Cosmology with CMB Anisotropy - iucaa

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Certificate of the Guide 

CERTIFIED that the work incorporated in the thesis "Cosmology with CMB Anisotropy" 

Submitted by Mr. Amir Hajian Forushani was carried out by the candidate under my 

supervision/guidance. Such material as has been obtained from other sources has been 

duly acknowledged in the thesis. 

Prof. Tarun Souradeep 

(Thesis Supervisor) 

Date: 

iii

Declaration 

I declare that the thesis entitled "Cosmology with CMB Anisotropy" submitted by 

me for the degree of Doctor of Philosophy is the record of work carried out by me 

during the period from January 2002 to January 2006 under the guidance of Prof. 

Tarun Souradeep and has not formed the basis for the award of any degree, diploma, 

associateship, fellowship, titles in this or any other University or other institution of 

Higher learning. 

I further declare that the material obtained from other sources has been duly acknowledged 

in the thesis. 

Amir Hajian Forushani 

Date: 

v

I gratefully dedicate this thesis to my parents, 

whose love and support I will forever be thankful. 

And to my wife, Sanaz, 

for her love, support, encouragement and for giving me a happy and 

complete life. 

vii

Abstract 

One of the main challenges in the study of the Cosmic Microwave Background (CMB) 

anisotropy is to develop methods to extract maximum information from the rich source 

of information provided by this random field on the sky. When the condition of statistical 

isotropy (SI) holds, the widely used angular power spectrum has proved to be a very 

useful quantity which contains the whole information encoded in the CMB anisotropy 

field. However, there are certain mechanisms that violate the SI condition. This thesis 

is dedicated to the study of this fundamental assumption in CMB studies. We have 

formulated the problem mathematically, constructed a quantitative method to describe 

SI violation (bipolar power spectrum), studied number of sources of violation of SI 

condition and finally examined this fundamental assumption using the most recent full 

sky CMB anisotropy maps provided by the first-year data of Wilkinson Microwave 

Anisotropy Probe (WMAP). The main conclusions are: 

• Statistical isotropy of CMB anisotropy is an important property of CMB anisotropy 

field which can be quantified and checked in distinct from Gaussianity of the field. 

Bipolar Power Spectrum (BiPS) is an orientation independent quantity which 

is sensitive to structures and patterns in the underlying two-point correlation function 

and is a very useful tool for searching for departures from SI. 

• We find no strong evidence for SI violation in the full sky CMB anisotropy maps 

based on the first-year data of WMAP up to l ∼ 60 − 70. 

• The possibility of compact spaces in cosmic topology is a theoretically well motivated 

possibility that has also been observationally targeted. The cleanest signaix

ture of the topology of the universe is written on the microwave sky. The existence 

of preferred directions along the highly correlated spots is the smoking gun of compact 

spaces and can be exploited to detect them. We propose BiPS as a fast and 

orientation independent method to look for such signatures in CMB anisotropy 

map. 

• Hiding an anisotropic template such as those of Bianchi models in a ΛCDM CMB 

anisotropy temperature map, will lead to observable effects on bipolar power spectrum. 

The null BiPS of WMAP can be used to put tight constraints on the 

existence of such hidden patterns in the WMAP data. 

• Observational artifacts such as non-circular beam, inhomogeneous noise correlation, 

residual striping patterns and foreground residuals are potential sources of SI 

breakdown. Our null BiPS results confirm that these artifacts do not significantly 

contribute to the first-year data of WMAP up to l ∼ 60 − 70. 

x

List of Publications 

This thesis is based on the following publications. 

Chapter 2: A. Hajian, T. Souradeep, “Measuring Statistical Isotropy of CMB”, 

Astrophys.J. 597 (2003) L5-L8. 

A. Hajian, T. Souradeep, “The Cosmic Microwave Background Bipolar Power 

Spectrum: Basic Formalism and Applications” (astro-ph/0501001) (Accepted for 

publication in ApJ). 

T. Souradeep, A. Hajian, “Statistical Isotropy of Cosmic Microwave Background 

Radiation Anisotropy”, Pramana, vol. 62, no. 3, 793-796; 2004. 

Chapter 3: A. Hajian, T. Souradeep, “Statistical Isotropy of CMB and Cosmic 

Topology”(astro-ph/0301590), to be submitted to PRL. 

Chapter 4: A. Hajian, D. Pogosyan, T. Souradeep, C. Contaldi and R. Bond, 2004 

in preparation; Proc. 20th IAP Colloquium on Cosmic Microwave Background 

physics and observation, 2004. 

Chapter 5: T. Ghosh, T. Souradeep and A. Hajian, 2006, in preparation. 

Chapter 6: R. Saha, A. Hajian, T. Souradeep and P. Jain, 2006, in preparation. 

Chapter 7: A. Hajian, T. Souradeep, N. Cornish, “Statistical Isotropy in the WMAP 

Data: A Bipolar Power Spectrum Analysis”, Astrophys.J. 618 (2004) L63-L66. 

T. Souradeep, A. Hajian, “Statistical isotropy of CMB anisotropy from WMAP”, 

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Proc. of General relativity and Gravitation (JGRG-14), Yukawa Institute for 

Theoretical Physics, Nov 29-Dec 3, 2004. 

During my PhD at IUCAA I also worked on another project, (J.V. Narlikar, R.G. Vishwakarma, 

A. Hajian, T. Souradeep, G. Burbidge, F. Hoyle, “Inhomogeneities in the 

Microwave Background Radiation interpreted within the framework of the Quasi-Steady 

State Cosmology”, Astrophys.J. 585 (2003) 1-11), but that research is not included in 

this thesis. 

xii

Acknowledgements 

In constant action are the clouds, wind, sun, and universe, 

So acknowledge your sustenance upon savoring each morsel. 

Golestan (The Rose Garden) 

Sheikh Mosleheddin Sádi Shirazi (1210-1290) 

This thesis, although is formally a report of my work during the last four years, is in 

fact a result of collective efforts and help from many people in the past 27 years. In 

the beginning I would like to thank everyone who have helped me through out my life. 

I wish to thank those whose support, help and encouragement have been invaluable to 

me, although I may not name all. 

First and foremost I should start by thanking my parents, to whom I dedicate this 

thesis. During the long difficult years of war and in the poverty induced by revolution, 

imposed war, international sanctions and embargo against Iran, my parents worked hard 

and sacrificed their future, to protect us and give us the best one could have in those 

years. They taught us to work hard, aim high and not to give up even if everything 

seems to be going wrong. Their support constantly continued till now. They put their 

house in pledge to get me exempted from the military service when I wanted to go out of 

Iran to continue my studies. I am forever grateful to them for every bit of their support 

and care all through my life. 

I owe special thanks to my brother and sister, Hamid and Zahra, who have always 

been there for me at all the important times of my life. Hamid has always inspired me 

xiii

through his passion for science, and showed me there are much more things to learn 

than I thought. He has been a wonderful brother and a great supporter for me. Zahra 

has given me every reason to live happily. Her vitality and energy has brought a lot of 

joy to our family. I hope and pray for the happiness and success of both of them. I 

am also grateful to my grandmother for seeing me and helping me in my life specially 

during my study at Sharif. She gave me the best books I have ever read. “Principles of 

Thermodynamics” that she gifted me changed my life and helped me realize my passion 

for physics in the early years of high school. 

I believe the credit of the beginning of my interest in research belongs to my wonderful 

high school, the National Organization for Development of Exceptional Talents. The 

infinite degree of freedom they gave us, the interesting workshops, the exciting research 

projects and very good teachers are the major things I can remember from my high 

school days. I am specially indebted to Mr. Ejei and two of my teachers Mr. Razi Zadeh 

and Mr. Mollabashi for providing us an excellent place to live, learn and experiment. 

I owe great thanks to Houshang Karimi whose excellent courses on electromagnetism 

and calculus introduced me to a new phase of learning physics and Ali Sedighi with 

whom I shared my fascinations and joys of early days of adventuring with physics. In 

my final year at high school, the first students’ conference of physics was organized in 

Sanandaj. For many years, this conference gave me great motivation to study physics. 

I was specially impressed by Ali Nayeri and his lecture on the fractal universe. He has 

always inspired me and answered my mails very kindly. I remember that he discovered 

IUCAA and told us about that when we were very young. 

From my undergraduate years I am most of all indebted to Yousef Sobouti, director 

of the Institute for Advanced Studies in Basic Sciences. His idea of giving everyone 

a chance to do well has helped many students including me. The unique scientific 

atmosphere of IASBS gave me a great opportunity to learn from everyone. Specially 

I learned a great deal from my Masters’ thesis supervisor Amir H. Fathollahi, a rolemodel 

both as a man and a very bright scientist. Two years at IASBS were perhaps the 

happiest of my life. I am specially grateful to my undergraduate friends Mohammad 

Charsooghi, Negar Aliakbarzadeh, Farshid Mohammad Rafiee, Abouzar Shirzad, Ali 

Bagheri Nejad, Ali Tabeai, Afshin Khezerloo, Said Paktinat, Mohammad Ashoori and 

other classmates at Sharif and IASBS who remained very good friends of mine for a 

xiv

long time and over long distances. 

When I was at IASBS, Ajit Kembhavi visited us and invited me to visit IUCAA 

for the first time. That was my introduction to IUCAA. He has always impressed me 

with the care, attention and hospitality. Considering my time in India, which is most 

relevant to this thesis, first and foremost I would like to thank my thesis supervisor 

Tarun Souradeep for his constant support, encouragement and caring during the whole 

period of my PhD. He was (and remains) my only reason to move to India and the 

greatest credit of every success that I might have had in these years belongs to him and 

his wonderful guidance. Tarun was much more than just a thesis supervisor for me. 

He knew when to let me struggle alone and when to help me out of the situations. He 

introduced me to all the right people and very patiently helped me step by step to gain 

exposure in the field. I should never forget the great help when I was absolutely stuck 

for my registration at the university. His heartwarming words helped me a lot during 

frustrations of the early years. I am specially thankful to him for the belief he showed 

in me even when I made terrible mistakes. When I made my first CMB map, he gave 

it much more credit than it probably deserved. I offer my deepest gratitude to him and 

his family, Sucheta and Sourath, for their warm hospitality in all these years. I shall 

never forget memorable moments I spent at their house, and may be I have never told 

them how much our dinner conversations, picnics, Tarun’s readings of Indian mythology, 

Sucheta’s advices, Sourath’s poems and songs and every moment of being with them 

meant to me. Upon my arrival at IUCAA, Jayant Narlikar welcomed me very warmly 

and invited me to his home. He, as the director of IUCAA, supported me very strongly 

during the first two years of my PhD and showed me a way out of several complicated 

bureaucratic situations. When my visa took time and I couldn’t make it for one term 

at the graduate school, he gave me the chance to study on my own and just give the 

exams. It was a great experience for me, at both intellectual and personal levels, to 

communicate with him and his wife. I should thank IUCAA for providing me a calm 

atmosphere to work. I am specially thankful to the director of IUCAA, Naresh Dadhich, 

for his kind support during the last couple of years. I will always remember the night 

he invited all Iranian students to his house to celebrate the occasion of winning Nobel 

Peace Prize by Shirin Ebadi. Many IUCAA staff helped me in these years. I wish to 

thank all of them. Neelima Pagonkar always helped me with the paper work, specially 

xv

when I needed a long leave for my wedding. Susan Kuriakose was not only helpful in the 

official work but also a very good friend of ours. Rajesh Pardeshi saved me from missing 

many deadlines by his speed in getting things done. Lata Shankar with her ability in 

planning a trip helped me a lot in my long pre-doctoral journey. I am also thankful 

to many other friends at IUCAA who helped me in these four years. Anand Sengupta 

helped me out of many Linux-related troubles and gave me his thesis template. Sanjit 

Mitra was a very nice officemate. Subharti Ray cured my Microsoft windows several 

times and helped me in many situations. Neeraj Gupta came to my rescue when I 

had serious troubles understanding the ISM. He also helped me a lot with AIPS during 

our Radio Astronomy course. I enjoyed working with him on cleaning maps and the 

memorable neutron star hunting experience during the graduate school. Ayesha Begum 

let me pay the SEVIS fees with her credit card when no bank gave foreigners a credit 

card in India. Mahdi Bazargan saved us from cold nights of IUCAA. Ali Dariush with 

his extreme vitality brought joys to my life for the short time he stayed at IUCAA. 

Arman Shafieloo introduced me to interesting people. Meysam Namayandeh gave me a 

thorough example of a trust worthy friend, I am very happy he finally recovered from the 

dengue fever. Elisabeth, Maelys, Ondine and Emannuel Rollinde provided a very warm 

family atmosphere for me during the short time I knew them, my first joyful experiences 

of baby sitting dates back to the days they were with us. Anil, Nikita and Mammad 

Sandwichi helped us in their own ways. Vida Rahiminejad took the trouble of helping 

me through the paper work in my absence and has always been a very good friend of 

ours. My very special thanks belongs to Dr. Mohammad Kafi, science counselor and 

director of science, technology and education in the Embassy of I.R. of Iran in New 

Delhi. In very difficult situations he helped and supported me. I am forever grateful to 

him for his invaluable supports and encouragements which I will never forget. 

Being an Iranian student in India I always had the trouble of finding funds for my 

travel. Tarun always helped me greatly to find kind people to fund me. I am deeply 

grateful to all of them for their support and hospitality which allowed me meet brilliant 

people and learn many things from them in my trips. In my first year Norma Sanchez 

covered all my expenses to attend the Chalonge School. I can never forget the excitement 

of being there. Farhad Ardalan invited me to give a talk on WMAP at the Institute 

for Theoretical Physics and Mathematics (IPM). In that trip I met the young society 

xvi

of cosmologists of Iran and it helped me to strengthen my ties to my home country. In 

my predoctoral trip I met Simon Prunet very briefly, but in the short time I learned 

a lot from him. The Chapter on foregrounds in my thesis is mostly the result of my 

discussion with him. He also helped me out when I was stuck with the computation 

of Clebsch-Gordan coefficients. Mr and Mrs Rollinde let me stay with them during 

my visit to Paris and took care of me like their own son. I am extremely grateful to 

them and I don’t know what I could have done in Paris if they did not accommodate 

me in their house. At ICTP Seif Randjbar-Daemi and Uros Seljak hosted me very 

kindly. Maria Jesus Pons helped me a great deal to attend the school on data analysis 

in cosmology which was very useful for me. Andrea Tartari and Georgio Sironi hosted 

me in Milan where for the first time I could see the horns with which CMB measurments 

are made (they were much smaller than what I thought!). Their hospitality was superb 

and Milan is always a memorable city for me since then. At Oxford I enjoyed very warm 

hospitality of Joe Silk, Sadegh Kouchfar and Asim Mahmood. My visit to Oxford was 

very fruitful specially because the Bianchi models were hot at that time and I could 

learn a lot about them from Joe Silk and Pedro Ferrera. Our discussions gave me the 

idea of what I later did on Bianchi models. It was a great experience to work at CITA 

and I am greatly indebtful to Dick Bond for inviting me and giving me full support 

to be there for a long period. Nazli and Houshang Karimi made my stay at Toronto 

wonderful. I can never thank them enough for their warm and kind Iranian hospitality. 

I am also thankful to Subhabrata Majumdar, Faranak and Yaser Kerachian for their 

kind help and hospitality. My other great experience was the period I spent with Dmitry 

Pogosyan. It was a wonderful time of excitingly hard work. Dmitry accommodated me 

at his own house. That was a great opportunity for me to discuss all through the day 

and night and learn a lot from him at every moment. He gave me the co-added noise 

covariance matrix of WMAP and the work I did on Poincare Dodecahedral Spaces and 

also the BiPS of anisotropic noise was done during this period. My trip to Harvard was 

very useful for me for which I am indebted to Niayesh Afshordi. I enjoyed discussions 

with Niayesh, Ramesh Narayan and Avi Loeb. I am also thankful to Banibrata, Ghazal 

and Niayesh for their kind hospitality. At UC Davis Lloyd Knox hosted me and I had a 

great chance of learning from him. He helped me understand some issues on statistics 

of power spectrum and also taught me how to write the equations in powerpoint in 

xvii

color! My greatest experience was visiting Princeton. I enjoyed discussions with David 

Spergel, Lyman Page and Olivier Dore. For a long time I had worked on the WMAP 

data and it was a thrill for me to present my work for the WMAP team members. I am 

also thankful to Lyman Page and David Spergel who helped me in getting my US visa 

and moving to Princeton. At Princeton I enjoyed very warm hospitality of Mojgan and 

Hormoz. They made me feel at home for the whole period of my stay and I will never 

forget the great fun and good memories of the cold evenings of New Jersey warmed with 

their kind Iranian hospitality. 

During four years of my PhD, Yahoo Messenger was my only way of remaining in 

touch with friends and family. I wish to thank Yahoo for such a wonderful facility and 

also for Yahoo Mail which is the most reliable mail server I have ever seen. Blogger 

gave me the possibility of writing, reading and publishing in Persian. This, along with 

Orkut, Gmail and Google search have been great inventions of Google and I can not 

express how much they have affected my life. All these were impossible to use if I did 

not have Windows XP which makes life very easy, allows you to use voice, video and 

graphics and lets you read and write in your own script. 

Last, but by no means least, my love, admiration, and gratitude goes to my wife, 

Sanaz. During the last three years, she has always been my closest and best friend. She 

came into my life in a terrible time and protected me in all the dark periods by her 

precious love. Sanaz bore with me through ups and downs of my life in IUCAA, and 

patiently tolerated years of living far apart. She listened to my ideas carefully, asked 

very difficult questions which motivated me to learn more, helped me in preparation of 

this thesis and designed fancy headers for my synopsis. I am thankful to her for her 

support and faith she showed in me and for the great patience she showed in the difficult 

times of our life. 

xviii

Contents 

Abstract 

Acknowledgements 

List of Figures 

List of Tables 

ix 

xiii 

xxi 

xxiii 

1 Introduction 1 

2 Statistical Properties of the CMB Anisotropy 6 

2.1 Statistical Isotropy of Temperature Anisotropies . . . . . . . . . . . . . . 8 

2.2 Statistical Isotropy and Gaussianity . . . . . . . . . . . . . . . . . . . . . 10 

2.3 The Bipolar Power Spectrum (BiPS) . . . . . . . . . . . . . . . . . . . . 11 

2.4 BiPS Representation in Real Space . . . . . . . . . . . . . . . . . . . . . 13 

2.5 Unbiased Estimator of BiPS . . . . . . . . . . . . . . . . . . . . . . . . . 16 

2.6 Cosmic Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 

2.7 Sources of Breakdown of Statistical Isotropy . . . . . . . . . . . . . . . . 20 

2.8 Primordial Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . 21 

3 SI Violation: Analytic Models 23 

3.1 One Preferred Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

3.2 Three Preferred Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 

3.3 Oblique Preferred Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 

3.4 Toy Model With Cross Terms . . . . . . . . . . . . . . . . . . . . . . . . 29 

3.5 Fourth Order Toy Models With Three Axes . . . . . . . . . . . . . . . . 30 

3.6 Toy Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 

3.7 Toroidal Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 

4 Signatures of Cosmic Topology 33 

4.1 Statistical Isotropy Violation in Compact Spaces . . . . . . . . . . . . . . 34 

xix

4.2 Method of images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 

4.3 A Bipolar Power Spectrum Study of Cosmic Topology . . . . . . . . . . . 41 

4.3.1 Flat Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 

4.3.2 Spherical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 

5 Homogeneous Cosmological Models 75 

5.1 Classification of Cosmological Models . . . . . . . . . . . . . . . . . . . . 77 

5.2 Modern Classification of Bianchi Models . . . . . . . . . . . . . . . . . . 80 

5.3 The Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 

5.4 Approaching Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 

5.5 Energy Momentum Tensor and Vorticity Vector . . . . . . . . . . . . . . 85 

5.6 Null Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 

5.7 Patterns of Anisotropic Models on the CMB Sky . . . . . . . . . . . . . . 89 

5.7.1 CMB Patterns in Type I Bianchi Models . . . . . . . . . . . . . . 91 

5.7.2 CMB Patterns in Type V Bianchi Models . . . . . . . . . . . . . 91 

5.7.3 CMB Patterns in Type VII 0 Bianchi Models . . . . . . . . . . . . 92 

5.7.4 CMB Patterns in Type VII h Bianchi Models . . . . . . . . . . . . 92 

5.7.5 CMB Patterns in Type IX Bianchi Models . . . . . . . . . . . . . 95 

5.8 Unveiling Hidden Patterns of Bianchi VII h . . . . . . . . . . . . . . . . . 96 

6 Observational Artifacts 105 

6.1 Anisotropic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 

6.2 The effect of non-circular beam . . . . . . . . . . . . . . . . . . . . . . . 108 

6.3 Mask effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 

6.4 Residuals from foreground removal . . . . . . . . . . . . . . . . . . . . . 110 

7 Statistical isotropy of CMB anisotropy from WMAP 111 

8 Summary and future directions 127 

A Detailed Steps of Calculations 131 

B Cosmic Variance of BiPS, Handling Gaussian 8 Point Functions 139 

C Search for Planar Symmetries in CMB Ansiotropy Maps 148 

D Symmetries of the Space 153 

E Bianchi Models: Bianchi’s Original Classification 158 

F Simulating CMB maps for non-SI correlation functions 165 

G Useful Standard Integrals and Summation Rules 172 

Bibliography 174 

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List of Figures 

2.1 BipoSH coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 

2.2 CMB temperature correlation function . . . . . . . . . . . . . . . . . . . 14 

2.3 Comparing analytically calculated bias with numerical computations . . . 17 

2.4 Analytically calculated cosmic variance vs. numerical computations . . . 19 

2.5 Effect of magnetic fields on covariance matrix . . . . . . . . . . . . . . . 21 

2.6 BiPS of homogeneous magnetic fields . . . . . . . . . . . . . . . . . . . . 22 

4.1 Fundamental domains for compact flat spaces . . . . . . . . . . . . . . . 44 

4.2 The chimney space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 

4.3 The slab space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 

4.4 Two point correlation in T 2 . . . . . . . . . . . . . . . . . . . . . . . . . 47 

4.5 Folded circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 

4.6 Correlations along folded circle . . . . . . . . . . . . . . . . . . . . . . . 48 

4.7 Circles in the sky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 

4.8 Dirichlet domains of squeezed tori . . . . . . . . . . . . . . . . . . . . . . 50 

4.9 BiPS of toroidal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 

4.10 Geometry of the distance between two points . . . . . . . . . . . . . . . . 53 

4.11 CMB temperature correlation function. . . . . . . . . . . . . . . . . . . . 64 

4.12 Fundamental domain for the regular dodecahedron . . . . . . . . . . . . 68 

4.13 Two point correlations in Poincaré dodecahedral spaces . . . . . . . . . . 69 

4.14 BiPS of Poincaré dodecahedron . . . . . . . . . . . . . . . . . . . . . . . 72 

4.15 Simulated CMB maps of Poincaré dodecahedral spaces . . . . . . . . . . 73 

4.16 BiPS of simulated CMB maps of Poincaré dodecahedral spaces . . . . . . 74 

5.1 BiPS of Bianchi VII h templates . . . . . . . . . . . . . . . . . . . . . . . 78 

5.2 Bianchi VII h patterns for x = 0.1 . . . . . . . . . . . . . . . . . . . . . . 96 

5.3 Bianchi VII h patterns for x = 0.55 . . . . . . . . . . . . . . . . . . . . . . 98 

5.4 Bianchi VII h patterns for x = 1 . . . . . . . . . . . . . . . . . . . . . . . 98 

5.5 Bianchi VII h patterns for x = 5 . . . . . . . . . . . . . . . . . . . . . . . 98 

5.6 Bianchi corrected map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 

5.7 Comparison of power spectra . . . . . . . . . . . . . . . . . . . . . . . . 99 

xxi

5.8 Adding a template to the background map . . . . . . . . . . . . . . . . . 100 

5.9 Bianchi added maps, Ω 0 = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . 101 

5.10 BiPS of Bianchi added maps, Ω 0 = 0.5 . . . . . . . . . . . . . . . . . . . 102 

5.11 Probability distribution of χ 2 for low density Bianchi template . . . . . . 103 

5.12 Bianchi added maps, Ω 0 = 0.99 . . . . . . . . . . . . . . . . . . . . . . . 103 

5.13 BiPS of Bianchi added maps, Ω 0 = 0.99 . . . . . . . . . . . . . . . . . . . 104 

5.14 Probability distribution of χ 2 for nearly flat Bianchi template . . . . . . 104 

6.1 Coadded noise covariance matrix of WMAP . . . . . . . . . . . . . . . . 106 

6.2 BiPS of diagonal noise covariance matrix . . . . . . . . . . . . . . . . . . 107 

6.3 Estimating BiPS from simulations of noise maps . . . . . . . . . . . . . . 108 

7.1 Window functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 

7.2 BiPS of WMAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 

7.3 BiPS of WMAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 

7.4 Effect of Galactic mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 

7.5 Probability distribution function of BiPS . . . . . . . . . . . . . . . . . . 120 

7.6 Probability of CMB maps being SI . . . . . . . . . . . . . . . . . . . . . 121 

7.7 SI violation vanishes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 

7.8 Probability of CMB maps being SI . . . . . . . . . . . . . . . . . . . . . 122 

7.9 BiPS of WMAP filtered with bandpass filters . . . . . . . . . . . . . . . 123 

7.10 BiPS of WMAP filtered with Gaussian filters . . . . . . . . . . . . . . . . 124 

7.11 PDF of κ l for a Gaussian filter Wl G (40) . . . . . . . . . . . . . . . . . . . 125 

7.12 PDF of κ l for a band pass filter Wl S (20, 30) . . . . . . . . . . . . . . . . . 126 

C.1 S-MAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 

C.2 S 0 statistics for ILC map . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 

F.1 Simulated noise map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 

F.2 Simulated map of CMB temperature anisotropy in a toroidal universe . . 171 

xxii

List of Tables 

4.1 Classification of three-dimensional flat spaces . . . . . . . . . . . . . . . . 43 

4.2 Finite subgroups of Q 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 

5.1 Classification of Cosmological Models with Respect to Their Symmetries. 79 

5.2 Classification of Homogeneous Cosmological Models into Ten Equivalence 

Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 

xxiii

Chapter 1 

Introduction 

This thesis is a study of statistics of the CMB anisotropy as a Gaussian random field 

on a sphere. We study statistical isotropy (SI) of CMB temperature anisotropy and use 

that to draw interesting conclusions on the physics, topology and large scale structure 

of the universe as well as properties of the systematics in CMB experiments. 

