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SECONDARY ANISOTROPIES OF THE COSMIC MICROWAVEBACKGROUND RADIATION

— A CROSS CORRELATION STUDY OF THE REES-SCIAMA EFFECT WITH

WEAK LENSING

AND THE IMPLICATIONS FOR DARK ENERGY —

——————————————————————————————–

A DISSERTATION SUBMITTED TO

DEPARTMENT OF PHYSICS AND ASTROPHYSICS

NAGOYA UNIVERSITY

IN CANDIDACY FOR DEGREE OF DOCTOR OF PHILOSOPHY

——————————————————————————————–

Nishizawa Atsushi

DEPARTMENT OF PHYSICS AND ASTROPHYSICS NAGOYA UNIVERSITY

FUROCHO CHIKUSA NAGOYA CITY 464-8602, AICHI JAPAN

NAGOYA, DECEMBER 14 2007

Supervisor: Professor. Dr. SUGIYAMA NAOSHI

Co-Supervisor: Professor. Dr. SHIBAI HIROSHI

3

Abstract

Recent astronomical observations revealed a number of mysteries of our Universe. One

of the most prominent results is the accurate measurement of temperature fluctuations of

Cosmic Microwave Background(CMB) byWilkinson Microwave Anisotropy Probe (WMAP)

satellite. The observation suggests that our universe consists of 22% dark matter, 4%

baryons and that the spacial curvature is almost equal to zero, i.e. space is flat. This

implies the existence of 74% unknown energy components in the universe. In the late

1990’s, supernovae(SNe) observations were actively pursued. SNe observations measured

the luminosity distance, which depends on a cosmological model, specifically the expan-

sion history of the universe, between SNe and us. The SNe results suggest that our

universe has been undergoing accelerating expansion since 6 Gyr ago. Combination of

the CMB and SNe results suggest that the 74% unknown energy component causes the

apparent accelerated expansion of our universe. The cosmological constant, introduced

originally by Einstein to keep the universe static, well accounts for this phenomenon in

the simplest manner. Another important result of recent observations is large scale dis-

tribution of galaxies. Wide field and deep galaxy observations, such as 2dF and SDSS

surveys, measured the 3-dimensional distribution of galaxies. One of the most important

results from these observations is that the dark matter component is cold, i.e. the velocity

is small. Since the major component of the universe is Cold Dark Matter(CDM) and Λ,

the standard model of cosmology is called as ΛCDM.

Now an outstanding issue in cosmology may be ”What is dark energy ?” One ap-

proach is observating the integrated Sachs Wolfe (ISW) effect, or its non-linear extension,

the Rees-Sciama (RS) effect. When CMB photons pass through the large scale struc-

ture(LSS) of the universe, their energies are slightly shifted; When a photon falls into the

gravitational potential of LSS, it gains the energy, or blueshifts. Whereas when it climbs

up the potential well, it loses its energy, or redshifts. If the potential would not vary with

time during the photon passage, the amount of blueshift and redshift is equal and thus

the net effects are cancelled out. However, since the potential wells in general evolve with

time, the net energy gain or loss is expected. On large scales, the existence of dark energy

dilutes the potential well (ISW), while on small scales, gravitational collapse concentrates

the potential well (RS). Thus the temperatures in the direction where potentials exist

along the line of sight and in the direction where potentials do not exist along the line

of sight are different. The temperature fluctuations caused by the RS effect are gener-

ally very small, and direct detection is extremely difficult. However, it is expected that

the temperature fluctuations caused by the RS effect and underlying matter distribution

must have some correlation. We propose to use a statistical method; the cross-correlation

between the CMB and the matter distribution traced by weak lensing.

4

We calculate the cross-correlation power spectrum using large cosmological N -body

simulations and analytical models. There is no exact theory to describe the non-linearity

of the potential: We show that the higher order perturbation theory can be applicable

and a few phenomenological models show good agreement with the result of the N -body

simulations.

We find the cross-correlation of the RS and weak lensing provides a unique probe for

investigating non-linear evolution of density fluctuations on small scales. Moreover we

discover an interesting feature in the cross-correlation angular power spectrum. It shows

correlation on large scales while it shows anti-correlation on small scales. There exists

cross over point in the intermediate scale. This cross over point is sensitive to the energy

density and the nature of dark energy, especially to its equation of state. We conclude

that the cross over point can be a unique probe of dark energy.

5

Acknowledgments

I am grateful to professor Naoshi Sugiyama for the careful supervising. He always points outthe physically or mathematically unclear things in my statements. This largely helps me tounderstand cosmology and to improve my way of thinking. I think the encounter with Sugiyama-san is one of the happiest happening in my research life. I would like to express gratitude toprofessor Hiroshi Shibai. He is supportive for me to join the Akari FIS team. With his greathelps, I undergo many precious experiences in the Akari satellite operation and data reduction.I would never experience such an important tasks without Shibai-san’s kind supports. I shouldalso acknowledge professor Takahiko Matsubara. He used to be my supervisor for first threeyears in my graduated course at Nagoya University. He taught me a lot of knowledge aboutcosmology and mathematics. His statements are always mathematically exactly correct, andhe solves all the problems immediately that I was troubled. I learned from Matsubara-santhe importance and interests of mathematical derivations or understanding of the physics. Ipartially learn from professor Satoru Ikeuchi. I also acknowledge Ikeuchi-san. He gives me a lotof precious episodes on physics and natural science. Though these are not directly related tomy research, the way of thinking or way of see the nature becomes very good lessons for me.

I am also grateful to Professor Naoki Yoshida. He is always amiable for my discussion andquestion. Also I leaned from Yoshida-san much about the N -body simulation and numericalcalculation technique. Some of the result of this thesis and the paper is achieved with thesupport of Yoshida-san. He also gives me good opportunities to get to know the internationalresearchers and an encouragement of the collaboration work. Eiichiro Komatsu, the collaboratorhelps many times me to understand the non-linear theory, halo model and related statistics. Iacknowledge him so much. He always happily has discussions with me about any issues on thecosmology. Also his passion and faculty to the cosmology stimulates my motivation to pursue thePhD study. Ryuichi Takahashi, also one of my important collaborator helps my work throughthe N -body simulation. I also acknowledge him.

I would like to acknowledge David Spergel, Olivier Dore, Joseph Henawii. They give manyvaluable comments on our study and we can improve the paper significantly. Ravi Sheth gives mean unpublished his draft about momentum power spectrum using halo model. This greatly helpsme to understand the halo model descriptions. I am grateful to Ravi Sheth. I also acknowledgeAsantha Cooray. I discussed him at Nagoya and London and in e-mail many times. I learnedfrom him much about the integrated Sachs Wolfe effect and how to write so many papers in oneyear.

I would like to show my gratitude to the researchers, Yasushi Suto, Atsushi Taruya, MasahiroTakada, Takashi Hamana, Kouji Yoshikawa, Tsutomu Takeuchi, Hiroyuki Tashiro, and DavidPerkinson. Although I could not discuss them constantly but when I met them at the conferencesand workshops, they welcome my simple and honest questions.

Matsubara-san gives me chance to participate the Prioritised Study of Akari (ASTRO-F)related to the ISW analysis. I am also grateful to Richard Savage, the PI of this study. Shibai-san

6

also invites me to the Akari-FIS team and Akari-FIS data reduction team. I had an invaluableexperience through the Akari satellite operation and part of the data analysis. I would like toshow my gratitude to the project manager of Akari, Hiroshi Murakami, and professors HideoMatsuhara, Issei Yamamura, Shuji Matsuura, Tsuneo Kii, and the researchers, Mai Shirahata,Shin’ichiro Makiuchi and Shinki Oyabu at ISAS/JAXA. I should say special thanks to theYamamura-san. He arranges all my business journey and teach me the way of data analysisand IDL programing. I also acknowledge Mitsunobu Kawada, Noriko Murakami and TakahumiOhtsubo. They helps the Akari operation and data analysis at Nagoya University.

I would like say the special thanks to Issha Kayo, Chiaki Hikage. They support the com-putational technique and maintenance and solve the problems about computers and software aswell as the difficulties on cosmology. Toshikazu Ohnishi and Akiko Kawamura greatly supportthe computational technique and maintenance. The very fine and comfortable environment ofworkstation, network system and any other equipments in the laboratory would be never realizedwithout them. I acknowledge them.

I have got to know many folks in Japan through the conferences, meeting, workshop, labora-tory and the summer school for young astronomer. They are Akihito Shirata, Kazuhiro Yahata,Shiou Kawagoe, Takahiro Nishimichi, Shun Saito, Masakazu. A. R. Kobayashi, Takeshi Oda,Norita Kawanaka, Daisuke Kato, Mikio Kurita, Joel F. Koerwer, Eiji Mitsuda, Kouhei Onda,Hiroaki Menjo, Kouki Kamiya, Hiroaki Yamamoto, Yozo Kawano, Yoshitaka Murata, ShinichiHikida, Teppei Okumura, Tomotake Takeuchi, Shingo Ito, Akiko Hayashi, Yoshiyuki Enomoto,Sachiko Kuroyanagi, Kouhei Ohtsubo, Midori Tokutani, Naoki Umemoto, Hiroyuki Hayashi,and Tomonori Furukawa.

I use all the calculation by cluster PC (star) and workstations (beta-system) at our Theo-retical Astrophysics group. I acknowledge the NASA WMAP team to provide publicly availablesoftware Healpix, and the WMAP data.

Finally I would like to say a special thanks to my family, Hideo Nishizawa, Misako Nishizawa,and Takashi Nishizawa to encourage and agree to pursue the PhD student. This thesis wouldnever exist without their constant support and the agreement.

All of my PhD student days are partially supported by the grant of 21st century COEprogram at Nagoya University. I am grateful to the professor Yasuo Fukui, the organizer of thisprogram as well as the Japan Society for the Promotion of Science (JSPS). The scholarship ofJApan Student Services Organization (JASSO) economically support rest of my PhD studentdays. The work is also supported by Grant-in-Aid for Scientific Research on Priority AreasNo. 467 ”Probing the Dark Energy through an Extremely Wide & Deep Survey with SubaruTelescope” and by The Mitsubishi Foundation.

Dec. 14. 2007 Atsushi Nishizawa

Contents

1 Introduction 11

1.1 The Cosmic Microwave Background Radiation . . . . . . . . . . . . . . . . 11

1.2 CMB Secondary Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 The Standard Cosmological Model 17

2.1 Relativistic Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Friedmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.2 Cosmological Distance . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Structure formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Primeval Density Fluctuations . . . . . . . . . . . . . . . . . . . . . 22

2.2.2 Density Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Statistics of Large-Scale-Structure . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.1 Gaussian Random Field . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.2 Power Spectrum and 2PCF . . . . . . . . . . . . . . . . . . . . . . 28

2.3.3 Angular Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.4 Higher Order Moments . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Observational Probes 39

3.1 The Rees-Sciama Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.1 Why using Cross Correlation ? . . . . . . . . . . . . . . . . . . . . 40

3.2 Weak Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Light Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.2 Statistical Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.3 Convergence Power Spectrum . . . . . . . . . . . . . . . . . . . . . 50

4 Non-linear Treatment 53

4.1 Third Order Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Derivative of a Fitting Model . . . . . . . . . . . . . . . . . . . . . . . . . 60

7

8 CONTENTS

4.3 Dark Matter Halo Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.1 Matter Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.2 Momentum Power Spectrum . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Potential Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5 N -body Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.5.1 Calculus of PΦΦ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.5.2 Consistency check between PΦΦ′ and ∂τPΦΦ . . . . . . . . . . . . . 76

4.6 Non-linear Evolution of Gravitational Potential . . . . . . . . . . . . . . . 80

5 Search for Rees Sciama 87

5.1 Possible Correlating Sources with the LSS . . . . . . . . . . . . . . . . . . 87

5.2 Angular Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2.1 The impact of non-linearity on CCAPS . . . . . . . . . . . . . . . . 94

5.3 Detecting the signature of the Rees–Sciama Effect . . . . . . . . . . . . . . 95

5.4 Sensitivities to the Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4.1 CCAPS Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4.2 Fisher Matrix Forecast . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Concluding Remark 109

A Transfer Function 123

A.1 Peebles 1982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

A.2 Bond Efstathiou 1984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

A.3 BBKS 1986 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A.4 Eisenstein & Hu 1998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A.5 Eisenstein & Hu nowiggle 1998 . . . . . . . . . . . . . . . . . . . . . . . . 125

A.6 Effect of Neutrino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

B Solving Poisson Equation 131

B.1 From Particle to Density Field – Gridding . . . . . . . . . . . . . . . . . . 131

B.2 Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B.3 FFT Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

C Derivation of ISW 137

List of Figures

1.1 CMB Temperature Angular Power Spectrum . . . . . . . . . . . . . . . . . 13

1.2 Simulated CMB map for three angular resolutions . . . . . . . . . . . . . . 13

1.3 Energy Content in the universe . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 geometry of 2-dimensional hyper surface . . . . . . . . . . . . . . . . . . . 18

2.2 three different definitions of cosmological distance . . . . . . . . . . . . . . 23

2.3 growth of density contrast in linear theory . . . . . . . . . . . . . . . . . . 26

2.4 power spectrum measurement of 2dF and SDSS . . . . . . . . . . . . . . . 30

2.5 Spherical Bessel function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6 Contribution scale to the angular power spectrum . . . . . . . . . . . . . . 34

2.7 Contribution scale to the angular power spectrum . . . . . . . . . . . . . . 35

3.1 Multiple images : MG J0414+0534 . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Gravitational lens image: Abell 1689 . . . . . . . . . . . . . . . . . . . . . 43

3.3 geometry of lens system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 shear component mapping to real image . . . . . . . . . . . . . . . . . . . 47

3.5 light path around the gravitational lens system . . . . . . . . . . . . . . . . 48

4.1 Non-linear evolution of matter power spectrum . . . . . . . . . . . . . . . . 54

4.2 PΦΦ′(k) from 3PT of two prescriptions . . . . . . . . . . . . . . . . . . . . 58

4.3 Dark matter halo: 1halo and 2halo contributions . . . . . . . . . . . . . . . 62

4.4 Mass contribution to the halo model power spectrum . . . . . . . . . . . . 64

4.5 Mass dependence of velocity dispersion of and within halo. . . . . . . . . . 66

4.6 Momentum power spectrum div and curl component . . . . . . . . . . . . 68

4.7 Momentum power spectrum using Halo model . . . . . . . . . . . . . . . . 68

4.8 Momentum–Density 3D correlation coefficient . . . . . . . . . . . . . . . . 70

4.9 Momentum–Density power spectrum using Halo model . . . . . . . . . . . 71

4.10 ΦΦ′ cross power spectrum with N -body simulations (I) . . . . . . . . . . . 73

4.11 ΦΦ′ cross power spectrum with N -body simulations (II) . . . . . . . . . . 74

4.12 Flow chart of the N -body simulation procedures . . . . . . . . . . . . . . . 76

9

10 LIST OF FIGURES

4.13 Numerical test of commutativity: 250 h−1Mpc simulation . . . . . . . . . . 77

4.14 Numerical test of commutativity: 40 h−1Mpc simulation . . . . . . . . . . 78

4.15 The non-linear evolution of Gravitational Potential with 3PT . . . . . . . . 82

4.16 Correlation coefficient of Φ and Φ′ . . . . . . . . . . . . . . . . . . . . . . . 84

4.17 Redshift Evolution of non-linearity of PΦΦ′ (I) . . . . . . . . . . . . . . . . 85

4.18 Redshift Evolution of non-linearity of PΦΦ′ (II) . . . . . . . . . . . . . . . . 86

5.1 Simulated SZ maps for three frequency band. . . . . . . . . . . . . . . . . 88

5.2 Frequency dependence of thermal and kinetic SZ . . . . . . . . . . . . . . . 88

5.3 Radial distribution of source galaxies for weak lensing . . . . . . . . . . . . 92

5.4 Angular power spectrum of RS-κ . . . . . . . . . . . . . . . . . . . . . . . 93

5.5 The distribution of correlation in the redshift . . . . . . . . . . . . . . . . . 96

5.6 Cumulative signal to noise ratio of RS-κ correlation . . . . . . . . . . . . . 97

5.7 signal to noise ratio of RS-κ correlation at each multipole . . . . . . . . . . 100

5.8 Dark Energy sensitivities of CRS−κ . . . . . . . . . . . . . . . . . . . . . . 101

5.9 Derivative of CRS−κ w.r.t Dark Energy parameters . . . . . . . . . . . . . . 103

5.10 Schematic illustration for the dependency of CRS−κ on ΩΛ0 . . . . . . . . . 104

5.11 expected 1σ contour obtained from RS cross correlation . . . . . . . . . . . 106

5.12 expected 1σ contour obtained from RS cross correlation . . . . . . . . . . . 107

A.1 no wiggle transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

B.1 The gridding schemes:NGP,CIC and TSC . . . . . . . . . . . . . . . . . . . 132

B.2 2D density map for various gridding schemes . . . . . . . . . . . . . . . . . 135

B.3 1D density map for various gridding schemes . . . . . . . . . . . . . . . . . 136

Chapter 1

Introduction

1.1 The Cosmic Microwave Background Radiation

In the standard Big Bang scenario, the universe is highly ionized until redshift z 1100 (about 40 thousand years after big-bang), the recombination epoch. In the ionized

universe, there exists a vast amount of free electrons which are tightly coupled to photons

through the Thomson scattering. Once the universe is cooled to ∼ 3000K due to the

adiabatic expansion, these electrons are instanteneously captured by protons. Thus the

number density of electrons which scatter photons drastically decreases and photons begin

to stream freely: the universe becomes transparent to photons; so-called decoupling. Since

these free streaming photons are propagating to us along the null-geodesic with almost

no interaction to matter content, we can observe the snapshot of the universe as of 40,000

years old as the Cosmic Microwave Background (CMB) radiation.

As CMB photons have been scattered by electrons many times, the spectral distribu-

tion of photons maintains a black body shape. The COBE satellite observed the almost

perfect black body signature in the CMB with its black body temperature being 2.725 K

(Mather et al., 1994, 1999), which is the proof of the Big Bang paradigm, whereas it also

observed tiny fluctuations in the CMB temperature, which are less than 10µK (Smoot

et al., 1992). These fluctuations are believed to be generated during the inflation epoch in

the early universe together with density fluctuations of matter which evolve into the com-

plex structure of the universe such as galaxies, clusters of galaxies, large scale structure

and so on. Since we are looking at the decoupling epoch through CMB, the temperature

fluctuations show us the fossil structure of the universe at 40,000 years old. However,

it is known that the temperature fluctuations are also generated after the decoupling

epoch via various interaction processes with the structure of the universe. Here we de-

note the former, i.e., fluctuations at decoupling, as the primary anisotropy and the latter

as the secondary anisotropy. Physical processes which induce the primary temperature

11

12 CHAPTER 1. INTRODUCTION

anisotropy are classified into roughly two types. One is due to the potential fluctuations

:Sachs Wolfe Effect. Temperature fluctuations approximately larger than 1, which isthe angular scale of the horizon at the decoupling epoch, are caused by the gravitational

redshift as (Sachs & Wolfe, 1967),

δTSWT

1

3φ (1.1)

where φ is a fluctuation of the gravitational potential. The Sachs Wolfe effect is impor-

tant since it directly reflects the potential fluctuation of the early universe. The other one

is temperature anisotropies caused by the sound waves prior to recombination: Baryon

Acoustic Oscillation. Within the horizon, the plasma fluid, consist mainly of baryon in

photon bath, were causally connected and oscillated like as sound waves. The density en-

hancement gathers the plasma fluid by the gravitational attracting force, the compression.

While once the plasma is compressed, the radiation pressure repulses it, rarefaction. The

successive iteration brings the baryon oscillation. In figure 1.2 we show the angular power

spectrum of CMB temperature fluctuations. The prominent acoustic peak seen at 200

indicates the maximum distance that the sound wave can reach until the recombination

epoch. The angular scale of this peak must be shifted as the geometry changes. Thus the

accurate measure reveals the spacial geometry of the universe. The data of WMAP 3yr

combined with large scale structure and supernovae data (e.g., Perlmutter et al., 1999)

prefer a slightly open geometry while the flat universe with cosmological constant is still

consistent within the 2 σ (Spergel et al., 2007).

The observational developments of CMB are partially appeared in the angular reso-

lution and temperature sensitivity. At the COBE era, the angular resolution is about 7

degrees (see top left of figure 1.2). It has an enough ability to probe the Sachs-Wolfe effect

which appears on low . However it hardly reaches to 200, the location of the acous-

tic peak. One of the most excellent achievements of WMAP is to measure the acoustic

peak location and its height so accurately. The symbols in figure 1.1 denote the binned

data of WMAP 3yr angular power spectrum. The dramatic agreement to the ΛCDM

prediction (solid line) can be seen. In figure 1.1 we also show the instrumental noise with

the shaded region. The gray region is for the WMAP 4 year observation and light blue

region is for the forthcoming Planck single year observation. WMAP can probe the CMB

anisotropies to 800 whereas Planck can reach to 2000, which corresponds to a few

arc minutes. In such small scales, the secondary CMB temperature anisotropies which are

caused by the foreground density distributions of large scale structure through thermal

or gravitational interactions provide dominant contribution.

1.1. THE COSMIC MICROWAVE BACKGROUND RADIATION 13

WMAP 3yr data

Figure 1.1: CMB Angular Power Spectrum for the temperature anisotropy. Symbols with error bars

are bin averaged WMAP 3yr data and solid blue line is best fit ΛCDM theoretical prediction. The gray

shaded region is expected 1 sigma error for WMAP 4 year observation and light blue shaded region is that

for PLANCK 1 year observation. Also plotted with black doted and solid lines are linear and non-linear

RS effect. On the largest scale, linear ISW is likely dominated the total CMB temperature fluctuations

while on small scales, the RS effect is less than primary by order of three.

θfwhm = 420 arcmin

θfwhm = 30 arcmin

θfwhm = 2 arcmin

Figure 1.2: Shown are simulated CMB temperature fluctuation maps for three angular resolutions with

420, 30 and 2 arcmin FWHM of the beam. These values are approximations of the COBE, WMAP

and ACT resolutions. The simulated maps are generated by synfast in Healpix ver 2.00 software with

Nside = 1024 (12,582,912 pixels) with same random seed.


1.2 CMB Secondary Anisotropies

After the photon-baryon decoupling, photons propagate along the null geodesic to us

through the large scale structure that lies at lower redshift than CMB. Since the large scale

structure has complex distributions of gravitational potentials and hot gases in the clusters

of galaxy that is neither isotropic nor homogeneous, the large scale structure imprints

the gravitational and/or thermal signatures on CMB temperature fluctuations. We thus

must observe the temperature fluctuation induced by the large scale structure, so-called

secondary CMB temperature anisotropies as well as those generated at the decoupling

epoch. Both the satellite such as PLANCK (Lamarre et al., 2003) and ground-based

observations such as Atacama Cosmological Telescope (ACT: Kosowsky, 2003) or South-

Pole Telescope (SPT Ruhl et al., 2004) will measure the CMB temperature fluctuations at

arc-minute scales (l > 1000) with sensitivities of ∼ µK, and thus are expected to detect

the secondary anisotropies. It is one of the major goals of the next generation CMB

experiments to separate each component of them.

The secondary anisotropies of CMB provide invaluable information on the history of

structure formation in the universe. Sources of the anisotropies include the thermal and

kinetic Sunyaev-Zeldovich (t/kSZ) effects by galaxy clusters and by patchy reionization,

the integrated Sachs-Wolfe (ISW) effect and its non-linear extension, the Rees-Sciama

(RS) effect, and the deflection effect of CMB by the gravitational lensing, with the relative

amplitudes at arc-minute scales being approximately less than the order of ∆T/T ∼ 10−5.The thermal SZ effect is induced by the hot gas of cluster of galaxies through the inverse-

compton scattering. As the hot gas modifies the photon distribution function from the

2.725K black-body distribution, the black-body temperature at long wave-length region

(Rayleigh-Jeans) decreases while it increases at short wave-length region (Wien). The

kinetic SZ effect is induced by electrons associated to bulk motions of clusters of galaxies

relative to the CMB rest frame.

