c in terms of

dimensionless quantities,

R

c=qa

3λ

(

y−2/3dy

dx+ 2Hy1/3

)

, (B.29)

where q ≡ RS,iαni/c is the initial Stromgren radius light crossing time in units

of the initial recombination time, and H ≡ 3H(t)ti/2 is the dimensionless Hubble

parameter. Combining equations (B.28) and (B.29), we obtain

dy

dx= 3λ

1 − y/a3 + 2qaHy1/3/(3λ)

3 + qay−2/3. (B.30)

Since q/λ = RS,i/(cti) is the initial Stromgren radius divided by (roughly) the

horizon size at turn-on, we can take this ratio to be small and thus drop the third

term in the numerator,

dy

dx= 3λ

1 − y/a3

3 + qay−2/3. (B.31)

224

For the matter-dominated era (a = x2/3),

dy

dx= 3λ

1 − y/x2

3 + q(x/y)2/3. (B.32)

Shown in Figures B.3 and B.4 are profiles of the I-front velocity and position vs.

time.

B.4 Plane-stratified Medium

Given that sources tend to form in regions in which gravitational instability

leads to collapse, the medium may not only be centrally concentrated but may also

depart from spherical symmetry. Gravitational collapse tends to result in flattened

structures, either by a “pancake” instability or by the formation of a rotationally-

supported disk. It is therefore instructive to consider the I-front for a point source

in a static, plane-stratified medium, therefore. This problem was also considered

in the nonrelativistic limit by Franco et al. (1990).

The density profile is given by

n(z) = n0sech2(z/z0), (B.33)

where n0 is the central density, z is the height above the central plane at z = 0,

and z0 is the scale height. Once again, we define the dimensionless parameters

Y0 ≡z0

rS,0

(B.34)

and

q ≡ rS,0

ctrec,0, (B.35)

where rS,0 and trec,0 are the central Stromgren radius and recombination time,

respectively. Application of the relativistic I-front jump conditions to the stratified

225

Figure B.5 Two-dimensional, axisymmetric I-front surfaces (xI/rS,0, zI/rS,0) (inunits of the Stromgren radius at the central density) for a steady source in astatic, plane-stratified medium at different times after turn-on, t/trec,0 (where trec,0is recombination time at the origin), as labelled, for q = 0 (i.e. nonrelativisticsolution) (left panels) and q = 1 (relativistic) (right panels). Both curves assumeY0 = 1/2.

density profile of equation (B.33) leads to the following angle-dependent differential

equation for the evolution of the I-front radius,

dYI

dw=

1 − f(YI , θ)

3Y 2I g(YI , θ) + q(1 − f(YI , θ))

, (B.36)

where YI(θ) ≡ rI(θ)/rS,0, w ≡ t/trec,0,

f(Y, θ) =0.66Y 3

0

sin3(θ)tanh3

(

Y sin(θ)

0.88Y0

)

, (B.37)

and

g(Y, θ) = sech2 [Y/Y0 sin(θ)] . (B.38)

226

Figure B.6 Relativistic I-front radius (upper panel) (in units of Stromgren radiusrS,0 at central density) and velocity (lower panel) (in units of rS,0/trec,0) along thesymmetry axis (z-axis) versus time (in units of central recombination time trec,0)for same plane-stratified case as shown in Figure B.5. Each curve is for a differentvalue of q = rS,0/(ctrec,0) (as labelled in lower panel), with q = 0 corresponding tothe nonrelativistic solution (solid). In these units, the speed of light is q−1. Verticaldotted line marks the finite time t∞ at which the NR solution (q=0) reaches infiniteradius: zI(t∞) = ∞. All curves assume Y0 = 1/2.

In deriving equations (B.36)-(B.38), we have measured the polar angle θ of the

direction from the central source to a point on the I-front with respect to the

z = 0 plane, so θ = π/2 corresponds to the symmetry axis (i.e. the z-axis). We

have also followed Franco et al. (1990) in evaluating the recombination integral

along each direction by approximating the integral using

∫ p

0p2sech4(p)dp ' 0.22tanh3

(

p

0.88

)

, (B.39)

where p ≡ (r/z0) sin θ.