The Cosmic Microwave Background (CMB) anisotropy is proved to be a very powerful 

observational probe of cosmology. Detailed measurements of the anisotropies in 

the CMB can provide a wealth of information about the global properties, constituents, 

history and eventual fate of the Universe. In standard cosmology, the CMB anisotropy 

is expected to be statistically isotropic, i.e., statistical expectation values of the temperature 

fluctuations ∆T (ˆq) are preserved under rotations of the sky. In particular, 

the angular correlation function C(ˆq, ˆq ′ ) ≡ 〈∆T (ˆq)∆T (ˆq ′ )〉 is rotationally invariant for 

Gaussian fields. In spherical harmonic space, where ∆T (ˆq) = ∑ lm a lmY lm (ˆq) this translates 

to a diagonal 〈a lm a ∗ l ′ m ′〉 = C lδ ll ′δ mm ′ where C l is the widely used angular power 

spectrum of CMB anisotropy. Statistical isotropy (SI) is usually assumed in almost all 

CMB studies. But there are certain mechanisms that violate this condition. This thesis 

is dedicated to the study of this fundamental assumption in CMB. We have formulated 

the problem mathematically, constructed a quantitative method to describe SI violation, 

studied possible ways of violation of SI condition and finally examined this fundamental 

assumption using the most recent CMB anisotropy data. 

In February 2003, Wilkinson Microwave Anisotropy Probe (WMAP) released its first 

data (Bennett et al. , 2003; Hinshaw et al. , 2003; Kogut et al. , 2003; Page et al. , 2003; 

1

1.0 2 

Peiris et al. , 2003; Spergel et al. , 2003). The WMAP mission was designed to advance 

observational cosmology by making accurate full sky CMB maps with precision and 

reliability. WMAP heralds the dawn of precision cosmology tightly constraining many 

key parameters (Bridle et al., 2003). Determining the flatness of space, the near scaleinvariance, 

adiabaticity, and Gaussian distribution of the density perturbations, the 

density of baryons, the age of the universe, and the early formation of the first stars are 

some major results of WMAP and are consistent with the standard model of cosmology 

(for an interesting review on standard model see Scott (2005)). After the release of firstyear 

data of WMAP, statistical isotropy of the CMB anisotropy attracted considerable 

attention. Tantalizing evidence of SI breakdown (albeit, in very different guises) has 

mounted in the WMAP first year sky maps, using a variety of different statistics. It was 

pointed out that the suppression of power in the quadrupole and octopole are aligned 

(Tegmark et al. , 2004). Further “multipole-vector” directions associated with these 

multipoles (and some other low multipoles as well) appear to be anomalously correlated 

(Copi et al. , 2004; Schwarz et al. , 2004). There are indications of asymmetry in 

the power spectrum at low multipoles in opposite hemispheres (Eriksen et al. , 2004a; 

Hansen et al. , 2004; Naselsky et al. , 2004). Possibly related, are the results of tests 

of Gaussianity that show asymmetry in the amplitude of the measured genus amplitude 

(at about 2 to 3σ significance) between the north and south galactic hemispheres (Park 

, 2004; Eriksen et al. , 2004b,c). Analysis of the distribution of extrema in WMAP 

sky maps has indicated non-Gaussianity, and to some extent, violation of SI (Larson & 

Wandelt , 2004). However, what is missing is a common, well defined, mathematical 

language to quantify SI (as distinct from non Gaussianity) and the ability to ascribe 

statistical significance to the anomalies unambiguously. 

In order to extract more information from the rich source of information provided 

by present (and future) CMB maps, it is important to design as many independent 

statistical methods as possible to study deviations from standard statistics, such as 

statistical isotropy. Since SI can be violated in many different ways, various statistical 

methods can come to different conclusions. Because each method by design is more 

sensitive to a special kind of SI violation. 

We have constructed a novel statistical tool of searching for departures from SI.

1.0 3 

This method is now known as the bipolar power spectrum (BiPS) which is sensitive 

to structures and patterns in the underlying two-point correlation function (Hajian & 

Souradeep , 2005, 2003b; Souradeep & Hajian , 2003). The BiPS is particularly sensitive 

to real space correlation patterns (preferred directions, etc.) on characteristic angular 

scales. In harmonic space, the BiPS at multipole l sums power in off-diagonal elements 

of the covariance matrix, 〈a lm a l ′ m ′〉, in the same way that the ‘angular momentum’ 

addition of states lm, l ′ m ′ have non-zero overlap with a state with angular momentum 

|l−l ′ | < l < l+l ′ . Signatures, like a lm and a l+nm being correlated over a significant range 

l are ideal targets for BiPS. These are typical of SI violation due to cosmic topology 

and the predicted BiPS in these models have a strong spectral signature in the bipolar 

multipole l space (Hajian & Souradeep , 2003a). The orientation independence of BiPS 

is an advantage since one can constrain mechanisms that violate the SI independent of 

the unknown specific orientation of the pattern (e.g., preferred directions). The BiPS 

statistics has been applied to the most recent full sky maps of the CMB anisotropy. 

The null detection allows us to place observational constraints on exotic possibilities for 

cosmology such as compact spaces, anisotropic cosmological models, etc. 

Here I give a brief description of how the thesis is organized into chapters. 

Chapter 2 contains the basic material and is provided as a qualitative introduction 

to the statistical isotropy. In this chapter we introduce basic statistical properties of 

CMB and discuss the statistical isotropy as distinct from Gaussianity. Also we introduce 

BiPS in real as well as harmonic space to quantify statistical isotropy violations in a 

CMB map. We define an unbiased estimator for BiPS and compute its cosmic variance 

in details. We introduce homogeneous primordial magnetic fields as the first example of 

a theoretical model that does not respect the SI and study the signature of these models 

on the bipolar power spectrum. A homogeneous primordial magnetic field with fixed 

direction induces correlations between the a l−1,m and a l+1,m multipole coefficients of the 

CMB temperature anisotropy field (see e.g. Durrer et al. 

(1998)). This means that a 

homogeneous primordial magnetic field with fixed direction breaks down the statistical 

isotropy in its own manner and hence its bipolar power spectrum has a specific shape. 

In Chapter 3 we study toy models of non-SI correlations which are analytically 

solvable. These toy models are motivated by various mechanisms of SI violation such as

1.0 4 

multiconnected spaces. Besides helping us to come to a good understanding of violation 

of SI, they shed a light on how BiPS method works. We repeatedly refer to this chapter 

to explain several signatures of different models on BiPS. 

In Chapters 4 to 6 we study possible sources of breakdown of SI. These sources 

vary from pure theoretical models of ultra large scale structure of the universe to the 

systematics of CMB experiments. 

Chapter 4 is dedicated to the study of topology of the universe. The cleanest signature 

of the topology of the universe is written on the microwave sky. We use BiPS to 

look for these signatures in a CMB map. BiPS exploits the existence of preferred directions 

along the highly correlated spots in compact spaces. Number and configuration 

of these preferred directions differ in each compact space and cause different correlation 

patterns on the CMB sky. We have shown that bipolar power spectrum is very 

sensitive to these patterns and can be used to distinguish them. A detailed study of 

the flat spaces has been carried out in this chapter. It is shown that different multiconnected 

spaces have different BiPS signatures. These signatures are found analytically 

and verified by numerical computation of BiPS on multiply connected flat spaces. In 

the second half of this chapter we study Poincare Dodecahedral spaces suggested as a 

possible explanation of low quadrupole of the WMAP data by Luminet et al. (2003). 

We show that the Poincaré Dodecahedron with Ω k = 0.013 breaks down the statistical 

isotropy at an observable level which was not observed in the WMAP data. Our results 

are consistent with the complete Bayesian likelihood analysis of Poincaré Dodecahedral 

spaces [Hajian et al. (2006)]. 

Anisotropic universe is another example of violation of SI. In Chapter 5 we study 

homogeneous cosmological models which are initially anisotropic and as the time goes 

on they become more isotropic and will asymptotically tend to FRW models. Bianchi 

models provide a generic description of homogeneous anisotropic cosmologies, see e.g. 

Ellis & MacCallum (1969) and Collins & Hawking (1973b). Distinct features of Bianchi 

models studied by a number of authors (Barrow et al. , 1985; Bunn et al. , 1996; Kogut et 

al. , 1997) and were constrained. Recently they were revived after the first year WMAP 

data by Jaffe et al. (2005). The authors reported the discovery of a hidden pattern 

in the WMAP data which according to them was due to a rotating universe template.

1.0 5 

The model studied by Jaffe et al. (2005) was not self-consistent as a rotating universe 

model. But the existence of a hidden template with a preferred axis in the WMAP data 

is very well suited the BiPS method. We have shown that hiding a Bianchi template in 

a ΛCDM CMB anisotropy temperature map, will lead to measurable non-zero bipolar 

power spectra. The null BiPS of WMAP can be used to put tight constraints on the 

existence of such hidden patterns in the WMAP data. 

In addition to the theoretical models presented above, foregrounds and observational 

artifacts (such as non-circular beam, incomplete/non-uniform sky coverage and 

anisotropic noise) would also manifest themselves as violations of SI. Chapter 6 is 

devoted to the study of these observational effects. 

In Chapter 7 we discuss the results of applying the BiPS method to the first year 

WMAP data. We carried out measurement of the BiPS, on four full-sky CMB anisotropy 

maps. Using various window functions we isolated regions in multipole space and probed 

them independently. Our results indicate that there is no strong evidence for violation 

of statistical isotropy in the first-year WMAP data on angular scales larger than that 

corresponding to l ∼ 60. We have verified that our null results are consistent with 

simulated statistically isotropic maps (1000 realizations). These results also confirm the 

other observational artifacts that cause SI violation in the CMB maps of WMAP, such 

as, incomplete and non-uniform sky coverage, beam anisotropy, foreground residuals and 

statistically anisotropic noise do not significantly contribute to the map (up to l ∼ 60). 

In the end we have briefly discussed the use of BiPS in CMB polarization studies. 

In order to make this thesis self contained, we have included details of calculations, 

mathematical preliminaries and computational methods. However, these are given in 

the appendices to keep the thesis readable. In addition to the above appendices, we 

present two methods of generating CMB sky maps from anisotropic correlation functions 

which have been used throughout this work in Appendix F. These methods are 

diagonalization method and Cholesky decomposition method. In these methods we use 

the full covariance matrix to generate the CMB map. This is very important when we 

deal with statistically anisotropic models such as those discussed above. We also present 

a pedagogical map making recipe for the beginners to make their own maps very easily.

Chapter 2 

Statistical Properties of the CMB 

Anisotropy 

The temperature anisotropies of the CMB are described by a random field, ∆T (ˆn) = 

T (ˆn) − T 0 , on a 2-dimensional surface of a sphere (the sky), where ˆn = (θ, φ) is a unit 

vector on the sphere and T 0 = ∫ dΩˆn 

T (ˆn) represents the mean temperature of the CMB. 

4π 

The statistical properties of this field can be characterized by the n-point correlation 

functions 

〈∆T (ˆn 1 )∆T (ˆn 2 ) · · · ∆T (ˆn n )〉. (2.1) 

Here the bracket denotes the ensemble average, i.e. an average over all possible configurations 

of the field. If the field is Gaussian in nature, the connected part of the n-point 

functions disappears for n > 2. The non-zero (even-n)-point correlation functions can 

be expressed in terms of the 2-point correlation function. As a result, a Gaussian distribution 

is completely described by the two-point correlation function. 

The statistics of the temperature fluctuations is Gaussian because they are linearly 

connected to quantum vacuum fluctuations of a very weakly interacting scalar field (the 

inflaton). Perturbations in primordial gravitational potential, Φ 0 , generate the CMB 

anisotropy, ∆T . The linear perturbation theory gives a linear relation between Φ 0 

and ∆T . Φ 0 (x) is assumed to be Gaussian, because in inflation, quantum fluctuations 

produce Gaussian random fluctuations in the gravitational potential 

∫ d 3 k 

Φ 0 (x) = 

(2π) Φ 0(k) exp ik · x (2.2) 

3 

6

2.0 7 

with 〈Φ 0 (k)Φ ∗ 0 (k′ )〉 = P (k)δ(k − k ′ ), the power spectrum of potential fluctuations at 

that epoch. 

In the linear approximation in fluctuations, in a spatially flat universe, 

the temperature anisotropies at the surface of last scattering are linearly related to the 

potential fluctuations at that epoch 1 

∫ 

∆T (ˆn) = 

d 3 k 

(2π) 3 eik·ˆnτrec Φ 0 (k)g T (k), (2.3) 

where τ rec is the comoving conformal lookback time of the recombination epoch (τ is the 

conformal time τ = ∫ dt/a), and g T (k) is the linear radiation transfer function which 

describes the relationship between potential fluctuations and temperature fluctuations 

at recombination. 

For small k, g T (k) = 1/3, which is the well known fact that for 

temperature fluctuations on super-horizon scales at the decoupling epoch, the Sachs- 

Wolfe effect (Sachs and Wolfe , 1967) dominates and hence on large angular scales 

On sub-horizon scales, g T 

∆T 

T (ˆn) = 1 3 Φ LS(ˆnτ rec , τ 0 ) (2.4) 

oscillates (due to acoustic oscillations in the baryon-photon 

fluid), and we need to evolve the coupled matter-radiation fluid, i.e. solve the Boltzmann 

photon transport equations coupled with the Einstein equations for g T . 

CMBFAST 2 compute g T (k) for different cosmological models. 

If we expand the gravitational potential into a series in powers of a density enhancement 

Codes like 

Φ(x, t) = Φ (1) + Φ (2) + · · · (2.5) 

The linear term Φ 1 = Φ 0 (x) and is assumed to be Gaussian. Non-linearity in inflation 

makes Φ weakly non-Gaussian and will lead to non-linear terms, such as Φ (2) , in eqn. 

(2.5). However, since potential fluctuations in the early universe are small, 〈Φ 2 〉 ∼ 10 −9 , 

these non-linear effects are usually viewed as undetectable. 

Moreover starting from 

Gaussian initial condition, it has been shown that non-linear evolution does not produce 

seeds of non Gaussianity (Munshi et al. , 1995; Spergel et al. , 1999) (for a nice review 

on non-Gaussianity see Bartolo et al. (2004)). Hence, here we limit our study to 

1 This description is very simple but naive. It does not take the line of sight effects such as integrated 

Sachs-Wolfe effect into account. For a more careful description see Appendix A. 

2 Available at http://www.cmbfast.org/

2.1 STATISTICAL ISOTROPY OF TEMPERATURE ANISOTROPIES 8 

Gaussian CMB anisotropy field, where the two-point correlation function 

C(ˆn, ˆn ′ ) ≡ 〈∆T (ˆn)∆T (ˆn ′ )〉 (2.6) 

contains all the statistical information encoded in the field. 

It is convenient to expand the temperature anisotropy field into spherical harmonics, 

the orthonormal basis on the sphere, as 

∆T (ˆn) = ∑ l,m 

a lm Y lm (ˆn) , (2.7) 

where the complex quantities, a lm are given by 

∫ 

a lm = dΩˆn Ylm ∗ (ˆn)∆T (ˆn). (2.8) 

For a Gaussian CMB anisotropy, a lm are Gaussian random variables and hence, the 

covariance matrix, 〈a lm a ∗ l ′ m ′〉, fully describes the whole field. 

2.1 Statistical Isotropy of Temperature Anisotropies 

A random field Φ(r) is statistically homogeneous (in the wide sense) if the mean 

and the second moment (covariance) of that remain invariant under the coordinate 

transformation r → r + δr. That is to say, 

〈Φ(r)〉 = 〈Φ(r + δr)〉, (2.9) 

C Φ (r 1 , r 2 ) = C Φ (r 1 + δr, r 2 + δr). 

The first condition implies that 〈Φ〉 = const. Putting δr = −r 2 in the second condition 

we find that C Φ (r 1 , r 2 ) = C Φ (r 1 − r 2 , 0), i.e. the covariance of a statistically 

homogeneous field is dependent only on the difference r 1 − r 2 , and not on r 1 and r 2 

separately. So instead of C Φ (r 1 − r 2 , 0) we can simply write C Φ (r) which means: 

C Φ (r) = 〈˜Φ(r 1 )˜Φ(r 1 + r)〉, (2.10) 

where ˜Φ ≡ Φ − 〈Φ〉. 

Statistically homogeneous fields, with C Φ (r) dependent only on the magnitude(but 

not on the direction) of the vector r = r 2 − r 1 connecting the points r 1 and r 2 

C Φ (r) = C Φ (r), r = |r| = √ r · r, (2.11)

2.1 STATISTICAL ISOTROPY OF TEMPERATURE ANISOTROPIES 9 

are said to be statistically isotropic. The covariance of isotropic random fields is even 

and always real. 

For temperature anisotropies of the CMB, statistical isotropy is simply equivalent 

to rotational invariance of n-point correlation functions 

〈∆T (Rˆn 1 )∆T (Rˆn 2 ) · · · ∆T (Rˆn n )〉 = 〈∆T (ˆn 1 )∆T (ˆn 2 ) · · · ∆T (ˆn n )〉, (2.12) 

where ∆T (Rˆn 1 ) is the temperature anisotropy at Rˆn 1 which is pixel ˆn 1 rotated by the 

rotation matrix R(α, β, γ) for the Euler angles α, β and γ. For two point correlations 

of CMB anisotropy, the condition of statistical isotropy (eqn. 2.11) will translate into 

C(ˆn, ˆn ′ ) = C(θ), θ = arccos (ˆn · ˆn ′ ) (2.13) 

which implies that the two point correlation is just a function of separation angle, θ, 

between the two points on the sky. 

Legendre polynomials, 

C(θ) = 1 

4π 

It is then convenient to expand it in terms of 

∞∑ 

(2l + 1)C l P l (cos θ), (2.14) 

l=2 

where C l is the widely used angular power spectrum. The summation over l starts from 

2 because the l = 0 term is the monopole which in the case of statistical isotropy the 

monopole is constant, and it can be subtracted out. The dipole l = 1 is due to the 

local motion of the observer and is subtracted out as well. 

To obtain the statistical isotropy condition in harmonic space, we expand the left 

hand side of eqn. (2.12) in terms of spherical harmonics, each ∆T (Rˆn i ) may be expanded 

as 

∆T (Rˆn 1 ) = ∑ l,m 

= ∑ l,m 

a R lm Y lm(Rˆn 1 ) (2.15) 

a R lm 

m ′ 

∑ 

Y lm ′(ˆn 1 )Dm l m(R), 

′ 

in which a R lm are harmonic coefficients in rotated coordinates and Dl m ′ m (R) is the finite 

rotation matrix in the lm-representation, Wigner D-function Varshalovich et al. (1988). 

By substituting this into eqn.(2.12) and using the orthonormality law of spherical har-

2.2 STATISTICAL ISOTROPY AND GAUSSIANITY 10 

monics, eqn.(G.2), we obtain the statistical isotropy condition in harmonic space 

∑ 

m ′ 1 ,m′ 2···,m′ n 

〈a R l 1 m ′ aR 1 l 2 m · · · ′ aR 2 l nm ′ 〉Dl 1 

n m 1 m 

(R)D l ′ 2 

1 m 2 m 

· · · D 

2(R) ln ′ m (R) = 〈a nm ′ l 

n 1 m 1 

a l2 m 2 

· · · a lnm n 

〉. 

(2.16) 

For a Gaussian field, only the covariance, 〈a lm a ∗ l ′ m ′〉, is important. As it is shown in 

Appendix A, the condition for statistical isotropy translates to a diagonal covariance 

matrix, 

〈a lm a ∗ l ′ m ′〉 = C lδ ll ′δ mm ′, (2.17) 

which means that the angular power spectrum of CMB anisotropy, C l , tells us everything 

we need to know about the (Gaussian) CMB anisotropy. When 〈a lm a ∗ l ′ m ′〉 is not diagonal, 

the angular power spectrum C l does not have all the information of the field and one 

should also take the off-diagonal elements into account. 

2.2 Statistical Isotropy and Gaussianity 

Statistical isotropy and Gaussianity of CMB anisotropies are two independent and 

distinct concepts. The Gaussianity of the primordial fluctuations is a key assumption 

of modern cosmology, motivated by simple models of inflation. And since the statistical 

properties of the primordial fluctuations are closely related to those of the cosmic microwave 

background (CMB) radiation anisotropy, a measurement of non-Gaussianity of 

the CMB will be a direct test of the inflation paradigm. If CMB anisotropy is Gaussian, 

then the two point correlation fully specifies the statistical properties. In the case of 

non-Gaussian fields, one must take the higher moments of the field into account in order 

to fully describe the whole field. Statistical isotropy is a condition which guarantees 

that statistical properties of CMB sky (e.g. n-point correlations) are invariant under 

rotation. Hence, in a SI model, the iso-contours of two point correlations where one 

point is fixed and the other scans the whole sky, are concentric circles around the given 

point. Any breakdown of statistical isotropy will make these iso-contours non-circular. 

In addition to cosmological mechanisms, instrumental and environmental effects in observations 

easily produce spurious breakdown of statistical isotropy. 

A Gaussian field may or may not respect the statistical isotropy. In other words we

2.3 THE BIPOLAR POWER SPECTRUM (BIPS) 11 

can have 

1. Gaussian and SI models: In these models two point correlation and C l have all 

the information of the field and C(ˆn 1 , ˆn 2 ) = C(ˆn 1 · ˆn 2 ) 

2. Gaussian but not SI models: two point correlation contains all the information of 

the field, but C(ˆn 1 , ˆn 2 ) ≠ C(ˆn 1 · ˆn 2 ). So C l is not adequate to describe the field 

and off-diagonal elements of covariance matrix should be taken into account. 

3. Non-Gaussian but SI models: two point correlation alone is not sufficient to describe 

the whole field and we need to take higher moments into account but every 

n-point correlation of the field is invariant under rotation. 