The ISW effect is induced by the time variation of gravitational potentials of large

scale structure. As the CMB photon falls into the gravitational potentials, it gains the

energy while as the photon climbs up the potentials, it loses the energy. If the height of

gravitational potentials vary with time, the net energy chage would be expected. The RS

effect is induced by same mechanism. We use the term ISW when the underlying matter

density fluctuations of large scale structure are well described by the linear theory, while

the term RS is used when they need a non-linear treatment.

The Rees-Sciama effect, which is a main topic in this thesis, is, in principle, a unique

probe of the time-variation of gravitational potential, and thus provides information of

the rate of growth of large-scale structure on small scales (Rees & Sciama, 1968). In

addition the Rees-Sciama effect also occurres when a galaxy or a cluster of galaxies moves

1.3. OVERVIEW OF THE THESIS 15

ΩΛ0 = 0.74ΩM0 = 0.26

Ωb0 = 0.04

ΩDM0 = 0.22

Figure 1.3: Energy content of the universe in the standard ΛCDM model.

across the line of sight. Typically, moving galaxy imprints a dipole pattern in the CMB

temperature map (Tuluie et al., 1996). Thus the dipole component in the CMB fluctuation

would enable us to measure the velocity of galaxy perpendicular to the line of sight.

In practice, however, direct detection of the RS effect is extremely difficult. This can

be easily appreciated by noting that the dominant source of arc-minute scale anisotropies

is the tSZ effect, of which the expected fluctuation is of order ∆T tSZ/T ∼ 10−5 (Sunyaev& Zeldovich, 1980), whereas the RS effect typically generates fluctuations of ∆TRS/T ∼10−8 (Seljak, 1996; Tuluie et al., 1996). The tSZ effect can be, ideally, removed usingmultifrequency data since the temperature fluctuation induced by tSZ effect depends on

the onbserved frequency. Whereas other effects such as the kSZ and CMB lensing as well

as the RS effect do not have the spectral dependence, and thus they are indistinguishable

by CMB observations alone. In order to extract the RS effect from the total CMB

temperature fluctuation, we need to use statistical methods. We propose to measure

the cross-correlation between the CMB temperature fluctuations and matter distribution

probed by weak lensing surveys.

1.3 Overview of the Thesis

The final goal of this thesis is to investigate the non-linear evolution of gravitational

potential and its time derivative in a standard ΛCDM universe. To evaluate the non-

linear evolution of fluctuations, we make an realistic estimation of the angular power

spectrum of cross correlation between convergence and temperature fluctuations induced

by ISW (and RS) using both third order perturbation theory and N -body numerical

simulations. The convergence is the effect that the elliptical galaxy image is magnified

due to the gravitational lensing of foreground matter distribution keeping its ellipticity

and orientation unchanged. This quantity trace the underlying matter distribution of

large scale structure. Then we discuss the possibility of the detection of it by on-going

or forthcoming observation sets. We also examine the sensitivity of this angular power

spectrum to dark energy parameters.

We describe below the structure of this thesis. In chapter 2, we give a review of the

basic theory of the standard cosmology is provided here. In chapter 3, we describe why and


how we should take the cross correlation of RS and weak lensing. Some reviews related to

the theoretical calculation of convergence, which is one of the most important statistics in

the weak lens theory, are also given here. In chapter 4, We develop the non-linear theory

to describe the non-linear gravitational potential and its time derivative in the analytical

and numerical methods. We calculate the cross correlation of Φ(k) and Φ′(k) using theN -body simulations for the first time. We will show some results of the non-linear power

spectrum of N -body simulations and compare them to theoretically derived calculations.

In chapter 5, we discuss about the detectability of the Rees-Sciama effect in the context

of cross correlation of convergence and RS effect. At first, we make a list of the possible

candidates that could correlate with convergence, and show that the dominant correlation

is induced by the Rees-Sciama effect after subtracting the spectral dependent sources such

as the SZ effect. We also discuss here about the interesting behavior that is sensitive to

dark energy and suggest that the Rees-Sciama vs convergence angular cross correlation

power spectrum can be a unique probe to unveil the nature of dark energy. Chapter 6 is

a summary of this thesis. Some useful formulae or technical details are collected in the

Appendices.

Throughout this thesis, the standard ΛCDMmodel consistent with result fromWMAP

(Spergel et al., 2003; Spergel et al., 2007) is adopted and a spatially flat universe with

matter density Ωm = 0.26 and dark energy ΩΛ = 0.74 (figure 1.3) are assumed. Other

related cosmological parameters take value as σ8 = 0.76, ns = 1 and h = 0.7. Also, the

convention of natural unit, c = 1 is applied throughout in this thesis.

Chapter 2

The Standard Cosmological Model

Recent cosmological observations reveal a number of mysteries of our universe. The

observations include temperature fluctuations of Cosmic Microwave Background (CMB),

luminosity distance from supernovae or gamma ray bursts, the cluster abundance from

wide field and deep galaxy surveys, distorted shapes of galaxy images by gravitational

lensing, the signature of baryon acoustic oscillation in the galaxy correlation and the

CMB temperature fluctuations generated at the large scale structure of the universe. The

combination of these observations suggests that our universe is accelerated expanding

since roughly about 6 billion years ago, the redshift z 1. In the framework of standard

theory of general relativity, all the results of these observations suggest the existence of

unknown energy component in a stress-energy tensor, the Λ term.

In this chapter, we will give a review of the standard theory of cosmology based on

the general relativity.

2.1 Relativistic Cosmology

In the zero-th approximation the universe is uniform and isotropic; there are any special

points and directions in the universe, the Cosmological Principle. It can be translated

as the generalization of Copernicus Principle. The most general spacetime metric that

satisfies the Cosmological Principle is given by Robertson-Walker metric. It can be written

in the spherical polar coordinate as

ds2 = gµνdxµdxν (2.1)

= a2[−dτ 2 + dr2

1− kr2+ r2(dθ2 + sin2 θdφ2)], (2.2)

where a is a scale factor, which is in general normalized to unity at today, and τ is

conformal time, related to cosmic time as adτ = dt. K is spacial curvature, andK >,<,=

17

18 CHAPTER 2. THE STANDARD COSMOLOGICAL MODEL

Negative Positive Flat

Figure 2.1: Geometries of 2-dimensional hyper surface. If we denote the figure that three

points connected by three geodesic lines as triangle, the sum of inner angles of this triangle

is lesser/greater than π for negative/positive geometry and equal to π for flat geometry.

This property is exactly same as in the 3-dimensional hyper surface, and thus used to

measure the spacial curvature of the universe.

0 correspond to closed, open and flat geometry respectively. We can understand the curved

three dimensional space as the analogy of two dimensional curved space embedded in the

imaginary three dimensional space. Figure 2.1 illustrates the curved two dimensional

space embedded in 3D space. The property of curved space appears in a sum of inner

angle of triangle. If we set arbitrary three points in a space, and connect them along the

geodesic lines, this figure defines triangle in a curved, of course in a flat space. In a open

space, which has negative spacial curvature, the sum of inner angles must be less than π,

and in a closed space, which has positive curvature it must be greater than π and in a

flat space, it must equal to π. This can be easily seen in a 2D case and can be extended

to three dimensional space.

2.1.1 Friedmann equation

The Einstein equation, which describes the dynamics of background spacetime in the

presence of energy component including cosmological constant is

Gµν = 8πGTµν + Λgµν (2.3)

where the Gµν = Rµν − gµνR/2 is Einstein tensor with Rµν and R being Ricci tensor and

Ricci scalar. Specifying the energy contents and the metric yields explicit form of this

equation. For the perfect fluid, the energy-momentum tensor is

Tµν = (p+ ρ)uµuν − pgµν (2.4)

2.1. RELATIVISTIC COSMOLOGY 19

where p is the pressure, ρ is the energy density and uµ is the fluid four velocity defined as

uµ = gµνdxν

ds(2.5)

In the universe described by Robertson-Walker metric, the Einstein equation yields(a

a

)2

=8πG

3ρ− K

a2+Λ

3(2.6)

a

a= −4πG

3(ρ+ 3p) +

Λ

3(2.7)

Equation (2.6) is 00 component of the Einstein equation and called as Friedmann equation

while the equation (2.7) is spacial part of equation (2.3), which is also derived from the

spacial part of energy conservation law, T µν;µ = 0. The Friedmann equation can be

rewritten as

H2(a) = H20 (Ωm0a

−3 + ΩK0a−2 + ΩΛ0) (2.8)

where H = a/a is Hubble parameter. Ωs are density parameters, which are normalized

by critical density, ρc0 = 3H20/8πG, and defined as

Ωm0 =8πGρ03H2

0

, ΩK0 = − K

H20

, ΩΛ0 =Λ

3H20

(2.9)

Note that the subscript 0 denotes the value of today and also that this notation of cur-

vature parameter is opposite in sign to usual definition. Now we introduce the equation

of state for each energy content that is defined as pressure to density ratio, w := p/ρ.

wm = 0 for a pressureless matter such as dark matter, wΛ = −1 for the cosmologicalconstant and wr = 1/3 for radiation such as photon. For the curvature, corresponding

equation of state is wK = −1/3. Then the Friedmann equation again rewritten as

H2(a) = H20

∑i

Ωi0Ei(a) (2.10)

I(a) = −H20

2

∑i

Ωi0Ei(a)[1 + 3wi(a)] (2.11)

Ei(a) = exp

[3

∫ 1

a

1 + wi(a′)

a′da′

](2.12)

where subscript i denotes the species of energy components and I(a) = a/a. Considering

the present epoch, following equation must be hold from equation (2.10),

1 = Ωm0 + Ωr0 + ΩΛ0 + ΩK0. (2.13)


This relation must have equality for any time in the history of universe as

1 = Ωm + Ωr + ΩΛ + ΩK , (2.14)

with the density parameter at given time being

Ωi(a) = Ωi0Ei(a)H20

H2(a)(2.15)

2.1.2 Cosmological Distance

In an expanding universe, the distance can be defined in various manner. To this end,

it is convenient to introduce the idea of redshift. Suppose that the light is emitted with

wavelength λe, propagates to us along the null geodesic and is observed with λo. The

wavelength of light is elongated both due to the cosmological expansion and peculiar

motion of source against the observer, the Doppler shift. The redshift is defined as ratio

of two wavelength,

1 + z :=λoλe

(2.16)

In the absence of any peculiar motion, i.e. the emitting source is fixed on the comoving

frame, this exactly represents the expansion rate of universe thus

1 + z = a−1 (2.17)

is hold with normalization being a(t0) = 1

Comoving Distance

Imagine that the photon is emitted at (r, τ) = (r1, τ1), propagates along the null geodesic,

ds = 0 and is observed by us at (r, τ) = (0, t0). From the fact that the light propagates

isotropically, dθ = 0, dφ = 0. Then the metric given by equation (2.2) yeilds the relation,

dτ = − dr√1−Kr2

. (2.18)

Integrating along the line of sight gives the comoving distance, χ

χ =

∫ τ0

τ1

dτ =

∫ a0

a1

da

a2H(a)=

∫ z1

0

dz

H(z). (2.19)

In the second equality, dτ = dt/a = a/(aa2)da is used, and in the last equality, equation

(2.17) is used. The explicit relation between comoving distance and coordinate r is given

by integrating left hand side of equation (2.18). This yields immediately,

r =

sin(

√Kχ)/

√K, K > 0

χ, K = 0

sinh(√−Kχ)/

√−K, K < 0

(2.20)

2.1. RELATIVISTIC COSMOLOGY 21

Luminosity Distance

If one knows the luminosity of the light source, observed flux (or magnitude) can be a

measure of distance. Let me first consider the static Euclidean space. Imagine that the

light is emitted at (r, t) = (r1, t1) with luminosity L. The observed flux at (r, t) = (0, t0)

is reduced proportionally to distance squared,

f =L

4πr21(2.21)

The number of photons emitted between t1 and t1 + δt1 with frequency range between ν1and ν1 + δν1 is set to δN . The luminosity in this frequency band is then

δL = hν1δt1δN

(2.22)

Since the photon number must be conserved, the photon energy (flux) that passes through

the unit area perpendicular to the photon trajectory between t0 and t0+δt0 with frequency

range between ν0 and ν0 + δν0 is

δf =hν04πr2

δN

δt0. (2.23)

In a static Euclidean space, ν1 = ν0 and δt1 = δt0, while in an expanding spacetime

frequency and time interval are modified as ν1 = ν0(1 + z), δt1 = δt0/(1 + z). In an

expanding universe, equating δN of equations (2.22) and (2.23) and integrating over the

frequency yield

f =L

4πr2(1 + z)2(2.24)

Of course equating δN and integrating in a Euclidean space results in equation (2.21). If

we define luminosity distance as dL := r(1 + z) the expression is exactly same as that in

the Euclidean space (eq. (2.21))

f =L

4πd2L. (2.25)

Note that here f and L are bolometoric values. When the bolometoric information is not

available, one should have the K-correction. We do not mention to it in this thesis.

Angular Diameter Distance


Again let me first consider the case for the Euclidean space. If there is a celestial object

whose size is known. Observing the subtended angle gives another measure of distance,

D = r∆θ, (2.26)

where D is size of the object perpendicular to the line of sight, ∆θ is its subtended angle

in the sky and r is distance to the object. In an expanding universe, integrating null

geodesic over the angle from right edge of the object to left edge equals to the physical

separation of the object, with dr = dφ = 0,

D =

∫ θL

θR

ar dθ = ar∆θ (2.27)

Compared with equation (2.26), if we define angular diameter distance as dA := r/(1+z),

the relation becomes same as Euclidean space.

Figure 2.2 shows these three definitions of cosmological distance in standard ΛCDM

universe.

2.2 Structure formation

2.2.1 Primeval Density Fluctuations

In our universe, there are various structures that are bounded by gravitational force;

galaxy, cluster of galaxy and large-scale structure of the universe and so on. These

structure is thought to be constructed by the gravitational instability with the begining

of initial tiny fluctuations. The initial condition is given by power low matter power

spectrum as,

P (k) = Askn, (2.28)

where As is the amplitude of initial matter power spectrum and n is spectral index. If n =

1, initial matter power spectrum is scale invariant, or Zel’dovich spectrum. The WMAP

3yr data imposes the constraints on n and As as, n = 0.958 ± 0.016 and log(1010As) =

3.156± 0.056 (Spergel et al., 2007). This implies that the initial matter power spectrumalmost equals to scale invariant, but slightly deviate from it. So, the values of n and As

above are conventionally measured at k = 0.002h/Mpc.

The CDM density fluctuations that are smaller than horizon scale interact with other

species such as photon, baryon and neutrino. While the fluctuations that are larger than

Hubble radius does not interact with these species and conserve the information of initial

density fluctuations. The physical effects, including Silk damping, damping due to baryon

2.2. STRUCTURE FORMATION 23

Figure 2.2: Plotted are cosmological distances with three different definitions. Assumed

cosmology is flat ΛCDM with Ωm0 = 0.26, ΩΛ0 = 0.74, h = 0.7.


oscillation, expansion law of background universe, damping due to the massive neutrino

and so on are all encoded in a transfer function, that connects the initial matter power

spectrum to that of today,

T (k) =δ(k, z = 0)

δ(k, z =∞)δ(0, z =∞)δ(0, z = 0)

(2.29)

The transfer function is analytically solved in the small scale limit by Hu & Sugiyama

(1996) and useful fitting formulae can be found with and without wiggle of baryon oscil-

lation (Eisenstein & Hu, 1998). The suppression of matter power spectrum due to the

free streaming of massive neutrino is included in (Eisenstein & Hu, 1999). The set of

fitting formulae are summarized in Appendix A. Alternatively, one can obtain the trans-

fer function by solving Boltzmann equation numerically that can contain multispecies

(Bond & Efstathiou, 1984; Holtzman, 1989; Hu et al., 1995; Seljak & Zaldarriaga, 1996).

CMBFast 1 is the most famous publicly available Boltzmann code to compute power spec-

trum of matter and CMB. Enormous modified versions of CMBFast such as CAMB 2,

KINKFAST(Corasaniti et al., 2004)are contrived.

The dimensionless power spectrum, which corresponds to the variance of fluctuation

of matter per logarithmic wave number is

∆2(k) :=4π

(2π)3k3P (k) =

1

2π2AsT

2(k)kn+3. (2.30)

2.2.2 Density Evolution

Regarding the matter as the continuous fluid, the evolution of matter is governed by the

continuity and Euler equations, which is given by respectively,

δ′ +∇ · [v(1 + δ)] = 0, (2.31)

and

v′ + 2H v + (v · ∇)v = ∇Φ, (2.32)

where v = ∂x/∂τ is the conformal velocity, H = d ln a/dτ is the conformal Hubble

constant. Prime denotes differentiation w.r.t the conformal time, τ . The potential fluc-

tuations are determined from density perturbations via Poisson equation in the comoving

frame as,

∇2[Φ(x, τ)−Ψ(x, τ)] = −8πGa2ρδ(x, τ) (2.33)

1http://cfa-www.harvard.edu/ mzaldarr/CMBFAST/cmbfast.html2http://camb.info/

2.2. STRUCTURE FORMATION 25

where Φ is Bardeen’s curvature perturbation during the matter-dominated era and related

to the trace of metric as gii = 3a2(1 + 2Φ), whereas Ψ is given by the g00 component,

g00 = −a2(1+2Ψ) (Kodama & Sasaki, 1984; Bardeen, 1980). In the absence of significant

sources of anisotropic stress, which is always the case, these two quantities are related to

each other by Φ = −Ψ.In the linear perturbation theory, the evolution of the density fluctuations does not

depend on scales so that the density fluctuations can be factorized as δ(x, τ) = D(τ)δ(x),

where δ(x) is the initial density fluctuation. Then combining equations (2.31)-(2.33)

yields the evolution equation for the matter,

D′′ + 2H D′ +3

2Ωm0H

20 D = 0. (2.34)

Since this equation is second order differential equation, it has two special solutions,

growing mode and decaying mode. However the magnitude of decaying mode decrease

rapidly thus we can ignore the contribution from decaying mode. In the matter-dominated

era, the solution D(a) is given by the scale factor, a (e.g., Peacock, 1999) and one can use

the fitting formula in the ΛCDM case(Carroll et al., 1992) when 0.03 ≤ ΩM ≤ 2,−5 ≤ΩΛ ≤ 5. For general case, equation (2.34) can be integrated numerically to obtain the

linear growth factor. To this end, it is convenient to rewrite equation (2.34) as (Matsubara

& Szalay, 2003)

f =d lnD

d ln a, (2.35)

d f

d ln a= −f 2 − (1− d lnH

d ln a)f +

3

2ΩM(a) (2.36)

Here f is the growth factor of the linear velocity field.

It is convenient to work with the Fourier transforms of the above equations: For

continuity equation,

δ′(k) = −iθ(k)− i

∫d3q

(2π)3δ(k − q)θ(q)

k · qq2

(2.37)

where θ is divergence of velocity, k · v. For Poisson equation,

k2Φ(k) =3ΩmH

20

2

δ(k)

a. (2.38)

Figure 2.3 shows growth factor of linear density fluctuations, the solution of equation

(2.34) (left) and that of linear velocity field given by equation (2.35) (right). Since the

linear growth rate is normalized to unity at present, shift from EdS case to the upper

side at past time means larger amplitude of initial density fluctuations than EdS. In other


0.01 0.1 1 10 1000

0.2

0.4

0.6

0.8

1

redshit z0.01 0.1 1 10 1000

0.2

0.4

0.6

0.8

1

redshit z

EdS

Figure 2.3: (Left) Plotted are growth factor of linear density fluctuation normalized to

unity at present epoch. The solid(black) and long-dashed(green) lines are ΛCDM model

with ΩΛ0 = 0.74, 0.5 respectively. The short-dashed(red) and dashed-dotted(blue) lines

are QCDMmodel, which means energy density of dark energy is not constant, equivalently

w = −1, with ΩΛ0 = 0.74, 0.5 respectively. We assume w = −1/3 for latter two QCDMmodels. Also plotted with thin dotted(cyan) line corresponds to EdS case in which the

D is proportional to scale factor a. (Right) Plotted are growth of linear velocity field,

f = d lnD/d ln a. All lines describe same models as left panel.

2.3. STATISTICS OF LARGE-SCALE-STRUCTURE 27

words, the larger initial fluctuations are required to realize today’s amplitude of density

fluctuations because the growth suppression should be occurred due to the dark energy.

The effect of dark energy appears more prominently in the velocity field (right panel of

figure 2.3). For the EdS universe, f is always unity. Dark energy also works to suppress

the evolution of velocity field. The amplitude of suppression at present epoch is sensitive

to the present value of dark energy while the epoch of deviation from EdS depends largely

on the equation of state. This can be explained that the larger value of equation of state

makes the dark energy significant at earlier time.

2.3 Statistics of Large-Scale-Structure

2.3.1 Gaussian Random Field

If the field is a random Gaussian, all the statistical properties can be described by second

order statistics, e.g., the quadratic quantities of the field such as the power spectrum and

the two point correlation function (2PCF). The density fluctuation is defined by

δ(x) :=ρ(x)− ρ

ρ, (2.39)

where the ρ(x) is the density at position x and ρ denotes the averaged density over the

whole universe at same epoch. The Fourier coefficients of density fluctuation in a volume

V , δ(x) is in general complex number

δk = Re δk + iIm δk = |δk|eiθk , (2.40)

i.e. the amplitude |δk| and phase θk. The reality of δ(x) constraints the condition on δk

as

δ∗k = δ−k (2.41)

The term, random Gaussian field, means that the phase of δk is random, or not correlated

and the probability of real and imaginary part of δk obeys Gaussian distribution function,

P(w) =√

V

2πσ2kexp

[−w2V

2σ2k

](2.42)

where the variance σ2k = δ2k/2. Thus the probability distribution function of moduli

becomes Rayleigh distribution,

P(|δk|, θk) d|δk| dθk =|δk|V2πδ2k

exp

[−|δk|

2V

2δ2k

]d|δk| dθk (2.43)


If the Fourier quantity |δk| obeys this Rayleigh distribution, real space density field δ(x)

has Gaussian distribution,

P(δ) dδ = 1√2πσ2

exp

[− δ2

2σ2

]dδ (2.44)

2.3.2 Power Spectrum and 2PCF

As can be seen from the previous section, the density fluctuations take various values:

density in some place is larger than average and smaller in other place. Thus in the

cosmological context, the values of density fluctuations themselves have no meaningful

information. Then the statistical treatment enables us to extract invaluable information

about the cosmology. By the definition of density fluctuation, the expectation value of δ

must be vanish,

〈δ(x)〉 = 0 (2.45)

We now focus on the arbitrary one position x1 in which the density fluctuation is δ1and measure the density fluctuation around there with position x2. Let it denotes δ2. If

the universe is perfectly isotropic and homogeneous, δ1 and δ2 is identical independent

from the selection of position : there are no correlations between δ1 and δ2. Even in the

inhomogeneous and anisotropic universe, if the separation of two positions is infinitely

large then statistically δ1 and δ2 can not be thought to have a correlation. In the finite

separation, there must be correlation between two due to the clustering induced by gravi-

tational interaction or other physical effects. Thus the product of two density fluctuations

with the separation being x1 − x2 averaged over the universe must have an cosmological

information, the clustering, or the degree of localized concentration of the field. (see e.g.,

Peebles, 1980)

These intuitions are quantified by the 2PCF, defined as

〈δ(x1) δ(x2)〉 := ξ(x12), (2.46)

where x12 = |x2−x1| is the distance between x1 and x2. Since the universe is statistically

isotropic, correlation function does not depend on the angle. In addition since the universe

is statistically homogeneous, it also does not depend on the position but only on the

separation of two points. Such a statistical treatment can be performed also in the Fourier

space. The Fourier counterpart of density fluctuation is

δ(k) =

∫d3x δ(x)e−ik·x (2.47)


The ensemble average of δ(k) can be calculated as

〈δ(k) δ∗(k′)〉 =∫∫

d3x1d3x2 〈δ(x1) δ(x2)〉 e−i(k·x1−k′·x2) (2.48a)

=

∫∫d3x1d

3x12 ξ(x12)e−i(k−k′)·x1+ik

′·x12 (2.48b)

= (2π)3δD(k − k′)∫

d3x12 ξ(x12)eik′·x12 (2.48c)

= (2π)3δD(k − k′)∫

x2dx4π sin(kx)

kxξ(x) (2.48d)

Thus if we define the power spectrum as

〈δ(k)δ∗(k′)〉 := (2π)3δD(k − k′)P (k), (2.49)

then the well known Wiener-Khintchine relation is obtained. Again from the statistical

isotropy, the power spectrum does not depend on the angles of wave number vectors but

only on the modulus of them.