227

Illustrative solutions of equations (B.36)-(B.38) for relativistic and nonrela-

tivistic I-fronts in a plane-stratified medium are plotted in Figures B.5 and B.6. We

adopt the value Y0 = 1/2 in all cases. Figure B.5 shows the two-dimensional, ax-

isymmetric I-front surfaces at different times for the nonrelativistic solution (q = 0)

and for the relativistic solution for q = 1. The NR I-front solution starts out at

superluminal speeds and decelerates in all directions. Along the symmetry axis,

the NR I-front eventually reaches a minimum velocity and, thereafter, accelerates

upward. This acceleration leads to a superluminal “blow out” in which the NR

I-front reaches infinite height in a finite time t∞. In the perpendicular direction,

however, along the plane of symmetry at z = 0, the NR I-front decelerates con-

tinuously and approaches the Stromgren radius of a uniform sphere of the same

density. The relativistic I-front also starts out decelerating but remains close to

the speed of light in all directions, so its shape is initially quite spherical. Like

the NR I-front, the relativistic I-front also reaches a minimum speed along the

symmetry axis, before it accelerates upward once again and approaches the speed

of light. Since its speed is always finite, however, the relativistic I-front cannot

“blow-out” as the NR front does in a finite time. Since, in the central plane, the

relativistic I-front slows to approach the same Stromgren radius as does the NR

solution, above this plane it must balloon upward and outward, confined at the

waist by this “Stromgren belt”.

228

Bibliography

Abel, T., Bryan, G. L., & Norman, M. L. 2002, Sci, 295, 93

Abel T., Norman M. L., Madau P., 1999, ApJ, 523, 66

Abel, T., Wise, J. H., & Bryan, G. L. 2006, ApJ, submitted, astro-ph/0606019

Ahn, K. & Shapiro, P.R. 2002, MNRAS, 363, 1092

Ahn, K. & Shapiro, P.R. 2006, ApJ, submitted, astro-ph/0607642

Alvarez, M. A., Bromm, V., & Shapiro, P. R. 2006a, ApJ, 639, 621

Alvarez, M. A., Komatsu, E., Dore, O., & Shapiro, P. R. 2006b, ApJ, 647, 840

Alvarez, M. A., Shapiro, P. R., Ahn, K., & Iliev I. T. 2006c, ApJ, 644, L101

Alvarez, M.A., Shapiro, P.R., & Martel, H., 2003, RevMexAA SC, 17, 39

Arad, I., Dekel, A., & Klypin, A., 2004, MNRAS, 353, 15

Ascasibar, Y., Yepes, G., Gottlober, S., & Muller, V., 2004, MNRAS, 352, 1109

Avila-Reese, V., Firmani, C., Hernandez, X., 1998, ApJ, 505, 37

Avila-Reese, V., Firmani, C., Klypin, A., & Kravtsov, V. 1999, MNRAS, 310, 527

Avila-Reese, V., Colin, P., Valenzuela, O., D’Onghia E., Firmani C., 2001, ApJ,

559, 516

Barkana, R., & Loeb, A. 2001, Phys. Rep., 349, 125

229

Barkana, R., & Loeb, A. 2004, ApJ, 609, 474

Barkana, R., & Loeb, A. 2005, ApJ, 624, L65

Becker, R. H., et al. 2001, AJ, 122, 2850

Bennett, C. L, et al. 2003, ApJS, 148, 1

Bertschinger, E. 1985, ApJS, 58, 39

Binney, J., Tremaine, S., 1994, Galactic Dynamics. (Princeton, Princeton Univ.