4. Non-Gaussian and non-SI models: neither two point correlation has all the information 

nor it is invariant under rotation. 

Here we confine ourselves to Gaussian fields where we only need to consider two point 

correlations. We will present a general way of describing fields of the kind (1) and (2) 

in the above list. 

2.3 The Bipolar Power Spectrum (BiPS) 

Two point correlation of CMB anisotropies, C(ˆn 1 , ˆn 2 ), is a two point function on 

S 2 × S 2 , and hence can be expanded as 

C(ˆn 1 , ˆn 2 ) = 

∑ 

l 1 ,l 2 ,L,M 

A lM 

l 1 l 2 

{Y l1 (ˆn 1 ) ⊗ Y l2 (ˆn 2 )} lM , (2.18) 

where A lM 

l 1 l 2 

are coefficients of the expansion (hereafter BipoSH coefficients) and {Y l1 (ˆn 1 )⊗ 

Y l2 (ˆn 2 )} lM are the bipolar spherical harmonics which transform in the same manner as 

the spherical harmonic function with l, M with respect to rotations (Varshalovich et al. 

, 1988) given by 

{Y l1 (ˆn 1 ) ⊗ Y l2 (ˆn 2 )} lM = ∑ m 1 m 2 

C lM 

l 1 m 1 l 2 m 2 

Y l1 m 1 

(ˆn 2 )Y l2 m 2 

(ˆn 2 ), (2.19) 

in which C lM 

l 1 m 1 l 2 m 2 

are Clebsch-Gordan coefficients. We can inverse-transform C(ˆn 1 , ˆn 2 ) 

to get the A lM 

l 1 l 2 

by multiplying both sides of eqn.(2.18) by {Y l ′ 

1 

(ˆn 1 ) ⊗ Y l ′ 

2 

(ˆn 2 )} ∗ l ′ M ′ and

2.3 THE BIPOLAR POWER SPECTRUM (BIPS) 12 

integrating over all angles. Then the orthonormality of bipolar harmonics, eqn. (G.2), 

implies that 

A lM 

l 1 l 2 

= 

∫ 

dΩˆn1 

∫ 

dΩˆn2 C(ˆn 1 , ˆn 2 ) {Y l1 (ˆn 1 ) ⊗ Y l2 (ˆn 2 )} ∗ lM. (2.20) 

The above expression and the fact that C(ˆn 1 , ˆn 2 ) is symmetric under the exchange of 

ˆn 1 and ˆn 2 lead to the following symmetries of A lM 

l 1 l 2 

A lM 

l 2 l 1 

= (−1) (l 1+l 2 −L) A lM 

l 1 l 2 

, (2.21) 

A lM 

ll = A lM 

ll δ l,2k−1 , k = 0, 1, 2, · · · . 

We show in Appendix A that the Bipolar Spherical Harmonic (BipoSH) coefficients, 

A lM 

l 1 l 2 

, are in fact linear combinations of off-diagonal elements of the covariance matrix, 

A lM 

l 1 l 2 

= ∑ m 1 m 2 

〈a l1 m 1 

a ∗ l 2 m 2 

〉(−1) m 2 

C lM 

l 1 m 1 l 2 −m 2 

. (2.22) 

This clearly shows that A lM 

l 1 l 2 

completely represent the information of the covariance 

120 

120 

100 

100 

l(l+1)+m+1 

80 

60 

40 

l(l+1)+m+1 

80 

60 

40 

20 

20 

0 

0 20 40 60 80 100 120 

l(l+1)+m+1 

0 

0 20 40 60 80 100 120 

l(l+1)+m+1 

Figure 2.1 BipoSH coefficients are linear combinations of elements of the covariance 

matrix. 

Here A 2M 

ll ′ 

(left) and A4M ll ′ 

(right) are plotted to show how BiPS covers the 

off-diagonal elements of the covariance matrix in harmonic space. 

matrix. Fig. 2.1 shows how A 2M 

l 1 l 2 

and A 4M 

l 1 l 2 

matrix. When SI holds, the covariance matrix is diagonal, hence 

combine the elements of the covariance 

A lM 

ll ′ = (−1)l C l (2l + 1) 1/2 δ ll ′ δ l0 δ M0 , (2.23)

2.4 BIPS REPRESENTATION IN REAL SPACE 13 

BipoSH expansion is the most general way of studying two point correlation functions 

of CMB anisotropy. The well known angular power spectrum, C l is in fact a subset of 

BipoSH coefficients. 

It is impossible to measure all A lM 

l 1 l 2 

A 00 

l 1 l 2 

= (−1) l 1√ 

2l1 + 1 C l1 δ l1 l 2 

. (2.24) 

individually because of cosmic variance. Combining 

BipoSH coefficients into Bipolar Power Spectrum reduces the cosmic variance 3 . 

BiPS of CMB anisotropy is defined as a convenient contraction of the BipoSH coefficients 

κ l = ∑ 

l,l ′ ,M 

|A lM 

ll ′ |2 ≥ 0. (2.25) 

The BiPS has interesting properties. It is orientation independent, i.e. invariant under 

rotations of the sky. For models in which statistical isotropy is valid, BipoSH coefficients 

are given by eqn (2.24), and therefore SI condition implies a null BiPS, i.e. κ l = 0 for 

every l > 0, 

κ l = κ 0 δ l0 . (2.26) 

Non-zero components of BiPS imply the break down of statistical isotropy, and this 

introduces BiPS as a measure of statistical isotropy, 

STATISTICAL ISOTROPY =⇒ κ l = 0 ∀l ≠ 0. (2.27) 

2.4 BiPS Representation in Real Space 

Two point correlation of the CMB anisotropy is given by ensemble average over 

many universes (eqn.(2.6)). But in reality, there is only one universe and the ensemble 

average is meaningless unless the CMB sky is SI, where the two point correlation function 

C(θ) can be well estimated by the average product of temperature fluctuations in two 

directions ˆn and ˆn ′ whose angular separation is θ (see Fig. 2.2), this can be written as 

˜C(θ) = 

∫ 

dΩˆn 

4π 

∫ 

dΩˆn ′ 

4π δ(ˆn · ˆn′ − cos θ)∆T (ˆn)∆T (ˆn ′ ) (2.28) 

3 This is similar to combining a lm to construct the angular power spectrum, C l = 1 

2l+1 

∑ 

m |a lm| 2 , 

to reduce the cosmic variance


Figure 2.2 C(θ) ‘recovered’ from a map of a statistically isotropic model matches the 

original C(θ) shown as a solid line from which we made the map. The error bars shown 

are computed from 10 realizations. Note that the size of error bar depends on the size 

of θ bins. 

If the statistical isotropy is violated the estimate of the correlation function from a 

sky map given by a single temperature product 

˜C(ˆn 1 , ˆn 2 ) = ∆T (ˆn 1 )∆T (ˆn 2 ) (2.29) 

is poorly determined. 

Although it is not possible to estimate each element of the full correlation function 

C(ˆn 1 , ˆn 2 ), some measures of statistical isotropy of the CMB map can be estimated 

through suitably weighted angular averages of ∆T (ˆn 1 )∆T (ˆn 2 ). The angular averaging 

procedure should be such that the measure involves averaging over sufficient number of 

independent measurements , but should ensure that the averaging does not erase all the 

signature of statistical anisotropy. Another important desirable property is that measure


be independent of the overall orientation of the sky. Based on these considerations, we 

have proposed a set of measures of statistical isotropy Hajian & Souradeep 

(2003b) 

which can be shown that is equivalent to the one in eqn.(2.25) 

∫ ∫ ∫ 

κ l = (2l + 1) 2 1 

dΩ n1 dΩ n2 [ dRχ l (R) C(Rˆn 

8π 2 1 , Rˆn 2 )] 2 . (2.30) 

In the above expression, C(Rˆn 1 , Rˆn 2 ) is the two point correlation at Rˆn 1 and Rˆn 2 

which are the coordinates of the two pixels ˆn 1 

and ˆn 2 after rotating the coordinate 

system through an angle ω where (0 ≤ ω ≤ π) about the axis n(Θ, Φ). The direction of 

this rotation axis n is defined by the polar angles Θ where (0 ≤ Θ ≤ π) and Φ, where 

(0 ≤ Φ ≤ 2π). χ l is the trace of the finite rotation matrix in the lM-representation 

χ l (R) = 

l∑ 

M=−l 

DMM l (R), (2.31) 

which is called the characteristic function, or the character of the irreducible representation 

of rank l. It is invariant under rotations of the coordinate systems. Explicit forms 

of χ l (R) are simple when R is specified by rotation by ω about an axis n(Θ, Φ), then 

χ l (R) is completely determined by the rotation angle ω and it is independent of the 

rotation axis n(Θ, Φ), 

χ l (R) = χ l (ω) (2.32) 

= 

sin [(2l + 1)ω/2] 

. 

sin [ω/2] 

And finally dR in eqn(2.30) is the volume element of the three-dimensional rotation 

group and is given by 

dR = 4 sin 2 ω dω sin Θ dΘ dΦ . (2.33) 

2 

As it is shown in Appendix (A), this expression for BiPS may be simplified further as 

κ l = 2l + 1 ∫ ∫ 

∫ 

dΩ 

8π 2 1 dΩ 2 C(ˆq 1 , ˆq 2 ) dRχ l (R)C(Rˆq 1 , Rˆq 2 ). (2.34) 

For a statistically isotropic model C(ˆn 1 , ˆn 2 ) is invariant under rotation, and therefore 

C(Rˆn 1 , Rˆn 2 ) = C(ˆn 1 , ˆn 2 ). 

If we use this property to substitute in eqn.(2.34) for 

C(Rˆn 1 , Rˆn 2 ), and if we use the orthonormality of χ l (ω) (see eqn. G.7), we will recover 

the condition for SI, 

κ l = κ 0 δ l0 . (2.35)

2.5 UNBIASED ESTIMATOR OF BIPS 16 

The proof of equivalence of BiPS definition in eqns.(2.30) and (2.25) is as follows. In 

eqn.(2.30), we substitute (2.31) for χ l (R) and from 

C(Rˆn 1 , Rˆn 2 ) = 

∑lM ′ 

A lM ′ 

l 1 l 2 

l 1 l 2 

∑ 

DMM l ′(R){Y l 1 

(ˆn 1 ) ⊗ Y l2 (ˆn 2 )} lM , (2.36) 

M 

we substitute for C(Rˆn 1 , Rˆn 2 ) and use the orthonormality of DMM l (R) (see eqn.(G.8)), 

we will obtain eq.(2.25). Real-space representation of BiPS is very suitable for analytical 

computation of BiPS for theoretical models where we know the analytical expression for 

the two point correlation of the model, such as theoretical models in Hajian & Souradeep 

(2003a). This formalism would immensely simplify analytical calculations. On the other 

hand, the harmonic representation of BiPS allows computationally rapid methods for 

BiPS estimation from a given CMB map. 

2.5 Unbiased Estimator of BiPS 

In statistics, an estimator is a function of the known data that is used to estimate 

an unknown parameter; an estimate is the result from the actual application of the 

function to a particular set of data. Many different estimators may be possible for any 

given parameter. The above theory can be use to construct an estimator for measuring 

BipoSH coefficients from a given CMB map 

Ã lM 

ll ′ = ∑ mm ′ √ 

Wl W l ′a lm a l ′ m ′ ClM lml ′ m ′ , (2.37) 

where W l is the Legendre transform of the window function. The above estimator is 

un-biased. Bias is the mismatch between ensemble average of the estimator and the 

true value. Bias for the BiPS is defined as B l = 〈˜κ l 〉 − κ l and is equal to 

B l = ∑ l 1 ,l 2 

W l1 W l2 

∑ 

∑ 

[ 

〈a 

∗ 

l1 m 1 

a l1 m ′ 1 〉〈a∗ l 2 m 2 

a l2 m ′ 2 〉 + 〈a∗ l 1 m 1 

a l2 m ′ 2 〉〈a∗ l 2 m 2 

a l1 m ′ 1 〉] 

m 1 ,m ′ 1 m 2 ,m ′ 2 

× ∑ M 

C lM 

l 1 m 1 l 2 m 2 

C lM 

l 1 m ′ 1 l 2m ′ 2 . (2.38) 

Therefore an un-biased estimator of BiPS is given by 

∣ 

∣ ∣∣ 

2 ∣ÃlM ll − ′ Bl , (2.39) 

˜κ l = ∑ ll ′ M

2.5 UNBIASED ESTIMATOR OF BIPS 17 

The above expression for B l is obtained by assuming Gaussian statistics of the temperature 

fluctuations. The procedure is very similar to computing cosmic variance (which 

is discussed in the next section), but much simpler. However, we can not measure the 

ensemble average in the above expression and as a result, elements of the covariance 

matrix (obtained from a single map) are poorly determined due to the cosmic variance. 

The best we can do is to compute the bias for the SI component of a map 

B l ≡ 〈˜κ B l 〉 SI 

= (2l + 1) ∑ ∑l+l 1 

l 1 

l 2 =|l−l 1 | 

C l1 C l2 W l1 W l2 

[ 

1 + (−1) l δ l1 l 2 

] 

. (2.40) 

Note , the estimator ˜κ l is unbiased, only for SI correlation,i.e., 〈˜κ l 〉 = 0. Consequently, 

for SI correlation, the measured ˜κ l will be consistent with zero within the error bars given 

by σ SI 

Hajian & Souradeep (2003b). We simulated 1000 SI CMB maps and computed 

BiPS for them using different filters. The average BiPS of SI maps is an estimation of 

the bias which can be compared to our analytical estimation. Fig. 2.3 shows that the 

theoretical bias (computed from average C l ) match the numerical estimations of average 

κ l of the 1000 realizations of the SI maps. 

Bias 

Average 

Bias 

Average 

0 

0 

5 10 15 20 

5 10 15 20 

Figure 2.3 Analytical bias for (left) a two beam window function with W S 

l (20, 30) and 

(right) a Gaussian window function with Wl 

G (40) computed from the average C l from 

1000 realizations of a SI CMB map compared with 〈κ realization 

l 〉 (the average κ l from 

1000 realizations). This shows that the theoretical bias is a very good estimation of the 

bias for a statistically isotropic map. 

It is important to note that bias cannot be correctly subtracted for non-SI maps.

2.6 COSMIC VARIANCE 18 

Non-zero ˜κ l estimated from a non-SI map will have contribution from the non-SI terms 

in full bias given in eq. (2.38). It is not inconceivable that for strong SI violation, B l 

over-corrects for the bias leading to negative values of ˜κ l . What is important is whether 

the measured ˜κ l differs from zero at a statistically significant level. 

2.6 Cosmic Variance 

Cosmic variance is defined as the variance of the estimator of the BiPS 

σ 2 = < ˜κ 2 l > − < ˜κ l > 2 (2.41) 

We can analytically compute the variance of ˜κ l using the Gaussianity of ∆T . Looking 

back at the eq.(2.30) we can see, we will have to calculate the eighth moment of the 

field 

〈∆T (ˆn 1 )∆T (ˆn 2 )∆T (ˆn 3 )∆T (ˆn 4 )∆T (ˆn 5 )∆T (ˆn 6 )∆T (ˆn 7 )∆T (ˆn 8 )〉. (2.42) 

Assuming Gaussianity of the field we can rewrite the eight point correlation in terms 

of two point correlations. One can write a simple code to do that 4 . This will give us 

(8 − 1)!! = 7 × 5 × 3 = 105 terms. These 105 terms consist of terms like: 

〈∆T (ˆn 1 )∆T (ˆn 2 )〉〈∆T (ˆn 3 )∆T (ˆn 4 )〉〈∆T (ˆn 5 )∆T (ˆn 6 )〉〈∆T (ˆn 7 )∆T (ˆn 8 )〉, (2.43) 

and all other permutations of them. On the other hand 〈˜κ l 〉 has a 4 point correlation 

in it which can also be expanded versus two point correlation functions. If we form 

〈˜κ 2 l 〉 − 〈˜κ l〉 2 , only 96 terms will be left which are in the following form 

( 2l + 1 ∫ 

) 2 dΩ 

8π 2 

1 · · · dΩ 4 dRdR ′ χ l (R)χ l (R ′ )C(ˆn 1 , R ′ˆn 4 )C(ˆn 2 , R ′ˆn 3 )C(Rˆn 1 , ˆn 4 )C(Rˆn 2 , ˆn 3 ) 

(2.44) 

and all other permutations. In Appendix A we explain how we can handle these 96 

terms to compute the following analytical expression for cosmic variance (details of the 

above calculations are presented in Appendix B.), 

σ 2 (˜κ SI 

l) = ∑ 

] 

4 Cl 4 W l 

[2 4 (2l + 1)2 

2l + 1 

+ (−1)l (2l + 1) + (1 + 2(−1) l )Fll 

l 

l:2l≥l 

4 F90 software implementing this is available from the authors upon request

2.7 COSMIC VARIANCE 19 

+ ∑ ∑l+l 1 

l 1 

l 2 =|l−l 1 | 

4 C 2 l 1 

C 2 l 2 

W 2 

l 1 

W 2 

l 2 

[ 

(2l + 1) + F 

l 

⎡ 

+ 8 ∑ (2l + 1) 2 

2l 1 + 1 C2 l 1 

W 2 ⎣ 

l 1 

l 1 

+ 16 (−1) l ∑ 

l 1 :2l 1 ≥l 

Numerical computation of σ 2 SI 

(2l + 1) 2 

2l 1 + 1 

∑l+l 1 

l 2 =|l−l 1 | 

∑l+l 1 

l 2 =|l−l 1 | 

⎤2 

C l2 W l2 

⎦ 

l1 l 2 

] 

C 3 l 1 

C l2 W 3 

l 1 

W l2 . (2.45) 

is fast. But the challenge is to compute Clebsch-Gordan 

coefficients for large quantum numbers. We use drc3j subroutine of netlib 5 in order to 

compute the Clebsch-Gordan coefficients in our codes. Again we can check the accuracy 

of our analytical estimation of cosmic variance by comparing it against the standard 

deviation of BiPS of 1000 simulations of SI CMB sky. The result is shown in Fig. 2.4 

and shows a very good agreement between the two. 

5 

100 

50 

0 

0 

-50 

-5 

-100 

5 10 15 20 

5 10 15 20 

Figure 2.4 Bias corrected ‘measurement’ of BiPS, ˜κ l for SI CMB maps with a best fit 

LCDM power spectrum smoothed by (left:) a two beam function W S 

l (20, 30) and (right:) 

a Gaussian beam Wl 

G (40). The cosmic error, σ(κ l ), obtained using 1000 independent 

realizations of CMB (full) sky map matches the analytical results shown by dotted curve 

with triangles . This shows a much better fit to the theoretical cosmic variance compared 

to what was obtained for 100 realizations Hajian & Souradeep (2003b) 

5 http://www.netlib.org/slatec/src/

2.7 SOURCES OF BREAKDOWN OF STATISTICAL ISOTROPY 20 

2.7 Sources of Breakdown of Statistical Isotropy 

An observed map of CMB anisotropy, ∆T obs (ˆn), is an n-dimensional vector (n = 

12 × 512 2 for the WMAP data at HEALPix resolution N side = 512). This observed 

map contains the true CMB temperature fluctuations, ∆T (ˆn), convolved with the beam 

and buried into noise and foreground contaminations. The observed map is related to 

the true map through this relation 

∆T obs = B∆T + n, (2.46) 

in which B is a matrix that contains the information about the beam smoothing effect 

and n is the contribution from instrumental noise and foreground contamination. Hence, 

the observed map is a realization of a Gaussian process with covariance C = C T + C N + 

C res where C T is the theoretical covariance of the CMB temperature fluctuations, C N 

is the noise covariance matrix and C res is the covariance of residuals of foregrounds. 

Breakdown of statistical isotropy can occur in any of these parts. In general we can 

divide these effects into two kinds: 

• Theoretical effects: These effects are theoretically motivated and are intrinsic to 

the true CMB sky, ∆T . We discuss three examples of these effects, i.e. non-trivial 

cosmic topology (Chapter 4), hidden patterns in the CMB maps which can be 

explained by models of homogeneous spaces known as Bianchi models (Chapter 5) 

and primordial magnetic fields, in the next section. 

• Observational artifacts: In an ideally cleaned CMB map, the true CMB temperature 

fluctuations are completely extracted from the observed map. But this 

is not always true. Sometimes there are some artifacts (related to B or n) left in 

the cleaned map which may in principle violate the SI. These effects are explained 

in Chapter 6.

l’ 

l’ 

2.8 PRIMORDIAL MAGNETIC FIELDS 21 

2.8 Primordial Magnetic Fields 

As the first example of theoretical motivations for models of breakdown of SI, we 

study the simplest configuration of cosmological magnetic fields. It has been shown 

that a cosmological magnetic field, generated during an early epoch of inflation [Ratra 

(1992); Bamba et al. (2004)], can generate CMB anisotropies [Durrer et al. (1998)]. 

The presence of a preferred direction due to a homogeneous magnetic field background 

leads to non-zero off-diagonal elements in the covariance matrix [Chen et al. (2004)]. 

This induces correlations between a l+1,m and a l−1,m multipole coefficients of the CMB 

temperature anisotropy field in the following manner (see Fig. 2.5) 

20 

x 10 −3 

20 

x 10 −3 

18 

2.5 

18 

2.5 

16 

16 

2 

2 

14 

14 

12 

1.5 

12 

1.5 

10 

10 

8 

1 

8 

1 

6 

6 

4 

0.5 

4 

0.5 

2 

2 

2 4 6 8 10 12 14 16 18 20 

l 

0 

2 4 6 8 10 12 14 16 18 20 

l 

0 

Figure 2.5 Effect of a homogeneuos magnetic field on covariance matrix is the introduction 

of off-diagonal elements (right). In the absence of magnetic fields the covariance 

matrix is diagonal (left). 

〈a lm a ∗ l ′ m ′〉 = δ m,m ′[δ l,l ′C l + (δ l+1,l ′ −1 + δ l−1,l ′ +1D l )], (2.47) 

where D l is the power spectrum of off-diagonal elements of the covariance matrix. For 

a Harrison-Peebles-Yu-Zel’dovich scale-invariant spectrum, D l behaves as l −2 . More 

precisely, it is given by 

D l = 4 × 10 −16 l −2 ( B 

1nG )4 . (2.48) 

This clearly violates the statistical isotropy. If we substitute the above 〈a lm a ∗ l ′ m ′〉 into 

eqn. (2.22), we will get 

A lM 

l 1 l 2 

= δ M,0 [(−1) l 1√ 

(2l1 + 1)C l1 δ l1 ,l 2 

δ l,0 (2.49)

2.8 PRIMORDIAL MAGNETIC FIELDS 22 

+ D l1 

∑ 

m1=−l1,l1 

+ D l1 

∑ 

m1=−l1,l1 

(−1) m 1 

C lM 

l 1 m 1 l 2 −m 1 

δ l1+1 ,l 2−1 

(−1) m 1 

Cl lM 

1 m 1 l 2 −m 1 

δ l1−1 ,l 2+1 

]. 

The first line is the statistically isotropic part and it just contributes to κ 0 . But the 

interesting parts are the two other lines. We can substitute this expression into eqn. 

(2.25) and compute the BiPS predictions for magnetic fields (see fig. 2.6). 

7 

6 

B=16nG 

B=14nG 

B=12nG 

5 

4 

κ l 

3 

2 

1 

0 

1 2 3 4 5 6 7 

l 

Figure 2.6 BiPS predictions for primordial homogeneous magnetic fields.

Chapter 3 

SI Violation: Analytic Models 

In this chapter we model the SI violation using toy models of non-SI correlations 

which are analytically solvable. These toy models are motivated by various mechanisms 

of SI violation such as multiply connected spaces. Besides helping us to come to a good 

understanding of violation of SI, they shed a light on how BiPS method works. We will 

repeatedly refer to this chapter in the rest of this thesis to explain several signatures of 

different mechanisms of SI violation on BiPS. 