The cross correlation of two kinds of field, δa, δb can be defined in the same manner,

ξab(x) = 〈δa(x′) δb(x+ x′)〉 (2.50)

Pab(k) = (2π)3δD(k − k′) 〈δa(k′) δb(k′)〉 (2.51)

estimation of power spectrum

In order to evaluate three dimensional correlation function or power spectrum, the

information of three dimensional position is required. For us, it is easy to measure the 2

dimensional angular position in the celestial sphere however the radial information is hard

to obtain since we are fixed at the origin of spherical coordinate. Though we have various

measures to determine the radial distance appropriate for its distance: annual parallax

method, variable star method for the distance to neighborhood objects or measurement

of redshift for cosmologically distant objects. For galaxies with cosmological distance,

identification of radial positions always requires the measurement of redshift. Since the

measurement of redshift requires spectroscopic observation of object, which is in general

difficult, it should be challenging to measure a three dimensional correlation function or

power spectrum.

Recent large spectroscopic galaxy survey such as 2 degree Field Galaxy Redshift

Survey (2dFGRS) or Sloan Digital Sky Survey (SDSS) measures the three dimensional

matter power spectrum using the spectroscopic observation of galaxies. For 2dF, the 3D

power spectrum is measured for galaxies (Peacock, 2001; Percival et al., 2001; Tegmark

et al., 2002; Percival, 2005; Cole et al., 2005) in a complementary way, and for the QSO


Figure 2.4: This figure is from (Cole et al., 2005). The 2dF power spectrum has larger power on large

scale. In other words, the SDSS power spectrum has larger power at small scale normalized on large

scale. Thus the shape parameter of SDSS is significantly larger than that of 2dF. The solid line shows a

model linear power spectrum with Ωmh = 0.168, Ωb/Ωm = 0.17, h = 0.72, ns = 1

(Hoyle et al., 2002; Outram et al., 2003). Many related works are done, e.g. the constraints

on the cosmological parameters combined with CMB data (e.g., Lewis & Bridle, 2002;

Percival et al., 2002; Colombo & Gervasi, 2006). Using Luminous Red Galaxies and main

galaxies of SDSS, the power spectrum is also measured with spectroscopic and photometric

redshift (Tegmark et al., 2004; Hutsi, 2006; Tegmark et al., 2006; Percival et al., 2007b;

Padmanabhan et al., 2007).

As pointed out in Cole et al. (2005); Sanchez et al. (2006), the power spectra of SDSS

(Tegmark et al., 2004) and that of 2dFGRS(Cole et al., 2005) show apparent disagreement

thus the derived shape parameters are out of square: Ωmh = 0.213±0.023 for SDSS, whileΩmh = 0.168± 0.016 for 2dF. Sanchez & Cole (2007) re-analyzed the power spectrum of

both 2dF and SDSS essentially in the identical method to account for the larger power

of 2dF on large scales or equivalently the lager power of SDSS on small scales. They find

out that the SDSS r′ band selectively picked up strongly clustered red galaxies, whichmay have strongly scale dependent bias and also that the existing scale dependent bias

model (Q-model Cole et al., 2005),

Pgal(k) = b21 +Qk2

1 + AkPlin(k) (2.52)

does not work to reconcile the discrepancy. They conclude that for the unbiased estimate


of cosmological parameters, the better understandings about the physical or at least

empirical processes which shape the power spectrum are desired.

2.3.3 Angular Power Spectrum

An angular correlation is useful when we can not obtain accurate measurement of radial

distances to the objects or when it is essentially unavailable. For example, the galaxy

distribution with poor measure of redshift is projected to the celestial sphere with appro-

priately forecasted radial selection function should be measured with angular function.

As an another example, intrinsically CMB temperature fluctuations are projected photon

distribution at the last scattering surface thus we need to measure them with the angular

function.

Any functions f(θ, φ) defined on the sphere can be expanded on the basis of the

spherical harmonics (Laplace’s series).

f(n) =∑,m

amYm(n), (2.53)

The unit direction vector n is set to point the orientation of (θ, φ) thus the angular

dependency can be replaced by n. If f is known, the coefficient am can be immediately

found by the orthogonality integral,

am =

∫dΩ f(n)Ym(n), (2.54)

where dΩ = sin2 θdθdφ is the solid angle element. Note that the sum of equation (2.53)

runs over −(+1) ≤ m ≤ +1 and 0 ≤ ≤ ∞ for the mathematical definition. Practically

the monopole, = 0, determines the overall amplitude and does not contribute to the

anisotropies. The angular power spectrum is defined as

〈ama∗′m′〉 = C δ′δmm′ (2.55)

Consider that the function on the sphere is a consequence of the projection of the field

which actually has three dimensional distribution in the universe. It can be represented

by the line of sight integral with appropriate weight kernel,

f(n) =

∫ rH

0

dr W (r)F (nr), (2.56)

where F is any field defined in a 3-dimensional hyper-surface, and W is the radial weight.

Fourier transforming F and expanding plane wave in a series of spherical wave (Rayleigh


expansion) yields

F (nr) = 4π∑m

(−i)∫

d3k

(2π)3F (k)j(kr)Ym(k)Ym(n)

∗ (2.57)

Thus the angular power spectrum of any combination of two fields is calculated as

δ′δmm′Cab = (4π)2

∑λµ

∑λ′µ′

∫∫dΩ1dΩ2

∫∫dr1dr2

∫∫d3k1(2π)3

d3k2(2π)3

×Wa(r1)Wb(r2)⟨Fa(k1)F

∗b (k2)

⟩jλ(k1r1)jλ′(k2r2)

× Ym(n1)Y∗′m′(n2)Yλµ(k1)Y

∗λ′µ′(k2)Y

∗λµ(n1)Yλ′µ′(n2) (2.58a)

= (4π)2∫∫

dr1dr2

∫∫d3k1(2π)3

d3k2(2π)3

×Wa(r1)Wb(r2)(2π)3δD(k1 − k2)Pab(k1)j(k1r1)j′(k2r2)

× Ym(k1)Y∗′m′(k2) (2.58b)

= (4π)2∫∫

dr1dr2

∫d3k

(2π)3

×Wa(r1)Wb(r2)Pab(k)j(kr1)j′(kr2)Ym(k)Y∗′m′(k) (2.58c)

= δmδ′m′8π

∫∫dr1dr2

∫k2dk Wa(r1)Wb(r2)Pab(k)j(kr1)j′(kr2) (2.58d)

For the multipoles larger than 10 , which roughly corresponds to angular scale of 20

degree, the small angle approximation or flat sky approximation can be applied (Limber,

1954; Peebles, 1980; Afshordi et al., 2004) since the spherical Bessel function picks up the

mode at scale kr. Thus the spherical Bessel function can be replaced by Dirac delta

function as

j(kr) √

π

2+ 1

[δD

−

(kr − 1

2

)+O(−2)

]. (2.59)

Application of this approximation extremely advances the performance of calculation

time because the multiple integral of equation (2.58d) is replaced with the integrand in

the value of most significant contribution. Figure 2.5 shows spherical Bessel functions of

order . In the upper panel, the multipoles are less than 10. It can be seen that the peaks

at kr ∼ have wide spread wings thus tend to mix various scales. While in the lower

panel, the multipoles are greater than 10. As the multipole becomes larger the shape of

peak becomes sharper. Thus it likely curves out the unmixed scale.

Applying equation (2.59), the angular power spectrum is rewritten as

Cab = 4π2

∫d ln r Wa(r)Wb(r)Pab(k)|k=/r. (2.60)


Figure 2.5: Shown are spherical Bessel functions. In the upper panel, the multipoles are

less than 10. It can be seen that the peaks at kr ∼ has wide spread wing thus tends

to mix various scale. While in the lower panel, the multipoles are grater than 10. As

the multipole become larger the shape of peak becomes sharper. Thus it curves out the

unmixed scale.


Figure 2.6: (upper) Solid lines show the contributions from each k mode of logarithmic

interval ln k to the angular power spectrum whose line of sight integral is exactly carried

out(without Limber’s approximation) normalized by C itself. Thus integral of this func-

tion over ln k lands up unity. Dotted lines are that for Limber’s approximation. (Bottom)

The relative error of Limber’s approximation.


multipole l

l(l+

1)C

l /

2π

∆

ΩΛ = 0.74

Ωm = 0.26

h = 0.7

Figure 2.7: Resulting angular power spectrum. Limber’s approximation and equation

(2.58d) differ by less than 10% which is well below the cosmic variance error.

The explicit form is given at §3.2.3 for the weak lensing convergence and given at §5.1 forthe ISW (or RS).

In the figure 2.6 the contributions of integrand from each logarithmic scale interval

ln k to the angular power spectrum C normalized by the resulting power C itself. Note

that the integral these curves over the ln k yields unity. The solid lines and dotted lines

denote the d lnC/d ln k without and with Limber’s approximation respectively. The

weight function here is what for the ISW which is provided at section 5. As can be seen,

the largest mode, = 1, shows at most 10% relative error of Limber’s approximation and

for 10 the error is within 1%. The resulting angular power spectrum is shown in

figure 2.7. As can be seen from this figure, the difference between two is well below the

cosmic variance (light-gray area). The relative error is again within 10% for < 10 and

within 1% for > 10.

2.3.4 Higher Order Moments

Though the inflationary scenario predicts the gaussian initial density fluctuations, it still

remain the possibility of existance of primordial non-gaussianity in the density fluctua-


tions. Moreover the gravitational instability does generate the non-gaussianity due to the

non-linearity of density fluctuations on small scales. The three point correlation func-

tion and its Fourier transformation, bispectrum or the higher point correlations and their

Fourier transformations are suitable tools for probing the non-Gaussianity of the field.

The projected angular three point function is transformed in the analogy of two point

function as,

ξabc(n1, n2, n3) := 〈fa(n1)fb(n2)fc(n3)〉=

∑i,mi

⟨aa1,m1

ab2,m2ac3,m3

⟩Y1m1(n1)Y2m2(n2)Y3m3(n3) (2.61)

where 〈· · ·〉 of the last equality defines the bispectrum,

Babc(m1m2m3123

) :=⟨aa1,m1

ab2,m2ac3,m3

⟩= (4π)3(−i)1+2+3

∫d3k1d

3k2d3k3Y

∗1m1

(k1)Y∗2m2

(k2)Y∗3m3

(k3)

× δD(k1 + k2 + k3)Babc(k1, k2, k3) (2.62)

Note that the bispectrum must satisfy the triangle conditions and selection rules: m1 +

m2+m3 = 0, 1+ 2+ 3 =even, and |i− j| ≤ k ≤ i+ j for all permutations of indices.

From the fact of the rotational invariance of the universe, it is more common to use angle

averaged bispectrum,

Babc123

:=∑

m1,m2,m3

Babc(m1m2m3123

)

(1 2 3m1 m2 m3

)

= (8π)3√(21 + 1)(22 + 1)(23 + 1)

4π

(1 2 30 0 0

)

×∫

k21dk1k22dk2k

23dk3J123(k1, k2, k3)Babc(k1, k2, k3), (2.63)

where (· · · ) are the Wigner 3-j symbol related to the Clebsch-Gordan coefficients (seee.g., Sakurai, 1994; Marinucci, 2005), Babc(k1, k2, k3) is 3-dimensional spacial bispectrum,

and J123 is the integral of product of three spherical Bessel functions,

J123(k1, k2, k3) =

∫x2dxj1(k1x)j2(k2x)j3(k3x). (2.64)

The computation of this badly oscillating integral is relatively rapidly performed with the

use of recurrence relation of Bessel functions

(Friedberg & Martin, 1987; Wang & Kamionkowski, 2000). To obtain equation (2.63),


the Gaunt integral,

Gm1m2m3123

:=

∫dΩY1m1Y2m2Y3m3

=

√(21 + 1)(22 + 1)(23 + 1)

4π

(1 2 30 0 0

)(1 2 3m1 m2 m3

)(2.65)

and the identity

∑m1,m2,m3

(1 2 3m1 m2 m3

)Gm1m2m3123

=

√(21 + 1)(22 + 1)(23 + 1)

4π

(1 2 30 0 0

)

(2.66)

are used. The reduced bispectrum is used in the literature(Komatsu & Spergel, 2001),

which contains all the physical information in the bispectrum,

b123 := Babc123

√4π

(21 + 1)(22 + 1)(23 + 1)

(1 2 30 0 0

)−1(2.67)

or similar quantity to this can be found as Bl1l2l3 in (Magueijo, 2000).


Chapter 3

Observational Probes

3.1 The Rees-Sciama Effect

The Rees-Sciama Effect is one of the mechanisms that generates the temperature fluctu-

ations of the CMB. Most of the temperature fluctuations are generated at the decoupling

epoch, z 1100 however on small scales, they could arise from the inhomogeneous dis-

tribution of foreground large scale structure (LSS). The RS effect brings temperature

fluctuations due to the time variation of the gravitational potential of the large scale

structure, which is induced on large scales by an accelerating expansion due to dark en-

ergy and on small scales by a gravitational collapse of bounded objects, such as clusters

of galaxy. When the CMB photon falls into the gravitational potential of LSS, it gains

the energy, or blueshifts. Whereas when it climbs up the potential well, it loses its en-

ergy, or redshifts. If the potential would not vary with time during the photon passage,

the amount of blueshift and redshift is balanced and the net effects are cancelled out.

However, since the potential in general evolves with time, the net energy gain or loss is

expected.

In the flat universe, temperature fluctuations induced by the RS effect can be ex-

pressed by the line of sight integral of time derivative of gravitational potential,

∆TRST

(n) = −2∫

dr∂Φ(nr; r)

∂r(3.1)

Readers should refer to the Appendix C for the derivation. Below an arcmin scale, this

brings the temperature fluctuations of order ∆TRS/T ∼ 10−8 (Seljak, 1996; Tuluie et al.,1996), which is smaller than that of primary CMB or that of thermal Sunyaev Zel’dovich

effect by order of magnitude three.

The CMB temperature fluctuations we observe are the total fluctuations including

primary fluctuations at decoupling epoch, thermal/kinetic SZ, RS and so on. Thus it is

39

40 CHAPTER 3. OBSERVATIONAL PROBES

likely impossible to isolate the temperature fluctuations generated by the RS effect from

the total CMB temperature fluctuations. We show here only the conceptual idea and will

give a detail in section 5.

3.1.1 Why using Cross Correlation ?

Now our goal is the detachment of the tiny fluctuations of RS effect in the total CMB

temperature fluctuations. As we mentioned above, the RS effect is smaller than other

sources of fluctuations by order of three. We propose to use the statistical method, the

cross correlation. Because the RS effect is generated by the large scale structure, it must

correlate with the matter distribution of LSS. The candidates of quantity that measures

the matter distribution could be the galaxy distribution and statistics of weak lensing. Of

which, the galaxy distribution has crucial disadvantage, the galaxy bias problem. The nat-

ural interpretation of CDM structure formation theory claims that the galaxies are born

at high density regions of the underlying dark matter. Thus the simplest consideration

results in the so called linear bias,

δgal = bδdm (3.2)

i.e. the density contrast of galaxy is proportional to that of dark matter. This idea

is acceptable on large scales however on small scales, where the density contrast is not

so small, δ 1, this relation must no more valid and any predictions of the galaxy

bias do not have the universality, it depends largely on the survey thus on the galaxy

population. Therefore the galaxy distribution is not a reliable or robust tracer for the

matter distribution. Whereas the convergence of weak lensing never suffer from the bias

problem. The convergence is one of the statistics which we explain later in this chapter.

It probes the mass lying between the lensed galaxy and us along the line of sight. Working

in the harmonic space, the observable quantity is the Cross Correlation Angular Power

Spectrum (CCAPS) between the RS and convergence. The expression of CCAPS is

written as

CRS−κ = −22

∫dzs n(zs)

∫ rs

0

drrs − r

r3rsPΦΦ′(k; r)|k=/r (3.3)

Thus for the theoretical prediction of CCAPS, PΦΦ′ , the power spectrum of gravitational

potential and its time derivative, is required. It can be rewritten as

PΦΦ′(k, τ) =

(3

2

Ωm0H20

ak2

)2

[Pδδ′(k, τ)−H Pδδ(k, τ)] . (3.4)

3.2. WEAK GRAVITATIONAL LENSING 41

It is straightforward to calculate this power spectrum within the linear regime since δ′(z) =D′(z)δ(0) = fH δ(z), thus

P linΦΦ′(k, z) =

(3

2

Ωm0

a

H20

k2

)2

[H (z)f(z)− 1]P linδδ (k, z). (3.5)

However we need the CCAPS below the arcmin scale where the power spectrum requires

the non-linear treatment. This is a main problem in this thesis to estimate the CCAPS

on these scales. We tried to evaluate CCAPS in both analytical and numerical manner.

The details of these treatments can be found in the section 4.

The rest of this chapter is devoted to develop the weak lensing theory and statistical

treatment.

3.2 Weak Gravitational Lensing

The light bundle emitted at source object travels along the geodesic line of space time.

Since the geometry around the gravitationally bounded object is locally curved, the pho-

ton path near the object seems to be bent. As can be seen below, since the gravitational

lens is sensitive both to angular diameter distance and density growth, the lens system

with cosmological distance can be used to measure the cosmological information.

Figure 3.1 shows an example of strong lensing event 1. Four multiple images are

clearly appeared. And one can also see the large arc between image A2 and B. For

the weak lensing there never appears such a giant arc or multiple images but just exists

slight distortion in their images or ellipticities. Because in general, the strong lensing

occur only when the location of lens and source is very close in the 2 dimensional sky, the

probability of strong lensing event is small. While for the weak lens, since the light should

be deflected more or less by the large scale structure of the universe during propagation

and its impact parameter must be large, the probability of weak lens event is enormous. In

other words, all of the light we observe is lensed, and distorted at least the inhomogeneous

structures of the universe, the cosmic shear. Figure 3.2 shows Abell 1689 cluster, one of

the predominant examples of the gravitational lens event. This image is taken by Hubble

Space Telescope with Advanced Camera of Survey. The lensed galaxy images are faint

and elongated along the direction tangential to the circle centered at cluster center.

1Images are available at CASTLE : http://cfa-www.harvard.edu/glensdata/


Figure 3.1: An example of strong lens : MG J0414+0534. Source object is quasar at

z = 2.64 and lens is early type galaxy at z = 0.96. This figure is taken by Green Bank

Telescope and cited from Curran et al. (2007). The data of lens systems including this

figure is available at CASTLE

3.2.1 Light Deflection

set up

The situation considered here is summarized in figure 3.3. The mass concentration as a

lens at redshift zd or angular diameter distance dL deflects the light from the source at

redshift zs or angular diameter distance dS. Note that the angular diameter distance dLSis

dLS =1

(1 + zs)

∫ zs

zl

dz′

H(z′). (3.6)

If there is no other deflectors near the light path from the source to us, and if the extent

of deflector mass distribution along the line of sight is quite small compared to dL and

dLS, then the light path which is in fact inflected gradually in the neighbourhood of

deflector can be replaced by two straight lines with a kink at the lens plane. Source

image has in general the finite size, i.e. the extended source. In figure 3.3, subscript 0

denotes the position of center: β0 is 2 dimensional vector pointing to center of source, θ0is that to center of image. Ignoring the extent of image for a while, we can focus on one

representative photon path emitted at source (β), deflected at lens (by α) and observed

by us at the direction θ.


Figure 3.2: Galaxy image is distorted by the gravitational lens, Abell 1689. This

image is observed by Hubble Space Telescope(HST) with Advanced Camera of Sur-

vey(ACS)(Benitez et al., 2002).Note that the bright blurred spheres are lens objects, the

member galaxies of the cluster Abell 1689 and the lensed objects are faint and stretched

along the tangential direction of circle centered at the cluster center.


β

θ

extended source

observer

dL

β 0

θ 0

α0

α dLS

d S

lens

source plane

lens plane

observer plane

distorted image

ξ

η

ζ

Figure 3.3: Schematic illustration of geometry of gravitational lens system. The observer

is set to bottom point, lens is set as light-green circle, source is located at light-blue circle

The line of sight


lens equation

First we consider the light deflection by single point mass M with the separation in the

lens plane being ξ = |ξ|. From the theory of general relativity, the deflection angle is

α(ξ) = 4GMξ

ξ2(3.7)

if the impact parameter ξ is enough larger than Schwarzschild radius, ξ Rs := 2GM .

This quantity is just twice of Newtonian theory’s one (e.g., Schneider et al., 1999).

We then extend the deflection by single mass to multiple point mass deflection. Imag-

ine that N point masses with their masses mi are putted in the lens plane with their loca-

tions ξi. Then the impact parameter vectors for each deflector masses are (ξ−ξi)/|ξ−ξi|2.In fact the mass distribution could be extended along the direction of line of sight how-

ever the protruberance is enough small compared to the distances, dLS, dL. Thus we can

approximate the mass distribution is confined within the lens plane. The total deflection

angle is then represented as the sum of contribution from each masses,

α(ξ) = 4GN∑i

miξ − ξi|ξ − ξi|2

(3.8)

Then we can take the continuum limit of equation (3.8), replacing the sum by an

integral. This is conveniently done by defining dm = Σ(ξ)d2ξ where d2ξ is surface ele-

ment in the lens plane, and Σ(ξ) is surface mass density, which is a projection of mass

distribution integrated along the line of sight onto the lens plane. Thus our assumption

made above that the mass distribution is confined within the lens plane is relaxed. Note

that this treatment still requires the condition that the extent of mass distribution to the

line of sight direction is small. Then the deflection angle becomes

α(ξ) = 4G

∫R2

d2ξ′ Σ(ξ′)ξ − ξ′

|ξ − ξ′|2 (3.9)

This equation is valid if the following conditions are satisfied.

• The gravitational field under consideration is weak, hence the deflection angle mustbe small.

• The mass distribution of lens along the line of sight is small compared to the angulardiameter distances from observer to lens and lens to source.

• The mass distribution of lens must be nearly stationary, i.e. the velocity of matterin the lens must be significantly smaller than the speed of light.


For the astrophysical situations, all these conditions are well satisfied and for the weak

lens analysis one should not be care about the violation of equation (3.9) as is the case

e.g. for the deflection by black holes.