Press)

Bharadwaj, S. & Ali, S. S. 2004, MNRAS, 352, 142

Blumenthal, G. R., Pagels, H., & Primack, J. R. 1982, Nature, 299, 37

Bode, P., Ostriker, J.P., Turok, N., 2001, ApJ, 556, 93

Bolton, J., Meiksin, A., White, M., 2004, MNRAS, 348, L43

Bond, J. R., Arnett, W. D., & Carr, B J. 1984, ApJ, 280, 825

Bondi, H., & Hoyle, F. 1944, MNRAS, 104, 273

Bromm, V., Coppi, P. S., & Larson, R. B. 1999, ApJ, 527, L5

Bromm, V., Coppi, P. S., & Larson, R. B. 2002, ApJ, 564, 23

Bromm, V., Kudritzki, R. P., & Loeb, A. 2001, ApJ, 552, 464

Bromm, V., & Larson, R. B. 2004, ARA&A, 42, 79

Bromm, V., & Loeb, A. 2003, Nature, 425, 812

230

Bromm, V., & Loeb, A. 2004, NewA, 9, 353

Bromm, V., Yoshida, N., & Hernquist, L. 2003, ApJ, 596, L135

Bullock, J.S., Kolatt, T.S., Sigad, Y., Somerville, R.S., Kravtsov, A.V., Klypin,

A.A., Primack, J.R., & Dekel, A., 2001, MNRAS, 321, 559

Carilli, C. L., Gnedin, N. Y., & Owen, F. 2002, ApJ, 577, 22

Cen, R., 2002, ApJS, 141, 211

Cen, R. 2003, ApJ, 591, 12

Cen, R. 2006, ApJ, submitted, astro-ph/0601010

Chen, X., & Miralda-Escude, J. 2004, ApJ, 602, 1

Chieze, J., Teyssier, R., Alimi, J., 1997, ApJ, 484, 40

Chiu, W. A., & Ostriker, J. P. 2000, ApJ, 534, 507

Choudhury, T. R., & Ferrara, A. 2005, MNRAS, 361, 577

Choudhury, T. R., & Ferrara, A. 2006, MNRAS, submitted, astro-ph/0603617

Chuzhoy, L., Alvarez, M. A., & Shapiro, P. R. 2006, ApJ, submitted, astro-

ph/0605511

Chuzhoy, L. & Shapiro, P. R. 2006, ApJ, submitted, astro-ph/0512206

Ciardi, B. & Ferrara, A. 2005, Space Science Reviews, 116, 625

Ciardi, B., Ferrara, A., & White, S. D. M. 2003, MNRAS, 344, L7

Ciardi, B., Ferrara, A., Governato, F., Jenkins, A., 2000, MNRAS, 314, 611

231

Ciardi, B. & Madau, P. 2003, ApJ, 596, 1

Colin, P., Avila-Reese, V., Valenzuela, O., 2000, ApJ, 542, 622

Colin, P., Klypin, A.A., Kravtsov, A.V. 2000, ApJ, 539, 561

Cooray, A. 2004, Phys. Rev. D, 70, 063509

Cooray, A. & Yoshida, N. 2004, MNRAS, 351, L71

Cooray, A., Bock, J. J., Keatin, B., Lange, A. E., & Matsumoto, T. 2004, ApJ,

606, 611

Couchman, H. M. P., & Rees, M. J. 1986, MNRAS, 221, 53

Davis, M., Efstathiou, G., Frenk, C. S., & White, S. D. M. 1985, ApJ, 292, 371

de Blok, W. J. G., & Bosma, A. 2002, A&A, 385, 816

de Blok, W. J. G., McGaugh, S. S., Bosma, A., Rubin, V. C., 2001, ApJ, 552, L23

Dieman, J., Zemp, M., Moore, B., Stadel, J., & Carollo, C. 2005, MNRAS, 364, 2

Dekel, A., Arad, I., Devor J., & Birnboim, Y. 2003, 588, 680

Eisenstein, D. J., & Hu, W. 1999, ApJ, 511, 5

Eke, V.R., Navarro, J.F., Frenk, C.S., 1998, ApJ, 503, 569

El-Zant, A., Shlosman, I., Hoffman, Y. 2001, ApJ, 560, 636

Fan, X., et al. 2002, AJ, 123, 1247

Fernandez, E. R., & Komatsu, E. 2005, ApJ, submitted (astro-ph/0508174)