3.1 One Preferred Axis 

The simplest way of having an aniostropic correlation function is to introduce a 

preferred axis in the correlation space. The toy model we first consider has a preferred 

axis in the z direction and its two point correlation is given by 

C(ˆq 1 , ˆq 2 ) = 1 − ɛ (ˆq 1 · ẑ − ˆq 2 · ẑ) 2 , (3.1) 

where ɛ < 1/4 to ensure positivity of C(ˆq 1 , ˆq 2 ). This is an interesting case where the 

BiPS can be computed analytically. To get the analytical expression for BiPS in this 

toy model, we use eqn.(2.30) and evaluate the following integral 

∫ ∫ ∫ 

κ l = (2l + 1) 2 1 

dΩ q1 dΩ q2 [ dR χ l (R) C(Rˆq 

8π 2 1 , Rˆq 2 )] 2 (3.2) 

∫ ∫ ∫ 

= (2l + 1) 2 1 

dΩ q1 dΩ q2 [ dR χ l (R) {1 − ɛ (ˆq ′ 

8π 2 1 · ẑ − ˆq′ 2 · ẑ)2 }] 2 . 

23

3.1 ONE PREFERRED AXIS 24 

In the above expression ˆq ′ i = R ωˆn ˆq i are the coordinates of the pixels after rotating them 

through an angle ω about an axis ˆn(Θ, Φ). Correlation only contains dot products of 

the pixel coordinates and the z axis, so one can equivalently calculate the function by 

keeping the directions ˆq 1 and ˆq 2 fixed and rotating the axes. In particular the ẑ axis 

rotated by an angle −ω about the same axis ˆn(Θ, Φ) goes to 

ẑ ′ = ẑ cos ω + ẑ(ˆn · ẑ)(1 − cos ω) − (ˆn × ẑ) sin ω (3.3) 

= (sin Φ cos Θ sin Θ(1 − cos ω) − sin Φ sin Θ sin ω, 

cos Θ sin Φ sin Θ(1 − cos ω) + cos Φ sin Θ sin ω, 

cos 2 Θ(1 − cos ω) + cos ω) . 

After substituting this in the correlation function we will get 

1 − ɛ (ˆq 1 · ẑ ′ − ˆq 2 · ẑ ′ ) 2 = 1 − ɛ((q 1x − q 2x ) 2 Z ′ 2 

x + (q 1y − q 2y ) 2 Z ′ 2 

y (3.4) 

+ (q 1z − q 2z ) 2 Z ′ 2 

z ) 

− 

− 

− 

2ɛ (q 1x − q 2x )(q 1y − q 2y )Z ′ x Z ′ y 

2ɛ (q 1x − q 2x )(q 1z − q 2z )Z ′ xZ ′ z 

2ɛ (q 1y − q 2y )(q 1z − q 2z )Z ′ yZ ′ z 

Where Z i are components of vector ẑ, i. e. ẑ = (Z x , Z y , Z z ). We also introduce following 

shorthands for simplicity 

δ x = q 1x − q 2x (3.5) 

δ y = q 1y − q 2y 

δ z = q 1z − q 2z . 

The correlation function then becomes 

1 − ɛ (ˆq 1 · ẑ ′ − ˆq 2 · ẑ ′ ) 2 = 1 − ɛ(δ 2 xZ ′ 2 

x + δ 2 yZ ′ 2 

y + δ 2 zZ ′ 2 

z ) (3.6) 

− 

− 

− 

2ɛ δ x δ y Z ′ x Z ′ y 

2ɛ δ x δ z Z ′ x Z ′ z 

2ɛ δ y δ z Z ′ y Z ′ z

3.1 ONE PREFERRED AXIS 25 

Now we should substitute for z ′ 

The ω dependence is only in z ′ 

from eqn.(3.3) and evaluate the integrals of eqn(3.2). 

so it is better to do the ω integration first. We find that 

the cross terms like Z ′ x Z ′ z will vanish and of the non zero terms, the first term, 1, would 

only contribute to κ 0 because 

∫ 

∫ π 

dR 1 χ l (R) = 4π dωχ l (ω) sin 2 ω/2 (3.7) 

0 

= 2π 2 δ l0 

which is a direct result of orthonormality of the characteristic function χ l (ω). The above 

result is in fact expected because the first term is the SI part of the correlation function 

and it sould only contribute to κ 0 . The anisotropic part would have a component of κ 2 

as well 

∫ 

dR χ l (R) (ˆq 1 · ẑ ′ − ˆq 2 · ẑ ′ ) 2 = ξ(ˆq 1 , ˆq 2 )δ l0 + ζ(ˆq 1 , ˆq 2 )δ l2 (3.8) 

Putting everything together and computing the inner-most integral, we would ge 

∫ 

dR χ l (R) C(Rˆq 1 , Rˆq 2 ) = 2π 2 δ l0 − ɛ(ξ(ˆq 1 , ˆq 2 )δ l0 + ζ(ˆq 1 , ˆq 2 )δ l2 ) (3.9) 

W here : ξ(ˆq 1 , ˆq 2 ) = 2π2 

3 (δ2 x + δ2 y + δ2 z ) 

ζ(ˆq 1 , ˆq 2 ) = 2π2 

15 (2δ2 x − δ2 y − δ2 z ). 

And using these, we can compute the analytical expression for BiPS by carrying out 

this integral 

κ l = 

∫ 

(2l + 1)2 

(8π 2 ) 2 

dΩ q1 

∫ 

dΩ q2 [2π 2 δ l0 − ɛ(ξ(ˆq 1 , ˆq 2 )δ l0 + ζ(ˆq 1 , ˆq 2 )δ l2 )] 2 . (3.10) 

which will give 

where 

κ l = 

(2l + 1)2 

(8π 2 ) 2 [κ 0 δ l0 + κ 2 δ l2 ] , (3.11) 

κ 0 = 64 

27 π6 (27 − 36ɛ + 16ɛ 2 ) (3.12) 

κ 2 = 4096ɛ2 π 6 

. 

3375 

This exercise shows us the sensitivity of BiPS to the characteristic patterns of correlation. 

Any SI violating mechanism which causes a preferred direction in the correlation space,

3.2 THREE PREFERRED AXES 26 

will have a non-zero κ 2 . 

Examples of this are anisotropic toroidal topologies of the 

Universe and noise covariance matrices like WMAP noise which are studied in the next 

chapters and are shown to have a non-zero κ 2 . 

3.2 Three Preferred Axes 

Having seen the effect of one preferred axis on BiPS, we can consider a more general 

case in which we have three preferred directions in the correlation function. Each axis 

will cause a non-zero l = 2 component in BiPS. We attribute three strength factors ɛ 1 , 

ɛ 2 and ɛ 3 to these directions to have a control on importance of each of them 

C(ˆq 1 , ˆq 2 ) = 1 − (ɛ 1 ((ˆq 1 − ˆq 2 ) · ˆx) 2 + ɛ 2 ((ˆq 1 − ˆq 2 ) · ŷ) 2 + ɛ 3 ((ˆq 1 − ˆq 2 ) · ẑ) 2 ) (3.13) 

to ensure positivity, we restrict our interest to models with ɛ i < 1/4. We can repeat the 

steps of our calculations in the last section to get the analytical expression for BiPS in 

this model. To avoid the repetition, we will only present the results of two important 

steps which will be useful later. The first is the integral over rotations which determines 

the non-zero l components 

∫ 

dR χ l (R) C(Rˆq 1 , Rˆq 2 ) = 2π 2 δ l0 − (ξ(ˆq 1 , ˆq 2 )δ l0 + ζ(ˆq 1 , ˆq 2 )δ l2 ) (3.14) 

where 

ξ(ˆq 1 , ˆq 2 ) = 2π2 

3 (ɛ 1 + ɛ 2 + ɛ 3 )(δ 2 x + δ2 y + δ2 z ) (3.15) 

ζ(ˆq 1 , ˆq 2 ) = − 2π2 

15 ((−2ɛ 1 + ɛ 2 + ɛ 3 )δ 2 x + (ɛ 1 − 2ɛ 2 + ɛ 3 )δ 2 y + (ɛ 1 + ɛ 2 − 2ɛ 3 )δ 2 z ). 

Then we integrate over Ωˆq1 and Ωˆq2 and we get the BiPS for this toy model 

κ l = 

and κ 0 and κ 2 are given by the strength factors 

(2l + 1)2 

(8π 2 ) 2 [κ 0 δ l0 + κ 2 δ l2 ] , (3.16) 

κ 0 = 64 

27 π6 (27 + 16(ɛ 2 1 + ɛ2 2 + ɛ2 3 ) + 32(ɛ 1ɛ 2 + ɛ 1 ɛ 3 + ɛ 2 ɛ 3 ) − 36(ɛ 1 + ɛ 2 + ɛ 3 ))(3.17) 

= 64 

27 π6 (27 + 16Σ ɛ 2 + 32Π ɛ − 36Σ ɛ ) 

κ 2 = 4096π6 

3375 (ɛ2 1 + ɛ 2 2 + ɛ 2 3 − ɛ 1 ɛ 2 − ɛ 1 ɛ 3 − ɛ 2 ɛ 3 ) 

= 4096π6 

3375 (Σ ɛ 2 − Π ɛ)

3.3 OBLIQUE PREFERRED AXIS 27 

In the above expressions, we have used these conventions for simplicity 

Σ ɛ 2 = ɛ 2 1 + ɛ 2 2 + ɛ 2 3 (3.18) 

Σ ɛ = ɛ 1 + ɛ 2 + ɛ 3 

Π ɛ = ɛ 1 ɛ 2 + ɛ 1 ɛ 3 + ɛ 2 ɛ 3 

Interestingly we can see that the only non-zero component is l = 2. If the strength 

factors are the same for three three preferred axes, κ 2 will vanish and we will get a null 

BiPS. This is expectable because this is in fact a statistically isotropic model which can 

be seen explicitly from the two point correlation 

C(ˆq 1 , ˆq 2 ) = 1 − ɛ [ ((ˆq 1 − ˆq 2 ) · ˆx) 2 + ((ˆq 1 − ˆq 2 ) · ŷ) 2 + ((ˆq 1 − ˆq 2 ) · ẑ) 2] (3.19) 

= 1 − ɛ(ˆq 1 − ˆq 2 ) · (ˆq 1 − ˆq 2 ) 

= (1 + 2ɛ) − 2ɛ(ˆq 1 · ˆq 2 ) 

= C(ˆq 1 · ˆq 2 ). 

3.3 Oblique Preferred Axis 

Three axes in the previous toy model where perpendicular to each other. A more 

general case is where the axes make arbitrary angles to each other. This model can 

be obtained from the previous one by the act of a linear transform that changes the 

direction of the preferred axes; 

⎛ ⎞ 

x 

⎜ 

⎝ y ⎟ 

⎠ −→R 

z 

⎛ ⎞ 

x R 

⎜ 

⎝ y R ⎟ 

⎠ (3.20) 

z R 

A convenient parametrization of this transform matrix is 

R = 

⎛ 

⎞ 

1 cos α 1 cos α 2 

⎜ 

⎝ 0 sin α 1 cos α 3 sin α ⎟ 2 ⎠ (3.21) 

0 0 sin α 2 sin α 3

3.3 OBLIQUE PREFERRED AXIS 28 

After this transformation, three preferred directions will make angles α 1 , α 2 and α 3 with 

each other. The toy correlation fucnation with these preferred directions may be given 

as 

C(ˆq 1 , ˆq 2 ) = 1 − [ ɛ 1 ((ˆq 1 − ˆq 2 ) · ˆx R ) 2 + ɛ 2 ((ˆq 1 − ˆq 2 ) · ŷ R ) 2 + ɛ 3 ((ˆq 1 − ˆq 2 ) · ẑ R ) 2] (3.22) 

Integral over the rotation group elements would tell us l = 2 is the (nontrivial) non-zer 

component 

∫ 

dR χ l (R) C(Rˆq 1 , Rˆq 2 ) = 2π 2 δ l0 − (ξ(ˆq 1 , ˆq 2 )δ l0 + ζ(ˆq 1 , ˆq 2 )δ l2 ) (3.23) 

in which 

ξ(ˆq 1 , ˆq 2 ) = 2π2 

3 Σ ɛ(δ 2 x + δ2 y + δ2 z ) (3.24) 

ζ(ˆq 1 , ˆq 2 ) = π2 

15 [Σ ɛ + 3(ɛ 1 + ɛ 2 cos 2α 1 + ɛ 3 cos 2α 2 )] δ 2 x 

− 

− 

π2 

60 [8ɛ 1 − 4ɛ 2 (1 − 3 cos 2α 1 ) + 6ɛ 3 (cos 2α 2 − cos 2α 3 + cos 2α 2 cos 2α 3 )] δ 2 y 

π2 

60 [8Σ ɛ + 6ɛ 3 (−1 + cos 2α 2 + cos 2α 3 − cos 2α 2 cos 2α 3 )] δ 2 z 

+ 4π2 

5 [ɛ 2 cos α 1 sin α 1 + ɛ 3 cos α 2 cos α 3 sin α 2 ] δ x δ y 

+ 4π2 [ ] 

ɛ3 cos α 3 sin 2 α 2 sin α 3 δy δ z + 2π2 

5 

5 [ɛ 3 sin 2α 2 sin α 3 ] δ x δ z 

This is the most general case. For simplicity we can confine ourselves to models with 

only one oblique axis and two orthogonal ones, 

α 2 = α 3 = π/2. (3.25) 

BiPS for this model can then be computed by integrating the square of the above 

expression over all possible directions on the sky 

κ l = 

(2l + 1)2 

(8π 2 ) 2 [κ 0 δ l0 + κ 2 δ l2 ] , (3.26) 

which shows that a non-zero κ 2 is the signature of these models, where 

κ 0 = 64 

27 π6 (27 + 16Σ ɛ 2 + 32Π ɛ − 36Σ ɛ ) (3.27) 

κ 2 = 2048π6 

3375 (2Σ ɛ 2 − 2Π ɛ + 3ɛ 1 ɛ 2 (cos α 1 + 1))

3.5 TOY MODEL WITH CROSS TERMS 29 

κ 2 varies as cos α 1 . Interestingly, if we set all three strength factors to be equal, say 

ɛ 3 = ɛ 2 = ɛ 1 (3.28) 

we will still have a non-zero l = 2 components in BiPS which is given as 

κ 0 = 64 3 π6 (3 − 12ɛ 1 + 16ɛ 2 1) (3.29) 

κ 2 = 4096π6 

1125 ɛ2 1 cos 2 α 1 . 

κ 2 given above is proportional to cos 2 α 1 and tends to zero when α 1 → π/2 and the 

correlation function tends to a SI correlation function. 

3.4 Toy Model With Cross Terms 

form 

Sometimes in our analytical calculations of BiPS, we encounter cross terms of thsi 

f(ˆq 1 , ˆq 2 ) = ((ˆq 1 − ˆq 2 ) · ˆx)((ˆq 1 − ˆq 2 ) · ŷ) (3.30) 

Which has two preferred axes in ˆx and ŷ directions and is of the first order in each of 

them. Although this is not an example of a physical two point function, it is useful to 

know the analytical BiPS for that (we will use this in our later calcularions). Integral 

over rotations of the above function will give us a non-zero l = 2 component 

∫ 

dR χ l (R) f(Rˆq 1 , Rˆq 2 ) = 2π2 

5 δ xδ y δ l2 (3.31) 

This tells us that each cross term will cause a non-zero l = 2 component. Using this, 

we can conclude that if there are three cross terms with strength factors ɛ 1 , ɛ 2 and ɛ 3 , 

the result of the above integral would be a superposition of three terms 

∫ 

dR χ l (R) f(Rˆq 1 , Rˆq 2 ) = 2π2 

5 (ɛ 1δ x δ y + ɛ 2 δ x δ y + ɛ 3 δ y δ z )δ l2 (3.32) 

This has no l = 0 component because there is no SI part in our toy model. BiPS of this 

model can be calculated to be 

κ l = 

(2l + 1)2 1024π 6 Σ ɛ 2 

δ 

(8π 2 ) 2 l2 , (3.33) 

1125 

This is always non-zero even for equal strength factors, and tends to zero as Σ ɛ 2 → 0.

3.5 FOURTH ORDER TOY MODELS WITH THREE AXES 30 

3.5 Fourth Order Toy Models With Three Axes 

We showed that all anisotropic second order correlation functions lead to a non-zero 

κ 2 in BiPS. Next toy model we consider is a fourth order toy model with three preferred 

directions along the ˆx, ŷ and ẑ directions with strength factors ɛ 1 , ɛ 2 and ɛ 3 respectively, 

C(ˆq 1 , ˆq 2 ) = 1 − (ɛ 1 ((ˆq 1 − ˆq 2 ) · ˆx) 4 + ɛ 2 ((ˆq 1 − ˆq 2 ) · ŷ) 4 + ɛ 3 ((ˆq 1 − ˆq 2 ) · ẑ) 4 ) (3.34) 

Integration over rotations of this correlation function will cause a non-zero l = 2 component 

as well as a non-zero l = 4 one, 

∫ 

dR χ l (R) C(Rˆq 1 , Rˆq 2 ) = 2π 2 δ l0 − (ξ(ˆq 1 , ˆq 2 )δ l0 + ζ(ˆq 1 , ˆq 2 )δ l2 + η(ˆq 1 , ˆq 2 )δ l4 ) (3.35) 

W here : ξ(ˆq 1 , ˆq 2 ) = 2π2 

5 Σ ɛ(δ 2 x + δ 2 y + δ 2 z) 2 

ζ(ˆq 1 , ˆq 2 ) = − 4π2 

35 ((−2ɛ 1 + ɛ 2 + ɛ 3 )δ 2 x + (ɛ 1 − 2ɛ 2 + ɛ 3 )δ 2 y + (ɛ 1 + ɛ 2 − 2ɛ 3 )δ 2 z), 

η(ˆq 1 , ˆq 2 ) = 2π2 

315 (δ4 x(8ɛ 1 + 3ɛ 2 + 3ɛ 3 ) + δ 4 y(3ɛ 1 + 8ɛ 2 + 3ɛ 3 ) 

+ δ 4 z (3ɛ 1 + 3ɛ 2 + 8ɛ 3 ) − 6δ 2 x δ2 y (4ɛ 1 + 4ɛ 2 − ɛ 3 ) 

− 6δ 2 y δ2 z (−ɛ 1 + 4ɛ 2 + 4ɛ 3 ) − 6δ 2 x δ2 z (4ɛ 1 − ɛ 2 + 4ɛ 3 )). 

Integrating the above expression is a tedious job. To simlify our calculations we can 

consider a special case in which three strength factors are the same 

ɛ 3 = ɛ 2 = ɛ 1 (3.36) 

This simplifies the calculations immensely and we can easily get the BiPS for this class 

of toy models to be 

κ l = 

(2l + 1)2 

(8π 2 ) 2 [κ 0 δ l0 + κ 2 δ l2 + κ 4 δ l4 ] , (3.37) 

where 

κ 0 = 64 

125 π6 (125 − 800ɛ 1 + 2304ɛ 2 1) (3.38) 

κ 2 = 0 

κ 4 = 262144 

212625 π6 ɛ 2 1

3.7 TOY MODEL II 31 

In this case, the l = 2 component of the power spectrum vanishes. This result along 

with what we obtained in 3.17 will hint us an interesting property of BiPS. Three 

perpendicular preferred directions in the correlation function lead to a zero κ 2 , but if 

they appear in higher orders in the correlation function, they will cause non-zero κ 4 and 

as we will see later in higher orders may cause higher l components non-zero. 

3.6 Toy Model II 

We can consider another toy correlation which is analytically solvable and has preferred 

directions in the following way 

C(ˆq 1 , ˆq 2 ) = (1 + ɛˆq 1 · ẑ)(1 + ɛˆq 2 · ẑ), (3.39) 

where ɛ < 1 ensures positivity of the correlation function. 

Integrating over rotations of the two point correlation will lead to the following result 

∫ 

dR χ l (R) C(Rˆq 1 , Rˆq 2 ) = 2π 2 δ l0 − ɛ(ξ(ˆq 1 , ˆq 2 )δ l0 + ζ(ˆq 1 , ˆq 2 )δ l2 ) (3.40) 

W here : ξ(ˆq 1 , ˆq 2 ) = 2π2 

3 (δ2 x + δ2 y + δ2 z ) 

ζ(ˆq 1 , ˆq 2 ) = 2π2 

15 (2δ2 x − δ2 y − δ2 z ). 

And BiPS is computed by integrating the square of the above expression over all possible 

pairs of pixels on the sky, 

κ l = 

(2l + 1)2 

(8π 2 ) 2 [κ 0 δ l0 + κ 1 δ l1 + κ 2 δ l2 ]. (3.41) 

Where , κ 0 κ 1 and κ 2 are given by 

κ 0 = 16 

27 4π6 (27 + ɛ 4 ), (3.42) 

κ 1 = 32 

27 π6 ɛ 2 , 

κ 2 = 32 

675 π6 ɛ 4 . 

These show that BiPS should have three non-zero components which are κ 0 , κ 1 and κ 2 . 

The rest of the components are zero. This is the first (and only) analytical toy model 

which has a non-zero odd-l component of BiPS.

3.7 TOROIDAL SPACES 32 

3.7 Toroidal Spaces 

A family of interesting models in which the SI condition is violated is the family of 

compact spaces. As we will see in Chapter 4, this breakdown of statistical isotropy is 

equivalent to having strongly correlated spots which are physically very far from each 

other. This is a hint of the existance of preferred directions in the correlation spaces of 

these models. In Chapter 4 we compute C(ˆq, ˆq ′ ) for CMB anisotropy in toroidal spaces 

using regularized method of images Bond, Pogosyan & Souradeep (1998, 2000). The 

BiPS can be numerically computed for these models from their two point correlation 

matrices. The results can be understood (and predicted) using the leading order terms 

of the correlation function in a torus where κ l can be calculated analytically. These are 

discussed in the next chapter. And as we will see, they show that BiPS is a very useful 

tool in studying the topology of the universe.

Chapter 4 

Signatures of Cosmic Topology 

The question of size and the shape of our universe are very old problems that have 

received increasing attention over the past few years, Ellis (1971); Lachieze-Rey & 

Luminet (1995); Levin et al. (1998); de Oliveira-Costa et al. (1996); Bond, Pogosyan & 

Souradeep (1998, 2000); Gomero et al. (2002); de Oliveira-Costa et al. (2003); Dineen 

et al. (2004); Copi et al. (2004); Mota et al. (2004). Although a multiply connected 

universe sounds non-trivial, but there are theoretical motivations, Linde (2004); Levin 

(2002), to favor a spatially compact universe. One possibility to have a compact flat 

universe is the consideration of multiply connected spaces. The oldest way of searching 

for global structure of the universe is by identifying ghost images of local galaxies and 

clusters or quasars at higher redshifts, Gomero et al. (1998, 2000, 2001) and Lachieze- 

Rey & Luminet (1995). This method can probe the topology of the universe only 

on scales substantially smaller than the apparent radius of the observable universe. 

Another method to search for the shape of the universe is through the effect on the 

power spectrum of cosmic density perturbation fields. In principle, this effect can be 

observed in the distribution of matter in the universe and the CMB anisotropy. Over the 

past few years, many independent methods have been proposed to search for evidence of 

a finite universe in CMB maps. These methods can be classified in three main groups. 

• Using the angular power spectrum of CMB anisotropies to probe the topology of 

the Universe. The angular power spectrum, however, is inadequate to characterize 

the peculiar form of the anisotropy manifest in small universes of this type. It has 

33

4.1 STATISTICAL ISOTROPY VIOLATION IN COMPACT SPACES 34 

been shown that a nontrivial topology breaks the SI down and as a result, there 

is more information in a map of temperature fluctuations than just the angular 

power spectrum, Levin et al. (1998); Bond, Pogosyan & Souradeep (1998, 2000); 

Hajian & Souradeep (2003a). 

• The second class of methods are direct methods which rely on multiple imaging of 

the CMB sky. The most well known methods amongst these methods are S-map 

statistics, de Oliveira-Costa et al. (1996, 2003) and the search for circles-in-thesky, 

Cornish, Spergel & Starkman (1998); Cornish et al. (2003). In Appendix C 

we study the search for planar symmetries in CMB anisotropy maps. 

• Third class of methods are indirect probes which deal with the correlation patterns 

of the CMB anisotropy field by using an appropriate combination of coefficients 

of the harmonic expansion of the field, Dineen et al. (2004); Donoghue et al. 