Now we can derive the lens equation which relates the deflection angle to the source

image position. As illustrated in figure 3.3, let η and ξ denote the true position vector

in the source plane and image position in the lens plane respectively. In addition, let

ζ denote the image position projected on the source plane. Then from the definition

of angular diameter distance, ζ = dSθ and ζ = η + αdLS are satisfied. Therefore two

quantities expressing the angular vector θ must be equal, θ = ξ/dL = ζ/dS. This relation

is rewritten as

η =dSdL

ξ − dLSα (3.10)

Again recalling the definition of angular diameter distance, η = dSβ and ξ = dLθ yields

the lens equation

β = θ − dLSdS

α(dLθ) := θ −α(θ) (3.11)

For the system with multiple images this equation has more than one solutions, θ for fixed

β as is the case for strong lensing while for the system with single image, the solution is

unique. This can be conveniently quantified by the dimensionless surface mass density,

κ(θ) :=Σ(dLθ)

Σcr(3.12)

where the critical surface mass density is Σcr = dS/(4πGdLdLS). If κ ≥ 1 or equivalently

Σ ≥ Σcr somewhere in the lens plane, the system has multiple images. This is a sufficient

condition for producing multiple images but not necessary one. The scaled deflection

angle is then rewritten in terms of κ as

α(θ) =1

π

∫R2

d2θ′ κ(θ′)θ − θ′

|θ − θ′|2 (3.13)

As an analogical inference of Poisson equation in 3 dimensional space, we can consider

the 2 dimensional Poisson equation, ∆2Dψ(θ) = 2κ(θ), where ∆2D is 2 dimensional

Laplacian. Integral this Poisson equation over the lens plane, R2 yields

ψ(θ) =1

π

∫R2

d2θ′ κ(θ′) ln |θ − θ′| (3.14)

Equations (3.13), (3.14) imply that the deflection angle must satisfy α = ∇2Dψ.


γ1

γ2

θ1 θ1

θ2 θ2

π/2

Figure 3.4: The complex shear component can be translated into the real elliptical image

rotated by angle φ, a half of complex phase of shear. Plotted are elliptical images to which

the circular source is distorted with shear γ = γ1 (left) and γ = iγ2 (right) respectively.

3.2.2 Statistical Treatment

In the previous section, we are not concerned about the extent of the source or image.

We now deal with the shape of image distorted by tidal field of gravity of the lens object.

In the weak lens analysis the distortion of shape must be treated statistically. In this

section summaries of statistical treatment of distorted shape are given and some statistical

quantities, which are used in the subsequent sections are introduced.

distortion of image

Light rays emitted at different position in the extended source must trace the different

paths, hence the deflection angles must be different. This results in the distortion of

shape of the intrinsic source. According to the Liouville’s theorem, the surface brightness

at source and image must be conserved supposed that ideally there is no scatter and

absorption of photon during the deflection. Let I(s)(β) denote the surface brightness

distribution in the source plane, the observed surface brightness in the lens plane is then

I(θ) = I(s)[β(θ)] (3.15)

Here assuming that the source is much smaller than the angular scale on which the

lens properties change, the lens equation can be locally linearized. Let θ0 denote a vector

pointing the position within the image or correspondingly β0 = β(θ0) is within the source.

We can expand the lens equation into Taylor series around the θ0 as

β = β(θ0) +A(θ0) · (θ − θ0), (3.16)

where A is a Jacobian matrix

A(θ) := ∂β

∂θ=

(δij − ∂2ψ(θ)

∂θi∂θj

)=

(1− κ− γ1 −γ2−γ2 1− κ+ γ1

)(3.17)


source distorted image

convergence only

convergence

and shear

β2

β1 θ1

θ2 O

φA

-1

source plane lens plane

Figure 3.5: Light bundle emitted at source object propagate to us through the locally

curved spacetime by lens object.

For the last step, complex components of shear given below have been introduced.

γ := γ1 + iγ2 = |γ|e2iφ (3.18)

γ1 = (ψ,11−ψ,22 )/2, γ2 = ψ,12 (3.19)

Note that the phase of complex shear is defined as 2φ because we always regard the image

as an ellipsoid hence the real image rotated by φ = π is identical to the original image

(see figure 3.4 and 3.5).

It is more convenient to introduce the reduced shear, g which is also the complex

quantity, g = g1 + ig2 = |g|e2iφ (Bartelmann et al., 2001). The reduced shear is relatedto shear γ and convergence κ as g = γ/(1 − κ). For the perfect circular source, the

eigenvalues of Jacobian matrix (equation (3.17)) give the ratios of radius of source and

semi-axes of elliptical image, as (1−κ)(1±|g|). Thus once the shape of image is measured,in other words the major and minor axes are identified, the amplitude of reduced shear

|g| and orientation of major axis φ can be obtained immediately by

|g| = 1− b/a

1 + b/a(3.20)

with a and b are lengths of major and minor axes.

definition of ellipticities

The center of image is defined by the intensity weighted average of position vector,

θ :=

∫d2θ I(θ)WI [I(θ)]θ∫d2θ I(θ)WI [I(θ)]

, (3.21)


where the example of window functionWI is top-hat function with the threshold being de-

termined by the appropriately chosen brightness intensity. Then the quadrupole moment

of the image is defined as

Qij :=

∫d2θ I(θ)WI [I(θ)](θi − θi)(θj − θi)∫

d2θ I(θ)WI [I(θ)], (3.22)

For a perfect circular image, Q11 = Q22 and Q12 = Q21 = 0 are fulfilled. From this

quadrupole moment, ellipticities can be defined with various manners. One possible

definition is

ε :=Q11 −Q22 + 2iQ12

Q11 +Q22 + 2√Q11Q22 −Q2

12

. (3.23)

Another ways of defining the ellipticities can be found in the literature (Kochanek, 1990;

Miralda-Escude, 1991; Bonnet & Mellier, 1995; Bartelmann et al., 2001).

For the unlensed source object, which in principle can not be observed, quadrupole

moment, Q(s)ij and ellipticity, ε(s) can be defined in a same way by replacing I(θ) and θ by

I(s)(β) and β respectively. For the sources such as galaxies must have intrinsic ellipticities

of themselves so the quadrupole moments and ellipticities for the sources have non zero

values. Recalling that d2β = detA(θ)d2θ, β− β = detA(θ) · (θ− θ), and I(s)(β) = I(θ),

quadrupole moment can be transformed as

Q(s) = AQAT . (3.24)

Using this transformation relation, the ellipticity of source can be mapped as

ε(s) =ε− g

1− g∗ε(3.25)

for |g| ≤ 1(Schneider & Seitz, 1995; Seitz & Schneider, 1997).

estimation of shear

Suppose that ellipticities, or orientations of source galaxies are random. This assumption

is quite natural because there are no special direction in the universe, the isotropy of the

universe. Thus the expectation value of source ellipticity is vanish,

〈ε(s)〉 = 0 (3.26)

Based upon this assumption, the expectation value of observed ellipticities can be thought

to be a net quantity of the shear that is generated by the gravitational lensing.


The inversion of transformation (equation (3.25)) is straightforward

ε =ε(s) + g

1 + g∗ε(s). (3.27)

Let pε(s) ydydϑ be the probability that the source ellipticity ε(s) = ye2iϑ is within ydydϑ

about ε(s). Then for |g| ≤ 1, the expectation value of the n-th moment is given

〈εn〉εs :=∫ 1

0

y dy pεs(y)

∫ 2π

0

dϑ εn = gn (3.28)

by carrying out the contour integral in the complex plane with gn being the residue(Seitz

& Schneider, 1997). Reader should refer to Appendix 1 of Seitz & Schneider (1997) for the

derivation. The advanced property is that the expectation value of any n−th momentsof ellipticity of image does not depend on the source ellipticity probability distribution

function. The equation (3.28) shows that each image ellipticity g provides an unbiased

estimator of the local shear. The uncertainty induced by intrinsic ellipticities is

σε =√〈εsεs∗〉 (3.29)

If we observe N galaxies, the dispersion involved in the expectation value decreases by√N .

3.2.3 Convergence Power Spectrum

In this section, the second order statistics, i.e. the quadratic of shear is introduced. The

flat sky approximation is quite reasonable for the local weak lens analysis. For the weak

lens survey of nearly whole sky, however, the sphericity of the sky can not be ignored

and the flat sky approximation breaks down: the multipole expansion is required. The

multipole expansion is introduced in §3.2.3.

shear – convergence relation

For the local weak lens analysis e.g. mass reconstruction of clusters of galaxy, the sphere

of sky can be regarded as a flat plane. From the equations (3.13) (3.19) the relation

between shear and convergence would be given as the convolution integral,

γ(θ) =1

π

∫R2

d2θ′ D(θ − θ′)κ(θ) (3.30)

where the convolution kernel is

D(θ) =θ21 − θ22 − 2iθ1θ2

θ4(3.31)


(see e.g. Bartelmann et al., 2001) Thus in the Fourier space, this relation is simply ex-

pressed in the multiplication form. Based on the flat sky assumption (see the next section

for the spherical sky treatment), the convergence and shear which are defined on the 2

dimensional plane, is Fourier transformed as

κ() =

∫d2θ κ(θ)ei·θ, γ() =

∫d2θ γ(θ)ei·θ. (3.32)

Thus using the equation (3.30), the relation between complex shear and convergence can

be represented as

γ() =

(21 − 22 + 2i12

2

)κ() = e2iβκ() (3.33)

The coefficient of κ is Fourier counterpart of convolution kernel. Since its amplitude is

unity, convergence and shear is same amplitude and difference between them is only the

complex phase. Recalling the fact that the power spectrum discards the information of

phase, the power spectrum of convergence is identical to that of shear,

〈γ()γ∗(′)〉 = 〈κ()κ∗(′)〉 = (2π)2δD(− ′)P κ() (3.34)

As in the three dimensional power spectrum of matter, the power spectrum of convergence

depends only on the amplitude of wave vector , not on the angle β due to the isotropy

of the universe.

angular correlation

While in the previous, the convergence is a function of 2 dimensional plane, it must

be defined on a sphere of sky. Thus the convergence or shear should be expanded in a

spherical harmonic series. Henceforth the κ and γ are the function of polar and azimuthal

angles of the spherical coordinate.

κ(n) =∑,m

aκmYm(n) (3.35)

Following Jain & Seljak (1997), explicit form of angular power spectrum is derived (see

also Kaiser, 1998; Seljak, 1996, for another derivation). Recall that for the weak lens,

the deflection angle is small and the Jacobian matrix can be expressed with the Born

approximation. This yields that the deflection potential ψ is related to the Newtonian

potential as

ψ(n, r) = 2

∫ r

0

dr′r − r′

rr′Φ(r′n, r′) (3.36)


Here we introduce the argument r of ψ to stress the radial position of the source. This

expression is obtained by the first order perturbation from the non deflected trajectory

in the Born successive approximation. Higher order corrections are made by (Schneider

et al., 1998). Thus the convergence are related to the Newtonian potential as

κ(n, r) =

∫ r

0

dr′r − r′

rr′∇2Φ(rn, r′). (3.37)

Note that although originally κ is defined as the 2D Laplacian of ψ, it can be replaced

by the 3D Laplacian by just adding Φ,33 with 3 denoting the radial coordinate because

the radial curvature of the Newtonian potential does nothing with light deflection after

integrating along the line of sight. Substituting the expression of inverse Fourier transform

of Φ and using the Rayleigh expansion of plane wave (the basis of Fourier expansion), one

yields the angular power spectrum of the convergence,

Cκ (r1, r2) = 4π

∫dk

k

∫ r1

0

dr′1

∫ r2

0

dr′2 ∆2Φ(k, r

′1, r

′2)g(r

′1)g(r

′2)j(kr

′1)j(kr

′2), (3.38)

where the ∆2Φ is dimensionless power spectrum of gravitational potential, g(r) is the kernel

appeared in equation (3.37) and j is spherical Bessel function of order . This is a angular

power spectrum of sources lied at r1 and r2. For practical, the source must have some

sort of distribution pr(r). The angular power spectrum including source distribution is

then number density weighted projection.

Cκ =

∫dr1

∫dr2 pr(r1)pr(r2)C

κ (r1, r2) (3.39)

with the normalization being∫dr pr(r) =

∫dz n(z) = 1

Chapter 4

Non-linear Treatment

In the inflationary scenario, the initial fluctuations of matter, δ are generated such that

the explosive expansion of inflation elongates the quantum fluctuation and turns it to

classical density fluctuation (Kolb & Turner, 1990; Dvali et al., 2004). Since the amplitude

of initial fluctuation is very small, δ 1, one can treat the evolution of δ using the

linear perturbation theory. Here, the term linear stands for the first order deviation from

isotropic and homogeneous background density field.

As time goes on, the gravitational instability makes δ ∼ 1 and even δ 1. The

typical value of δ in the virialized cluster, is of order δvir 180 (Eke et al., 1996). For

such a case, the linear perturbation theory breaks down and some theories describing non-

linearity are required. The one approach involves higher order terms in the perturbation

theory (§4.1). Another approach is to consult a fitting formula to the large set of N -bodysimulations developed by Peacock & Dodds (1996); Smith et al. (2003) (§4.2). The otheris so-called halo approach (§4.3). Figure 4.1 shows non-linear evolution of matter powerspectrum with time. Plotted lines are calculated using the Smith et al.’s fitting formula

(solid line). The dashed lines are result of linear perturbation theory, and symbols show

the N -body simulations by Gadget-2 (Springel, 2005). It can be seen that the linear

theory can no more trace the true amplitude of matter clustering on small scales but the

fitting formula of Smith et al. well describes the non-linearity of matter power spectrum.

We refer to the term, non linearity as the deviation of any quantities from those of linear

theory predictions.

In this chapter, we develop the calculus of non-linear evolution of cross correlation

power spectrum between gravitational potential and its time derivative by applying a

couple of existing non-linear theories, partially including the review of existing theories;

the third order perturbation theory (3PT), fitting formula for the dark matter N -body

simulation, and the halo model. As we described at section 3.1.1, all we need to estimate

is the cross power spectrum of density and its time derivative, Pδδ′(k, z) involving the

53

54 CHAPTER 4. NON-LINEAR TREATMENT

k [h/Mpc]

∆2(k

)

z = 0z = 0.5

z = 2

z = 5

z = 10

Smith et al.2003

linear theory

N-body

Figure 4.1: Non-linear evolution of matter power spectrum.

non-linear treatment. We found that the results from N -body simulations show very

good agreement with the analytical predictions. We will discuss about the non linear

properties of the ΦΦ′ in the last section of this chapter.

4.1 Third Order Perturbation Theory

As a natural extension of linear perturbation theory, one might be tempted to use higher-

order perturbation theory to describe the non-linearity of evolution of matter. The evo-

lution of matter in an Einstein de-Sitter(EdS) background is solved analytically using

the third order perturbation theory (Suto & Sasaki, 1991; Makino et al., 1992; Jain &

Bertschinger, 1994). In a perturbative approach, one should keep a general assumption in

mind that the matter density can be treated as a fluid. Thus we can track the evolution

of matter by solving the continuity equation and Euler equation together with Poisson

equation in an expanding background spacetime. Therefore one should solve δ with θ, a

divergence of velocity in simultaneous equations order by order.

The density fluctuation and divergence of velocity is expanded in a series as,

δ(k, τ) =∞∑n=1

δn(k, τ), θ(k, τ) =∞∑n=1

θn(k, τ). (4.1)

4.1. THIRD ORDER PERTURBATION THEORY 55

If the linear growth of velocity, f(= d lnD/d ln a) is equal to Ω1/2m (z), the solution of

time dependency and scale dependency in any order can be exactly separable (Martel &

Freudling, 1991; Bernardeau et al., 2002, for a review). In the EdS universe, f and Ωm is

entirely equal to unity, thus fulfills the above condition and the solutions for δ and θ are,

δ(k, τ) =∞∑n=1

an(τ)δn(k), θ(k, τ) =∞∑n=1

a′an−1(τ)θn(k). (4.2)

where n−th variables δn and θn are,

δn(k) =

∫d3q1(2π)3

· · · d3qn

(2π)3δD

(n∑i=1

qi − k

)F (s)n (q1, · · · , qn)δ1(q1) · · · δ1(qn) (4.3a)

θn(k) = −∫

d3q1(2π)3

· · · d3qn

(2π)3δD

(n∑i=1

qi − k

)G(s)n (q1, · · · , qn)δ1(q1) · · · δ1(qn) (4.3b)

where F(s)n and G

(s)n are mode coupling kernels which are symmetric for interchanging any

arguments. The next order to the linear regime for P (k) is quartic of δ1, since the power

spectrum is the quadratic quantity of δ or θ. Thus we need to expand δ and θ to the third

order to obtain the next order of power spectrum.

In the ΛCDM model, similar expansions as equation (4.2) can be taken,

δ(k, τ) =∞∑n=1

Dn(τ)δn(k), θ(k, τ) =∞∑n=1

D′Dn−1(τ)θn(k). (4.4)

This expansion is simple extension from EdS to ΛCDM cosmology, replacing the time

dependency from scale factor with the linear growth factor. It is not exactly correct in

the ΛCDM universe however Jeong & Komatsu (2006) has reported that for the density

perturbation, the largest difference in the perturbation variables from the cosmology is

almost entirely encoded into the linear growth factor, and the contribution from other

terms can be less than one percent. Bernardeau (1994) provides exact solutions of density

and divergence of velocity up to third order perturbation using spherical collapse theory.

However we use the equation (4.4) since it is a good approximation and the is simple for

calculation. Suggested the gaussianity of δ1, all the odd order moment is vanish. Thus

the matter power spectrum can be written as,

P (k, τ) = D2(τ)P 11δδ (k) +D4(k)[P 22

δδ (k) + 2P13δδ (k)] (4.5)

where P 11δδ is linear power spectrum defined as 〈δ1(k)δ∗1(k)〉, and P 22

δδ and P 13δδ are quartic


for δ1,

P 22δδ (k) ≡ 〈δ2(k)δ∗2(k)〉

= 2

∫d3q

(2π)3P 11δδ (q)P

11δδ (|k − q|)[F (s)

2 (q,k − q)]2, (4.6a)

2P 13δδ (k) ≡ 〈δ1(k)δ∗3(k)〉

= 6P 11δδ (k)

∫d3q

(2π)3P 11δδ (k)F

(s)3 (q,−q,k). (4.6b)

We give later an explicit form to be integrated.

Differentiation Prescription

For the power spectrum, Pδδ′ one might be tempted to use

Pδδ′ =1

2

∂

∂τPδδ. (4.7)

Using equation (4.5) yields

Pδδ′(k, z) = D′DP11(k, 0) + 2D′D3(P22(k, 0) + P13(k, 0))

= fH P11(k, z) + 2fH (P22(k, z) + P13(k, z)) (4.8)

This leads to the power spectrum of ΦΦ′through the equation

PΦΦ′(k, τ) =

(3

2

Ωm0H20

ak2

)2

[Pδδ′(k, τ)−H Pδδ(k, τ)] . (4.9)

We refer this method to differentiation prescription for the 3PT. We discuss about the

validity of this prescription in the next section.

Momentum Prescription

In order to verify the differentiation prescription, we attempt to calculate Pδδ′ in another

way in the 3PT: using the momentum information. Since the time derivative of density

field is related to the momentum field (velocity field in the linear theory) through the

continuity equation (2.37). This leads the cross correlation between δ and δ′ as,

Pδδ′(k) := 〈δ′(k)δ(k′)〉 = −〈θ(k)δ(k′)〉 −∫

d3q

(2π)3〈δ(k − q)θ(q)δ(k′)〉k · q

q2. (4.10)

= D(τ)D′(τ)P 11δδ (k) +D3(τ)D′(τ)[P 13

θδ (k) + P 31θδ (k) + P 22

θδ (k)+∫d3q

(2π)3B112

δθδ (k, q) +B121δθδ (k, q) +B211

δθδ (k, q)k · qq2

]. (4.11)

Again this yields the cross power spectrum PΦΦ′ using equation (4.9). We call this method

as momentum prescription for 3PT. The explicit forms of each terms are given below.


Comparison

It can be seen analytically that the Pδδ′ from two prescriptions are consistent within

the linear regime. The resulting potential and its time derivative cross power spectrum

obtained by above two prescriptions are plotted in figure 4.2. For the simplicity we plot

only the differentiation prescription power spectrum in the top panel. The ratio of two

is plotted at bottom panel instead of plotting the power from momentum prescription.

Shaded regions denote the relative differences of 10% (dark gray) and 20% (light gray).

It can be seen that on scales larger than k 1h/Mpc, where may be the limitation of

the 3PT, two methods show the good agreement with the relative difference being within

10%. This implies that the differentiation prescription is reliable in the mildly non-linear

scale.

Calculus Detail

In what follows, we show the explicit calculation for two 3PT methods described above.

All the power spectrum and bispectrum to calculate the Pδδ and Pδδ′ are explicitly written

as,

P 13θδ (k) ≡ 〈θ1(k)δ∗3(k′)〉

= −∫∫∫

d3q1(2π)3

d3q2(2π)3

d3q3(2π)3

δD(q1 + q2 + q3 − k′)δD(k − k′)

× 〈δ1(k)δ1(q1)δ1(q2)δ1(q3)〉F (s)3 (q1, q2, q3)

= −P11(k)∫

d3q

(2π)3F(s)3 (q,−q,k)P11(q) (4.12a)

P 31θδ (k) ≡ 〈θ3(k)δ1(k′)〉

= −∫∫∫

d3q1(2π)3

d3q2(2π)3

d3q3(2π)3

δD(q1 + q2 + q3 − k)δD(k − k′)

× 〈δ1(q1)δ1(q2)δ1(q3)δ1(k′)〉G(s)3 (q1, q2, q3)

= −3P11(k)∫

d3q

(2π)3G(s)3 (q,−q,k)P11(q) (4.12b)

P 22θδ (k) ≡ 〈θ2(k)δ2(k′)〉

= −∫∫∫∫

d3q1(2π)3

d3q2(2π)3

d3q3(2π)3

d3q4(2π)3

δD(q1 + q2 − k)δD(q3 + q4 − k′)

× 〈δ1(q1)δ1(q2)δ1(q3)δ1(q4)〉G(s)2 (q1, q2)F

(s)2 (q3, q4)

= −2∫

d3q

(2π)3F(s)2 (q,k − q)G

(s)2 (q,k − q)P11(q)P11(|k − q|) (4.12c)


k [h/Mpc]

Log [

∆2 (

k)]

ΦΦ

'

Figure 4.2: Comparison of PΦΦ′(k) from 3PT of two prescriptions for 4 redshifts. Top panel shows

those from differentiation prescription. The difference of two prescriptions is plotted at bottom panel as

the ratio of two. Shaded regions denote the relative differences of 10% (dark gray) and 20% (light gray).

They agree within 10% at k 1h/Mpc.


B112δθδ (k, q) ≡ 〈δ1(k − q)θ1(q)δ2(k

′)〉

= −∫

d3q1(2π)3

F(s)2 (q1,k

′ − q1)〈δ1(q1)δ1(k′ − q1)δ1(k − q)δ1(q)〉

= −2F (s)2 (q,k − q)P11(q)P11(|k − q|) (4.13a)

B121δθδ (k, q) ≡ 〈δ1(k − q)θ2(q)δ1(k

′)〉

= −∫

d3q1(2π)3

G(s)2 (q1, q − q1)〈δ1(k − q)δ1(q1)δ1(q − q1)δ1(k

′)〉

= −2G(s)2 (k, q − k)P11(k)P11(|k − q|) (4.13b)

B211δθδ (k, q) ≡ 〈δ2(k − q)θ1(q)δ1(k

′)〉

= −∫

d3q1(2π)3

F(s)2 (q1, q − q1)〈δ1(q1)δ1(k − q − q1)δ1(q)δ1(k

′)〉

= −2F (s)2 (k,−q)P11(k)P11(q). (4.13c)

where we use the symmetry of G(s) and F (s) under the exchange of any arguments. These

kernels are given recursively by solving perturbation equation order by order (Fry, 1984).