232

Ferrara, A. 1998, ApJ, 499, L17

Field, G. B. 1959, ApJ, 129, 536

Fillmore, J.A. & Goldreich, P., 1984, ApJ, 281, 1

Flores R. A., Primack J. R., 1994, ApJ, 427, L1

Franco, J., Tenorio-Tagle, G., & Bodenheimer, P. 1990, ApJ, 349, 126

Fukushige, T., Makino, J., 2001, ApJ, 557, 533

Furlanetto, S. R. 2006, Physics Reports, astro-ph/0608032

Furlanetto, S. R., Hernquist, L., & Zaldarriaga, M. 2004, MNRAS, 354, 695

Furlanetto, S. R. & Loeb, A. 2004, ApJ, 611, 642

Furlanetto, S. R. & Loeb, A. 2005, ApJ, 634, 1

Furlanetto, S. R., Oh, S. P., 2005, MNRAS, 363, 1031

Furlanetto, S. R. & Pritchard, J. R. 2006, MNRAS, submitted, astro-ph/0605680

Furlanetto, S. R., Sokasian, A., & Hernquist, L. 2004, ApJ, 347, 187

Furlanetto, S. R., Zaldarriaga, M., & Hernquist, L. 2004b, ApJ, 613, 16

Furlanetto, S. R., Zaldarriaga, M., Hernquist, L., 2004a, ApJ, 613, 1

Ghigna, S., Moore, B., Governato, F., Lake, G., Quinn, T., Stadel, J., 2000, ApJ,

544, 616

Gleser, L., Nusser, A., Ciardi, B., & Desjacques, V. 2006, MNRAS, in press, astro-