(2004); Hajian & Souradeep (2003a); Copi et al. (2004); Ferreira & Magueijo 

(1997). 

In the next section we show that SI violation is the smoking gun of compact spaces and 

can be used to detect and constrain these spaces. The strength of this method lies in the 

fact that it works even if the fundamental domain is slightly bigger than the diameter 

of the surface of last scattering, in which case the identified circles do not exist. 

4.1 Statistical Isotropy Violation in Compact Spaces 

One of the most important properties of compact spaces is that statistical isotropy 

is broken in them. Here we follow Riazuelo et al. (2004a) to calculate the covariance 

matrix in the universal cover and the quotient spaces in general. We use this result to 

show how one can use this property to search for compact spaces with the use of BiPS.


Two point correlations in the universal cover – a general case 

The universal cover is a space of constant curvature and hence can be E 3 , H 3 or S 3 . 

The metric of the universal cover is a FLRW metric 

ds 2 = −c 2 dt 2 + a 2 (t) [ dχ 2 + f 2 K(χ)dΩ 2] , (4.1) 

where dΩ 2 = dθ 2 + sin 2 θ dϕ 2 , and χ is the comoving radial distance. f K (χ) for different 

geometries is given by 

⎧ 

⎪⎨ 

sinh( √ |K| χ)/ √ |K| (hyperbolic case, K < 0) 

f K (χ) = χ (flat case, K = 0) 

⎪⎩ 

sin( √ K χ)/ √ K (spherical case, K > 0) 

(4.2) 

We usually work in Fourier space because it is more convenient to work with the individual 

modes of our system which evolve independently. The allowed modes are the 

eigenmodes of the space, Y [X] 

k lm 

(where superscript [X] shows the space of interest) given 

by (for example) Laplace equation 

∇ 2 Y [X] 

k lm + (k2 − K)Y [X] 

k lm 

= 0. (4.3) 

Eigenmodes Y [X] 

k lm 

can be decomposed into a radial part and an angular part like this 

Y [X] 

k lm = R[X] kl 

(χ)Y lm (ˆn) (4.4) 

To compute the observables of CMB anisotropy, we start from a set of initial conditions 

which is given to us by an early universe scenario. Modes that we study, evolve before 

they enter the Hubble radius till now. From this we can compute the ∆T/T at each 

point on the sky which is a random variable, and hence we must study its statistics 

which is assumed to be Gaussian. To do this, here we will first consider a flat case and 

then we will generalize our result to all cases. In a flat case the computation of the 

CMB anisotropies will contain three major steps 

(i) 

eigenmodes of Laplacian are plain waves given by 

e i(k·x) , 

hence the Fourier modes of temperature fluctuations are 

∫ 

∆T 

T (ˆn) = d 3 k ∆T 

ei(k·x) 

(2π) 

3/2 

T (ˆk)


(ii) 

initial conditions are given by the power spectrum P Φ (k) of primordial fluctuations 

in the gravitational potential field Φ. These are then mapped to 

CMB anisotropies in k-space, 

∆T 

T (ˆk) ∝ √ P Φ (k)ê k , 

where in the simplest inflationary models, ê k is a three dimensional Gaussian 

random variable, which satisfies 

〈ê k ê ∗ k〉 = δ(k − k ′ ) 

In simply connected spaces, reality condition of the temperature field dictates 

that 

ê −k = ê ∗ k. 

(iii) 

(iv) 

initial condition are mapped (related) to the present temperature fluctuations 

in a given direction on the sky, ˆn, by a linear convolution operator 

O [E3 ] 

k 

the result of all this can be summarized as 

∫ 

∆T 

T (ˆn) = d 3 k ] 

(2π) 3/2 O[E3 k 

e i(k·x)√ P Φ (k)ê k . (4.5) 

Using the addition theorem 

and 

l∑ 

m=−l 

Y lm (ˆn)Y ∗ 

lm (ˆn′ ) = 2l + 1 

4π P l(cos α), (4.6) 

e ik·r = 

∞∑ 

(2l + 1)i l j l (kr)P l (θ k,r ), (4.7) 

l=0 

eqn. (4.5) can be written as 

∫ 

∆T 

T (ˆn) = k 2 dk √ P Φ (k) O [E3 ] 1 ∑ 

k 

i l j l (kχ)Y lm (ˆn)Ylm ∗ 

4π 

(ˆk)ê k (4.8) 

l,m 

= ∑ ∫ 

i l k 2 dk √ (√ ) [∫ ] 

P Φ (k) O [E3 ] 2 

k 

π j l(kχ) dΩ k Ylm ∗ (ˆk)ê k Y lm (ˆn) 

l,m 

= ∑ ∫ 

i l k 2 dk √ ( ) 

P Φ (k) O [E3 ] 

k 

Y [E3 ] 

k lm 

ê lm (k), (4.9) 

l,m


where ê lm (k) is the spherical harmonic transform of ê k and hence obeys the isotropy 

condition 

〈ê lm (k)ê ∗ l ′ m ′(k′ )〉 = 1 k 2 δ(k − k′ )δ ll ′δmm ′ . (4.10) 

Eqn. (4.9) can be written as 

∆T 

T (ˆn) = ∑ ∫ 

i l 

l,m 

k 2 dk √ P Φ (k) O [E3 ] 

k 

R [E3 ] 

k l 

ê lm (k)Y lm (ˆn), (4.11) 

and a lm can be read from that to be 

a lm = i l ∫ 

k 2 dk √ P Φ (k) G l (k)ê lm (k) (4.12) 

where 

G l (k) = O [E3 ] 

k 

R [E3 ] 

k l 

and is computed from the local physics of the fluctuations. 

This was all for flat spaces. But for non-flat cases, everything is exactly the same, 

except that the eigenmodes are different. Therefor while working with non-flat spaces, 

there are two points to be taken care of. First, j l must be replaced by ultra-spherical 

Bessel functions Zaldarriaga et al. (1998). And second, in closed spaces, the modes are 

discrete, so we should replace the integral by a discrete sum over eigenmodes. 

However, the observables of CMB anisotropy are not a lm but their variance. In the 

case of above spaces (E 3 , H 3 and S 3 ), which are statistically isotropic, the covariance 

matrix turns out to be diagonal with angular power spectrum on its diagonals 

〈a lm a ∗ l ′ m ′〉 = C lδ ll ′δ mm ′ (4.13) 

which means that the angular power spectrum C l is the trace of the covariance matrix, 

C ll′ 

mm ′ = 〈a lma ∗ l ′ m ′〉, C l = 1 

2l + 1 

that is 

C l = 

∫ 

∑ 

Cmm ll (4.14) 

m 

k 2 dk 

( 

G [X] 

l 

(k)) 2 

PΦ (k). (4.15)


Effect of topology 

Imposing the topology, will not change the local physics of our problem. Because all 

physical equations (including cosmological equations which are derived from Einstein 

equations) are local differential equations and are untouched by the global properties of 

the space. This implies that implementing the topology will not alter O [X] 

k 

in the above 

calculation. But it will change the modes that can exist in the universe. One should 

solve the Laplace equation in the compact space X/Γ in order to find the eigenmodes 

of Laplacian, Ψ [Γ] 

ks , 

∇ 2 Ψ [Γ] 

ks + (k2 − K)Ψ [Γ] 

ks 

= 0. (4.16) 

These eigenmodes are given by the eigenmodes of the universal covering space as 

Ψ [Γ] 

ks = ∞ 

∑ 

l∑ 

l=0 m=−l 

ξ [Γ]s 

klm Y[X] k lm 

(4.17) 

where s describes the degenerate eigenmodes and k is discrete. One should then use 

these eigenmodes and use a discrete sum instead of integral in eqn. (4.9) in order to 

calculate the temperature anisotropies in compact spaces. In a quotient space of volume 

V this becomes 

∆T 

T 

(ˆn) = 

(2π)3 

V 

∑ 

k,s 

( ) √ 

O [X] 

k 

Ψ [Γ] 

ks PΦ (k)ê k (4.18) 

we can denote ê k by ê ks and the orthogonality conditition will become 

〈ê k ê ∗ k 〉 = V 

(2π) 3 δ kk ′δ ss ′ (4.19) 

Substituting eqn. (4.17) in eqn. (4.18), we will find the analytical expression for temperature 

fluctuations in a compact space 

∆T 

T 

(ˆn) = 

(2π)3 

V 

from which a lm can be read off to be 

a lm = (2π)3 

V 

∑ ∑ 

∑ 

k 

k,s 

l,m 

( √ 

ξ [Γ]s 

klm O[X] k 

Y [X] 

k lm) 

PΦ (k)ê ks (4.20) 

√ 

PΦ (k)O [X] 

k 

( 

R [X] 

kl 

) ∑ 

s 

ξ [Γ]s 

klmêks (4.21) 

The covariance matrix is calculated using the above expression, and is given by 

〈a lm a ∗ (2π)3 ∑ ( ) ( ) ∑ 

l ′ m ′〉 = P Φ (k)O [X] 

k 

R [X] 

kl 

O [X] 

k 

R [X] 

kl 

ξ [Γ]s 

V 

′ klm ξ[Γ]s kl ′ m 

(4.22) 

′ 

k 

s

4.2 METHOD OF IMAGES 39 

We should notice the existence of the term ∑ s ξ[Γ]s klm ξ[Γ]s kl ′ m ′ which is not necessarily diagonal 

in l and m. 

This is the statement of breakdown of statistical isotropy in compact 

spaces. As we mentioned earlier, the angular power spectrum which is given by C l = 

1 

2l+1 

∑ 

m Cll mm 

does not have the whole information of the temperature field. Also as 

we showed in A, breakdown of SI imposes large cosmic variance on the angular power 

spectrum. All together, these make the angular power spectrum not a convenient probe 

of the cosmic topology. We use the BiPS which exploits all the information in the field 

by combining the off-diagonal elements in the covariance matrix into a power spectrum 

A lM 

ll ′ 

= ∑ mm ′ (−1) m′ C lM 

lml ′ −m ′Cll′ mm ′, 

κ l = ∑ 

l,l ′ ,M 

|A lM 

ll ′ |2 . (4.23) 

However this is not the fastest way of computing two point correlations in compact 

spaces. 

In the next chapter we study another method which is very convenient for 

computing two point correlations and generating simulated CMB skies in any compact 

space. 

4.2 Method of images 

Another method of computing two point correlations in compact spaces is the regularized 

method of images which was introduced and developed by Bond, Pogosyan & 

Souradeep (1998, 2000). The idea of the method of images is that since compact spaces 

are obtained from tilings of the fundamental domain, one should take all the images of 

each point into account while computing the two point correlations. This method saves 

us from computing the eigenmodes in compact spaces which is usually a numerically 

tedious job. Method of images has a nice mathematical origin which is given below. 

Like previous chapter we expand the temperature anisotropies in any space in terms 

of a complete, orthonormal set of functions in that space, e.g. eigenfunctions of the 

Laplacian operator denoted by Ψ [Γ] 

ks 

(∇ 2 + k 2 − K)Ψ [Γ] 

ks = 0. 

The spectrum of the Laplacian on a compact manifold is discrete. Hence the two point

4.2 METHOD OF IMAGES 40 

correlations will be given by 

ξ(x, x ′ ) = ∑ k,s 

Ψ [Γ] 

ks (x)Ψ[Γ] ks (x′ )P Φ (k). (4.24) 

Except in simple spaces, neither the spectrum of Laplacian {k}, nor the eigenfunctions 

Ψ [Γ] 

ks 

(x) are known and thus eqn. 4.24 can not be used to compute the two point correlation 

on compact spaces in general. In contrast the universal cover, M [X] , of the 

compact space is usually a very simple manifold whose eigenmodes are very well studied 

and two point correlations on these spaces (which are E 3 , S 3 or H 3 ) are usually very easily 

computable. The regularized method of images allows computation of the correlation 

function ξ [Γ] (x, x ′ ) on a compact space 1 M [Γ] = M [X] /Γ. The eigenfunctions Ψ [Γ] 

ks (x) 

on M [Γ] are eigenfunctions on the universal cover, M [X] too. So, they simultaneously 

satisfy the following integral equations for eigenfunctions on manifolds M [Γ] and M [X] 

∫ 

dx ′ ξ [X] (x, x ′ )Ψ [Γ] 

ks (x′ ) = P Φ (k)Ψ [Γ] 

ks 

(x), (4.25) 

∫M [X] dx ′ ξ [Γ] (x, x ′ )Ψ [Γ] 

ks (x′ ) = P Φ (k)Ψ [Γ] 

ks (x). 

M [Γ] 

The eigenfunctions Ψ [Γ] 

ks 

(x) are automorphic with respect to Γ 

Ψ [Γ] 

ks 

(γx) = Ψ[Γ] (x) ∀γ ∈ Γ. (4.26) 

ks 

Using eqns (4.25) and (4.26) and the fact that M [Γ] tessellates M [X] , we get 

∫ 

∫ 

dx ′ ξ [Γ] (x, x ′ )Ψ [Γ] 

ks (x′ ) = dx ′ ξ [X] (x, x ′ )Ψ [Γ] 

ks (x′ ) (4.27) 

M [Γ] M [X] 

= ∑ ∫ 

dx ′ ξ [Γ] (x, γx ′ )Ψ [Γ] 

ks (x′ ) 

γ∈Γ 

M [Γ] 

∫ [ ] 

∑ 

= dx ′ ξ [Γ] (x, γx ′ ) Ψ [Γ] 

ks (x′ ). 

M [Γ] γ∈Γ 

Since Ψ [Γ] 

ks 

(x) are orthonormal functions and the above equation should be satisfied for 

every x, we conclude that 

ξ [Γ] (x, x ′ ) = ∑ γ i ∈Γ 

ξ [X] (x, γ i x ′ ), (4.28) 

1 The compact manifold M [Γ] can be regarded as a quotient of a simply connected manifold M [X] 

by a fixed point free discontinuous subgroup Γ of the isometry group of M [X] . The elements γ ∈ Γ 

generate images of M [Γ] which tessellate the universal cover M u .

4.3 A BIPOLAR POWER SPECTRUM STUDY OF COSMIC TOPOLOGY 41 

This is the main equation of the method of images, which implies that the correlation 

function on a compact space is given by the sum over the correlation function on its 

universal cover calculated between x and the images γ i x ′ . But the sum in this equation 

can be obtained analytically in a closed form only in a few cases, e.g. the simple flat 

torus (which we have already seen it). For a numerical implementation of the sum over 

images, it is useful to present the formal expression (4.28) as the limit of a sequence of 

partial sums 

ξ [Γ] (x, x ′ ) = lim 

N→∞ 

N∑ [ 

ξ [X] (x, γ i x ′ ) ] , (4.29) 

i=0 

with the discrete motions γ i sorted in increasing separation, d(x, γ i x ′ ) ≤ d(x, γ i+1 x ′ ). 

In practice, the summation over images is carried out over a sufficiently large but finite 

number of images, from which the limit N → ∞ is estimated. Accuracy is enhanced 

if the γ i used correspond to a tessellation of M [X] with the base-point of the Dirichlet 

domain shifted to x. In this case the Nth partial term of eq.(4.28) is equal to 

ξ [Γ] (x, x ′ ) |N = 

N∑ 

ξ [X] (x, γ i x ′ ). (4.30) 

i=0 

A more effective and simpler limiting procedure for implementing the regularized 

method of images is to explicitly sum images up to a radius r ∗ . 

This further eliminates 

the need to know the precise shape of Dirichlet domains (and integrate ξ [X] over 

potentially complicated shapes). 

⎡ 

⎤ 

ξ [Γ] = lim ⎣ ∑ 

ξ [X] (r j ) ⎦ , r j = d(x, γ j x ′ ) ≤ r j+1 . (4.31) 

r ∗→∞ 

r j


one could use the bipolar power spectrum as a probe to detect the topology of the 

universe. 

When CMB anisotropy is multiply imaged, the bipolar power spectrum corresponds 

to a correlation pattern of matched pairs of circles. Since it is orientation independent, 

the BiPS has to be computed only once for the CMB map irrespective of the orientation 

of the DD with respect to the sky. 

This method is very fast because there are fast 

spherical harmonic transform methods of the CMB sky maps. 

Using these methods 

it is very fast and efficient to compute the BiPS even for maps in WMAP resolution 

(HEALPix, N side = 512) in a few minutes on a single processor. We have used BiPS to 

put constraints on the topology of the Universe, Hajian et al. (2006). 

The A lM 

l 1 l 2 

signature, which is not discussed here, contains more details of orientation 

of the SI violation. It may be possible to detect the A lM 

l 1 l 2 

too for strong violation of SI. 

In what follows we use the method of images to compute the two point correlations 

in compact spaces. BiPS is then computed from two point correlations. What we are 

interested in is the signature of compact spaces on BiPS and its measurability from a 

single map. We study two families of multi-connected spaces which are presented below. 

4.3.1 Flat Spaces 

Among all multi-connected three-dimensional spaces, “flat spaces” have been studied 

the most extensively in the cosmological context. This is because of the computational 

simplicity of these spaces. There are 18 multi-connected flat spaces, given as the quotient 

E 3 /Γ of three-dimensional Euclidean space E 3 by a group Γ of symmetries of E 3 

that is discrete and fixed point free. The classification of these spaces has long been 

known. Among these 18 spaces, 10 are quotients of the 3-torus, 6 of them are orientable 

and 4 are non-orientable. Three of these spaces have a cubic fundamental domain, 6 

have hexagonal fundamental domains and one of them has a rhombic dodecahedron circumscribed 

about a rectangular box as its fundamental polyhedron. These fundamental 

polyhedra are shown in Fig. 4.1. These spaces are compact in three dimensions. There 

are two partially compact families as well; the chimney space and its quotients which


Symbol Name NC Volume Orientable 

E 1 3-torus 3 Finite Yes 

E 2 half turn space 3 Finite Yes 

E 3 quarter turn space 3 Finite Yes 

E 4 third turn space 3 Finite Yes 

E 5 sixth turn space 3 Finite Yes 

E 6 Hantzsche-Wendt space 3 Finite Yes 

E 7 Klein space 3 Finite No 

E 8 Klein space with horizontal flip 3 Finite No 

E 9 Klein space with vertical flip 3 Finite No 

E 10 Klein space with half turn 3 Finite No 

E 11 chimney space 2 ∞ Yes 

E 12 chimney space with half turn 2 ∞ Yes 

E 13 chimney space with vertical flip 2 ∞ No 

E 14 chimney space with horizontal flip 2 ∞ No 

E 15 chimney space with half turn and flip 2 ∞ No 

E 16 slab space 1 ∞ Yes 

E 17 slab space with flip 1 ∞ No 

E 18 Euclidean space 0 ∞ Yes 

Table 4.1 Classification of the 18 three-dimensional flat spaces (NC means number of 

compact dimensions). Table reproduced from Riazuelo et al. 

(2004b).


3-Torus Quarter Turn Space Half Turn Space 

Sixth Turn Space Third Turn Space Hantzsche-Wendt Space 

Klein Space 

Klein Space 

with Horizontal Flip 

Klein Space 

with Vertical Flip 

Klein Space 

with Half Turn 

Figure 4.1 Fundamental domains for the compact flat three-manifolds. The unmarked 

walls are glued straight across. Figure from Riazuelo et al. 

(2004b). 

have two compact dimensions (fundamental polyhedra are shown in Fig. 4.2), and the 

slab space with its quotient with only one compact dimension (figure 4.3). Summary of 

properties of these spaces is given in table 4.1. A detailed study on the signature of flat 

spaces on CMB has been done by Riazuelo et al. (2004b). 

The simple torus, T 3 , is the simplest compact space and one can in fact calculate 

most of its properties analytically. Here we will study this simple case which indeed has 

very interesting properties.


Chimney Space 

Chimney Space with 

Half Turn 


Vertical Flip 


Horizontal Flip 


Half Turn and Flip 

Figure 4.2 The chimney space is made from an infinitely tall rectangular chimney with 

front and back faces (resp. left and right faces) glued straight across. The four variations 

on the chimney space space glue the front face to the back as indicated by the doors. In 

all variations except the last the left and right faces are glued straight across; in the last 

variation they are glued with a top-to-bottom flip so that the windows match. Figure 

from Riazuelo et al. (2004b). 

Slab Space 

Slab Space with Flip 

Figure 4.3 The slab space is made from an infinitely tall and wide slab of space with 

its front face glued to its back face straight across. The variation glues the faces with a 

flip. Figure from Riazuelo et al. (2004b).


Two point correlation 

In a simple T 3 space which is defined by identifying x → x + L, y → y + L and 

z → z + L, the eigenvalue spectrum is discretized as 

with 

k = 2π L (n 1ˆx + n 2 ŷ + n 3 ẑ) = 2π n = kˆn, (4.32) 

L 

n ≡ (n 1 , n 2 , n 3 ), (4.33) 

n ≡ √ n · n, (4.34) 

ˆn ≡ (n 1 , n 2 , n 3 )/n, (4.35) 

k ≡ √ k · k = 2π n, (4.36) 

L 

Considering only naive Sachs-Wolfe effect, the correlation function will become 

ξ [Γ] (x, x ′ ) = 

1 L 3 ∑ 

n 

P φ (k n ) exp (−i 2πn 

L · (x − x′ )). (4.37) 

The above expression is in fact a cosine transform of the primordial power spectrum 

P (k). To see this, we should break the summation over n into summation over positive 

and negative n. And the one can use the definition cos x = (exp (ix) + exp (−ix))/2 to 

obtain 

ξ [Γ] (x, x ′ ) = 23 

L 3 

= 23 

L 3 

= 23 

L 3 

= 23 

L 3 

∑ 

n x,n y,n z>0 

∑ 

∑ 

n x>0 n y>0 n z>0 

∑ 

∑ 

P φ (k n ) cos 2πn x∆x 

L 

∑ 

∑ 

n x>0 n y>0 n z>0 

∑ 

∑ 

∑ 

n x>0 n y>0 n z>0 

cos 2πn y∆y 

L 

cos 2πn z∆z 

L 

k 3 

k P φ(k 3 n ) cos 2πn x∆x 



L L L 

P φ (k n ) 

k 3 

cos 2πn x∆x 

L 


L 


L 

( L 

2πn )3 P φ (k n ) cos 2πn x∆x 


L L 


(4.38) . 

L 

This is nothing but discrete Fourier transform of P φ (k n ). One can use FFT methods 

to compute the two point correlations in T 3 spaces. A 2-dimensional example of this is 

shown in figure 4.4. 

In an Euclidean space when the separation between two points increases, the correlation 

between them decreases. For a compact space it is different. We know that all


correlation function using 3D FFT analysis (computed for 4 Dirichlet Domains) 

6 

4 

2 

0 

−2 

140 

120 

100 

80 

100 

120 

140 

60 

80 

40 

20 

0 

0 

20 

40 

60 

Figure 4.4 Two point correlations in a 2-dimensional Toroidal space of the size of 64 

units. The correlation is between the central point and other points on the surface and 

is computed using the FFT method. Figure from Kourkchi (2006). 

the light received from a specific time, e.g. the epoch of the last scattering, represents a 

sphere. But if this sphere is larger than the universe it will intersect itself. If the space 

is a torus, then the sphere will intersect the imaginary walls of a “periodic” universe, 

and hence will intersect itself. This intersection has a very important effect on the large 

scale properties of the sphere for it identifies parts of the sphere which are apparently 

very far from each other. This will show up in the two point correlation as unusually 

high correlation between widely separate points on the sky. 

Consider a great circle on the surface of last scattering. This circle is greater than 

the size of the fundamental domain so it will fold back on itself and make a pattern like 

what is shown in Fig.4.5. Intersection of surface of last scattering with itself in small 

universes defines identified circles in the sky (as it is shown in Fig. 4.7). Search for


4 

3 

2 

1 

y 

0 

-1 

-2 

-3 

-4 

-4 -3 -2 -1 0 1 2 3 4 

x 

Figure 4.5 A circle with a radius greater than the size of the fundamental domain will 

fold back on itself. 