The kernels for second order variables are given as

F(s)2 (k1,k2) =

5

7+2

7

(k1 · k2)2k21k

22

+k1 · k22

(1

k21+1

k22

), (4.14)

G(s)2 (k1,k2) =

3

7+4

7

(k1 · k2)2k21k

22

+k1 · k22

(1

k21+1

k22

). (4.15)

The kernels for third order variables, F(3)3 , G

(3)3 are rather complicated,

3!F(s)3 (k1,k2,k3) (4.16)

=7

9

k · k3k23

F(s)2 (k1,k2) +

(7

9

k · (k1 + k2)

|k1 + k2|2 +2

9

k2k3 · (k1 + k2)

k23|k1 + k2|2)G(s)2 (k1,k2) + cyclic,

3!G(s)3 (k1,k2,k3) (4.17)

=1

3

k · k3k23

F(s)2 (k1,k2) +

(1

3

k · (k1 + k2)

|k1 + k2|2 +2

3

k2k3 · (k1 + k2)

k23|k1 + k2|2)G(s)2 (k1,k2) + cyclic

They can be reduced to easy form once integrated over the angle (equations (4.18),(4.19)

below). The angular dependencies of all integrands can be put into polar angle: since the

direction of vector k is arbitrary one can always set its direction to z direction of vector

q. The integrands that contain F(s)2 or G

(s)2 include singular points at q → 0,k, so we

decompose the range of integrals in order to avoid the divergence as suggested by


Makino et al. (1992). The integrals that contain G(s)3 (q,−q,k) or F

(s)3 (q,−q,k), over

the angle can be done analytically (Suto & Sasaki, 1991), because the other terms of the

integrands including power spectra are independent of angles between k and q.

∫ 2π

0

dqφ

∫ 1

−1dµF

(s)3 (q,−q,k)

=2π

1512

[12r4 − 158r2 + 100− 42r−2 − 3

r3(r2 − 1)3(2r2 + 7) log

(1 + r

|1− r|)]

(4.18)

∫ 2π

0

dqφ

∫ 1

−1dµG

(s)3 (q,−q,k)

=2π

504

[12r4 − 82r2 + 4− 6r−2 − 3

r3(r2 − 1)3(2r2 + 1) log

(1 + r

|1− r|)]

(4.19)

where r = k/q and µ is cosine of polar angle of q. Using equations from (4.11)-(4.19)

yields a power spectrum of Pδδ′ .

4.2 Derivative of a Fitting Model

In the linear theory the density fluctuation evolves with time independently on scales.

This means that the density fluctuations are factorized as,

δ(k, z) = D(z)δ(k) (4.20)

Thus the δδ′ power spectrum is exactly written as

〈δ(k, z)δ′(k, z)〉 = D(z)D′(z) 〈δ(k)δ(k)〉 = 1

2

dD2(z)

dτ〈δ(k)δ(k)〉 (4.21)

In this case, the ensemble average and time derivative are commutable. On non-linear

scales, the evolution of the density fluctuations of k mode is affected by the neighboring

k′ modes, in principle by all k modes as is obvious from the mode coupling integral of

equation (4.6a),(4.6b). This likely makes impossible to hold the commutativity.

However the fact that the 3PT power spectrum Pδδ′ obtained from continuity equation

(eq. (4.10)) well agrees with that from time derivative (eq. (4.8)) strongly supports the

commutativity hence encourages us to take time derivative of non-linear density power

spectrum. This is one of the most important consequence in this thesis.

4.3. DARK MATTER HALO APPROACH 61

4.3 Dark Matter Halo Approach

In this section, we describe the method to compute the RS-κ cross correlation using the

so-called halo approach. We chose z as a time-variable instead of r in order to avoid

confusion in notation: two quantities are related by equation (2.19)-(2.20).

4.3.1 Matter Power Spectrum

In the halo model, all the dark matter particles are assumed to be encapsulated in the dark

matter halo. Further assumption that the dark matter particles have the spherically sym-

metric distribution in the halo is reasonable. Then the dark matter density distribution

in the halo is supposed to be written as

ρ(r) =ρs

(r/rs)−α(1 + r/rs)3+α. (4.22)

The density profile has r−3 skirt in the outer region, which is confirmed by the high-resolution N -body simulations, and rα cusp in the inner region with the transition oc-

curred at r ∼ rs. This equation assume that the mass profile scaled as r/rs does not

depend on the mass scale of the halo but its characteristic density ρs at r = rs does. The

value of parameter α is proposed α = −1, by Navarro et al. (1997) and further tested by(e.g., Cole & Lacey, 1996; Tormen et al., 1997; Huss et al., 1999). Other models are also

proposed with α = −1.5 (Moore et al., 1998, 1999; Fukushige & Makino, 1997) or α > −1(Colın et al., 1999). The other parametrisation is provided by (e.g., Hernquist, 1990).

For the complete description of halo model, it is commonly adopted to use mass

dependent concentration c = rv/rs as a free parameter, instead of rs. The virial radius rv is

obtained from virialized mass in the halo, M = 4π/3 r3vδvirρ. Note that the characteristic

density ρs is fixed by equating M to the integral of equation (4.22) over 0 ≤ r ≤ rv. For

α > −3, the ρs can be integrated as

ρs =(3 + α)M

4πcαr3v 2F1(3 + α, 3 + α, 4 + α;−c) , (4.23)

where 2F1 is a hypergeometric function. Thus the halo profile can be described totally

by the concentration parameter and the halo mass.

To evaluate the power spectrum, the knowledge of the number density of halo which

has mass M , i.e. dn/dM , a mass function, is required. This can be described as (Seljak,

2000)

dn

dMdM =

ρ

Mf(ν)dν. (4.24)


1halo 2halo

DM h DM particle

Figure 4.3: Dark matter halo: 1halo and 2halo contributions

The variable ν denotes a peak height, ν = [δc(z)σ0(m)]2 (Bardeen et al., 1986), where

δc(z) is the critical over-density for collapse at z. For flat universe, this is calculated as

(Nakamura & Suto, 1997; Eke et al., 1996)

δc(Ωm0, z) =3(12π)2/3

20[1− 0.0123 log(1 + x3)], (4.25)

where x :=(Ω−1

m0 − 1)1/31 + z

For the open, closed and flat case, the explicit form can be summarized in Henry (2000).

σ0(M) is the r.m.s. fluctuation in a sphere that contains mass M within radius R at an

initial time, extrapolated to z by linear theory,

σ2j (m, z) =4π

(2π)3

∫k2dkkjP lin(k, z)W 2(kR) (4.26)

whereW (kR) is a Fourier transform of window function. For the top-hat window function,

W (kR) = 3j1(kR)/kR. The functional form of f(ν) in equation (4.25) is proposed by

(Press & Schechter, 1974). Compared to N -body simulations, it overpredicts the halo

abundance by a factor of two. Sheth & Tormen (1999) modified their mass function as

νf(ν) = A(1 + ν ′−p)√ν ′e−ν′ (4.27)

where ν ′ = aν, and parameters a = 0.707 and p = 0.3 are determined by fitting the

abundance to that of N -body simulations. Note that a = 1, p = 0 correspond to the Press

& Shechter’s mass function. The matter correlation consists of two terms, 1halo term and

2halo term (see Fig. 4.3). 1 halo term is the two point correlation whose points are

selected within a halo, while the 2halo term is that whose points are selected two discrete

halos. The halos which are more massive than non-linear mass scale M∗(:= M(ν = 1))

are more strongly clustered than the matter, while the clustering of less massive halos

are weaker than the matter. This phenomenon causes the discrepancy of power spectra

between halo and matter, known as a halo bias (Cole & Kaiser, 1989; Mo & White, 1996).


The halo bias given by these authors is modified suitable for Sheth & Tormen’s mass

function as (Sheth & Tormen, 1999),

b(ν) = 1 +ν − 1δc

+2p

δc(1 + ν ′p)(4.28)

Since the halo is not point like but has finite extended profile. Thus the 2halo correlation

must be convolved with their halo profile of equation (4.22). Working in Fourier space

reduces the complications of the expression significantly (Scherrer & Bertschinger, 1991):

the convolution is replaced by multiplication. Thus the power spectrum of 2halo term is

the convolution of density fluctuations and Fourier transform of the halo profile with the

weight of mass function and halo bias integrated over all mass scales,

P 2hδδ (k; z) = P lin

δδ (k; z)

[∫dM n(M ; z)

M

ρu(k|M ; z)

]2, (4.29)

where u(k|M ; z) is Fourier transform of halo profile normalized by mass M ,

u(k|M ; z) =4π

M

∫r2dr ρ(r,M ; z)

sin(kr)

kr(4.30)

so that u(0|M ; z) = 1. This correlation become important on large scales and is identical

to linear theory prediction. Further correlation exists on small scales which comes from

the matter clustering within a same halo. This is given by the mass squared weighted

integral of mass function,

P 1hδδ (k; z) =

∫dM n(M ; z)

(M

ρ

)2

|u(k|M ; z)|2 (4.31)

The matter power spectrum is then given as a sum of two terms (Ma & Fry, 2000; Seljak,

2000),

Pδδ(k; z) = P 1hδδ (k; z) + P 2h

δδ (k; z) (4.32)

The three point function, and some third order statistics are found in (Ma & Fry, 2000)

using the halo model.

4.3.2 Momentum Power Spectrum

The momentum representation as in the 3PT is available also in the halo approach. The

momentum power spectrum is rather complex. The velocity of dark matter particle can

be decomposed into two terms : virial motion within a halo and bulk motion of the halo

itself (Sheth & Diaferio, 2001; Sheth et al., 2001),

v = vvir + vhalo. (4.33)


Figure 4.4: This figure is from (Seljak, 2000). Contribution to the P 1h(k) from different halo mass

intervals for the two models α = −1 (top) α = −1.5 (bottom). Short dashed lines from left to right are

1014h−1M < M , 1013h−1M < M < 1014h−1M, 1012h−1M < M < 1013h−1M and 1011h−1M <

M < 1012h−1M. Solid line is the total P (k), dotted the correlated term P 2h(k).


Each components have a rms fluctuation in a quadratic form

σ2v = σ2vir + σ2halo. (4.34)

on the assumption that the random motion within a halo is not correlated with the halo’s

streaming bulk motion. The momentum correlation function is (Sheth et al. unpublished)

ξpp = 〈ρ(x1) [vvir(x1) + vhalo(x1)] ρ(x2) [vvir(x2) + vhalo(x2)]〉 (4.35)

The corresponding power spectrum is given by the Fourier transform of the correlation

function as,

k2P 1hpp (k; z) = k2P 1h

vv (k; z) +D′2(z)∫

dMM2n(M)

ρk2

σ2halo(M ; z)

f2(Ω)H2|u(k|M ; z)|2 (4.36)

k2P 2hpp (k; z) = k2P 2h

vv (k; z) +D′2(z)k2σ2halo(M∗; z)f2(Ω)H2

P 2hδδ (4.37)

and the velocity spectra

k2P 1hvv (k; z) = D′2(z)

∫dM

M2n(M)

ρ2(∆nl/Ω)

k2σ2halo(M ; z)

f2(Ω)H2|W (krvir)|2 (4.38)

k2P 2halovv (k; z) = D′2(z)P lin

δδ (k; z)

[∫dM

Mn(M)

ρW (kR|M)

]2(4.39)

where ∆nl is the density contrast of the virialized halo, for which an accurate fit is given

by formula (Sheth & Diaferio, 2001)

∆nl = 18π2 + 60[Ω(z)− 1]− 32[Ω(z)− 1]2 (4.40)

for a ΛCDM universe.

The dispersion of halo peculiar velocities, σhalo, is assumed to be approximated with

that of peaks in an initial density field extrapolated to z and smoothed on the the relevant

scale, R ∝M1/3 using linear theory(Sheth & Diaferio, 2001),

σhalo(M) = Hf(Ω)σ−1√1− σ40/σ

21σ

2−1 (4.41)

where f(Ω) = d lnD/d lna. Provided that the mass dependence of σhalo is weak, which

can be seen in Fig.4.5, σhalo(M) can be replaced with its value at the non-linear mass

scale M∗. This replacement yields the expression of equation (4.37). Similarly, the crosscorrelation of δ and p are described as a sum of two terms(Cooray & Sheth, 2002). The

1-halo term is given by

kP 1hδp (k; z) = kP 1h

δv +

∫dM

M2n(M)

ρ2σhalo|u(k|M ; z)|2 (4.42)


Figure 4.5: This figure is from (Sheth & Diaferio, 2001). Dependence on halo mass of the nonlinear (σvir)

and linear theory (σhalo) terms in their model. Solid curves show the scaling they assume, and symbols

show the corresponding quantities measured in the z = 0 output time of the ΛCDM GIF simulation.

Error bars show the 90 percentile ranges in mass and velocity. Dashed curve in panel on right shows the

expected scaling after accounting for the finite size of the simulation box. Symbols and curves in the

bottom of the panel on the right show the predicted and actual velocities at z = 20.

and

kP 1hδv (k; z) =

∫dM

M2n(M)

ρ2√∆nl/Ω

σhalo|W (krvir)u(k|M ; z)|. (4.43)

In the linear regime, momentum and density fields are perfectly correlated since the

momentum is identical to velocity and divergence of velocity is scaled as D′δ(k) in thelinear regime. Hence the 2-halo term can be approximated as

kP 2hδp (k; z) =

√P 2hδδ (k; z)k

2P 2hpp (k; z). (4.44)

Note that the time derivative of density field is related to momentum via continuity

equation (eq. (2.37)), δ′ = ik · p. Thus the power spectrum of divergence of momentum

is required to evaluate the time derivative of density field. According to the Helmholtz

theorem, any vector field can be divided into the divergence free term and curl free term,

p = pE + pB, (4.45)

and ∇× pE = 0, ∇ · pB = 0.

In the Fourier space, the divergence of vector is transformed into the inner product with

wave number vector, and the curl of vector is transformed into the outer product. Thus the


divergence component is projected to the direction of wave vector, and curl component is

projected to the plane perpendicular to the wave vector. If the power of vector field on the

non-linear scale is distributed isotropically, then it can be divided into each components by

the ratio div:curl = 1 : 2. While in the linear regime, the momentum field, or equivalently

velocity field must be curl free since v ∝ δkk/k and ∇2Φ ∝ δ lead to v ∝ ∇Φ. Thus thepower of divergence of momentum is equal to the momentum power spectrum on large

scales. On non-linear scales, the power spectrum of divergence and curl component of

momentum can be calculated(Ma & Fry, 2000; Cooray, 2002) (see also Hu, 2000a),

div : k2Ppp(k) = k2P linvv (k) +

k2

3Pδδ(k)

∫dk′

2π2k′2Pvv(k′), (4.46a)

curl : k2Ppp(k) =2k2

3Pδδ(k)

∫dk′

2π2k′2Pvv(k′), (4.46b)

which well represent the intuitive suggestion made above. Figure 4.6 shows the momentum

power spectra obtained from the halo model calculations and the simulations (Cooray &

Sheth, 2002). The Bottom panel shows the contribution of divergence and curl component

to the total momentum power. The simulation support the theoretical calculation of

equation (4.46a) and (4.46b) though they slightly over estimate the curl component on

scales k 0.5h/Mpc.

Figure 4.7 shows the power spectrum of momentum, velocity and their cross correla-

tion calculated from above halo model (Cooray & Sheth, 2002). Bottom panels show the

correlation coefficient, Rpv := Ppv(k)/√Ppp(k)Pvv(k) which indicate that the momentum

and velocity are highly correlated from linear to the fully non-linear scale.

We calculated the momentum, density and their cross correlation power spectrum

which in linear theory are predicted to be perfectly correlated (figure 4.9). The plotted are

divergence component of momentum. Note that the plotted power spectrum is normalized

by f(z)H (z). This normalization is useful because in the linear theory, Pδδ = Pψδ = Pψψunder this normalization, where ψ := ik ·p the divergence of momentum. The correlationcoefficients are shown in figure 4.8, which are defined as

Rtot :=kPpδ(k)√

k2Ppp(k)Pδδ(k), (4.47a)

Rdiv :=Pψδ(k)√

Pψψ(k)Pδδ(k)(4.47b)

The power spectrum k2Ppp is given in equation (4.36), (4.37) and cross correlation kPpδ is

given in equation (4.42)-(4.44). The fact that on the fully non-linear scale the divergence

free component contribute twice as curl free component to the total momentum in the

quadratic form (as is seen from equation (4.46a)) can be described as p2B = 2p2E. Thus


Figure 4.6: This figure is from (Cooray & Sheth, 2002). Power spectrum of the momentum: k2 ∆pp(k)

with divergence and curl component. Filled circles show the sum of the power spectrum of the divergence

(open) and the curl (crosses) of the momentum fields in the simulations. Dashed curves show the linear

theory prediction, and solid curves show halo model prediction for the total power. Bottom panel shows

the fraction of the total power contributed by the divergence (open) and the curl (crosses) components.

This model over predict the divergence of momentum at trans-linear region, k 0.5h/Mpc.

Figure 4.7: This figure is from (Cooray & Sheth, 2002). Cross-spectrum of the momentum and the

velocity for EdS (left) and ΛCDM (right). Filled circles, triangles and stars show ∆pp, ∆vv and ∆pv

respectively. Dashed curves show the linear theory prediction, and dotted curves show the halo model

predictions for the momentum and the velocity, and solid curves are those for the cross correlation.

Bottom panel shows the correlation coefficient for these power spectrum. Solid lines show what their

model predicts.


provided that pB ∼√2pE, then the divergence component of cross correlation, Pψδ(k)

can be approximated as

Pψδ(k) (2−√2)kfH P lin

δδ (k) + (√2− 1)kPpδ(k) (4.48)

This approximation is reasonable only at high redshift where the non-linearity is weak.

While for lower redshifts it significantly over-estimates the cross correlation between δ and

ψ. It can be seen in figure 4.8 that the divergence of momentum and density field is poorly

correlated at high-k, though the halo model predicts that the total momentum vs density

are totally correlated to the fully non-linear scale. The weak correlation between ψ and

δ at high-k is partially accounted for the fact that the divergence component which may

be correlated with density field is one third of total momentum. However the simulation

results pointed out the much weaker correlation than expected. The reason of this has

not been resolved and further investigations are required about the halo model.


z = 9.5z = 4.8z = 2.7z = 1.0z = 0.5z = 0.05

Figure 4.8: (top) The correlation coefficient between total momentum and density given by equation

(4.47a). The total momentum and density is almost perfectly correlated over the all scale. (bottom)

The correlation coefficient between ψ and density (equation (4.47b)). The solid lines are halo model

predictions and symbols are result from N -body simulations. The redshifts are from top to bottom,

z = 9.5, 4.8, 2.7, 1.0, 0.5, 0.05.


z = 9.5 z = 4.5 z = 2.7

z = 1.0 z = 0.5 z = 0.05

k [h/Mpc]

Pδδ(k

), P

δϕ(k

), P

ϕϕ(k

)

Figure 4.9: Momentum–Density power spectrum using Halo model. The red(top), blue(middle) and

black(bottom) symbols for each panels denote the Pψψ, Pψδ and Pδδ normalized by fH The red(dashed),

blue(dotted) and black(solid) lines are predictions from halo model. Pψψ and Pδδ are well described by

the halo model while it overpredicts Pψδ significantly at low redshift.


4.4 Potential Power Spectrum

Here we introduce the notation to simplify the expressions below,

K(z) := 3

2

Ωm0

a

H20

k2, (4.49)

the coefficient of Poisson equation. Since a gravitational potential is related to a matter

density field via Poisson equation, differentiating it w.r.t τ yields

Φ′k(z) = K(z)[δ′k(z)−H δk(z)]. (4.50)

Thus the time derivative of the gravitational potential fluctuation is related to the density

fluctuation and its time derivative. Thus it can be rewritten as

Φ′k(z) = K(z)[ψ(k; z)−H δk(z)], (4.51)

where ψ is a divergence of momentum ik ·p. Then the power spectra of the potential andits time-derivative can all be calculated using the halo model,

P haloΦΦ (k; z) = K2(z)P halo

δδ (k; z) (4.52a)

P haloΦΦ′ (k; z) = K2(z)

[P haloψδ (k; z)−H P halo

δδ (k; z)]

(4.52b)

P haloΦ′Φ′(k; z) = K2(z) [P halo

ψψ (k; z)− 2H P haloψδ (k; z) +H 2P halo

δδ (k; z)]

(4.52c)

It is desired to compare them to other theoretical predictions in order to assure the

theoretical reliability. Only PΦΦ and PΦΦ′ are available for 3PT and Smith et al. fitting

models. These can be written for the 3PT model,

P 3PTΦΦ (k; z) = K2(z)P 3PT

δδ (k; z) (4.53a)

P 3PTΦΦ′ (k; z) = K2(z)

[P 3PTψδ (k; z)−H P 3PT

δδ (k; z)]

(4.53b)

= K2(z)[1

2

∂P 3PTδδ (k; z)

∂τ−H P 3PT

δδ (k; z)

]. (4.53c)

And for the fitting model,

P fitΦΦ(k; z) = K2(z)P fit

δδ (k; z) (4.54a)

P fitΦΦ′(k; z) = K2(z)

[1

2

∂P fitδδ (k; z)

∂τ−H P fit

δδ (k; z)

]. (4.54b)

4.4. POTENTIAL POWER SPECTRUM 73

3PT model

Halo modelSmith fit model

z = 4.8 z = 1.1

0.1 1

z = 0.52

0.1 1

z=0.05

Figure 4.10: Plotted are cross correlation power spectrum between potential and its conformal time

derivative for z = 4.8, 1.1, 0.5 and 0.05. The blue, green and red lines show the theoretical predictions

for 3PT, Smith fit and halo model respectively. For each line, the solid part has a positive sign and the

dotted part represents a negative sign. Linear theory predicts the cross correlation to be negative at all

length scales, which means that Φ and ∂τΦ is anti-correlated over the all scale (dashed lines). Symbols

are results from N -body simulations with boxsize being 250 Mpc/h. open and closed circles are used for

negative and positive values. The adopted cosmology is a Λ CDM with ΩΛ = 0.74. On scales k ∼ 0.1−1,there is a cross over point that the linear anti-correlation, Φ∂τΦ < 0 turns into non-linear correlation,

Φ∂τΦ > 0.


Halo model

PT3 model

Smith fit model

z = 4.8 z = 1.1

z = 0.52 z=0.05

Figure 4.11: Same as figure 4.10 but the boxsize of Nbody simulation is 40 Mpc/h. For 40 Mpc/h

simulation we make three realizations and error bars are 1 σ standard deviation for these 3 realizations.

4.5. N -BODY SIMULATIONS 75

4.5 N-body Simulations

We carry out the N -body simulations to evaluate the non-linear power spectrum of PΦΦ,

PΦΦ′ and PΦ′Φ′ . We employ 5123 dark matter particles in a volume of Lbox = 250h−1Mpcand 40h−1Mpc on a side. The simulations are performed by the GADGET-2 code

(Springel, 2005). The initial conditions are generated by a standard method using the

Zel’dovich approximation. We dump outputs from z = 0.01 to 10 uniformly sampled in

log(1 + z). For each output redshift, we dump adjacent two outputs in order to calculate

δ′ and Φ′.In Fig. 4.12, we summarize the procedures of calculation to obtain the power spectrum

of Φ, Φ′ and their cross correlation. The outputs from the GADGET-2 are position and

velocity of each particle. The density maps are produced from the particle position in the

way described in the Appendix B. The momentum maps are calculated in the same way

as the density maps but just multiply the velocity of a particle as a weight.

We have three methods to calculate ΦΦ′ cross correlation power spectrum and the

agreement of the results from these different schemes will enable us to assure the robust-

ness of our calculation.

4.5.1 Calculus of PΦΦ′

(A) Using Φ & Φ′ mapsThe method which does not involve uncertainties is using Φ and Φ′ maps. Thismethod corresponds rightmost flow in the figure 4.12. First we obtain three density

maps for z = z, z +∆z, z −∆z. For each map, solving Poisson equation in Fourier

space yields the maps of gravitational potential field for three neighboring redshifts.

Then we can take a derivative of gravitational potential by the central difference.