ph/0602616

233

Gnedin, N. Y., Abel, T., 2001, New Astronomy, 6, 437

Gnedin, N. Y., & Ostriker, J. P. 1997, ApJ, 486, 581

Gnedin, N. Y., & Shaver, P. A. 2004, ApJ, 608, 611

Gorski, K. M., Hivon E., Banday A. J., Wandelt, B. D., Hansen, F. K., Reinecke,

M., & Bartelmann, M. 2005 ApJ, 622, 759

Gott, J. R., Dickinson, M., & Melott, A. L. 1986, ApJ, 306, 341

Gunn, J.E., & Gott, J.R. 1972, ApJ, 176, 1

Gunn, J. E., & Peterson, B. A. 1965, ApJ, 142, 1633

Haiman, Z. 2002, ApJ, 576, 1

Haiman, Z., Abel, T., & Rees, M. J. 2000, ApJ, 534, 11

Haiman, Z., & Bryan, G. L. 2006, ApJ, submitted, astro-ph/0603541

Haiman, Z., Rees, M. J., & Loeb, A. 1997, ApJ, 476, 458

Haiman, Z., Holder, G. P., 2003, ApJ, 595, 1

Haiman, Z, & Cen, R. 2005, ApJ, 623, 627

Haiman, Z., Thoul, A. A., & Loeb, A. 1996, ApJ, 464, 523

Hayes J. C., Norman M. L., 2003, ApJS, 147, 197

Heger, A., Fryer, C. L., Woosley, S. E., Langer, N., & Hartmann D. H. 2003, ApJ,

591, 288

Hoffman, Y., & Shaham, J. 1985, ApJ, 297, 16

234

Hu, E. M., et al. 2002, ApJ, 568, L75

Huss, A., Jain, B., Steinmetz, M., 1999, ApJ, 517, 64

Iliev, I. T., Scannapieco E., Shapiro P. R., 2005a, ApJ, 624, 491

Iliev, I. T., Shapiro P. R., Raga A. C., 2005b, MNRAS, 361, 405

Iliev, I. T., Mellema, G., Pen, U.-L., Merz, H., Shapiro, P. R., Alvarez, M. A.,

2006a, MNRAS, 369, 1625

Iliev, I. T., Mellema, G., Shapiro, P. R., and Pen, U. 2006b, MNRAS, submitted,

astro-ph/0607642

Iliev, I. T., et al. 2006c, MNRAS, accepted, astro-ph/0603199

Iliev, I. T., Pen, U., Bond, J. R., Mellema, G., Shapiro, P. R. 2006d, MNRAS,

submitted, astro-ph/0607209

Iliev, I. T., Shapiro, P. R., Ferrara, A., & Martel, H. 2002, ApJ, 572, L123

Iliev, I. T., Scannapieco, E., Martel, H., & Shapiro, P. R. 2003, MNRAS, 341, 811

Iliev, I.T., & Shapiro, P.R., 2001, MNRAS, 325, 468

Kaiser, N. 1987, MNRAS, 227, 1

Kaplinghat, M., Chu, M., Haiman, Z., Holder, G. P., Knox, L., & Skordis, C. 2003,

ApJ, 583, 24

Kashlinsky, A., et al. 2004, ApJ, 608, 1

Kashlinsky, A., Arendt, R. G., Mather, J., Moseley, S. H. 2005, Nature, 438, 45

235

Kitayama, T., Yoshida, N., Susa, H., & Umemura, M. 2004, ApJ, 613, 631

Klypin, A., Kravtsov, A.V., Bullock, J.S., & Primack, J.R. 2001, ApJ, 554, 903

Knebe, A., Devriendt, J.E.G., Mahmood, A., Silk, J., 2002, MNRAS, 329, 813

Kogut, A., et al. 2003, ApJS, 148, 161

Kohler, K., Gnedin, N. Y., Hamilton, A. J. S., 2005a, ArXiv Astrophysics e-prints

(astro-ph/0511627)