1.1 

1 

0.9 

0.8 

Correlation 

0.7 

0.6 

0.5 

0.4 

0.3 

0 0.5 1 1.5 2 

Angular Separation/Pi 

Figure 4.6 Correlation of two points on a circle whose radius is equal to the size of the 

fundamental domain. Strong correlations between apparently far apart points are seen.


these identified circles has been going on in the last few years by Cornish, Spergel & 

Starkman (1998); Cornish et al. (2003) and is a nice and direct method to find the 

topology of the universe. We use the method of images Bond, Pogosyan & Souradeep 

Figure 4.7 Identified circles in the sky along the x, y and z axes, for a toroidal space 

with R SLS = 0.65L. 

(1998, 2000) to compute the two point correlations on this circle as a function of the 

angular separation between them. The result is interesting: correlation falls off and 

again rises as we increase the angular separation between the two points. This result 

is shown in Fig.4.6. If we choose another great circle on the sphere, the shape of the 

correlation function and its dependence to θ would be completely different because it 

depends on the way the circle has folded back on itself. This shows that in flat torus 

the statistical isotropy breaks down.

NONP 


ο 

α=75 

α=45 ο 

α=60 ο QRQ 

SRS 

Figure 4.8 The pattern in CMB correlation in a multi-connected universe, M [Γ] ≡ 

M [X] /Γ is related to the distribution of ‘images’ of the sphere of last scattering (SLS) on 

the universal cover. The three figures illustrate this for three values of α = 45 ◦ , 60 ◦ , 75 ◦ 

in a squeezed torus. The solid parallelepiped is the fundamental domain and the dashed 

show its images tessellate,M [X] . The solid circle is the SLS and dashed/dotted ones are 

its images. In each case, the radius of SLS is equal to the in-radius, R < of M [Γ] . The 

shaded polygon is the Dirichlet domain (DD). The closest SLS images which determine 

the DD are dashed (others are dotted). Note the hexagonal shape of DD when α ≠ π/2 

and is equal sided for α = π/3. For smaller α, the DD approximates an elongated 

cuboid. 

BiPS of Toroidal Spaces 

We restrict attention to the case where CMB anisotropy arises entirely at the sphere 

of last scattering (SLS) of radius R ∗ (nearly equal to observable horizon). 

Invoking 

method of images, the CMB correlation pattern on the SLS is known to be dictated by 

the distribution of nearest ‘images’ of the SLS on the universal cover Bond, Pogosyan 

& Souradeep (1998, 2000). The correlations are distorted even when the SLS and its 

images do not intersect (R ∗ < R < ). When SLS intersects its images the CMB sky is 

multiply imaged in characteristic correlation pattern of pairs of circles Cornish, Spergel 

& Starkman (1998). Fig. 4.8 illustrates the role of the nearest SLS images and related 

DD in defining the principal directions in the correlation for a few cases. 

We compute C(ˆq, ˆq ′ ) for CMB anisotropy in torus space using regularized method 

of images Bond, Pogosyan & Souradeep (1998, 2000). We can compute the κ l in real 

space using eq. (2.30) or from A lM 

ll ′ 

using eq. (2.20) in harmonic space. Fig. 4.9 plots 

the κ l spectrum for a number of cubic, cuboidal and squeezed torus spaces. We find the 

following interesting results :


2 

1.5 

1 

0.5 

Cubic 

(big) 

R=0.4L 

R=0.5L 

R=0.6L 

0 

1 2 3 4 5 6 7 8 9 

8 

L 2 

=5 L 

Cuboid 

1 

6 

eql. vol. 

50 Cubic R=1.0L 

(small) R=1.5L 

40 

R=2.0L 

30 

20 

10 

0 

1 2 3 4 5 6 7 8 9 10 11 12 13 

4 

L 2 

=5 L 

Cuboid 

1 

3 

R=R L 2 

=2 L < 

1 

L 2 

=2 L 1 

1 2 3 4 5 6 7 8 9 

4 

2 

2 

1 

0 

1 2 3 4 5 6 7 8 9 

0 

0.8 

2 

0.6 

Squeezed 

45 Deg 

eql. vol. 

60 Deg 

75 Deg 

1.5 

0.4 

1 

Squeezed 

R=R < 

45 Deg 

60 Deg 

75 Deg 

0.2 

0 

1 2 3 4 5 6 7 8 9 

0.5 

0 

1 2 3 4 5 6 7 8 9 

Figure 4.9 The κ l spectra for flat tori models are plotted. The top row panels are for 

cubic tori spaces. The left panel shows spaces of volume, V M [Γ], larger than the volume V ∗ 

contained in the sphere of last scattering (SLS) with V M [Γ]/V ∗ = 3.7, 1.9, 1.1, respectively. 

The right panel shows small spaces with V M [Γ]/V ∗ = 0.24, 0.07, 0.03, respectively. Note 

that κ 2 = 0 for cubic tori. The middle panels consider cuboid tori with 1 : 5 and 

1 : 2 ratio of identification lengths. The bottom panels show κ l for equal-sided squeezed 

tori with α = 45 ◦ , 60 ◦ and 75 ◦ . In the middle and bottom rows, the right panels show 

the case when radius of SLS, R ∗ = R < the in-radius of the space. Here, the SLS just 

touches its nearest images (see Fig. 4.8) which is at the threshold where CMB anisotropy 

is multiply imaged for larger R ∗ . The cases in the left panels of lower two rows have 

V M [Γ]/V ∗ = 1 and are at the divide between large and small spaces.


i. κ l = 0 for odd l for all torus models. This feature has been studied by Mukhopadhyay 

(2005) and can be explained by 2n-fold rotational symmetries of the space about 

an axis. This does not hold for compact hyperbolic spaces. 

ii. For cubic torus κ 2 = 0. κ 2 is non-zero for cuboidal and squeezed torus. This is a 

clear signature of non-cubic torus where the DD differs from the FD and has more than 

three principle axes. 

iii. For equal-sided squeezed torus, κ 4 , decreases as α decreases from 90 ◦ to 60 ◦ 

as R < increases. For α < 60 ◦ sharply increases with decreasing α as R < decreases 

sharply Bowen & Ferreira (2002). 

iv. κ 2 = 0 increases monotonically as α decreases from 90 ◦ . The trend is well fit by 

eq. (4.88). 

v. The peak of κ l shifts to larger l for small spaces. 

The results can be understood using the leading order terms of the correlation function 

in a torus where κ l can be calculated analytically. The spatial correlation function 

of gravitational potential in the periodic box implied by the T 3 topology with cubic 

fundamental domain is given by eqn. (4.37) 

ξ Φ (x, x ′ ) = L −3 ∑ n 

2πn 

−i 

P Φ (k n ) e L ·(x−x′) , (4.39) 

where n is 3-tuple of integers, k 2 n = (2π/L) 2 (n · n), and the term with n · n = 0 is 

excluded from the summation. We restrict attention to the case where CMB anisotropy 

arises entirely at the sphere of last scattering (SLS) of radius R SLS . Two point correlation 

function of CMB anisotropy is obtained from the spatial correlation function of 

gravitational potential on the SLS along the two directions ˆq 1 and ˆq 2 as 

C T 3 (ˆq 1 , ˆq 2 ) = ξ T 3 

Φ (R SLS ˆq 1 , R SLS ˆq 2 ) (4.40) 

= 1 ∑ 

P 

L 3 Φ (kˆn ) exp −iπR SLS 

(ˆn · ˆq 1 − ˆn · ˆq 2 ) 

L/2 

ˆn 

Figure 4.10 shows the relative configuration of ˆq 1 , ˆq 2 and ∆ˆq. Using the simple geometry 

of this configuration, we can find the vector subtraction which is needed to compute the 

above two point correlation; Using cosine rules in triangles, R SLS ∆ˆq is given by 

R SLS |∆ˆq| = R SLS 

√ 

2 − 2 cos θ = 2RSLS sin θ/2 (4.41) 

=⇒ R SLS (ˆq 1 − ˆq 2 ) = 2R SLS sin θ/2∆ˆq


R 

SLS ∆ q^ 

R SLS 

q^ 2 

θ 

R SLS 

q^ 1 

Figure 4.10 Geometry of the distance between two points along the two directions ˆq 1 

and ˆq 2 on the surface of last scattering; R SLS ∆ˆq = 2R SLS sin θ/2∆ˆq. 

Substituting this into the expression for two point correlation in toroidal spaces, eqn.(4.40), 

we get 

C T 3 (ˆq 1 , ˆq 2 ) = 

1 ∑ 

P 

L 3 Φ (kˆn )e −2iπ P i ɛ in i ∆q i 

(4.42) 

ˆn 

where the small parameter ɛˆq ≤ 1 is the physical distance to the SLS along ˆq in units of 

L i /2, where L i are sizes of three dimensions of the fundamental domain. 

ɛ i = 2R SLS 

L i 

sin θ/2 θ = cos −1 ˆq 1 · ˆq 2 (4.43) 

Contribution to C(ˆq, ˆq ′ ) from large wavenumbers, |n|ɛ ≫ 1 is expected to be approximately 

statistically isotropic. The SI violation is found in the low wavenumbers. 

When the SLS is contained with the fundamental domain around the observer, i.e., 

the CMB anisotropy is not multiply imaged on the sky, 2R SLS /L is a constant. When 

ɛ is a small constant, the leading order terms in the correlation function eq. (4.42) can 

be readily obtained in power series expansion in powers of ɛ. In what follows, we will 

compute the leading order terms in the correlation function in powers of ɛ for the lowest 

wavenumbers |n| 2 = 1, |n| 2 = 2 and |n| 2 = 3 in a cubic FD torus. 

ˆn · ˆn = 1 

The lowest wavenumbers are |n| 2 = 1 which means 

n x = ±1; n y = 0; n z = 0 (4.44)


with a cyclic relation between n x , n y and n z . The correlation function in a cubic torus 

space for these wavenumbers will be 

C (1) (ˆq 1 , ˆq 2 ) = 1 L 3 P Φ( 2π L ) [ exp 2iπɛ 1∆q 1 + exp −2iπɛ 1 ∆q 1 (4.45) 

This can be expanded in powers of ɛ i , 

+ exp 2iπɛ 2 ∆q 2 + exp −2iπɛ 2 ∆q 2 

+ exp 2iπɛ 3 ∆q 3 + exp −2iπɛ 3 ∆q 3 ] 

= 2 L P Φ( 2π 3∑ 

3 L ) cos 2πɛ i ∆q i 

i=1 

C (1) (ˆq 1 , ˆq 2 ) ≈ 

2 L 3 P Φ( 2π L 

)[3 − 

(2π)2 

2! 

3∑ 

i=1 

ɛ 2 i ∆q 2 i + (2π)4 

4! 

3∑ 

ɛ 4 i ∆qi 4 ]. (4.46) 

i=1 

The above expression is for an arbitrary Cuboid torus with un-equal sides. In a cubic 

torus, all three sides are equal, i.e. 

L i = L =⇒ ɛ 3 = ɛ 2 = ɛ 1 (4.47) 

And hence the leading order terms in the correlation function will be 

C 1 (ˆq 1 , ˆq 2 ) = C 0 [1 − ɛ2 

2! 

3∑ 

i=1 

∆q 2 i + 3ɛ4 

4! 

3∑ 

∆qi 4 ] (4.48) 

i=1 

Where 

C 0 = 6 L 3 P Φ( 2π L ) (4.49) 

ɛ = 2πɛ 1 

√ 

3 

. 

This correlation function can be written as a superposition of toy models we have studied 

so far. The second term is toy model I with three axis and third is the fourth order toy 

model, so we have 

C (1) 

T 3 = Toy Model I With Three Axes ( O(ɛ 2 ) ) + Fourth Order Toy Model ( O(ɛ 4 ) ) 

We retain O(ɛ 4 ) terms because as we have seen, Toy Model I up to the second order 

in ɛ, O(ɛ 2 ), is rotationally invariant and hence does not contribute to the violation of


SI. From the above estimations we should expect a non-zero κ 4 in BiPS. The non-zero 

κ l can be analytically computed to be 

κ l = 

(2l + 1)2 

(8π 2 ) 2 [κ 0 δ l0 + κ 4 δ l4 ] , (4.50) 

where 

κ 0 

C0 

2 

κ 4 

C0 

2 

= π 2 (1 − 4ɛ 2 + 368 

15 ɛ4 − 288 

5 ɛ6 + 20736 

125 ɛ8 ) 

= 12288π2 ɛ 8 (4.51) 

875 

As we expected, the first non zero κ l in cubic (equal-sided) torus occurs at l = 4. κ 4 ∼ ɛ 8 

falls off rapidly as ɛ → 0 for large spaces. 

ˆn · ˆn = 2 

The next lowest wavenumbers are those of |n| 2 = 2, which can be like 

n x = n y = ±1; n z = 0 (4.52) 

and any cyclic variant of them. The correlation function in a cubic torus space for these 

wavenumbers will be 

C (2) 

T 3 (ˆq 1 , ˆq 2 ) = 

1 L P Φ( 2√ 2π 

3 L ) [ exp −2iπ(ɛ 1∆q 1 + ɛ 2 ∆q 2 ) (4.53) 

+ exp −2iπ(ɛ 1 ∆q 1 − ɛ 2 ∆q 2 ) 

+ exp −2iπ(−ɛ 1 ∆q 1 + ɛ 2 ∆q 2 ) 

+ exp −2iπ(−ɛ 1 ∆q 1 − ɛ 2 ∆q 2 ) 

+ similar terms for ɛ 2 & ɛ 3 ] 

= 2 L P Φ( 2√ 2π 

3 L ) [ cos (2πɛ 1∆q 1 + 2πɛ 2 ∆q 2 ) + cos (2πɛ 1 ∆q 1 − 2πɛ 2 ∆q 2 ) 

+ cos (2πɛ 1 ∆q 1 + 2πɛ 3 ∆q 3 ) + cos (2πɛ 1 ∆q 1 − 2πɛ 3 ∆q 3 ) 

+ cos (2πɛ 2 ∆q 2 + 2πɛ 3 ∆q 3 ) + cos (2πɛ 2 ∆q 2 − 2πɛ 3 ∆q 3 ) ]. 

The above expressions can be simplified if we use the following rule of cosines 

cos (α + β) + cos (α − β) = 2 cos α cos β, (4.54)


We can then expand it in powers of ɛ i , 

C (2) 

T 3 (ˆq 1 , ˆq 2 ) = 

4 L P Φ( 2√ 2π 

3 L ) [ cos 2πɛ 1∆q 1 cos 2πɛ 2 ∆q 2 (4.55) 

+ cos 2πɛ 1 ∆q 1 cos 2πɛ 3 ∆q 3 

≈ 4 L 3 P Φ( 2√ 2π 

L 

= 4 L 3 P Φ( 2√ 2π 

+ cos 2πɛ 2 ∆q 2 cos 2πɛ 3 ∆q 3 ] 

) [ (1 − 

(2π)2 

2! 

(1 − (2π)2 

2! 

+ (1 − (2π)2 

2! 

(1 − (2π)2 

2! 

+ (1 − (2π)2 

2! 

(1 − (2π)2 

2! 

L ) [ 3 − (2π)2 3∑ 

+ (2π)4 

4! 

ɛ 2 1 ∆q2 1 + (2π)4 ɛ 4 1 

4! 

∆q4 1 ) 

ɛ 2 2∆q2 2 + (2π)4 ɛ 4 

4! 

2∆q2) 

4 

ɛ 2 1∆q1 2 + (2π)4 ɛ 4 

4! 

1∆q1) 

4 

ɛ 2 3∆q3 2 + (2π)4 ɛ 4 

4! 

3∆q3) 

4 

ɛ 2 2∆q2 2 + (2π)4 ɛ 4 

4! 

2∆q2) 

4 

ɛ 2 3 ∆q2 3 + (2π)4 ɛ 4 3 

4! 

∆q4 3 ) ] 

3∑ 

i=1 

ɛ 2 i ∆q2 i + 2(2π)4 

4! 

∑ 

ɛ 2 i ɛ2 j ∆q2 i ∆q2 j ] 

i,j 

i=1 

ɛ 4 i ∆q4 i 

And finally the two point correlation is 

[ 

3∑ 

C (2) 

T 

(ˆq 3 1 , ˆq 2 ) ≈ C 0 1 − ɛ 2 ∆qi 2 + 3 2 ɛ4 

in which 

i=1 

3∑ 

i=1 

∆q 4 i + 9 4 ɛ4 ∑ i,j 

∆q 2 i ∆q2 j 

] 

. (4.56) 

ɛ 2 = 8(2π) 2 ɛ 2 1 (4.57) 

C 0 = 

1 

18L P Φ( 2√ 2π 

3 L ). 

The above correlation function can be written as a superposition of toy models we have 

studied so far. 

C (2) 

T 3 = Toy Model I With Three Equal Axes ( O(ɛ 2 ) ) (4.58) 

+ Fourth Order Toy Model ( O(ɛ 4 ) ) 

+ Second Order Cross Terms ( O(ɛ 4 ) )


Among the toy models on the RHS of the above expression, the first one is rotationally 

invariant and does not contribute to non-trivial components (l ≠ 0) of BiPS. The second 

term as well as the third term will cause a non-zero κ 4 which will be of the order of 

O(ɛ 8 ). 

ˆn · ˆn = 3 

κ l = 

(2l + 1)2 

(8π 2 ) 2 [κ 0 δ l0 + κ 4 

( 

O(ɛ 8 ) ) δ l4 ] , (4.59) 

The next lowest wavenumbers are those of |n| 2 = 2, which are 

n x = ±1; n y = ±1; n z = ±1. (4.60) 

Substituting these in the expression for the correlation function, we see that the correlation 

function in a cubic torus space for these wavenumbers will again have a cosine 

structure 

C (3) 

T 3 (ˆq 1 , ˆq 2 ) = 

4 L P Φ( 2√ 2π 

3 L ) [ cos 2π(ɛ 1∆q 1 + ɛ 2 ∆q 2 + ɛ 3 ∆q 3 ) (4.61) 

+ cos 2π(ɛ 1 ∆q 1 − ɛ 2 ∆q 2 + ɛ 3 ∆q 3 ) 

+ cos 2π(ɛ 1 ∆q 1 + ɛ 2 ∆q 2 − ɛ 3 ∆q 3 ) 

+ cos 2π(−ɛ 1 ∆q 1 + ɛ 2 ∆q 2 + ɛ 3 ∆q 3 ) ] 

= 12 

L 3 P Φ( 2√ 2π 

L ) [ cos 2πɛ 1∆q 1 cos 2πɛ 2 ∆q 2 cos 2πɛ 3 ∆q 3 ]. 

Expanding the above expression in powers of ɛ i , we will obtain 

C (3) 

T 3 (ˆq 1 , ˆq 2 ) ≈ 12 

L P Φ( 2√ 2π 

3 L 

) [ (1 − 

(2π)2 

2! 

× (1 − (2π)2 

2! 

ɛ 2 1 ∆q2 1 + (2π)4 ɛ 4 1 

4! 

∆q4 1 ) (4.62) 

ɛ 2 2 ∆q2 2 + (2π)4 ɛ 4 2 

4! 

∆q4 2 ) 

× (1 − (2π)2 ɛ 2 2 

2! 

∆q2 2 + (2π)4 ɛ 4 2 

4! 

∆q4 2 ) ] 

= 12 

L P Φ( 2√ 2π 

3∑ 

3∑ 

3 L ) [ 3 − (2π)2 ɛ 2 i 

2! 

∆q2 i + (2π)4 ɛ 4 i 

4! 

∆q4 i 

+ (2π)4 

4! 

i=1 

∑ 

ɛ 2 i ɛ 2 j∆qi 2 ∆qj 2 ] 

For a cubic (equal-sided) torus, we have ɛ i = ɛ. And the two point correlation will be 

3∑ 

3∑ 

3∑ 

C (3) 

T 

(ˆq 3 1 , ˆq 2 ) ≈ C 0 [1 − ɛ 2 ∆qi 2 + ɛ4 ∆qi 4 4 

+ ɛ4 ∆qi 4 4 

q4 j ]. (4.63) 

i=1 

i=1 

i,j 

i≠j 

i=1


in which 

ɛ 2 = (2π)2 

3 ɛ2 1 (4.64) 

C 0 = 18 

L 3 P Φ( 2√ 2π 

L ). 

The above correlation function can be written as a superposition of toy models we have 

studied so far. 

C (3) 


+ Fourth Order Toy Model ( O(ɛ 4 ) ) 

+ Second Order Cross Terms ( O(ɛ 4 ) ) 

Among the toy models on the RHS of the above expression, the first one is rotationally 

invariant and does not contribute to non-trivial components (l ≠ 0) of BiPS. The second 

term as well as the third term will cause a non-zero κ 4 which will be of the order of 

O(ɛ 8 ). 

Cuboid Torus 

κ l = 

(2l + 1)2 

(8π 2 ) 2 [κ 0 δ l0 + κ 4 

( 

O(ɛ 8 ) ) δ l4 ] , (4.66) 

Till now we have studied the equal sided torus spaces in which the dimensions of the 

fundamental domain are the same and as a result, ɛ i = ɛ. A more general case is cuboidal 

(unequal-sided) torus in which ɛ i are different. This can be constructed from the equal 

sided torus by the act of a rescaling operator, ρ. This operator is a diagonal 3×3 matrix 

which re-scales the sizes of the fundamental domain by different identification scales in 

each direction. This matrix is given by 

ρ = 

⎛ 

⎞ 

R 1 0 0 

⎜ 

⎝ 0 R 2 0 ⎟ 

⎠ (4.67) 

0 0 R 3 

This transformation in the first place changes the volume of the fundamental domain 

by a factor of R 1 R 2 R 3 , and on the other hand it induces anisotropic scale differences in


the sides of the fundamental domain and changes it into a rectangular parallelepiped. 

In that case for wavenumbers of |n| 2 = 1, the correlation violates SI at order ɛ 2 and to 

that order is given as 2 C 1 (ˆq 1 , ˆq 2 ) = C 0 

[ 

1 − ɛ2 

2! 

] 

3∑ 

βi 2 ∆qi 

2 

i=1 

(4.68) 

in which ɛ is the physical distance to the SLS along ˆq in units of ¯L/2, where ¯L is the 

geometrical average of sizes of three dimensions of the fundamental domain, L i . And β i 

is the ratio of size of each dimension and the average size, 

ɛ 2 = (2π)2 

3 (R SLS 

¯L )2 (2 sin θ/2) 2 (4.69) 

β 2 i = ( ¯L 

L i 

) 2 . 

The above expression is the one for Toy Model I, with three different axes, 

C (3) 

T 3 = Toy Model I With Three Un − Equal Axes ( O(ɛ 2 ) ) 

and hence, should have a non-zero κ 2 of the order of O(ɛ 4 ). 

The non zero κ l corresponding to the correlation eqn. (4.68) can be analytically 

calculated to be 

κ 0 

C0 

2 

κ 2 

C0 

2 

= π 2 (1 − 4 3 β2 ɛ 2 + 16 

27 β4 ɛ 4 ) 

= 64π2 

135 

[ 

β 4 − 3(β 2 1β 2 2 + β 2 1β 2 3 + β 2 2β 2 3) ] ɛ 4 (4.70) 

where β 2 = ∑ 3 

i=1 β2 i . The κ 2 ∼ ɛ 4 signal falls off slower than the κ 4 signal in cubic space 

for large spaces (ɛ → 0). 

Squeezed Torus 

Next, we study the flat toroidal spaces in their most general form. We studied the 

cases of unequal-sided orthogonal cubes as the fundamental domain which were obtained 

2 This is in fact the |n| 2 = 1 result for cubical torus with un-equal ɛ i


from the simple cubic torus by linear transformations that did not keep the volume 

conserved, but kept the cube orthogonal. We can now allow for linear transformations 

that change the angles of the corner of the fundamental domain while preserving its 

volume. This can be done by a shape transformation matrix, S, 

S = ω 

⎛ 

⎞ 

1 cos α 1 cos α 2 

⎜ 

⎝ 0 sin α 1 cos α 3 sin α 2 

⎟ 

⎠ (4.71) 

0 0 sin α 2 sin α 3 

where ω is defined as 

ω = (sin α 1 sin α 2 sin α 3 ) 1/3 (4.72) 

and ensures the unity of the shape transformation matrix detS = 1. 