This yields a set of maps of gravitational potential and its time derivative. Fourier

transforming them simultaneously, we have the cross power spectrum of Φ and Φ′

together with the auto correlation power spectrum of them.

(B) Using δ & δ′ mapsNext we can use the equation (4.9) to obtain the PΦΦ′ . This method corresponds

to center flow branched from rightmost procedure in figure 4.12. From the three

density maps, we take a derivative to obtain the δ′ map. Same as in the previous,we have Pδδ, Pδδ and Pδ′δ′ . The equation (4.9) relates these quantities to the PΦΦ′

and corresponding auto correlation.

(C) Using δ & Momentum maps

This procedure corresponds to leftmost flow in the figure 4.12. The GADGET-


N-body outputs

Velocity

Gid averaged

Velocity

Particle

Positions

Density maps

Time deriv. of

Density map

Gid averaged

momentumsPotential maps

Time deriv. of

Potential map

Time deriv. of

Potential map

Time deriv. of

Potential map

FFT

multiply

griddinggridding

theoretically equivalent

(continuity equation)

in the absence of curl

=

PΦΦ'(k)

PΦΦ'(k) = Ωm0( )2[Pδδ'(k,z) - H(z)Pδδ(k,z)]

4

9

H0

ak2

2

differenciatePoisson eq.

FFT

differenciate Poisson eq.

(A)(B)(C)

(C)

(A)(B)

(B)

(C)

Figure 4.12: Flow chart of the N -body simulation procedure.

2 code also outputs the velocity information of all particles. Together with the

position of particles, we can make the vector maps of momentum, p = vρ/ρ by

weighting the velocities to the particle position when we assign the particles to the

lattice (griding).

The obtained power spectra calculated by above three ways show very good agree-

ments which imply our calculation of PΦΦ′ is reliable. Next we compare this reasonably

estimated power spectrum to those calculated by ∂τPΦΦ to test the commutativity of

ensemble average and time derivative, 〈ΦΦ′〉 = ∂τ 〈ΦΦ〉/2.

4.5.2 Consistency check between PΦΦ′ and ∂τPΦΦ

Now we can calculate the derivatives of power spectra, ∂τPΦΦ to compare it with PΦΦ′

obtained by three ways of calculation in the previous subsection. We take the conformal

time derivative of power spectrum numerically as,

1

2

∂

∂τPΦΦ(k, z0) 1

2

PΦΦ(k, z1)− PΦΦ(k, z2)

(z1 − z2)

dz

dτ

∣∣∣∣z=z0

(4.55)

where (z0 − z1)/z0, (z0 − z2)/z0 3%. As can be easily expected, on large scales where

the linear theory is valid, the commutativity is exactly true. The numerical calculation


0.19.3 2.4

Figure 4.13: (Top) Power spectrum of Φ and ∂τΦ calculated from 250h−1Mpc N -body simulations

at various epochs between z ∼ 10 to 0.01, which are logarithmically uniformly sampled; they are, from

black to blue, z = 9.3, 7.2, 5.6, 4.3, 3.2, 2.4, 1.7, 1.1, 0.71, 0.37 and 0.096. Lines with points denote

positive, the non-linear effect, and those without points denote negative, the linear effect. Plotted are

dimensionless power spectrum, k3P (k)/2π. (Bottom) Fractional differences between PΦΦ′ and ∂τPΦΦ/2.

This quantifies the commutativity of ensemble average and time derivative of gravitational potential

directly. The commutativity is guaranteed within 1% error on scales k 0.2h/Mpc, and within 10% on

scales k 1h/Mpc. Scales smaller than k 3h/Mpc, the differences are significant.


0.19.3 2.4

Figure 4.14: Same as figure 4.13, but use 40h−1 Mpc boxsize simulations.


verified this statement well. Figure 4.13, 4.14 show the power spectrum of ΦΦ′ andfractional difference between ΦΦ′ and ∂τPΦΦ. They are calculated using the 250h

−1MpcN -body simulations at various epochs between z ∼ 10 to 0.01, which are logarithmically

uniformly sampled; they are, from black to blue, z = 9.3, 7.2, 5.6, 4.3, 3.2, 2.4, 1.7, 1.1,

0.71, 0.37 and 0.096. In figure 4.13, lower redshifts show good agreement on large scale,

i.e. k < 0.7h/Mpc. On these scales, the non-linearity is significant at higher redshift

because of the suppression of the linear ISW effect. In other words, the non-linearity at

higher redshift is not so strong but linear effect is much smaller than it. On small scales,

i.e. k > 0.7h/Mpc, the non-linearity significantly evolves as redshift decrease. The linear

effect again becomes sub-dominant on these scales and on scales, k < 1h/Mpc the non-

linearity is so strong that the commutativity does not seems to hold. We thus infer that

the linear part of Φ and non-linear part of Φ evolve in different manners, which we will

provide further detailed investigation using third order perturbation calculation. Note

that on scales k < 1h/Mpc, we should keep in mind the fact that the N -body simulations

come under the influence of gridding. Since we sample the 2503h−1Mpc3 volume by 5123

grid points, the sampling critical frequency, Niquist frequency, is kN ∼ 6h/Mpc. Thus we

can trust the result on scales k 3h/Mpc.

In order to check this intuitive consideration, we examine the same quantity using

40h−1/Mpc boxsize simulations (figure 4.14). In this case, the Niquist frequency is kN ∼40h/Mpc, thus we can trust the result on scales k < 10h/Mpc. We find that on scales 1 <

k < 10, the fractional differences between 〈ΦΦ′〉 and ∂τ 〈ΦΦ〉 calculated from 250h−1Mpcsimulations are over-estimated by a factor of 3 to an order of magnitude compared to that

from 40h−1Mpc simulation. We summarize the maximum wavenumber below which the

In summary, the commutativity of ensemble average of time derivative is trivial in the

linear theory and is acceptable in mildly non-linear region as verified also by the 3rd order

PT. However on deeply non-linear scales, the assumption violates: the ensemble average

and time derivative does not commute each other. This suggests that the evolution of each

mode, k is independent in linear theory and is coupled with neighbouring modes though

they are still weak. However the evolution of fluctuations on deeply non-linear scales, the

mode-mode coupling is strong and the way of evolution of each k significantly differs from

mode to mode (We will see how the linear and non-linear part of gravitational potential

fluctuations evolve with time in the next subsection.). It is worth noting however that in

the context of calculating cross correlation of power spectrum between convergence and

Rees-Sciama effect up to ∼ 10000, deeply non-linear regime does not significantly suffer

from the violation of assumption as can be seen in section 5.2.


Table 4.1: Valid Range of Commutativity With 1 and 10 % Error.

redshift 9.3 7.2 5.6 4.3 3.2 2.4 1.7 1.1 0.71 0.37 0.096

kmax(1%)h/Mpc 4.2 3.3 3.2 3.0 2.5 2.1 1.8 1.5 1.4 1.2 1.2

kmax(10%)h/Mpc 23 19 15 12 11 6.5 5.2 4.7 2.2 3.5 4.2

4.6 Non-linear Evolution of Gravitational Potential

In this section, We discuss about the linear and non-linear behavior of PΦΦ′ seen in

figure 4.2. In particular, the scale of cross over point is of great interest. Since in the

EdS universe, the gravitational potential is constant with time and thus show the no

correlation of PΦΦ′ . Even in the ΛCDM universe, the universe is close to EdS at earlier

epoch. Thus the power on large scales at z = 10 is significantly suppressed (black dotted

line of fig. 4.2). Thus the power originates from the non-linear effect appears remarkably

on the large scale, k 0.1h/Mpc. As the redshift becomes smaller, dark energy becomes

dominant and thus on large scales, the linear ISW effect becomes prominent. Third

order perturbation theory enables us to study such an evolution of gravitational potential

rather quantitatively. Recalling that D′ = DfH the linear and non-linear parts of cross-

correlation power spectrum are

P linΦΦ′(k, z) = K2(z)D2(f − 1)H P 11

δδ (k, 0), (4.56)

PNLΦΦ′(k, z) = K2(z)D4(2f − 1)H P2(k, 0). (4.57)

The redshift evolution of each component is D2(f − 1)H (1 + z)2 := Ylin(z) and D4(2f −1)H (1 + z)2 := YNL(z), where the factor (1 + z)2 comes from time dependency of K2(z).This expression is useful in the sense that the non-linear evolution can be treated inde-

pendently of the fluctuation scale.

In a standard ΛCDM paradigm, Ylin is always negative and monotonically decreases

since the peculiar velocity f is slightly less than unity at high redshift, that is, almost

EdS universe, and decreases as redshift goes to zero: f never exceeds 1 and other terms

are all positive. This means the linear ISW effect brings Φ and Φ′ anti-correlation. Ifthe universe is EdS, the linear power spectrum of PΦΦ′ vanishes since f = 1. This is

a natural consequence of the absence of linear ISW effect in the EdS universe. The

non-linear evolution, YNL, is positive at high redshift. Figure 4.15 shows the behavior of

redshift evolution of YNL. As can be seen, YNL increases with time at high redshift and

YNL decreases monotonically after reaching to a peak. If f becomes less than 1/2, YNLdecreases with negative sign. Considered that 〈ΦΦ′〉 = 1/2∂τ 〈ΦΦ〉, PΦΦ′ is interpreted

as the first derivative of ΦΦ. Thus the increase phase of ΦΦ′ means accelerating collapse

4.6. NON-LINEAR EVOLUTION OF GRAVITATIONAL POTENTIAL 81

of the objects with time, the peak of ΦΦ′ is corresponding to the inflection point thatgravitational collapse begins to decelerate, and decrease phase of ΦΦ′ with positive signmeans decelerating collapse and the decrease phase with negative sign means accelerating

dilution of gravitational potential.

We found that the peak location of ΦΦ′ is exactly the point where the ΩΛ = 0.3, and

that the zero crossing of ΦΦ′ occurs when ΩΛ 0.7. This suggests that the accelerating

expansion due to the dark energy dilutes the non-linear gravitational potential as well

as the linear one. Note that in the context of 3PT, the physical interpretation of non-

linear is not straightforward; the non-linear term does not necessarily correspond to the

gravitationally bounded system.

Next we consider how the Φ and ∂τΦ are correlated. The correlation coefficient can

quantify this using the power spectrum as

R3D(k) =PΦΦ′(k)√

PΦΦ(k)PΦ′Φ′(k)(4.58)

Figure 4.16 shows the correlation coefficient for 5 redshifts, z = 9.5, 4.8, 1.1, 0.5 and 0.05

together with the results of N -body simulations for two boxsizes, 250 Mpc/h and 40

Mpc/h. In the linear theory R3D is expected to be -1, namely perfect anti-correlation and

that as the non-linearity is significant, the absolute value of correlation coefficient become

to small. Note however that, the suppression of correlation coefficient does not mean

the suppression of Φ′ but means that the origin of dilation or condensation of Phi is notpure linear effect. Therefore at higher redshift, ∂τΦ can be produced by the gravitational

collapse, a non-linear effect thus shows the weak correlation between Φ. If we look at

the given scale, this tendency is prominent for lower redshift. Put simply, at the scale

k = 0.1h/Mpc, R3D(k) is almost equal to −1 at z = 0.05 and the absolute value of R

becomes smaller. This means that at given scale the non-linearity is significant at earlier

times. This sounds to be unnatural since we know the non-linearity is induced by the

mass assembling due to the gravitational forth as clearly shown in Figure 4.1.

However in this case one should recall the fact that the Φ′ is exactly vanish in the EdSuniverse within the linear regime. In the standard ΛCDM model, the universe is close to

EdS universe at higher redshift and the relative contribution to the power at large scale

is likely dominated by the non-linear effect. If ∂τΦ is generated by the linear effect, i.e.

the accelerating expansion due to dark energy, then the correlation coefficient must show

perfect anti-correlation, R3D = −1. If one looks at scales k < 0.1h/Mpc, R3D is likely to

reach to −1 at lower redshift, which indicates that the linear ISW effect dominates at low

redshift on these scales, and this effect is suppressed as one goes to higher redshifts.

This point can be clearly figured out by figure 4.17 for ΩΛ0 = 0.74 and 4.18 for

ΩΛ0 = 0.1, close to EdS. These figure shows the ratio of PΦΦ′(k, z) and P linΦΦ′(k, z). If the


0.870.740.610.480.35

Tim

e dep

enden

cy o

f P

ΦΦ

'(k)

ΩΛ

(z)

redshift

Figure 4.15: (Top) The evolution of dark energy density, ΩΛ(z) as a function of redshift with 5 present

values. (Bottom) Redshift evolution of PNLΦΦ′ , that is YNL(z) with same 5 model as in top panel. The

triangle symbols denote the inflection points for PNLΦΦ (k, z) and closed circles denote the corresponding

ΩΛ.


power is dominated by the linear effect, they are identical to unity. For the comparison

we also plot the same quantity for Pδδ and Pδδ′ . It can be easily seen that the non

linear contribution to the PΦΦ′ is significant at large scale from early time while it is not

significant for Pδδ. As the time goes on and the ΩΛ become dominant in the universe, the

ISW effect is enhanced thus dominate the power on the large scale.


-1

-0.5

0

0.5

z=9.5

-1

-0.5

0

0.5

z=4.8

-1

-0.5

0

0.5

z=1.1

-1

-0.5

0

0.5

z=0.5

0.1 1 1 0-1

-0.5

0

0.5

z=0.05

k [h/Mpc]

R3

D(k

)

Figure 4.16: We show the correlation coefficient betwenn Φk and ∂τΦk defined by equation (4.58).

From top to bottom, redshift is 9.5, 4.8, 1.1, 0.5 and 0.05 respectively. The solid lines are calculated

based on a halo model following (Cooray & Sheth, 2002). We also plot the results of N−body simulationsof two boxsizes.


0.1

1

10

100z=10 z=9 z=5

0.1

1

10

100z=1 z=0.5 z=0.1

0.01 0.1 1

0.1

1

10

100z=0.01

0.01 0.1 1 10

z=0

k [/Mpc]

Figure 4.17: Plotted are ratio of PΦΦ′ and P linΦΦ′ for ΩΛ0 = 0.74 which means the redshift evolution

of non-linearity of PΦΦ′ . The solid line shows positive and dashed negative. Note that the linear power

spectrum is always negative.


0.1

1

10

100z=10 z=9 z=5

0.1

1

10

100z=1 z=0.5 z=0.1

0.01 0.1 1

0.1

1

10

100z=0.01

0.01 0.1 1 10

z=0

k [/Mpc]

Figure 4.18: Same as figure 4.18 but for the ΩΛ0 = 0.1, close to EdS universe. The non-linear

contribution to the power spectrum is more prominent than that of ΩΛ0 = 0.74 case.

Chapter 5

Search for Rees-Sciama effect:

Detection and implication for Dark

Energy

5.1 Possible Correlating Sources with the LSS

CMB temperature anisotropies on small angular scales, below a few arc-minutes, are

expected to be correlated with cosmic shear, the convergence field. The anisotropies

are generated by various sources including the thermal and kinetic Sunyaev-Zel’dovich

(t/kSZ), late time integrated Sachs Wolfe effect and its non-linear extension, Rees-Sciama

effect and deflection by gravitational lensing. The observed temperature correlations can

be decomposed as

aobsm = aPm + aΘm + atSZm + akSZm + aRSm, (5.1)

where aPm is the spherical harmonic coefficient of the primary anisotropies that are gener-

ated at the last scattering surface of CMB, aΘm is that of lensed CMB, ak/tSZm is from the SZ

effects, and aRSm is from the RS effect. Among these, the primary CMB and cosmic shear

must have no correlation because two anisotropies are generated at very different epochs,

one is induced at z 1100 and the other is at z 10. Thus the primary anisotropy of

CMB can be ignored in the cross correlation spectrum with the tracer of the large scale

structure. The tSZ can be distinguished since the amount of distortion in the photon

intensities depends on the frequency (Fig.5.15.2). Thus it can be in principle removed

from the temperature map using the multi-frequency observation. Henceforth we assume

that the thermal SZ effect can be ignored.

Since the kSZ, RS, and lensed CMB have the same spectral feature, i.e. they do not

depend on frequency, it is impossible to discriminated them using the frequency informa-

87

88 CHAPTER 5. SEARCH FOR REES SCIAMA

−100

−75

−50

−25

0

25

50

75

100

145 GHz 225 GHz 265 GHz

(a) (b) (c)

Figure 5.1: This figure is cited from Sehgal et al. (2005). SZ simulations of a simulated N -body+hydro

cluster before smoothing and adding detector noise. The cluster is about 1015M, has an average gas

temperature of about 9 keV, and is at z=0.43. Figures a, b, and c are of the 145, 225, and 265 GHz bands

respectively. Each figure is about 6’ x 6’ with a pixel size of 0.02’ x 0.02’. The images are converted

to temperature differences from the mean microwave background temperature. The color scale is from

-100µK to 100µK. Primary microwave background fluctuations and point source contamination are not

included. The temperature decreases at the Rayleigh-Jeans region (a), increase at the Wien region (c)

and shift vanishingly near the cross over frequency (b)see Fig.5.2.

1 10 100

-4

-2

0

2

4

6

8

200 400 600 800

frequency [GHz]

spec

tral

dep

enden

cies

thermal

kinetic

217 GHz

Figure 5.2: Solid (red) line and dashed (blue) line show frequency dependence of thermal and kinetic SZ

respectively. For thermal SZ, there are cross over point that the temperature fluctuation is not occurred

near ν = 217 GHz. By contrast, kinetic SZ has the peak at this frequency. Thus this frequency band is

useful to isolate kinetic SZ effect from thermal SZ. However in practical there occurs spectral dependency

shift due to the relativistic effect or non thermal comptonization and remain the residual signal in this

frequency band (Birkinshaw, 1999; Diego et al., 2003, e.g.,).

5.1. POSSIBLE CORRELATING SOURCES WITH THE LSS 89

tion as like tSZ. However, in the linear regime, the kSZ effect and the mass distribution are

not correlated because peculiar velocities of clusters detached from the Hubble flow are

random in their directions along the line-of-sight. However the non-linear cross-correlation

between the kSZ effect and the convergence must be correlated and is calculated by Dore

et al. (2004). They found that the amplitude of correlation is rather small, at most

∆T/T 10−14. As we show below, the RS-κ cross-correlation is much larger, by about

two orders of magnitudes over the whole range of angular scales we consider.

The remaining source of correlation to be considered is lensed CMB. The spherical

harmonic coefficients for the lensed CMB are given by Komatsu & Spergel (2001)

aΘm =∑′m′

∑′′m′′

(−1)m+m′+m′′G−mm′m′′′′′ (5.2)

× ′(′ + 1)− (+ 1) + ′′(′′ + 1)2

aP∗′−m′Θ∗

′′−m′′ ,

where, Θm is the spherical harmonic transform of the deflection angle, Θ,

Θ(n) = −2∫ r∗

0

drr∗ − r

rr∗Φ(nr, r), (5.3)

which is expressed explicitly as

Θm = −8π(−i)∫

d3k

(2π)3Y ∗m(k)

∫ r∗

0

drr∗ − r

rr∗Φk(r)j(kr), (5.4)

with G−mm′m′′′′′ in equation (5.2) being the Gaunt integral given by equation(2.65). This

equation implies that 〈aΘmaκm〉 becomes third order moment and exactly vanishes for theGaussian field. Although there should be some non-Gaussian features in Φ (Zaldarriaga,

2000; Zaldarriaga & Seljak, 1999), the amplitude of the correlation between the lensed

CMB temperature and κ is induced purely non-Gaussianity thus would be thought to

be quite small. In summary, the CMB-κ correlation at small angular scales is likely to

be dominated by the RS-κ correlation. However the strict estimates of the amount of

correlation of κ and lensed CMB is left for the future work.

Note that Θm is related to aκm(z∗) by

aκlm(z∗) = −(+ 1)

2Θm (5.5)

which gives

aκm(z∗) = 4π(+ 1)(−i)∫

d3k

(2π)3Y ∗m(k)

∫ r∗

0

drr∗ − r

rr∗Φk(r)j(kr). (5.6)


This expression agrees with equation (5.13b) when z∗ → zs and ( + 1) → (kr)2. The

latter substitution is known as the ”flat-sky approximation” (Hu, 2000b). It is then easy

to show that for 1

〈aRSmΘ∗lm〉 = 2

∫ rs

0

drrs − r

r3rs

∂PΦ(k; r)

∂r

∣∣∣∣k=/r

, (5.7)

which agrees with equation (10) of Verde & Spergel (2002).

5.2 Angular Correlation Function

The CMB temperature anisotropies induced by a time-varying gravitational potential is

given by

∆T (n)

T= −2

∫ r∗

r0

dr∂

∂rΦ[nSK(r), r], (5.8)

where n is a two dimensional unit direction vector, and r is the conformal lookback time,

with r0 and r∗ denoting the present time and the surface of last scattering, respectively.The distribution of total matter can be probed by gravitational lensing observations.

A useful quantity is the convergence, κ, that is essentially the projection of matter density

along a line-of-sight,

κ(n; zs) =3ΩmH

20

2

∫ rs

0

drSK(r)SK(rs − r)

SK(rs)

δ[SK(r)n; r]

a, (5.9)

where rs ≡ r(zs) is the comoving distance out to a source at z = zs (Bartelmann et al.,

2001). Note that the angular diameter distance is equal to the comoving distance for a

spatially flat Universe, so we can reduce the above expression to simpler form,

κ(n; zs) =3ΩmH

20

2

∫ rs

0

drr(rs − r)

rs

δ(rn; r)

a, (5.10)

To take into consideration a distribution of sources at various redshifts contributing to the

weak gravitational lensing, we convolve equation (5.9) with a source distribution function,

n(zs),

κ(n) =

∫ z∗

0

dzs n(zs)κ(n; zs). (5.11)

The distribution function is normalized such that its integration over redshifts equals

unity. Working in a harmonic space simplify the analysis below. We first expand equa-

5.2. ANGULAR CORRELATION FUNCTION 91

tion (5.8) and (5.10) using the spherical harmonics:

∆TRS(n)

T=

∑,m

aRSmYm(n), (5.12a)

κ(n) =∑,m

aκmYm(n) (5.12b)

with

aRSm = 8π(−i)l∫

d3k

(2π)3Y ∗m(k)

∫ r∗

0

dr∂Φk

∂rjl(kr), (5.13a)

aκm(zs) = 4π(−i)∫

d3k

(2π)3k2Y ∗

m(k)

∫ rs

0

drr(rs − r)

rsΦk(r)j(kr). (5.13b)

Φk is Fourier counterpart of gravitational potential and the power spectrum of Φk is

related to Pδ(k, r) by

PΦ(k, r) =9Ω2m4

H40

k4Pδ(k, r)

a2(r). (5.14)

The cross-correlation power spectrum is then obtained to be

CRS−κ (zs) =

⟨aRSma

κ∗m(zs)

⟩(5.15)

=4

π

∫dk k4

∫ r∗

0

dr

∫ rs

0

dr′r′(rs − r′)

rsPΦΦ′(k; r, r′)j(kr)j(kr′),

where we have used the orthonormality relation of Ym(k), and a relation⟨∂Φk(r)

∂rΦk′(r′)

⟩:= (2π)3PΦΦ′(k; r, r′)δD(k− k′). (5.16)

Finally, we use the Limber’s approximation (Limber, 1954; Afshordi et al., 2004),

2

π

∫dk k4PΦΦ′(k; r, r′)j(kr)j(kr′) (5.17)

≈ l2

r4PΦΦ′

(k =

r; r

)δD(r − r′),

where δD is a Dirac Delta function. The cross-correlation power spectrum is then simpli-

fied to be

CRS−κ (zs) = 22

∫ rs

0

drrs − r

r3rsPΦΦ′(k; r)|k=/r . (5.18)


Figure 5.3: Two models of weak lensing survey. The Model 1 has its peak at very high redshift but

galaxies are lying over the wide range of redshift thus tend to mix wide range of scale for a given multipole.