Kohler K., Gnedin N. Y., Miralda-Escude J., Shaver P. A., 2005b, ApJ, 633, 552

Komatsu, E., et al. 2003, ApJS, 148, 119

Kuhlen, M., & Madau, P. 2005, MNRAS, 363, 1069

Lacey, C., & Cole, S., 1993, MNRAS, 262, 627L

Larson, R.B. 1970, MNRAS, 147, 323

Lim, A. J., Mellema, G., 2003, A&A, 405, 189

Lokas, E.L., Mamon, G.A., 2001, MNRAS, 321, 155

Ma, C.-P., & Bertschinger, E. 1995, ApJ, 455, 7

Mackey, J., Bromm, V., & Hernquist, L. 2003, ApJ, 586, 1

Madau, P., Meiksin, A., & Rees, M. J. 1997, ApJ, 475, 429

Madau, P., & Rees, M. J. 2001, ApJ, 551, L27

Madau, P. & Silk, J. 2005, MNRAS, 359, 37

236

Magliocchetti, M., Salvaterra, R., & Ferrara, A. 2003, MNRAS, 342, L25

Malhotra, S. & Rhoads. J. E. 2004, ApJ, 617, L5

Marchesini, D. et al. 2002, ApJ, 575, 801

Maselli, A., Ferrara, A., Ciardi, B., 2003, MNRAS, 345, 379

Matsumoto, T., et al. 2005, ApJ, 626, 31

McQuinn M., Furlanetto S. R., Hernquist L., Zahn O., Zaldarriaga M. 2005, ApJ,

630, 643

Mecke, K. R., Buchert, T., Wagner, H. 1994, A&A, 288, 697

Mellema G., Iliev I. T., Alvarez M. A., Shapiro P. R., 2006, New Astronomy, 11,

374

Mellema, G., Raga, A. C., Canto, J., Lundqvist, P., Balick, B., Steffen, W.,

Noriega-Crespo, A., 1998, A&A, 331, 335

Merz H., Pen U.-L., Trac H., 2005, New Astronomy, 10, 393

Mo, H. J. & White, S. D. M. 1996, MNRAS, 282, 1096

Mobasher, B., et al. 2005, ApJ, 635, 832

Moore, B., Governato, F., Quinn, T., Stadel, J., & Lake, G. 1998, ApJ, 499, L5

Moore, B., Ghigna, S., Governato, F., Lake, G. Quinn, T., Tozzi, P., 1999, ApJ,

524, L19

Moore, B., Quinn, T., Governato, F., Stadel, J., Lake, G., 1999, MNRAS, 310,

1147

237

Moore, B., Gelato, S., Jenkins, A., Pearce, F.R., Quillis, V., 2000, ApJ, 535, L21

Morales, M. F. & Hewitt, J. 2004, ApJ, 615, 7

Nakamoto, T., Umemura, M., Susa, H., 2001, MNRAS, 321, 593

Nakamura, F. & Umemura, M. 2001, ApJ, 548, 1

Navarro, J.F., Frenk, C.S., & White, S.D.M. 1997, ApJ, 490, 493

Nusser, A., Sheth, R.K. 1999, MNRAS, 303, 685

O’Shea, B. W., Abel, T., Whalen, D., & Norman, M. L. 2005, ApJ, 626, L5

Oh, P., & Haiman, Z. 2003, MNRAS, 346, 456

Omukai, K., & Inutsuka, S. 2002, MNRAS, 332, 59

Omukai, K., & Palla, F. 2001, ApJ, 561, L55

Omukai, K., & Palla, F. 2003, ApJ, 589, 677

Ostriker, J. P. & Vishniac, E. T. 1986, Nature, 322, 804

Owen, J.M., Villumsen, J.V., Shapiro, P.R., Martel, H., 1998, ApJS, 116, 155

Page et al. 2006, ApJ, submitted, astro-ph0603450

Panagia, N., et al. 2005, ApJ, 633, 1

Peebles, P.J.E., 2000, ApJ, 534, L127

Power, C. et al. 2003, MNRAS, 338, 14

238

Press W. H., Flannery B. P., Teukolsky S. A., Vetterling W. T., 1992, Numerical

Recipes: The Art of Scientific Computing, 2nd edn. Cambridge University Press,

Cambridge (UK) and New York

Press, W. H., Schechter, P., 1974, ApJ, 187, 425

Purcell, E. M., & Field, G. B. 1956, ApJ, 124, 542

Rasia, E., Tormen, G., & Moscardini, L., 2004, MNRAS, 351, 237

Ricotti, M. & Ostriker, J. P. 2004, MNRAS, 350, 539

Razoumov, A. O., Norman, M. L., Abel, T., Scott, D., 2002, ApJ, 572, 695

Razoumov, A. O., Scott, D., 1999, MNRAS, 309, 287

Reed, D., Gardner, J., Quinn, T., Stadel, J., Fardal, M., Lake, G., Governato, F.,

2003, MNRAS, 346, 565

Rees, M. J. 1997, in The Hubble Space Telescope and the High Redshift Universe,

N. R. Tanvir, A. Aragon-Salamanca, & J. V. Wall, eds., (World Scientific: Sin-

gapore), 115

Ricotti, M. 2003, MNRAS, 244, 1237

Ricotti, M., Gnedin, N. Y., & Shull, J. M. 2002, ApJ, 575, 33

Ricotti M. & Ostriker J. P. 2004, MNRAS, 352, 547

Richstone, D.O. & Tremaine, S. 1984, ApJ, 286, 27

Ryu, D., Ostriker, J. P., Kang, H., & Cen, R. 1993, ApJ, 414, 1

Salvaterra, R. & Ferrara, A. 2003, MNRAS, 339, 973

239

Salvaterra, R., Ciardi, B., Ferrara, A., & Baccigalupi, C. 2005, preprint (astro-

ph/0502419)