If we apply this transformation on a unit cube defined by diag(1, 1, 1), we will get a 

unit parallelepiped in the following form 

S 

ˆx ŷ ẑ ˆx S ŷ S ẑ S 

⎛ ⎞ ⎛ 

⎞ 

1 0 0 ω ω cos α 1 ω cos α 2 

⎜ 

⎝ 0 1 0 ⎟ 

⎠ = ⎜ 

⎝ 0 ω sin α 1 ω cos α 3 sin α 2 

⎟ 

⎠ (4.73) 

0 0 1 0 0 ω sin α 2 sin α 3 

The geometrical interpretation of this shape transformation can be readily read from the 

above example: the angle between the unit vectors ˆx, ŷ and ẑ after this transformation 

becomes 

cos (ˆx S , ŷ S ) = cos α 1 

cos (ˆx S , ẑ S ) = cos α 2 

cos (ŷ S , ẑ S ) = cos α 1 cos α 2 + sin α 1 sin α 2 cos α 3 

We can now consider a transformation like this X ⃗ RS = RSX ⃗ This is a big general class 

of models and for simplicity, we will look at two special families following Bowen & 

Ferreira (2002). 

• Family One: this is when we squeeze the cube in one direction and keep the other 

two angles orthogonal. 

α 1 = α ∈ [0, π/2]; α 2 = α 3 = π/2 (4.74)


R 1 = R 2 = R 3 = R 

The transformation matrix then will be 

X RS = 

⎛ 

⎞ 

Rω Rω cos α 0 

⎜ 

⎝ 0 Rω sin α 0 ⎟ 

⎠ (4.75) 

0 0 Rω 

• Family Two, is the family of models in which all three angles are squeezed, but 

they are equal, 

α 1 = α 2 = α 3 = α ∈ [0, π/2]; (4.76) 

R 1 = R 2 = R 3 = R. 

The transformation matrix for this family is 

X S = 

⎛ 

⎞ 

Rω Rω cos α Rω cos α 

⎜ 

⎝ 0 Rω sin α Rω cos α sin α ⎟ 

⎠ (4.77) 

0 0 Rω sin α 2 

To calculate the analytical expression for BiPS for these models, we should repeat our 

previous steps. The correlation function for these models is the one for torus, but with 

different wave modes, 

where k ⃗n 

= 2π L 

C T 3 (ˆq 1 , ˆq 2 ) = 

up with the squeezing angles. 

1 ∑ 

P 

L 3 Φ (k ⃗n ) exp [−i ⃗ k ⃗n · (ˆq 1 − ˆq 2 )R SLS ] (4.78) 

⃗n 

⃗n. The wavenumbers are quantized and the effect of S is to mix them 

The ⃗ k vectors get transformed by the inverse of our 

transformation matrix, ⃗ k RS = S −1 R −1 ⃗ k, and hence is given by 

k 1 = 2π [ 

n 1 − n 2 cot α 1 + n 3 (cot α 1 cot α 3 − cot α ] 

2 

) , (4.79) 

R 1 ω 

sin α 3 

k 2 = 2π [ 

] 

1 cot α 3 

n 2 − n 3 , 

R 2 ω sin α 1 sin α 1 

k 3 = 2π [ 

] 

1 

n 3 . 

R 3 ω sin α 2 sin α 3


Our aim here is to compute the BiPS analytically for the lowest wavenumbers, say for 

⃗n · ⃗n = 1. 

which corresponds to the following values of ⃗ k 

n 1 = ±1 , n 2 = n 3 = 0 =⇒ k 1 = ± 2π 

R 1 ω , k 2 = k 3 = 0, (4.80) 

n 2 = ±1 , n 1 = n 3 = 0 =⇒ k 1 = ± 2π 

R 1 ω cot α 1 , k 2 = ± 2π 

R 1 ω 

1 

sin α 1 

, k 3 = 0, 

n 3 = ±1 , n 1 = n 2 = 0 =⇒ k 1 = ± 2π 

R 1 ω (cot α 1 cot α 3 − cot α 2 

) , k 2 = ± 2π 

sin α 3 R 2 ω 

1 

k 3 = ± 2π 

R 3 ω 

sin α 2 sin α 3 

cot α 3 

sin α 1 

, 

The full calculation will be very tedious, and we will limit ourselves to Family One for 

simplicity 

The correlation function is then given as 

R 1 = R 2 = R 3 = R (4.81) 

α 2 = α 3 = π/2; α 1 = α ∈ [0, π/2] 

C (1) (ˆq 1 , ˆq 2 ) = C 0 

6 

= C 0 

3 

[ exp (−iɛ∆q 1 ) + exp (iɛ∆q 1 ) (4.82) 

+ exp −iɛ(cot α∆q 1 + 1 

sin α ∆q 2) + exp iɛ(cot α∆q 1 + 1 

sin α ∆q 2) 

+ exp −iɛ∆q 3 + exp iɛ∆q 3 ] 

[ cos ɛ∆q 1 + cos (ɛ cot α∆q 1 + α 

sin α ∆q 2) + cos ɛ∆q 3 ]. 

where 

C 0 = 6 L 3 P Φ(k 1 ), (4.83) 

ɛ = 2π 

Rω 2R SLS sin θ/2 

And after expanding this expression in powers of ɛ, the leading order approximation to 

correlation C(ˆq, ˆq ′ ) turns out to have the form 

[ 

3∑ 

C (1) (ˆq 1 , ˆq 2 ) ≈ C 0 1 − ɛ2 

∆q 2 ɛ 2 

3 sin α 2 i + 

3 sin α cos α ∑ ] 

γ 2 ij ∆q i ∆q j . (4.84) 

i=1 

i≠j


in which 

γ ij = 

⎛ 

⎞ 

0 1 0 

⎜ 

⎝ 1 0 0 ⎟ 

⎠ (4.85) 

0 0 cos α 

We can compute the BiPS for this correlation function analytically, but it is interesting 

to speculate what to expect from the analytical calculations. 

correlation can be written in this simple way 

C (1) (ˆq 1 , ˆq 2 ) ≈ C 0 

[ 

1 − ɛ 2 |∆q|2 + cos 2 α∆q 2 3 

3 sin 2 α 

The above two point 

+ 2ɛ 2 cos α∆q 1 ∆q 2 

] 

. (4.86) 

The second term on the RHS of the above expression is our Toy Model I (with three 

equal axes), the third term is Toy Model I with one axes and the fourth term is the toy 

model with cross terms, 

C (3) 


+ Toy Model I With One Axes ( O(ɛ 2 ) ) 

+ Cross Terms ( O(ɛ 2 ) ) 

The first term is rotationally invariant and has no non-trivial component of BiPS, but 

second and third terms have a l = 2 components of BiPS which scales as O(ɛ 2 ). We 

can analytically calculate the BiPS for these models to be 

κ 0 

C0 

2 

κ 2 

C0 

2 

= π 2 [1 + 8 sin 2 α + 64 

27 cos4 α − 12ɛ 2 (1 − 16 

27 cos2 α)] 

= 64π2 

135 ɛ4 (cos 4 α + 3 cos2 α 

sin 4 ). (4.88) 

α 

BiPS has a non-zero l = 2 component as we expected. κ 2 vanishes for very large space 

(ɛ → 0). 

4.3.2 Spherical Spaces 

Flat spaces are not the only possibilities for the the topology of the universe. Recent 

observations suggest that the universe is “very close” to flat with a total density very 

close to one, Ω tot = 1.02 ± 0.02, Bennett et al. (2003); Spergel et al. (2003). This


leaves the possibility of having “nearly flat” compact spaces as a candidate for the 

topology of our universe. This means besides the 18 possible Euclidean spaceforms (see 

e.g. Riazuelo et al. (2004b)) we have a whole set of countable infinity of spherical 

spaceforms (see Gausmann et al. (2001)) and a non-countable infinity of hyperbolic 

spaceforms (see Weeks (2003)). Weeks et al. (2004) showed that only a special 

family of closed multi-connected spaces called “well-proportioned” spaces, make the 

best candidates for a topological explanation of the low CMB quadrupole observed by 

COBE and WMAP (Fig. 4.11). Well-proportioned spaces are spaces whose dimensions 

are approximately equal in all directions. We studied some of these spaces in the previous 

section. Recently spherical spaces have attracted a lot of attention, see e.g., Luminet 

et al. (2003); Aurich et al. (2005); Roukema et al. (2004); Lachieze-Rey (2004); 

Luminet (2005). Motivated by these, we study multi-connected spaces with spherical 

geometry in the following sections. The universal covering space of these topologies is 

the 3-sphere, S 3 . 

Figure 4.11 CMB temperature correlation function of the WMAP and COBE data (left) 

and of the Poincare dodecahedron (right). Curves correspond to Poincare dodecahedron 

spaces with Ω tot = 1.0001 (red), 1.013 (green), 1.03 (blue) and 1.2 (purple). Figures 

from Bennett et al. (2003) and Hajian et al. (2006).


Classification 

The isometry the 3-sphere is SO(4) and every isometry in SO(4) can be decomposed 

as the product of a right-handed and a left-handed Clifford translation. The 3-sphere 

S 3 enjoys a group structure as the set Q 3 of unit length quaternions. Each right(left)- 

handed Clifford translation corresponds to left (right) quaternion multiplication of Q 3 , so 

the group of right(left)-handed Clifford translations is isomorphic to Q 3 . It follows that 

SO(4) is isomorphic to Q 3 × Q 3 /{±(1, 1)} and thus the classification of the subgroups 

of SO(4) can be obtained from the classification of subgroups of Q 3 which are 

• the cyclic groups Z n of order n, 

• the binary 3 dihedral groups D ∗ m 

of order 4m, m ≥ 2, 

• the binary tetrahedral group T ∗ of order 24, 

• the binary octahedral group O ∗ of order 48, 

• the binary icosahedral group I ∗ of order 120 Riazuelo et al. 

(2004a), 

From the above classification, we can show that there are three categories of spherical 

three manifolds, these categories are briefly given below and in table 4.2. A detail 

description of spherical spaces is given in Gausmann et al. (2001). 

• The single action manifolds are those for which a subgroup R (resp. L) of Q 3 

acts as pure right-handed (resp. pure left-handed) Clifford translations. They 

are thus the simplest spherical manifolds and can all be written as S 3 /Γ with 

Γ = Z n , Dm, ∗ T ∗ , O ∗ , I ∗ . 

• The double-action manifolds are those for which subgroups R and L of Q 3 act 

simultaneously as right- and left-handed Clifford translations, and every element 

of R occurs with every element of L. There are obtained for the groups Γ = 

Γ 1 × Γ 2 with (Γ 1 , Γ 2 ) = (Z m , Z n ), (Dm ∗ , Z n), (T ∗ , Z n ), (O ∗ , Z n ), (I ∗ , Z n ), with 

gcd(m, n) = 1, gcd(4m, n) = 1, gcd(24, n) = 1, gcd(48, n) = 1, and gcd(120, n) = 

1, respectively. 

3 A binary group is the two-fold cover of the corresponding plain group.


Name of the manifold Symbol Order Fundamental domain 

cyclic Z n n lens 

binary dihedral Dm ∗ 4m 2m-sided prism 

binary tetrahedral T ∗ 24 octahedron 

binary octahedral O ∗ 48 truncated cube 

binary icosahedral I ∗ 120 dodecahedron 

Table 4.2 Finite subgroups of Q 3 . There is a two-to-one homomorphism from Q 3 to 

SO(3) so the finite subgroups of SO(3) – the cyclic, dihedral, tetrahedral, octahedral 

and icosahedral groups – lift to Q 3 . Each such group lifts two-to-one, giving the corresponding 

binary group. Only the cyclic groups of odd order lift one-to-one. That is, 

each even-order cyclic group Z 2n ⊂ Q 3 arises as the two-fold cover of Z n ⊂ SO(3), while 

each odd-order cyclic group Z n ⊂ Q 3 is the one-to-one lift of Z n ⊂ SO(3). The binary 

polyhedral groups are not merely the product of the original polyhedral groups with 

a Z 2 factor, nor are they isomorphic to the so-called extended polyhedral groups containing 

orientation-reversing as well as orientation-preserving isometries; rather they are 

something completely new. The plain dihedral, tetrahedral, octahedral and icosahedral 

groups do not occur as subgroups of Q 3 . From Weeks et al. (2003). 

• The linked-action manifolds are similar to the double action manifolds, except that 

each element of R occurs with only some of the elements of L. 

Single-action manifolds are the simplest class of spherical 3-manifolds and are directly 

given by finite subgroups of Q 3 . They are all given below 

• Each cyclic group Z n gives a lens space L(n, 1), whose fundamental domain is a 

lens shaped solid, n of which tile the 3–sphere. 

• Each binary dihedral group Dm ∗ gives a prism manifold, whose fundamental domain 

is a 2m–sided prism, 4m of which tile the 3–sphere. 

• The binary tetrahedral group T ∗ gives the octahedral space, whose fundamental 

domain is a regular octahedron, 24 of which tile the 3–sphere in the pattern of a 

regular 24–cell.


• The binary octahedral group O ∗ gives the truncated cube space, whose fundamental 

domain is a truncated cube, 48 of which tile the 3–sphere. 

• The binary icosahedral group I ∗ gives the Poincaré dodecahedral space, whose 

fundamental domain is a regular dodecahedron, 120 of which tile the 3–sphere in 

the pattern of a regular 120–cell. Poincaré discovered this manifold in a purely 

topological context, as the first example of a multiply connected homology sphere. 

A quarter century later Weber and Seifert glued opposite faces of a dodecahedron 

and showed that the resulting manifold was homeomorphic to Poincaré’s homology 

sphere Gausmann et al. (2001). 

The other two categories are also classified as given in Gausmann et al. (2001). But we 

are only interested in Poincaré dodecahedral spaces and hence will stop right here. What 

we need to compute two point correlations in these spaces is the matrix representation 

of the icosahedral group, I, which generates the Poincaré dodecahedral space. This 

matrix is given in the appendix B of Gausmann et al. (2001) and we quote them here. 

Matrix representation of Icosahedral group, I 

The icosahedron’s twelve vertices being at (±φ, ±1, 0), (0, ±φ, ±1), and (±1, 0, ±φ), 

where φ = ( √ 5 − 1)/2 is the golden ratio, the icosahedral group consists of 

• the identity, 

• order 2 rotations about the midpoints of the icosahedron’s edges, realized as π rotations 

about the fifteen axes (1,0,0), (0,1,0), (0,0,1), (±φ, ±(φ+1), 1), (1, ±φ, ±(φ+ 

1)), and (±(φ + 1), 1, ±φ), and 

• order 3 rotations about the centers of the icosahedron’s faces, realized as counterclockwise 

2π/3 rotations about the twenty axes (±1, ±1, ±1), (0, ±(φ + 2), ±1), 

(±1, 0, ±(φ + 2)), (±(φ + 2), ±1, 0), and 

• order 5 rotations about the icosahedron’s vertices, realized as both counterclockwise 

2π/5 rotations and counterclockwise 4π/5 rotations about the twelve axes 

(±φ, ±1, 0), (0, ±φ, ±1), and (±1, 0, ±φ).


Figure 4.12 Fundamental domain for the regular dodecahedron which corresponds to 

the spaces generated by the binary icosahedral group, I. All face-pairings are Clifford 

translations, which implies that each face is glued to the face opposite, with a twist 

equal to the translation distance. 

This defines a total of 1 + 15 + 20 + 24 = 60 = |I| distinct rotations. These yield the 

120 quaternions of the binary icosahedral group I ∗ . 

BiPS of Poincaré dodecahedral spaces 

Simulating the CMB temperature anisotropy in Poincaré dodecahedral spaces requires 

knowing the eigenmodes of the Laplace operator on the space. These eigenmodes 

are given in Lachieze-Rey (2004); Weeks (2005). An alternative method developed 

by Hipolito-Ricaldi & Gomero (2005) expresses the covariance matrix in terms of the 

elements of the covering group of the space, Γ, instead of computing the eigenmodes 

of the Laplacian operator. However, the universal cover, S 3 , of the Poincaré dodecahedral 

spaces is a very simple manifold whose eigenmodes are very well studied and two 

point correlations on this space is very easily computable. The regularized method of 

images developed by Bond, Pogosyan & Souradeep (1998, 2000), allows computation 

of the correlation function ξ [Γ] (x, x ′ ) on a Poincaré dodecahedral space where Γ is the


Figure 4.13 Two point correlations in Poincaré dodecahedral spaces, C(North P ole, ˆn). 

Correlations have been computed numerically using the method of images including 

only naive Sachs-Wolfe effect for Ω k = 0.0001 (top left), 0.013 (top right), 0.2 (bottom 

left). Hot spots represent the highly correlated regions. Inclusion of the integrated 

Sachs-Wolfe effect weakens the correlations, the right panel on bottom shows the two 

point correlation computed with nSW+ISW for a space with Ω k = 0.013. 

Icosahedral group discussed in the previous section. Right panel of Fig. 4.11 shows the 

two point correlations of Poincaré dodecahedral spaces computed for four different sizes 

of the space. The power gets suppressed on large angular scales as we go to smaller 

spaces. This feature of Poincaré dodecahedral spaces has been used to explain the low 

quadrupole in the WMAP data by some authors, (see e.g. Luminet et al. (2003)). But 

this suppression of the quadrupole happens at the cost of introducing highly correlated 

spots which are spatially far apart in the CMB sky. Some examples of this have been 

presented in Fig. 4.13, in which we have plotted the correlations for a fixed point of 

the sky (North Galactic Pole) and the rest of the sky in Poincaré dodecahedral spaces. 

When the space is very big (when it is very close to flat, e.g. Ω tot = 1.0001), the correlation 

falls off rapidly as we go further from the Pole. But in smaller spaces such as 

the one corresponding to Ω tot = 1.013, regions with strong correlations appear on the


map. These spots are signatures of small spaces and as we discussed in the section on 

flat spaces, they are signs of violation of statistical isotropy in compact spaces. These 

correlations are suppressed as we add the integrated Sachs-Wolfe effect to the computation 

of two point correlations, but they do not disappear (right panel on the bottom of 

Fig. 4.13). Using the fact that statistical isotropy is violated in Poincaré dodecahedral 

spaces one could use the bipolar power spectrum as a probe to look for the topology of 

the universe. We compute the BiPS from the correlation functions in real space using 

eqn. (2.30). Fig. 4.14 plots the BiPS for a number of Poincaré dodecahedral spaces. We 

find the following interesting results: 

(a) 

(b) 

(c) 

κ l = 0 for odd l for all Poincaré dodecahedral spaces. This feature has been 

studied by Mukhopadhyay (2005) and can be explained by 2n-fold rotational 

symmetries of the space about an axis. 

The first non-zero BiPS component is the l = 6 component. This is a clear 

signature of Poincaré dodecahedral spaces. 

Other components of BiPS become non-zero as the inner radius, R < , decreases 

and BiPS signature shifts to larger l for small spaces. 

Above we presented the theoretical predictions of BiPS signatures of Poincaré dodecahedral 

spaces. To investigate the detectability of non-zero BiPS from the data, we 

simulated 1000 CMB temperature anisotropy maps for various Poincaré dodecahedral 

spaces at HEALPix resolution of N side = 16 corresponding to l max = 47. Methods 

which use the angular power spectrum to simulate CMB maps (like HEALPix) can not 

be used in simulating CMB maps for compact spaces. This can instead be done using 

the methods presented in Appendix F to simulate the CMB anisotropy maps from theoretical 

two point correlation matrices. We added noise maps to these CMB simulated 

maps. Noise maps were simulated from the coadded noise covariance matrix of WMAP. 

Since BiPS is orientation independent, we don’t need to worry about the orientation of 

the Dirichlet domain with respect to the CMB map. We computed the BiPS for these 

simulated CMB anisotropy maps for various Poincaré dodecahedral spaces using the 

estimator given in eqn. (2.37). We average these 1000 BiPS and compute the dispersion 

in them. The dispersion is an estimate of 1σ error bars. At last we use the theoretical


angular power spectrum of simulated maps to estimate the bias and correct for it. We 

have plotted the BiPS of three models in Fig. 4.16. Which shows what we should expect 

to get from the data if the observed universe was a Poincaré dodecahedral space. This 

is used by Hajian et al. (2006) to put constraints on Poincaré dodecahedral topologies.


3.5 x 1010 l 

7 x 1010 l 

3 

6 

2.5 

5 

2 

4 

κ l 

κ l 

1.5 

3 

1 

2 

0.5 

1 

0 

0 2 4 6 8 10 12 14 16 18 20 

0 

0 2 4 6 8 10 12 14 16 18 20 

6 x 1010 l 

3 x 1010 l 

5 

2.5 

4 

2 

κ l 

3 

κ l 

1.5 

2 

1 

1 

0.5 

0 

0 2 4 6 8 10 12 14 16 18 20 

0 

0 2 4 6 8 10 12 14 16 18 20 

Figure 4.14 BiPS of Poincaré dodecahedron spaces for Ω k = 0.013 (top left), Ω k = 0.03 

(top right), Ω k = 0.1 (bottom left) and Ω k = 0.2 (bottom right). Computed from a 

theoretical two point correlation at HEALPix resolution N side = 16. Figures from Hajian 

et al. (2006).


Figure 4.15 Simulated CMB maps of Poincaré dodecahedral spaces for Ω k = 

0.013, 0.03 and 0.1.


0 

5 10 15 

0 

5 10 15 

0 

5 10 15 

Figure 4.16 Average of BiPS of simulated CMB maps of Poincaré dodecahedral spaces 

obtained from 1000 simulations for Ω k = 0.013, 0.03 and 0.1, Smoothed with W G (l s = 

18). Simulated noise maps from coadded noise covariance matrix of WMAP has been 

added to the CMB maps. Figures from Hajian et al. (2006).

Chapter 5 

Homogeneous Cosmological Models 

Friedmann-Robertson-Walker (FRW) models are simplest models of the expanding 

Universe which are spatially homogeneous and isotropic. Although the principal 

justification for studying these models was mathematical tractability, rather than observational 

evidence. But now we have strong observational evidence from isotropy of 

CMB and large scale structure of the universe that the structure of the universe at 

large scales must be very close to that of a FRW model. But there is still a freedom to 

choose homogeneous models which are initially anisotropic and as the time goes on they 

become more isotropic and will asymptotically tend to FRW models. Bianchi models 

provide a generic description of homogeneous anisotropic cosmologies and they are classified 

into 10 equivalence classes Ellis & MacCallum (1969). Among these models it is 

reasonable to consider only those Bianchi types which admit FRW models. These are 

types I and VII 0 in the case of k = 0, V and VII h in the case of k = −1, and IX in 

the case of k = +1. Bianchi models which do not admit FRW solutions become highly 

anisotropic at large times. Type IX models recollapse after a finite time and hence will 

not approach arbitrarily near to isotropy. Also models of type VII h will not in general 

approach isotropy Collins & Hawking (1973b). The most general Bianchi types which 

admit FRW models are Bianchi types VII h and IX. These two types contain types I, V, 

VII 0 as special subcases. An interesting feature of these models is that they resemble 

a universe with a vorticity. It is interesting to determine bounds on universal rotation 

from cosmological observations because the absence of such a rotation is a prediction of 

inflation. 

75

5.0 76 

CMB anisotropy is a strong tool to study the evolution of vorticity in the universe 

because such models have clear CMB profiles. These patterns which are imprinted 

by the unperturbed anisotropic expansion are roughly constructed out of two parts: 

production of pure quadrupole variations (or focusing of the quadrupole pattern into a 

hotspot in open models) and a spiral pattern which is the signature of VII 0 and VII h 

models Collins & Hawking (1973b); Doroshkevich et al. (1975); Barrow et al. (1985). 

This temperature anisotropy pattern for Bianchi VII h universes is of the form 

∆T B 

(θ, φ) = f 1 (θ) sin φ + f 2 (θ) cos φ. (5.1) 

T 0 

The functions f 1 (θ) and f 2 (θ) depend on details of the particular Bianchi model and 

must be computed numerically Barrow et al. (1985). There are distinct features in 

each Bianchi model. In a pioneering work analytical arguments were used to find upper 

bounds on the amount of shear and vorticity in the universe today, from the absence 

of any detected CMB anisotropy Collins & Hawking (1973b). A detailed numerical 

analysis of such models used experimental limits on the dipole and quadrupole to refine 

limits on universal rotation Barrow et al. (1985). After the first detection of CMB 

anisotropy by COBE-DMR, Bianchi models were again studied by fitting the full spiral 

pattern from models with global rotation to the 4-year DMR data to constrain the 

allowed parameters of a Bianchi model of type VII h Bunn et al. (1996); Kogut et al. 