The model 2 has its steeper peak at relatively lower redshift thus likely to isolate the scale at peak redshift

for a given multipole.

We compute the total cross-correlation, CRS−κ , by integrating equation (5.18) with a

wight of source distribution for a given weak-lensing survey,

CRS−κ =

∫ z∗

0

dzs n(zs)CRS−κ (zs). (5.19)

We refer CCAPS(Cross Correlation Angular Power Spectrum) to CRS−κ henceforth. As in

the literature, the radial distribution of source galaxies can be parametrized as (Efstathiou

et al., 1991),

n(z) = Az2 exp[−(z/z0)β] (5.20)

The values for realistic observation designs are provided by Semboloni et al. (2006) for

CFHTLS and Hennawi & Spergel (2005) for LSST. The normalization factor A is deter-

mined by integrating∫ ∞0

n(z)dz = 1. Thus the distribution n(z)dz can be thought as a

probability of finding the galaxy between z and z + dz. Another parametrization can be

found in the reference (e.g., Hu & Scranton, 2004). For the definiteness we adopted the

two models, (Model 1) Deep Survey, (β, z0) = (0.7, 0.5), whose n(z) peaks at z ∼ 2.2 with

a broad distribution, and (Model 2) Shallow Survey, (β, z0) = (2, 0.9), which peaks at

z 0.9 with a narrower distribution. Figure 5.3 shows normalized distribution function.

5.2. ANGULAR CORRELATION FUNCTION 93

100 1000

0.1

10000

Figure 5.4: CMB-WL cross-correlation spectra, C, for two WL survey designs: Deep (Model 1) and

Shallow (Model 2). (Top left) The symbols show C from the N -body simulation. The open and filled

symbols show C < 0 and C > 0, respectively. The solid and long-dashed lines show the fully non-linear

model for Model 1 and 2, respectively, while the dotted lines show the linear theory predictions. Note

that the linear theory predicts C < 0 at all . (Bottom left) CMB-WL cross-correlation coefficients, R.

(Top right) Third order PT predictions. The solid and dashed lines show two ways of calculating C,

based upon Eq. (4.11) and (4.7), respectively. The fully non-linear calculations are also shown as the

dotted lines (which are exactly the same as the solid and long-dashed lines on the top-left panel). (Bottom

right) Difference between two ways of calculating C. The fractional differences are at most ∼ 10%.


In the top left panel of Figure 5.4, we show the analytical model of C with Smith

et al.’s Pδδ (solid and long-dashed show Model 1 and 2, respectively) and the simulation

results (open and filled symbols show negative and positive values, respectively). The

agreement is good: the model describes the amplitude, shape, and cross-over at ∼ 800

of C that are measured in the simulations. We therefore conclude that the ansatz,

Pδδ′ ≈ P ′δδ/2, is indeed accurate, up to = 5000 where we can trust resolution of our

N -body simulations.

How well are RS and WL correlated? In the bottom left panel of Figure 5.4, we show

the 2-d correlation coefficient, R = C/√Cκ C

RS . We have used the same non-linear Pδδ

to calculate the power spectrum of convergence, Cκ , while we have used the halo model

approach (Cooray & Sheth, 2002) to calculate that of RS, CRS . The RS and κ are strongly

correlated, R 1, in the linear regime. (It is not exactly 1 because of the projection

effect.) The correlation weakens as non-linearity becomes important: R −0.05 ∼ 0.1

at 103 104 for Deep (Model 1), and R −0.01 ∼ 0.03 for Shallow (Model 2).

The weak correlation of C is due to the fact that Φ and Φ′ are not correlated very well

in the non-linear regime: the 3-d correlation coefficient, R3D(k) = PΦΦ′(k)/√PΦ(k)PΦ′(k),

reaches the maximal value, R3D(0.5/Mpc) 0.2, at z = 1, where the RS effect becomes

largest (see figure 4.16). The 2-d correlation, R, is even weaker than R3D(k) because of

the projection effect and a mismatch between the redshift at which WL becomes largest

(z 0.5) and that for RS (z 1).

The weak correlation makes it challenging to measure the CMB-WL correlation from

non-linearity. We quantify the significance of detection of the non-linear Rees-Sciama

effect as well as the linear ISW effect in the section 5.3.

5.2.1 The impact of non-linearity on CCAPS

In the top panel of figure 5.5, we plot the distribution of correlation in the redshift,

the integrand of CCAPS for 5 multipoles which are normalized by the integrated value,

CCAPSa themselves thus integrating these curves by d ln z yields unity. = 10, 109

(solid and dashed) lines show positive correlation thus can be interpreted as a linear ISW

effect. If we look at smaller scale, = 878, long-dashed line, the negative correlation

from low redshifts and positive correlation from high redshifts are cancelled out. Going

on further smaller scales, = 2492, 10015, dot-dashed and long-dot-dashed lines show

negative correlation over the all scale thus can be interpreted as the effect of non-linear

Rees-Sciama. In the bottom panel of figure 5.5 we also plot relations between redshifts

and scales which are traced during integrals. These relations arise due to the projection

effect. Under the Limber’s approximation, relations are exactly bijective, once determined

a multipole. Colored regions denote the scales and redshifts which are affecting to the

5.3. DETECTING THE SIGNATURE OF THE REES–SCIAMA EFFECT 95

total correlation. The widths are determined such that the correlation signal becomes

one third of its peak. Also shown are confidence region within which the commutativity

assumption is valid with blue and red solid lines labeled 10% and 1% respectively. Regions

below those lines are valid with fractional difference being 10% and 1%.

We can conclude that when we calculate cross correlation angular power spectrum

between convergence and Rees Sciama effect, we can use the fitting formular differenciated

with respect to time with accuracy being better than 1% for 2500 and 10% for 104

though the commutativity is invalid on deeply non-linear regime.

5.3 Detecting the signature of the Rees–Sciama Ef-

fect

In the section 5.1, we showed that the RS effect is the dominant source of temperature

anisotropy that contributes to the CMB-κ cross-correlation. We thus need to evaluate the

amplitude of the correlation to see if it can be detected by forthcoming/ideal observations.

The significance of detection can be quantified by the signal to noise ratio. The signal to

noise ratio for cross correlation at each multipole can be expressed as,

(S

N

)2

fskyCov−1

(CRS−κ

)2(5.21)

Cov =CCMB Cκ

+(CRS−κ

)22+ 1

, (5.22)

where fsky is the fraction of sky which both CMB and weak lensing are observed. The

factor 2 + 1 of covariance matrix is coming from the fact that for each multipoles there

are 2+1 harmonic m modes of ams. C is the expected angular power spectrum which is

a sum of the true angular correlation power spectrum and a noise spectrum, C = C+N.

The noise spectra of CMB and convergence are modeled as in (Knox, 1995; Schneider,

2005):

NCMB = σ2pixθ

2fwhm exp[(+ 1)θfwhm/8 ln 2], (5.23)

Nκ = σ2γ/ngal (5.24)

where σpix is the sensitivity to CMB temperature fluctuations in units of background

temperature, θfwhm is the full width half maximum of the gaussian beam size, ngal is the

number density of galaxies observed by the weak lensing survey per unit steradian, and

σγ is the dispersion on the intrinsic ellipticities of lensed galaxies. There exist no Noise

spectrum for the cross correlation since all of the noises associated to each observations


k [

Mp

c-1

]dC

l /

dln

z /

|Cl|

redshift z

l=10015

l=2492

l=878l=109

l=10

1/3

10%1%

Figure 5.5: (Top) The distribution of correlation in the redshift, the integrand of CCAPS for 5 multi-

poles which are normalized by the integrated value, CCAPSa themselves thus integrating these curves by

d ln z yields unity. = 10, 109 (solid and dashed) lines show positive correlation thus can be interpreted

as a linear ISW effect. If we look at smaller scale, = 878, long-dashed line, the negative correlation

from low redshifts and positive correlation from high redshifts are cancelled out. Going on further smaller

scales, = 2492, 10015, dot-dashed and long-dot-dashed lines show negative correlation over the all scale

thus can be interpreted as the effect of non-linear Rees-Sciama. (Bottom) Plotted are relations between

redshifts and scale which are traced during integrals. These relations arise due to the projection effect.

Under the Limber’s approximation, relations are exactly bijective, once determined a multipole. Colored

regions denote the scales and redshifts which are affecting to the total correlation. The widths are deter-

mined such that the correlation signal becomes one third of its peak. Also shown are confidence region

within which the commutativity assumption is valid with blue and red solid lines labeled 10% and 1%

respectively. Regions below those lines are valid with fractional difference being 10% and 1%.

5.3. DETECTING THE SIGNATURE OF THE REES–SCIAMA EFFECT 97

PLANCK nl

ACT nl

Figure 5.6: We plot the cumulative signal-to-noise ratio against angular multipole. The solid line is

calculated from the Smith et al. fitting formula, and the dotted line is for the halo approach with an

assumed observational error. The dashed line is a case without observational errors; this is an ideal,

or the most optimistic case. We consider ACT for CMB and CFHTLS for WL (left), and PLANCK 3

channels for CMB and LSST for WL (right).


Table 5.1: SURVEY PARAMETERSModel 106σPIX θFWHM fsky σγ ng

(arcmin) (/arcmin2)

CFHTLS - - 2.4e-2 0.3 20

LSST - - 0.8 0.1 100

ACT 4.4 1.7 2.4e-2 - -

PLANCK 1.7/2/4.3 10.7/8/5.5 0.8 - -

are never correlated. It is worth noting that, while detection of the integrated Sachs-Wolfe

effect on the linear scale is hampered by cosmic variance, the signal-to-noise (SN) ratio of

the Rees–Sciama effect on the non-linear scale is largely determined by the instrumental

capabilities and shot noise as well as the fraction of sky (to be) observed.

The cumulative signal-to-noise ratio is written as the sum of the S/N at each multipole,

(S

N

)2

=max∑=min

(S

N

)2

. (5.25)

In Figure 5.6, we show the predicted cumulative S/N as a function of the maximum

multipole, max, for cosmic-variance limited CMB and WL (Deep and Shallow) surveys,

as well as for the forthcoming surveys: Planck correlated with LSST, and ACT correlated

with CFHTLS (Nishizawa et al.2007 ApJL submitted). The observational strategies are

summarized at table 5.1. For the Planck, we assumed three lowest channels of High

Frequency Instrument (HFI) detector, which are 100, 143 and 217 GHz.

We find that S/N is totally dominated by the linear contribution (dotted lines) at

3000, and then becomes dominated by the non-linear contribution (dashed lines) at

higher . All-sky CMB and WL surveys can yield S/N ∼ 50 (10) for Deep (Shallow)

WL Survey, whereas 1000 deg2 surveys can only yield S/N ∼ 7 (1). Once noise of the

forthcoming surveys is included, however, S/N from the non-linear contribution becomes

small compared to the linear contribution. For Planck+LSST we find S/N ∼ 1.5 for the

non-linear, and 6 for the linear. For ACT+CFHTLS we find S/N ∼ 0.1 for the non-

linear, and 0.7 for the linear. Therefore, we conclude that these forthcoming surveys are

not expected to yield significant detection of non-linear RS effect.

The right bottom panel of figure 5.7 shows signal to noise ratio squared at each

multipoles , C2Cov−1. Blue lines are those of cosmic variance limited surveys with all

sky(dashed line) and 1000 deg2 sky (solid line) to be observed. We assume the deep weak

lens survey, Model 1 for all sky survey and shallow weak lens survey, Model 2 as described

by equation (5.20). Red lines are same as blue lines with realistic observational error for

5.4. SENSITIVITIES TO THE DARK ENERGY 99

combinations of CFHTLS with ACT (solid) and LSST with Planck (dashed). As can be

seen from this figure, LSST with Planck almost reaches the cosmic variance limited survey

in large scales, thus the detection of linear ISW effect using cross-correlation with mass

distribution enters a phase of its limitation at the Planck era, a next decade. However,

on small scales, the signal to noise ratio is suppressed in particular by the detector noise

of CMB observation at 1000.

5.4 Sensitivities to the Dark Energy

As the linear ISW effect vanishes during the matter dominated era, it is sensitive to dark

energy (DE) (e.g., Boughn & Crittenden, 2004; Nolta et al., 2004; Afshordi et al., 2004).

The non-linear RS effect measures the structure growth, which is also sensitive to DE

(Verde & Spergel, 2002). The cross-over at ∼ 800, at which the linear and non-linear

contributions cancel, is particularly a unique probe of DE. This is one of the main findings

of this thesis.

5.4.1 CCAPS Behavior

It is interesting to see the CCAPS behaviors as the parameters of dark energy vary.

Figure 5.8 shows how the CCAPS behaves as the dark energy parameters are varied.

The top panel show the dependency on ΩΛ0. The left and right panel correspond to the

two weak lensing survey models mentioned above. Each lines from top to bottom are

ΩΛ0 = 0.95, 0.8, 0.74, 0.65, 0.5, 0.35, 0.2, 0.05. The lowest (red) curve is close to the EdS

universe thus the significant suppression of linear ISW can be seen. Whereas on the small

scale (high ) the large matter density aid the accelerating structure evolution thus causes

the larger Φ′ at the non-linear stage. This brings the enhancement of non-linear RS effect.As the ΩΛ0 increases, the linear ISW is boosted and the non-linear structure growth is

suppressed. Note that the cross-over point does not depend largely on the ΩΛ0. This is

the result of somewhat artificial effect, the flat universe assumption, which we will give

an explanation below.

Figure 5.9 show sensitivity of C to DE parameters: ∂C/∂ΩΛ0, ∂C/∂w0, and ∂C/∂w1.

We use a simple form of DE equation of state, w(a) = pΛ(a)/ρΛ(a), given by w(a) =

w0+w1(1− a), the Taylor expansion with respect to scale factor around today (e.g., Lin-

der, 2003). The fiducial values are set to w0 = −1 and w1 = 0, the cosmological constant

model. We find ∂C/∂w0 < 0 and ∂C/∂w1 < 0 at all and that ∂C/∂ΩΛ0 < 0 at almost

entire but has positive part at intermediate scale.

In Fig. 5.10, we illustrate this enigmatic behaviors with the schematic sketch. The

solid black line is the CCAPS for the fiducial cosmological model. If the dark energy


0.1

1

10

100

10 100 1000 10000

0.1

1

10

10 100 1000 10000

(dCl / dΩΛ)2 Cov-1

(dCl / dw0)2 Cov-1

(dCl / dw1)2 Cov-1

(Cl)2 Cov-1

multipole l multipole l

Figure 5.7: (right-bottom) Signal to noise ratio from each multipole: equation (5.25) before taking

sum over multipole. The combination of LSST and Planck likely reaches to cosmic variance limited on

large scales while it loses S/N significantly on smaller scales because of the finite angular resolution of

Planck. The blue solid line shows cosmic variance limited (CVL) survey with 1000 deg2 observation, and

corresponding weak lens survey is shallow one. The blue dashed line shows CVL with fsky = 1, and deep

weak lens survey. The red solid and dashed lines show the combination of CFHTLS with ACT and LSST

with Planck respectively. (left-top) Contributions to fisher matrix given by equation (5.27) from each

multipoles for ΩΛ0. All lines denotes same observational sets as in the right-bottom panel. (left-bottom)

Same as previous but for w0. All lines denotes same observational sets as in the right-bottom panel.

(right-top) Same as previous but for w1. All lines denotes same observational sets as in the right-bottom

panel.


10 100 1000 10 100 1000multipole l multipole l

ΩΛ0 ΩΛ0

w0 w0

w1 w1

Figure 5.8: Plotted are CCAPS for various dark energy parameters. Top, middle and bottom panel

are CCAPS for changing ΩΛ0, w0 and w1 respectively. The values of dark energy parameters are varying

from top to bottom as ΩΛ0 = 0.95, 0.8, 0.74, 0.65, 0.5, 0.35, 0.2, 0.05, w0 = −0.1, −0.25,−0.5,−0.75,−1,−1.5,−2, and w1 = 2, 1, 0.5, 0,−0.5,−1,−2 respectively. The left panel is for the weak lens model 1(deep survey) and right for the model 2 (shallow survey). The discussions on this figure can be found in

the text.


density increases keeping Ωm0 unchanged, the linear ISW is enhanced on the large scale,

i.e. more positive, and non-linear evolution of structure is suppressed due to the dark

energy, i.e. less negative. Thus the signal moves upward over the all scale. While if the

matter density, Ωm0 decreases keeping ΩΛ0 its fiducial value, the radiation–matter equality

time is impeded thus the horizon scale at the equality time must be larger than fiducial

case since keq ∝ Ωm0h2/Mpc. This changes the shape of matter power spectrum to the

leftward below the scale of keq. Assuming the flat universe, as the case of our analysis,

the increase of ΩΛ0 is equivalent to the decrease of Ωm0. Therefore if we increase the ΩΛ0,

Ωm0 decrease by the same amount to keep the flatness of universe, thus CCAPS moves

upward as well as leftward simultaneously.

These views can account for the behavior of ∂C/∂w as well. By increasing w0 or

w1, one makes w(a) less negative which, in turn, makes DE more important at earlier

times. This again makes two things. One is the linear ISW enhancement and the another

is the suppression of non-linear growth of structure. Thus the increase of w is effectively

equivalent to the increase of ΩΛ0. However in this case, the value of w never change the

amount of matter density today, thus does not affect the shape of matter power spectrum.

This is the reason that ∂C/∂w0 and ∂C/∂w1 is positive over the all scale.

5.4.2 Fisher Matrix Forecast

Fisher Matrix is defined as an ensemble average of the second derivative of likelihood

function, thus represents the curvature of likelihood function in the parameter space.

Fαβ :=

⟨∂2 lnL∂θα∂θβ

⟩(5.26)

According to the Cramer-Rao’s theorem (see e.g., O’Hagan & Forster, 2004; Matsubara

& Szalay, 2002), the unbiased estimator must have a finite minimum covariance and it is

identical to the inverse of the Fisher matrix if one use the maximum likelihood estimate.

Thus the Fisher Matrix tells us the limitation of the statistical errors that in principle we

can attain.

Having evaluated the derivatives, we use the Fisher matrix analysis to calculate the

expected constraints on DE parameters, θα = (ΩΛ0, w0, w1), from the CMB-WL correla-

tion. The Fisher matrix for the cross correlation is given by

Fαβ =max∑=2

∂C

∂θαCov−1

∂C

∂θβ(5.27)

where Cov is the covariance matrix (Eq. (5.22)) and max = 104. As can be seen

in equation (5.27), if the ∂C/∂θα is large in other words the C largely depends on the


Figure 5.9: This figure shows the derivatives of CCAPS with respect to dark energy parameters, which

also imply the sensitivities of CCPAS to the dark energy parameters. The sign of these spectrum is

almost positive but has negative part at intermediate scale for ΩΛ0 and all positive for w0 and w1. The

dotted lines are predictions from linear theory. The first turn-up of ΩΛ0 appears also in the linear theory

thus it is a linear theory effect which we give a more detail in the text.


multipole l

ΩΛ increase

Ωm decrease

∆Ωm+∆ΩΛ = 0ISW boost

suppression of

structure formation

Horizon scale at

equality time

become larger

Figure 5.10: This is a schematic illustration how the CCAPS depends on the ΩΛ0. The solid black

line is the CCAPS for fiducial cosmological model. Red dotted line is that for the model with slightly

increased ΩΛ0 keeping Ωm0 unchanged. Blue dotted line is that for model with slightly decreased Ωm0

keeping ΩΛ0 at its fiducial value. If two are occurred at the same time, the resulting CCAPS can be

expected to be a dotted magenta line.

parameter θα, corresponding element of Fisher Matrix becomes large. Thus the constraint

on the θα is expected to be tight.

Figure 5.11 shows the expected 1σ constraints on w0 and w1 from cross correlation

Cl between ISW(+RS) and weak lensing. For left top panel, we assumed cosmic-variance

limited survey with shallow weak lensing galaxy distribution and the sky observed is 1000

deg2. Top right panel also shows constraints from cosmic-variance limited survey with

deep weak lensing galaxy distribution and the all-sky survey. In the bottom panel, we

plot the expected constraints from the forthcoming observational sets, CFHTLS with ACT

and LSST with Planck. For each panel, outer blue lines do not contain the non-linear RS

information. We use linear theory, do not include non-linear theory and set lmax to 300.

Inner red circles include non-linear RS effect and we chose lmax to 10000. If we extend lmaxfurther, we would gain more tight constraints on w0, w1 for the cosmic variance limited

survey, however there is a possibility that on further smaller scales, the kinetic SZ effect

might dominate the cross-correlation signal. Thus we truncate lmax at 10000. For all

panels, we fixed the value of ΩΛ0 to its fiducial value, 0.74 instead of marginalizing it.

When ΩΛ0 is fixed at the fiducial value, we find w0 = −1±1.1 (0.4) and w1 = 0±4.1 (1.1)for Shallow (Deep) WL Survey respectively. The constraints are not very strong however,

the direction of degeneracy line, w1 2(w0 + 1), is about twice as steep as that of Type

Ia supernovae observations (Seo & Eisenstein, 2003; Wood-Vasey et al., 2007). Thus it


may provide an independent cross-check of the results from the observation.

It is more prominent in the Ωm0 − w0 plane. Figure 5.12 gives a good sing of this.

Each lines represent 1,2 and 3 σ confidence region in the Ωm0 − w0 plane. Dotted lines

are constraints from type Ia supernovae (Wood-Vasey et al., 2007) and dashed lines are

those from baryon acoustic oscillation(BAO) (Eisenstein et al., 2005). Solid red contours

are joint constraints of these two observations. We overplot expected confidence regions

expected to be obtained from RS-WL correlation assuming the combination of LSST and

Planck. It can be seen that RS-WL cross-correlation gives direction of degeneracy between

Ωm0 − w0 that is almost orthogonal to that of SNe result, and also differs from that of

BAO result.


Figure 5.11: Expected 1 σ contours on time varying equation of state of dark energy calculated from

fisher matrix analysis, equation (5.26). Light blue contours show the constraints only from the linear ISW

effect; lmax = 300 while blue solid contour lines denote those including non-linear Rees-Sciama effect;

lmax = 10000. Each panel shows the different observational set shown in the panel.


Figure 5.12: Dotted and dashed lines are 1,2, and 3 σ confidence region obtained from SNe and

BAO observations (Wood-Vasey et al., 2007; Eisenstein et al., 2005) assuming a flat universe. All other

cosmological parameters are fixed. We over plot the 1,2, and 3 σ confidence region expected from the

RS-WL cross correlation using LSST and Planck. This can be expected to be an independent measure

of dark energy parameters.


Chapter 6

Concluding Remark

We have studied the cross-correlation between the RS effect and the large scale matter

distribution traced by weak lensing convergence. The RS effect generates tiny tempera-

ture fluctuations of order ∆T/T 10−8. For this reason, cross-correlating it with massdistribution is more promising. The matter distribution differs from galaxy distribution

because of non-linear and/or scale dependent galaxy bias (e.g., Cole et al., 2005), and it

is non-trivial to infer the true mass distribution from the galaxy distribution. Thus we

use the weak lensing convergence, which directly traces the mass distribution along the

line of sight and thus free from the bias problem.

We have developed analytic models to calculate the cross-correlation at non-linear

scales. To this end, we assumed that the time derivative and average over k, the ensemble

average, are commutative up to fully non-linear scale. In other words, the differentiation

prescription is effective. This is valid in the linear theory because the time dependency

and scale dependency can be factorized. However, on the non-linear scale, the evolution

of fluctuation depends on scale and so the validity of the factorization is not obvious.