Sand, D. J., Treu, T., Smith, G. P.,& Ellis, R. S. 2004, ApJ, 604, 88

Santos M. G., Cooray A., Haiman Z., Knox L., Ma C.-P., 2003, ApJ, 598, 756

Santos, M. R. 2004, MNRAS, 349, 1137

Santos, M. R., Bromm, V., & Kamionkowski, M. 2002, MNRAS, 336, 1082

Schaerer, D. 2002, A&A, 382, 28

Schmalzing, J. & Buchert, T. 1997, ApJ, 482, 1

Scott, D. & Rees, M. J. 1990, MNRAS, 247, 510

Seljak, U., Slosar, A., & McDonald, P. 2006, astro-ph/0604335

Seljak, U., Zaldarriaga, M., 1996, ApJ, 469, 437

Shapiro, P. R., Giroux, M. L., 1987, ApJ, 321, L107

Shapiro, P. R., Giroux, M. L., & Babul, A. 1994, ApJ, 427, 5

Shapiro, P. R., Giroux, M. L., Babul, A., 1994, ApJ, 427, 25

Shapiro, P.R., Martel, H., Villumsen, J.V., Owen, J.M., 1996, ApJS, 103, 269

Shapiro, P.R., Iliev, I.T., & Raga, A.C., MNRAS, 307, 203

Shapiro, P.R., Iliev, I.T., 2000, ApJ, 542, L1

Shapiro, P. R., Iliev, I. T., & Raga, A. C. 1999, MNRAS, 307, 203

240

Shapiro, P. R., Iliev, I. T., Raga, A. C., 2004, MNRAS, 348, 753

Shapiro, P. R., Iliev, I. T., Scannapieco, E., & Alvarez, M. A. 2006a, ApJ, in press,

astro-ph/0507677

Shapiro, P. R., & Kang, H. 1987, ApJ, 318, 32

Shapiro, P. R., Ahn, K., Alvarez, M. A., Iliev, I. T., Martel, H., & Ryu, D. 2006b,

ApJ, 646, 681

Shapiro, P. R., Ahn, K., Alvarez, M. A., Iliev, I. T., & Martel, H. 2006c, EAS

publication series, 20, 5

Shaver, P. A., Windhorst, R. A., Madau, P., & de Bruyn, A. G. 1999, A&A, 345,

380

Sheth, R. K., Tormen, G., 2002, MNRAS, 329, 61

Shu, F. H., Lizano S., Galli, D., Canto, J., & Laughlin, G. 2002, ApJ, 580, 969

Sokasian, A., Abel, T., Hernquist, L. E., 2001, New Astronomy, 6, 359

Sokasian, A., Abel, T., Hernquist, L., & Springel, V. 2003, MNRAS, 344, 607

Sokasian, A., Yoshida, N., Abel, T., Hernquist, L., & Springel, V. 2004, MNRAS,

350, 47

Spergel, D. N., Steinhardt, P. J. 2000, PRL, 84, 3760

Spergel D. N., Verde L., Peiris H. V., Komatsu E., Nolta M. R., Bennett C. L.,

Halpern M., Hinshaw G., Jarosik N., Kogut A., Limon M., Meyer S. S., Page

L., Tucker G. S., Weiland J. L., Wollack E., Wright E. L., 2003, ApJS, 148, 175

241

Spergel, D. N. et al. 2006, ApJ, submitted, astro-ph/0603449

Spitzer, L. 1978, Physical Processes in the Interstellar Medium (New York: Wiley)

Springel, V., Di Matteo, T., & Hernquist, L. 2005, MNRAS, 361, 776

Springel, V., Yoshida, N., & White, S. D. M. 2001, NewA, 6, 79

Stern D., Yost S. A., Eckart M. E., Harrison F. A., Helfand D. J., Djorgovski S. G.,

Malhotra S., Rhoads J. E., 2005, ApJ, 619, 12

Subramanian, K., & Padmanabhan, T. 1993, MNRAS, 265, 101

Susa, H. 2006, PASJ, 58, 445

Susa, H. & Umemura, M. 2006, ApJ, 645, L93

Syer, D., & White, S.D.M. 1998, MNRAS, 293, 337

Tan, J. C., & McKee, C. F. 2004, ApJ, 603, 383

Taylor, J.E., Navarro, J.F., 2001, ApJ, 563, 483

Tegmark, M., Silk, J., Rees, M. J., Blanchard, A., Abel, T., & Palla, F. 1997, ApJ,

474, 1

Thomas, P.A. et al., 1998, MNRAS, 296, 1061

Thoul, A. A. & Weinberg, D.H. 1995, ApJ, 442, 480

Titley, E.R., Couchman, H.M.P., 1999, astro-ph/9911365

Tormen, G., Bouchet, F.R., White, S.D.M., 1997, MNRAS, 286, 865

Tozzi, P., Madau, P., Meiksin, A., & Rees, M. J. 2000, ApJ, 528, 597

242

Tyson, G. P., Kochanski, G. P., & dell’Antonio, I. P., ApJ, 498, L107

Valinia, A., Shapiro, P.R., Martel, H., Vishniac, E.T., 1997, ApJ, 479, 46

van den bosch, F.C., 2002, MNRAS, 331, 98

van den Bosch, F. C., Robertson, B. E., Dalcanton, J. J., de Blok, W. J. G., 2000,

ApJ, 119, 1579

van den Bosch, F.C., Swaters, R.A., 2001, MNRAS, 325, 1017

Venkatesan, A., Tumlinson, J., & Shull, J. M. 2003, ApJ, 584, 621

Viel, M., Haehnelt, M. G., & Lewis, A. 2006, MNRAS, submitted, astro-

ph/0604310

Whalen, D., Abel, T., & Norman, M. 2004, ApJ, 610, 14

Wechsler, R.H., Bullock, J.S., Primack, J.R., Kravtsov, A.V., & Dekel, A. 2002,

ApJ, 568, 52

Weinberg, M. D. & Katz, N. 2002, ApJ, 580, 627

White, M., Carlstrom, J. E., Dragovan, M., & Holzapfel, W. L. 1999, ApJ, 514, 12

Wouthuysen, S. A. 1952, AJ, 57, 31

Wyithe, J. S. B., Loeb, A., 2003, ApJ, 586, 693

Wyithe, S. & Loeb, A. 2005, ApJ, 625, 1

Yoshida, N., Abel, T., Hernquist, L., & Sugiyama, N. 2003, ApJ, 592, 645

Yoshida, N., Bromm, V., Hernquist, L. 2004, ApJ, 605, 579

243

Yu, Q., 2005, ApJ, 623, 683

Zaldarriaga, M. 1997, Phys. Rev. D, 55, 1822

Zaldarriaga, M., Furlanetto, S. R., & Hernquist, L. 2004, ApJ, 608, 622

Zhang, P., Pen, U.-L., Trac H., 2004, MNRAS, 355, 451

Zahn, O. et al. 2006, ArXiv Astrophysics e-prints (astro-ph/0604177)

Zentner, A. R. & Bullock, J. S. 2002, PRD, 66, 043033

244

Vita

Marcelo Alonso Alvarez was born in Norwalk, Connecticut on October 30

1977, the son of Marceliano Alvarez and Consuelo Molina. He grew up in Houston,

Texas, and graduated from North Shore Senior High School in 1995. He attended

The University of Texas at Austin from the fall of 1995 to the fall of 1999, where

he earned a Bachelor of Science in Physics. In the fall of 2000 he began graduate

work in the Astronomy Department at the University of Texas at Austin. In 2004

he married Shizuka Akiyama, and currently resides with her in Austin.

Permanent address: 13843 CrosshavenHouston, Texas 77015

This dissertation was typeset with LATEX† by the author.

†LATEX is a document preparation system developed by Leslie Lamport as a special version ofDonald Knuth’s TEX Program.

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