Several works on the RS effect in the context of the bispectrum (Goldberg & Spergel, 1999;

Spergel & Goldberg, 1999; Verde & Spergel, 2002) implicitly assume this commutativity

without explicitly proving. We have calculated the cross power spectrum using third order

perturbation theory by two different ways: differentiation prescription and momentum

one. The power spectrum obtained by these two method agree within 10% to the non-

linear scale. The obtained result supports the validity to take a time derivative of non-

linear power spectrum. However it remains unclear whether these two calculations are

consistently correct.

Then, in order to estimate the cross power spectrum, we use N -body simulations.

The results of simulations show remarkable agreement with the theoretical prediction

of differentiating the power spectrum obtained by Smith et al’s fitting formula, which

is more accurate than the third order PT in non-linear regime. We conclude that the

109

110 CHAPTER 6. CONCLUDING REMARK

differentiation prescription is valid over the all scales we traced by N -body simulations.

Moreover we have examined the detectability of the RS-κ cross-correlation by a com-

bination of forthcoming CMB and WL observations assuming an ideal observation set

for the cosmic variance limited survey. Overall, it appears rather difficult to establish

a significant detection because of large statistical noise. While next generation facilities

such as LSST and PLANCK are expected to detect non-linear RS effect with at most 2σ

confidence level, statistically we can achieve 50σ for an angular resolution of lmax = 10000.

The cross angular power spectrum shows negative sign, which means anti-correlation

between RS and κ on large scales, and positive sign on small scales. Thus there is a

cross over point at an intermediate scale 800. This scale is sensitive to the nature

of dark energy, in particular to equation of state of dark energy. We propose that the

scale can be used as a unique probe of dark energy. We found from the Fisher matrix

analysis that the expected constraints on w0–w1 is not so strong but the orientation

of degeneracy is twice steeper than that from the joint observation of baryon acoustic

oscillations and supernovae. It will provide an independent cross-check of the results

from these observations.

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Appendix A

Fitting Formulae of Transfer

Function

In this appendix, I summarize fitting formulae of transfer function.

A.1 Peebles 1982

The fitting formula of transfer function is first developed by (Peebles, 1982)

T (k) = (1 + αk + βk2)−1 (A.1)

with α = 6(Θ2.7/h)2Mpc, β = 2.65(Θ2.7/h)

4Mpc2. Θ2.7 is CMB temperature in the unit

of 2.7K.

A.2 Bond Efstathiou 1984

Bond & Efstathiou (1984) and Efstathiou et al. (1992) improve this formula as

T (k) =1 +

[ak + (bk)3/2 + (ck)2

]ν−1/ν(A.2)

with the best fit values, a = (6.4/Γ)h−1Mpc, b = (3/Γ)h−1Mpc, c = (1.7/Γ)h−1Mpc,ν = 1.13 respectively when measuring k in the unit of hMpc−1. In the Einstein de Sitteruniverse, Ωm0 = 1, this fitting function is very accurate, and even in the flat ΛCDM

universe, it remains reasonable. CDM with decaying neutrino with mass 10m10[keV] and

lifetime τd [yr] change the shape parameter as Γ = Ωm0h/[0.861 + 3.8(m10τd)3/2]1/2 and

equation (A.2) is still reasonable.

123

124 APPENDIX A. TRANSFER FUNCTION

A.3 BBKS 1986

Bardeen et al. (1986) provides the another fitting formula drawn from the works (Bond &

Szalay, 1983; Bond & Efstathiou, 1984; Efstathiou & Bond, 1986) classified into 4 types

of situation by the condition of perturbation and species. I refer here only for 2 cases,

CDM and neutrino dark matter without entropy fluctuations.

Case I : Cold Dark Matter

TCDM,DM(k) =ln(1 + 2.34q)

2.34q[1 + 3.89q + (16.1q)2 + (5.46q)3 + (6.71q)4]−1/4 (A.3)

q :=k

Γ hMpc−1(A.4)

for the case that there exists, as a relativistic particles, only neutrino whose flavor

is three and photon. For the baryon, transfer function is filtered due to the acoustic

oscillation,

TCDM,B(k) = TCDM,DM(k)[1 + (kRJ)2/2]−1, (A.5)

where RJ = 1.6/√Ωm0 h

−1kpc is the Jeans scale. For the ΩB, being not small, Bond

& Efstathiou (1984) also gives similar fitting formula for the baryon.

Case II : One species Massive Neutrino Dark Matter

Tν,ad(k) = exp[−0.16(kRf )− (kRf )2/2][1 + 1.6q + (4.0q)3/2 + (0.92q)2]−1, (A.6)

where

q :=k

Γν hMpc−1, Rf = 2.6(Γν)

−1 h−1Mpc. (A.7)

Γν := Ωνh is shape parameter for Massive Neutrino dark matter, and Rf is trans-

lated as characteristic scale of neutrino free streaming below which power spec-

trum causes dumping exponentially. Note that physical amount of neutrino Ωνh2 =

0.31mν/(30eV).

A.4 Eisenstein & Hu 1998

Eisenstein & Hu (1998) takes small scale physics more carefully into consideration based

upon the small scale analytic solution of Boltzmann equation (Hu & Sugiyama, 1996).

A.5. EISENSTEIN & HU NOWIGGLE 1998 125

Especially, they include the effects of baryon, Jeans oscillation and Silk dumping. First

the transfer function for CDM is separated as

TEH(k) =ΩbΩm0

Tb(k) +ΩcΩm0

Tc(k) (A.8)

Note that here Tb(k) and Tc(k) themselves are not true transfer function for their species.

For the CDM contribution,

Tc(k) = fT0(k, 1, βc) + (1− f)T0(k, αc, βc), (A.9)

where

f =1

1 + (ks/5.4)4, (A.10)

T0(k, αc, βc) =ln(e+ 1.8βcq)

ln(e+ 1.8βcq) + Cq2, (A.11)

C =14.2

αc+

386

1 + 69.0q1.08, (A.12)

q =k

hMpcΘ22.7Γ

−1 (A.13)

and s is the scale of sound horizon at recombination epoch. The drag effect, the baryon

contribution to CDM transfer function, becomes

Tb(k) =

[T0(k, 1, 1)

1 + (ks/5.2)2+

αb1 + (βb/ks)3

exp[−(k/kSilk)4]]j0(ks) (A.14)

Readers should refer to Eisenstein & Hu (1998) for the variables not explicitly appear in

this thesis. For the large scale limit, ks 1, k kSilk, Tb(k)→ 1. This reflects the fact

that baryon acts same as collision-less particle over the sound horizon scale.

A.5 Eisenstein & Hu nowiggle 1998

These set of fitting formulae provided by Eisenstein & Hu (1998) are very accurate to

describe the location of peak and trough of baryon acoustic oscillation and very fast to

calculate. In addition to this, they also provide a shape fitting of baryon oscillation which

almost exactly conserve the location of matter-radiation equality.

T0(k) =ln(2e+ 1.8q)

ln(2e+ 1.8q) + [14.2 + 731/(1 + 62.5q)]q2. (A.15)


Figure A.1: Figure is quoted from Eisenstein & Hu (1998). Plotted are relative difference

between transfer function of a series of fitting formulae and that of CMBFast. Black solid

line is fitting formula of EH98. The relative error of EH98 is less than 1% over the wide

range of k, [10−3, 10]h/Mpc.

Here q, is rescaled by the shape parameter, Γ as

Γeff = Γ

[αΓ +

1− αΓ1 + (0.43ks)4

], (A.16)

αΓ = 1− 0.328 ln(431Γh) Ωb

Ωm0

+ 0.38 ln(22.3Γh)

(Ωb

Ωm0

)2

(A.17)

Figure A.1 displays how much this fitting formula is accurate compared to the others.

The deviation from transfer function solved Boltzmann code numerically is less than 1%

over the wide range of k.

One of the most interesting applications of this accurate fitting formula can be found

in a context of analysis of baryon acoustic oscillation. It remains uncertainty on defining

the location of peak and trough of baryon oscillation in a matter power spectrum because

the signature of oscillation appears on a scale 0.1 k where the tilt of matter power,

d lnP/d ln k < 0, damping approximately with power law. In other word, the places

holding dP/dk = 0 do not necessarily correspond to the physical scale of peak and trough

of baryon oscillation. Alternatively one can define the peak location after dividing matte

power spectrum with baryon oscillation by one whose signature of baryon oscillation is

smoothed out (e.g., Hu & Haiman, 2003; Seo & Eisenstein, 2003; Percival et al., 2007a;

Nishimichi et al., 2007).

A.6. EFFECT OF NEUTRINO 127

A.6 Effect of Neutrino

Hu & Eisenstein (1998); Eisenstein & Hu (1999) encode the spacially dependent suppres-

sion due to the neutrino free streaming into the growth rate. So in this case the growth rate

depends both on time and scale. In such a situation, total matter is sum of CDM, baryon

and (massive) neutrino; Ωm = Ωc+Ωb+Ων , alternatively, 1 = fcb+ fν = fc+ fb+ fν with

fi = Ωi0/Ωm0. Focused on the specific species, the transfer function of CDM + baryon

(+ neutrino) can be decomposed into

Tcb(q, z) = Tmaster(q)Dcb(q, z)

D1(z), (A.18)

Tcbν(q, z) = Tmaster(q)Dcbν(q, z)

D1(z), (A.19)

where the wave number q is given at equation (A.13), D1 is growth factor without neutrino

free streaming normalized to unity at equality time, which evolves with time proportional

to a in a matter dominated universe. If dark energy is cosmological constant, D1 can be

expressed in an integral form (e.g., Heath, 1977), or given by useful fitting formula (Lahav

et al., 1991; Carroll et al., 1992) otherwise can be solved numerically (e.g., Matsubara &

Szalay, 2003). The scale dependencies after recombination are encapsulated into Dcb(q, z)

and Dcbν(q, z). Tmaster(q) is time independent universal transfer function; the physics of

scale dependency only before the recombination epoch is incorporated. With the neu-

trino free streaming, growth of density after recombination is logarithmic below the free

streaming scale(Bond et al., 1980) according to the factor,(Hu & Eisenstein, 1998)

pi := (5−√1 + 24fi)/4 (A.20)

where i = cb, c. Then the growth factors in the presence of free streaming are

Dcb(q, z) =

[1 +D1(z)

1 + yfs(q, fν)

]pcb/0.7

D1(z)1−pcb , (A.21)

Dcbν(q, z) =

f0.7/pcb

cb +

[1 +D1(z)

1 + yfs(q, fν)

]pcb/0.7

D1(z)1−pcb . (A.22)

Here, y is free streaming epoch as a function of scale,

yfs(q, fν) = 17.2fν(1 + 0.488f−7/6ν )(Nνq/fν)

2, (A.23)

where Nν is number of flavors of massive neutrino. In the absence of massive neutrinos,

master function Tmaster is exactly same as CDM + baryon case(equation (A.8)). This

agreement can be seen even in the presence of some species of neutrinos on the scale


larger than sound horizon at recombination epoch and than horizon at the epoch when the

significant neutrino become non-relativistic. Hu & Eisenstein (1998) solves the evolution

equations analytically in the small scale limit, and estimate the amount of suppression on

small scale.

αν(fν , fb, yd) =fcfcb

5− 2(pc + pcb)

5− 4pcb (A.24)

× 1− 0.553fνb + 0.126f 3νb1− 0.193√fνNν + 0.169fνN0.2

ν

(1 + yd)pcb−pc

×1 +

pc − pcb2

[1 +

1

(3− 4pc)(7− 4pcb)](1 + yd)

−1

The fitting formula is developed keeping these two limit solutions in mind (Eisenstein &

Hu, 1999). As an analogy of the case of no-wiggle fitting formula, they rescale the shape

parameter as

Γeff = Γ

[√αν +

1−√αν1 + (0.43ks)4

], (A.25)

qeff =Θ22.7Γeff

k

hMpc−1(A.26)

Using this rescaled wave number allows to fit the effective amount of suppression due to

the baryon and neutrino.

Tsup(k) =ln(e+ 1.84βc

√ανqeff)

ln(e+ 1.84βc√ανqeff) + [14.4 + 325/(60.5q1.11eff )]q2eff

(A.27)

where βc = (1 − 0.949fνb)−1. Although this fitting well represents a behavior of trans-

fer function suppressed by baryon and neutrino before the recombination, overestimate

should be occurred due to the incomplete treatment on the transition scale where neutrino

becomes gradually non-relativistic. For the baryon fraction, fb 0.3, this overshoot can

be compensated by the following correction term,

B(k) = 1 +1.2f 0.64ν N0.3+0.6fν

ν

q−1.6ν + q0.8ν, (A.28)

qν = 3.92√Nνq/fν (A.29)

Then the master function is

Tmaster(k) = Tsup(k)B(k). (A.30)

Standard model of element particle predicts the number of generations of neutrino is

three. The measurement of cross section of Z boson reveals that the number of flavors of

A.6. EFFECT OF NEUTRINO 129

light neutrino well agrees with 3 (e.g., Brandt et al., 2000). Recently the solar, atmospheric

man-made and long-baseline neutrino observations such as SuperKamiokande detect the

strong signal of neutrino oscillation, which can be direct consequence of the existence of

mass in neutrino by measuring the delta mass squared(Ahn et al., 2006) in a two flavor

oscillation scenario. This result well agrees with the number of flavors of neutrino. On

the contrary, LSND group (Athanassopoulos et al., 1995; Hill, 1995) report the extra

channel of neutrino oscillation,νµ → νe, which suggests the existence of 4th flavor of

neutrino, sterile neutrino. Okada & Yasuda (1997) gives a review of the constraint on

sterile neutrino from experiments of neutrino oscillation and the impacts on cosmology.

Viel et al. (2005) investigates the effect of sterile neutrino as a candidate of warm dark

matter on the matter power spectrum and give some constraints on mass of sterile neutrino

through the Ly-α forest.


Appendix B

Solving Poisson equation in Real and

Fourier Space : A numerical

instruction

In this chapter I summarize the technical methods of solving the Poisson equation in both

Real and Fourier space. Poisson equation should be solved to calculate the gravitational

potential field from the particle position in the N -body simulation. The Poisson equation

in an expanding spacetime is given by

∇2Φ =3Ωm0

2H20

δ

a(B.1)

The Laplacian operator is that of comoving frame, and δ is matter density in comoving

coordinate. Fourier transforming of this equation yields

Φk =3Ωm0

2

H20

k2δk

a(B.2)

B.1 From Particle to Density Field – Gridding

Suggested that one has an given Np particle position data, (xi, yi, zi) with i = 1, Np in

a comoving coordinate. Before solving Poisson equation, one should construct a density

field from position data. This procedure is called as gridding or assignment of particles

into the grid. Three types of gridding are conventionally used; they are Nearest Grid Point

(NGP), Cloud In Cell (CIC) and Triangular Shaped Cloud (TSC) methods (Hockney &

Eastwood, 1981; Efstathiou et al., 1985). Let Ng, n and x be the number of Grids, 3-

dimensional coordinate of grid and 3-dimensional particle position normalized by boxsize

131

132 APPENDIX B. SOLVING POISSON EQUATION

NGP CIC TSC

Figure B.1: Schematic illustration of three method of gridding. The mass particles con-

tributing to the right bottom point marked by green star is drawn by filled circles. Open

circles are not contributing to that point: the volumes contributing to that point is en-

closed by thick dashed line (and symmetric 8 volumes around the star). Apparently larger

box of TSC than NGP causes the smoothing of density distribution and results in a power

suppression on a small scale.

respectively. Then the density ρ as a function of grid position is

ρ(n/Ng) =N3g

Np

Np∑i=1

W (xi − n/Ng). (B.3)

The vector function W is decomposed into the product of window function of each dimen-

sions, W = wiwjwk. Defining the displacement vector that represents the deviation of

particle position from the center of the cell, as δx = x−n/Ng, each assignment methods

can be expressed as

NGP:

wi = 1, Ng|δxi| ≤ 1/2, (B.4)

CIC:

wi = 1−Ng|δxi|, Ng|δxi| ≤ 1 (B.5)

TSC:

wi =

3/4−N2

g |δxi|2, Ng|δxi| ≤ 1/2

(3/2−Ng|δxi|)2/2, 1/2 < Ng|δxi| ≤ 3/2(B.6)

These methods are schematically illustrated in figure B.1.

B.2. DIFFERENCE SCHEME 133

B.2 Difference Scheme

It is straightforward to calculate gravitational potential field in a real space once obtaining

the density field by means of above gridding methods. In the real space, the differential

equation can be solved numerically by Successive Over Relaxation(SOR) method well

known as the Gauss-Seidel’s method or Jacobi’s method for the special case. Let me

rewrite the Poisson equation as ∇2Φi,j,k = κδijk where κ = 3Ωm0H20/2a and i, j, k denote

the 3D-coordinate in the grid. Replacing the Laplacian by Central Difference yields

Φi,j,k =1

6(Φi−1,j,k + Φi+1,j,k + Φi,j−1,k + Φi,j+1,k + Φi,j,k−1 + Φi,j,k+1 − κδi,j,k) (B.7)

In principle, iterating this infinitely yields a convergence to the collect answer of Φi,j,k.

In practice, the sufficient large number of iteration can achieve a solution with a desired

accuracy. Denoting Φni,j,k the solution after the nth iteration, equation (B.7) can be

rewritten with preferred weight C(1 ≤ C ≤ 2) as

Φn+1i,j,k = Φn

i,j,k + C

[1

6(Φn

i−1,j,k + Φni+1,j,k+

Φni,j−1,k + Φ

ni,j+1,k + Φ

ni,j,k−1 + Φ

ni,j,k+1 − κδi,j,k)− Φn

i,j,k

]. (B.8)

The boundary condition is set to periodic which is consistent with that of Fourier space

solvent. C = 1.7 enables us to converge twice faster than the case of C = 1, which

corresponds to Gauss-Seidel’s method and gives a best performance for this calculation.

B.3 FFT Scheme

Although the SOR method with its weight being most efficient one gives faster convergence

for solving Poisson equation, it costs enormous calculation time compared to FFT scheme.

As seen in equation (B.2), Fourier transform reduce the differential equation into easier

form. All one should do is Fourier transforming the density field and multiplying κ/k2, and

then inverse Fourier transforming. Efficient way of this procedure is given by Efstathiou

et al. (1985) as

Φi,j,k = κ∑p,q,r

Gp,q,rδp,q,r exp[2πi(ip+ jq + kr)/Ng] (B.9)

where the Green function of Laplacian is given by seven point Finite-Difference Approxi-

mation(FDA)

Gp,q,r =

0, i = j = k = 0,

−π/Ng

[sin2(πp/Ng) + sin

2(πq/Ng) + sin2(πr/Ng)

], otherwise

(B.10)


I found both real and Fourier space solvents agree very well though real space method

takes 30 minutes for Ng = 643 while Fourier space method takes 10 seconds for same

number of grid. This might scales as ∼ N/ log(N).

B.3. FFT SCHEME 135

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

-1

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

NGP CIC

TSC

Figure B.2: Shown maps are same particle distribution assigned by NGP, CIC and TSC

scheme. One can see that the NGP keeps the small scale structure the most and TSC the

worst. TSC scheme smooth out the structure approximately smaller than the grid scale.


Figure B.3: This figure illustrate the fact described in the text and figure B.2 more

apparently. Higher order interpolating makes smooth out the small scale structure.

Appendix C

Derivation of integrated Sachs Wolfe

effect

In this appendix, I derive the Integrated Sachs Wolfe (Rees Sciama) effect mainly based

on (Sachs & Wolfe, 1967) in the conformal Newtonian gauge representation.

To begin with, consider the decomposition of metric perturbation into scalar, vector

and tensor mode. The vector quantity h0i ≡ Bi can be divided into the contribution from

scalar and others. The tensor quantity hij ≡ Cij can be divided into the contribution

from scalar, vector and the part proportional to the background spacial metric,

h00 ≡ −2a2A (C.1)

h0i ≡ −a2Bi = −a2(B(s)|i +B

(v)i ) (C.2)

hij ≡ 2a2Cij = 2a2(Dγij + C(s)|ij +

1

2[C

(v)i|j + C

(v)j|i ] + C

(T )ij ). (C.3)

Note that subscript | denotes the 3 dimensional covariant derivative, i.e.Xi|j = Xi,j +

(3)ΓkijXk, (C.4)

(3)Γijk =1

2γia(γja,k + γka,j − γjk,a)). (C.5)

Now one should remind the condition that the scalar quantity can not be generated from

the vector mode and that the vector and scalar quantity can not be generated from the

tensor mode,

B(v)|ii = 0, (C.6)

C(v)|ii = 0, C

(T )ii = 0, C

(T )|iij = 0. (C.7)

Since the Newtonian(longitudinal) gauge fixes variables as B(s) = C(s) = 0, the gauge in-

variant variables are Φ = A,Ψ = D. Here suppose that only the scalar mode is considered

137

138 APPENDIX C. DERIVATION OF ISW

and ignoring the anisotropic stress, Φ = −Ψ, then the metric becomes

ds2 = a2[(1 + 2Φ)dτ − (1− 2Φ)γijdxidxj]. (C.8)

Denoting the wave-vector which is tangential to the photon geodesic as

kµ :=∂xµ

∂w, (C.9)

the non perturbed geodesic equation becomes

− d2 0τ

dw2=1

2γ′ij 0k

i0k

j = 0 (C.10)

− d2 0xi

dw2= (3)Γijl 0k

j0k

l = 0. (C.11)

The solutions of this equation are

0τ = τ0 − w (C.12)

0xi = eiw, (C.13)

where w is affine parameter, and ei is a unit vector pointing to the ki. The subscript

0 in the left bottom of variable denotes the unperturbed quantity and 1 is first order

perturbation quantity. The first-order of geodesic equations are

d

dw

(d 1τ

dw− 2Φ

)=

∂Φ

∂τ(1 + γij 0k

i0k

j) (C.14)

d

dwγij

(2Φ 0k

j − 0kj)= Φ,i + (Φγjl),i + 0k

j0k

l. (C.15)

Integrating yields

d 1τ

dw= 1k

0 = 2

[Φ +

∫ w

0

∂Φ

∂τdw′

], (C.16)

d 1xi

dw= 1k

i = 2

[Φei −

∫ w

0

∂Φ

∂τdxi

]. (C.17)

Next, we will see how the four velocity can be expressed up to the first order. The uniform

component is

0uµ = a(1, 0, 0, 0), 0uµ = 0. (C.18)

The sum of zero-th and first order yields

uµ = a(1 +X, vi), (C.19)

139

where I put vi = ui/u0 is 3D velocity. Note that the inner product in any coordinate

system must be unity, uµuµ = 1. Thus we have

uµ = a(1− Φ, vi), (C.20)

uµ = a−1(−1− Φ, vi). (C.21)

Recalling that the redshift is defined as the ratio of photon energy at emitter and receiver

projected to the fluid, and keeping in mind that we are moving with the cosmological

fluid, then

z + 1 =(kµuµ)w=τ0−τ∗(kνuν)w=0

, (C.22)

where asterisk ∗ denotes the epoch of decoupling. Substituting equation (C.20) into

equation (C.22) yields

z + 1 =a0a∗

[1 + Φ(τ0)− Φ(τ∗) + 2

∫ τ0−τ∗

0

∂Φ

∂τdw

]. (C.23)

As the temperature of CMB redshifts as

T0 =1

1 + zT∗ = T∗

a∗a0

(1 +

∆T

T

)(C.24)

Thus the temperature anisotropy becomes

∆T

T= −Φ(τ0) + Φ(τ∗)− 2

∫ τ0−τ∗

0

∂Φ

∂τdτ. (C.25)

The first term is the monopole which we can in principle not observe and second term

represent the temperature anisotropy caused by the gravitational redshift due to the

potential fluctuations at last scattering surface, the Sachs Wolfe effect. The integral

term represent the temperature anisotropy caused by the time variation of gravitational

potential integrated along the line of sight, the integrated Sachs Wolfe effect.

-Ï È Õ department of physics and …member.ipmu.jp/atsushi.nishizawa/study/phd_thesis.pdfi am...

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