(1997). Recently Bianchi VII h models were compared to the first-year WMAP data on 

large scales and it was shown that the best fit Bianchi model corresponds to a highly 

hyperbolic model (Ω 0 = 0.5) with a right-handed vorticity ( ω ) H 0 = 4.3 × 10 −10 (Jaffe et 

al. (2005)). These models may be ruled out by their low density but it is curious to 

note that Bianchi models are examples of breakdown of statistical isotropy even without 

considering a perturbed universe. Hence tests of SI can be carried out to put constraints 

on various parameters of these models Ferreira & Magueijo (1997). As an example, 

let’s assume that a CMB map is constructed out of two ingredients 

∆T (ˆn) = ∆T I (ˆn) + ∆T B (ˆn), (5.2) 

where ∆T B is the Bianchi pattern and ∆T I represents the isotropic residual fluctuation 

caused by variations in the density and gravitational potential of a FRW universe.

5.1 CLASSIFICATION OF COSMOLOGICAL MODELS 77 

The BiPS due to the Bianchi template can be computed for different models. As it 

is shown in fig. 5.1, different Bianchi models have different characteristic BiPS that 

can be used to constrain them using the first-year WMAP data. In summary, if the 

CMB sky is a superposition of a SI part and a Bianchi pattern, the resultant CMB 

map should be statistically anisotropic and in principle, that should be seen by BiPS. 

Exact bounds on these models from BiPS based on detailed studies, are given at the 

final section of this chapter. Also as it was discovered by Jaffe et al. (2005), the 

deviations from Gaussianity in the kurtosis of spherical Mexican hat wavelet coefficients 

are eliminated once the data is corrected for the Bianchi template. In a recent work 

McEwen et al. (2005) investigated the effect of this Bianchi correction on the detections 

of non-Gaussianity in the WMAP data. According to McEwen et al. (2005) previous 

detections of non-Gaussianity observed in the skewness of spherical wavelet coefficients 

reported in McEwen et al. (2005a) are not reduced by the Bianchi correction and 

remain at a significant level of 98.4%. 

In the next section (and in Appendix D) we will study the symmetries of the space 

to show how spatially homogeneous models are constructed. Most of definitions and 

mathematical notations in the following sections are borrowed from Nakahara (1984), 

Wald (1984) and Ellis & van Elst (1999). 

5.1 Classification of Cosmological Models 

We can classify cosmological models by their symmetries. For any space, the dimension 

of the group of symmetries of the space (group of isometries), is given by the sum 

of dimensions of groups of rotational and translational symmetries 1 , i.e., 

dim group of isometries (r) = dim group of rotational symmetries (s) + dim group of 

translational symmetries (q). 

In a 4-dimensional cosmological model, r can pick up different values. These values 

can be obtained by a variety of q and s. The possibilities for the dimension of orbits, s, 

are 0, 1, 2, 3 and 4. For isotropies of the spatial dimensions, q can be 0, 1 or 3. q = 2 is 

excluded because in cosmology we consider non-empty perfect fluids in which ρ + p > 0 

1 For a detailed mathematical description of this see Appendix D.


Figure 5.1 BiPS of Bianchi VII h models (arbitrary units); for a hyperbolic model x = 

0.55, Ω 0 = 0.5 (left) and a nearly flat model x = 0.55, Ω 0 = 0.99 (right). If the observed 

CMB map contains a pattern in it, it will violate the statistical isotropy which would 

be seen in the BiPS. 

and hence there will be uniquely defined notions of the average velocity of matter, and 

corresponding preferred world-lines. Their 4-velocity in each case is given by 

u µ = dx µ /dτ, (5.3) 

where τ is proper time measured along the fundamental world-lines. We assume this 

4-velocity is unique; that is, there is a well-defined preferred motion of matter at each 

space-time event. This u µ is therefore invariant and hence allowed rotations are those 

which act orthogonally on u µ . There is no 2-dimensional subgroup of O(3) and hence 

q = 2 is excluded from the allowed values that q can choose. All over the space, r should 

be fixed. But q can vary. For example right at the axis center of symmetry, q can be 

bigger. But at that point the dimension of orbits is less and hence r remains fixed. 

Given these, we can have the following spaces: 

(i) 

Possibilities for isotropy; 

1. Isotropic: q = 3, 

2. Local Rotational Symmetry: q = 1, 

3. Anisotropic: q = 0;


Table 5.1. 

Classification of Cosmological Models with Respect to Their Symmetries. 

... inhomogeneous spatially homogeneous space-time homogeneous 

... (s=2) (s=3) (s=4) 

Anisotropic spatially self-similar, Bianchi Osvath/Kerr 

(q = 0) Admitting Abelian 

or non-Abelian G 2 

Local Rotational Symmetry Lemaitre-Tolman- Kantowski-Sachs Godel stationary 

(q = 1) Bondi family rotating universe 

Isotropic not allowed* FLRW Einstein static 

(q = 3) universe 

Note. — Table reproduced from Ellis & van Elst (1999). 

∗ inhomogeneous models cannot be isotropic about a general point because isotropy everywhere implies 

spatial homogeneity; 

(ii) 

Possibilities for homogeneity; 

1. Space-time homogeneous models: s = 4 

2. Spatially homogeneous universes: s = 3 

3. Spatially inhomogeneous universes: s ≤ 2. 

Using the above classes, one can make every cosmological model with a given symmetry. 

Table 5.2 shows different varieties of cosmological models that may be constructed from 

different combinations of symmetries of the space. A complete list with more discussions 

is given in Ellis & van Elst (1999). 

For example the family of FLRW spaces that model the standard cosmology, are 

isotropic and spatially homogeneous universes (q = 3, s = 3 =⇒ r = 6). Another 

interesting family is the spatially homogeneous but anisotropic case (q = 0, s = 3 =⇒ 

r = 3) which is the family of Bianchi universes. This family has a simply transitive 

group of isometries G 3 and hence no continuous isotropy group. In the rest of this 

chapter we study these models in detail.

5.2 MODERN CLASSIFICATION OF BIANCHI MODELS 80 

5.2 Modern Classification of Bianchi Models 

Today, the Bianchi classification of 3-dimensional Lie algebras is no longer presented 

by the original Bianchi method (see Appendix E). Instead, a simpler scheme for classifying 

the equivalence classes of 3-dimensional Lie algebras is used. This scheme uses the 

irreducible parts of the structure constant tensor under linear transformations rather 

than the more complicated derived group approach of Lie and Bianchi. Following Ellis 

and MacCallum Ellis & MacCallum (1969), we decompose the spatial commutation 

structure constants Cbc a (eqn. D.9) into a tensor, nab , and a vector, a b 

Cbc a = ɛ dbc n ad + δc a a b − δb a a c (5.4) 

where ɛ abc is the 3-dimensional antisymmetric tensor and n ab and a b are defined as 

a b = 1 2 Ca ba (5.5) 

The structure constants C a bc 

n ab = 1 2 C(a cd ɛb)cd 

expressed in this way, clearly satisfy the first Jacobi identity, 

eqn. (D.10). Second Jacobi identity, eqn. (D.11) shows that a b must have zero 

contraction with the symmetric 2-tensor n ab ; 

C a e[b Ce cd] = 0 =⇒ nab a b = 0 (5.6) 

We can choose convenient basis to diagonalize n ab to attain n ab = diag(n 1 , n 2 , n 3 ) and 

to set a b = (a, 0, 0). The Jacobi identities are then simply equivalent to n 1 a = 0. 

Consequently we can define two major classes of structure constants 

Class A : a = 0, 

Class B : a ≠ 0. 

One can then classify further by the sign of the eigenvalues of n ab (signs of n 1 , n 2 and 

n 3 ). This classification is given in Table 5.2. Parameter h in class B, where a is non-zero, 

is defined by the scalar constant of proportionality in the following relation 

a b a c = h 2 ɛ bikɛ cjl n ij n kl . (5.7) 

In the case of diagonal n ab , the h factor has a simple form h = a 2 /(n 2 n 3 ). 

In addition to these, we can have two different Bianchi models

5.3 THE FIELD EQUATIONS 81 

Table 5.2. 

Classification of Homogeneous Cosmological Models into Ten Equivalence 

Classes 

Group Class Group Type Eigenvalues of n ab a Dimension FLRW Type ... 

I 0, 0, 0 0 0 k=0 Abelian 

II +, 0, 0 0 3 — 

VI 0 0, +, − 0 5 — 

A VII 0 0, +, + 0 5 k=0 

VIII −, +, + 0 6 — 

IX +, +, + 0 6 k=+1 

V 0, 0, 0 + 3 k=-1 

IV 0, 0, + + 5 — 

B VI h 0, +, − + 6 — h < 0 

III 0, +, − n 2 n 3 6 VII h with h = −1 

VII h 0, +, + + 6 k=-1 h > 0 

Note. — Table from Ellis & van Elst (1999) & Collins & Hawking (1973a). 

Orthogonal models, with the fluid flow lines orthogonal to the surfaces of homogeneity; 

which means the fluid 4-velocity U µ is parallel to the normal vectors n µ . In this case, 

the matter variables will be just the fluid density and pressure. 

Tilted models, with the fluid flow lines not orthogonal to the surfaces of homogeneity; 

i.e. the fluid 4-velocity is not parallel to the normals. Hence we also need the peculiar 

velocity of the fluid relative to the normal vectors. These components enter as further 

variables. 

Rotating models must be tilted and have a more complex behavior Ellis & van Elst 

(1999). 

5.3 The Field Equations 

We follow Collins & Hawking (1973a) and Collins & Hawking (1973b) to obtain 

the Einstein field equations for Bianchi models. 

We saw that the spatially homogeneous models admit a group of symmetries, G,

E A µ satisfy the relation E A µ;ν − EA ν;µ = CA BC EB µ EC ν , (5.10) 


which is transitive on a family of 3-dimensional space-like hypersurfaces, S(t). We label 

these spaces by a time parameter t which is chosen to measure proper time along the 

geodesics normal to S(t). The normal to the hypersurface is given by n α = t ;α such that 

g αβ n α n β = −1 and ; means covariant derivative 2 . We define three invariant vector fields 

E µ A and its dual EA µ 

at each point p ∈ S and covectors E A µ 

basis satisfy 

We drag E µ A and EA µ 

the Lie derivative of the fields 

on the surfaces of homogeneity by dragging the tangent vectors E µ A 

(one-forms) under the group. The above dual 

E A µ Eµ B = δA B (5.8) 

and E A µ Eν A = δν µ + n µn ν . 

by the unit vector field n ν orthogonal to S(t), this is done by 

Now at every point p ∈ S(t), vector fields E µ A and EA µ 

L nν Eµ A = L n ν 

E µ A 

= 0. (5.9) 

are defined and are invariant 

under the 3-dimensional Lie group G (group of isometries of the space). Covector fields 

where C A BC 

are constants given by the structure constants of the Lie group G 

C A BC = Cµ νλ EA µ Eµ B Eν C (5.11) 

The metric of this model can be written in the following form 

g µν = −n µ n ν + g AB E A µ E B ν . (5.12) 

We split the matrix g AB 

(1968), 

into two parts: its volume and the distortion part,Misner 

g AB = e 2α ( e 2β) AB . (5.13) 

where the scalar α represents the volumetric expansion whilst β is symmetric, trace free 

3 × 3 matrix and hence e 2β is given by the series 

∞∑ 

e 2β 1 

= 

r! (2β)r 

r=0 

2 Greek indices run from 0 to 3, Latin indices from 1 to 3. Latin indices may be lowered and raised 

by the 3 × 3 matrix g AB = g µν E µ A Eν B and its inverse gAB = g µν E A µ E B ν which depend only on t.


. Now we define an orthonormal basis X ν µ 

such that 

X 0 µ = −n µ, X i µ = eα (e 2β ) iA E A µ . (5.14) 

where i, j, k, · · · = 1, 2, 3. The 0-th component of these basis changes sign on raising 

or lowering, but rest of the components remain unchanged. Ricci rotation coefficients 

Γ αβγ are obtained from the connection coefficients through δ αδ Γ δ βγ . For this basis these 

coefficients satisfy Γ δɛγ + Γ γɛδ and are given by 

Using these properties we will obtain 

Γ = 1 2 

X γ µ;ν = Γγ δɛ Xδ µ Xɛ ν (5.15) 

[ 

Xγ[µ;ν] X µ δ Xν ɛ + X ɛ[µ;ν]X µ γ Xν δ − X δ[µ;ν]X µ ɛ Xν γ 

] 

. (5.16) 

This with eqn. (5.10) will determine the components of Ricci rotation coefficients 

Γ ij0 = −Γ 0ij = (∂ 0 α)δ ij + σ ij , (5.17) 

Γ i00 = −Γ 00i = 0, 

Γ i0j = −ν ij 

Γ ijk = 1 2 e−α (e β ) F A (e −β ) GB (e −β ) HC C A BC [δ F iδ Gj δ Hk + δ F k δ Gi δ Hj − δ F j δ Gk δ Hi ] . 

In the above expressions, ∂ 0 denotes differentiation by t and σ ij represents the shear of 

the normals n µ and is given by anticommutator of (∂ 0 e β ) and e −β 

σ ij ≡ 1 2 {(∂ 0e β ), e −β } ij (5.18) 

= 1 2 

[ 

(∂0 e β )e −β + e −β (∂ 0 e β ) ] ij . 

The extent of commutativity of ∂ 0 e β and e −β is given by their commutator denoted by 

ν ij 

ν ij ≡ 1 2 

[ 

(∂0 e β ), e −β] ij 

(5.19) 

Ricci tensor of the space Sin the induced metric is 

Rij ∗ = − 1 4 e−2α { 2 CBCC A DA(e C −β ) Bi (e −β ) Dj (5.20) 

+ CBCC A EF D (e −2β ) BE [2(e 2β ) DA (e −β ) Ci (e −β ) F j − (e −2β ) CF (e β ) Ai (e β ) Dj ] 

+ CABC A DE(e C −2β ) BE [(e β ) Ci (e −β ) Dj + (e β ) Cj (e −β ) Di ]}.

5.4 APPROACHING ISOTROPY 84 

We can now write the field equations for this basis Collins & Hawking 

filed equations are 

(1973a). The 

3(∂ 0 α) 2 − 1 2 σ ijσ ij + 1 2 R∗ = 8πT 00 , (5.21) 

e −α [(e −β σe β ) BA CBC A (e−β ) Ci − σ ij (e −β ) jC CAC A ] = 8πT 0i , (5.22) 

∂ 0 σ ij + 3(∂ 0 α)σ ij + [σ, ν] ij + Rij ∗ − 1 3 R∗ δ ij = 8π(T ij − 1 3 T kkδ ij ) (5.23) 

−6∂0 2 α − 9(∂ 0α) 2 − 3 2 σ ijσ ij − 1 2 R∗ = 8πT kk . (5.24) 

5.4 Approaching Isotropy 

We say a model approaches isotropy if at arbitrary large time scales it has the 

following properties 

(a) 

The universe should continue to expand forever, which means 

t → +∞ =⇒ α → +∞ 

Also we need 

(b) 

(c) 

(d) 

Second condition is the isotropy condition. T 0i /T 00 represents an average 

velocity of the matter relative to the surface of homogeneity. The universe 

can only reach isotropy only if at large times this velocity vanishes 

t → +∞ =⇒ T 00 > 0 

and 

T 0i 

T 00 → 0 

Anisotropy in the locally measured Hubble parameter should tend to zero. 

If we define σ 2 = 1 2 σ ijσ ij , this condition will then translate into 

t → +∞ =⇒ 

σ 

∂ 0 α → 0 

If we notice that the anisotropy observed in CMB due to a specific Bianchi 

model is somehow related to the change of β between the time of last scattering 

and the time it is observed, we will conclude that the amazing degree 

of isototropy of the CMB, will tell us that β should be very small, 

t → +∞ =⇒ β → β 0 where β 0 = const.

5.5 ENERGY MOMENTUM TENSOR AND VORTICITY VECTOR 85 

The following theorem which was proved by Collins & Hawking (1973a) shows that 

only 4 Bianchi types approach isotropy. 

Theorem 5.4.1 If the dominant-energy condition 3 and the positive-pressure criterion 4 

are satisfied, the universe can approach isotropy only if it is one of the types I, V , V II 0 

and V II 5 h . 

Since we know that our universe is isotropic to a very high extent, from now on we will 

only be interested in these models. 

5.5 Energy Momentum Tensor and Vorticity Vector 

In a matter dominated universe the dominant contribution to the energy-momentum 

tensor comes from non-relativistic matter, 

T µν = ρ m U µ U ν , (5.25) 

where ρ m is the matter density and U µ is the flow-vector. The conservation equation 

tells us that flow-lines satisfy the geodesic equation, hence 

∂ t (ρ m U 0 e 3α ) = −ρ m e α C A ABU C (e −2β ) BC . (5.26) 

in models of class A, right hand side of the above expression vanishes and will result in 

Class A : ρ m = a m (U 0 e 3α ) −1 (5.27) 

where a m is a constant. Using this constant, T 00 can be written as a m e −3α (1 + E m ) 

where a m is constant in class A models and hence the kinetic energy associated with the 

peculiar velocity of the matter relative to the surfaces of homogeneity will be 

E m = U 0 − 1 = [ U A U B e −2α (e −2β ) AB + 1 ] 1/2 

− 1. (5.28) 

And the anisotropic part of the spatial components of the energy-momentum tensor will 

be 

T ij − 1 3 T kkδ ij = −a m e −3α ∂E m 

∂β ij 

. (5.29) 

3 T 00 ≥ |T αβ | 

4 T kk ≥ 0 

5 Spatially homogeneous cosmological models with cosmological constant were investigated by Wald 

(1983) and Goliath & Ellis (1998)

5.6 ENERGY MOMENTUM TENSOR AND VORTICITY VECTOR 86 

In a radiation dominated universe, the energy momentum tensor would be 

and the energy density of radiation obeys 

T µν = 4 3 ρ γU µ U ν + 1 3 ρ γg µν , (5.30) 

∂ t 

[ 

ργ (U 0 ) 4/3 e 4α] = − 4 3 ρ γ(U 0 ) 1/3 C A ABU C e 2α (e −2β ) B C. (5.31) 

Right hand side will vanish in class A models and as a result 

Class A : ρ γ (U 0 ) 4/3 e 4α = a γ (5.32) 

Flow lines feel an acceleration caused by the radiation pressure 

4 

3 ρ γU µ 

;ν U ν = − 1 3 ρ γ;νh µν , where h µν ≡ g µν + U µ U ν . (5.33) 

Because of homogeneity ρ γ;ν = −∂ t ρ γ n ν . Therefore 

∂ t (ρ 1/4 

γ U A ) = ρ 1/4 

γ (U 0 ) −1 C B CA U BU D e −2α (e −2β ) C D. (5.34) 

The kinetic energy and the anisotropic part of the spatial components of the energymomentum 

tensor are then given by Hawking (1968); 

E γ = (U 0 ) 2/3 − 1 = [ U A U B e −2α (e −2β ) AB + 1 ] 1/3 

− 1 (5.35) 

T ij − 1 3 T kkδ ij = −2a γ e −4α ∂E γ 

∂β ij 

. 

In these models we are mainly interested in the vorticity 

ω = (g AB ω A ω B ) 1/2 = (ω i ω i ) 1/2 (5.36) 

The vorticity vector of the matter is defined as ω µ = 1 2 ηµνλρ U ν U λ;rho . Thus 

ω A = 1 2 e−3α ɛ ABC 

[ 1 

2 CD BCU D U 0 + U B ∂ 0 U C 

] 

, (5.37) 

ω 0 = 1 2 e−3α ɛ ABC C D BC U AU D 

For small, non-relativistic velocities, U 0 ∼ 1 and U A is small, so 

ω A ≃ 1 4 e−3α ɛ ABC C D BC U D. (5.38)

5.6 NULL GEODESICS 87 

5.6 Null Geodesics 

In general, all of our observations take place on our past light cone, which will 

develop many cusps and caustics at early times because of gravitational lensing, but is 

still locally generated by geodesic null rays. The information we receive comes to us 

along these null rays, with tangent vector 

K µ K µ = 0 (5.39) 

where K µ is the tangent vector to the null geodesic from the observer in a given direction. 

In the orthonormal basis Xµ α which was defined in the previous section, the null geodesic 

condition will become 

K 0 K 0 + K A K A = 0 (5.40) 

As we saw, the 0-th component of these basis changes sign on raising or lowering. So, 

K 0 = −K 0 =⇒ K 0 = (K A K A ) 1/2 (5.41) 

and 

K A = g AB K B = e 2α (e 2β ) AB K B (5.42) 

with K A = (K cos θ, K sin θ cos φ, K sin θ sin φ) where K is constant. For small anisotropies 

Barrow et al. (1985), i.e. β ij ≪ 1, 

g AB ≃ e 2α δ AB =⇒ K 0 ≃ Ke −α (5.43) 

The null geodesics satisfy the geodesic equation 

In the orthonormal basis, the geodesic equations are 

K A ≃ e 2α δ AB K B 

K µ;ν K ν = 0, (5.44) 

∂ t K A = CCA 

B K B K C 

. (5.45) 

K 0 

Using the small anisotropy condition, eqn. (5.43), the geodesic equation to fist order 

will be 

∂ t K A = 1 K CB CA K BK C e −α . (5.46)

5.6 NULL GEODESICS 88 

We introduce a new time parameter by dτ = e −α dt and we denote the differentiation 

with respect to this time by prime 

K A ′ = 1 K CB CAK B K C . (5.47) 

We are interested in those Bianchi types which are containing FLRW models as special 

isotropic cases. These models are Bianchi types I, V , V II 0 , V II h and XI. Null geodesic 

equations and their solutions for these models were first derived by Barrow et al. (1985) 

and are given below 

Bianchi Type I 

All structure constants are zero, 

θ ′ = φ ′ = 0 (5.48) 

=⇒ θ = θ 0 = const. 

φ = φ 0 = const. 

Bianchi Type V 

Non-zero structure constants: 

C 2 21 = −C 2 12 = C 3 31 = −C 3 13 = 1. 

θ ′ = − sin θ, φ ′ = 0, (5.49) 

( ( ) 

θ θ0 

=⇒ tan = tan exp [−(τ − τ 0 )] 

2) 

2 


Bianchi Type V II 0 


C 2 31 = −C2 13 = C3 12 = −C3 21 = 1, 

θ ′ = 0, φ ′ = − cos θ, (5.50) 

=⇒ θ = θ 0 = const. 

φ = φ 0 − (τ − τ 0 ) cos θ 0 .

5.7 PATTERNS OF ANISOTROPIC MODELS ON THE CMB SKY 89 

Bianchi Type VII h 


C 2 31 = −C2 13 = C3 12 = −C3 21 = 1, 

C 2 12 = −C 2 21 = C 3 13 = −C 3 31 = √ h. 

θ ′ = − √ h sin θ, φ ′ = − cos θ, (5.51) 

( ( ) 

θ θ0 

=⇒ tan = tan exp [−(τ − τ 0 ) 

2) 

√ h] 

2 

φ = φ 0 + (τ − τ 0 ) − √ 1 ( ) ( ) 

ln {sin 2 θ0 

+ cos 2 θ0 

exp [2(τ − τ 0 ) √ h]}. 

h 2 

2 

Bianchi Type IX 

C A BC = ɛ ABC 

θ ′ = φ ′ = 0 (5.52) 

=⇒ θ = θ 0 = const. 


5.7 Patterns of Anisotropic Models on the CMB 

Sky 

In the previous section, we found the null geodesic solutions in Bianchi types that 

include FLRW model as an isotropic limit. Now we will compute the effect of these 

models on the mean temperature of CMB sky. In this section, we do our calculations 

with the assumption of small anisotropy which means 

β ij ≪ 1, |U i | ≪ 1, |σ ij | ≃ |∂ t β ij | ≪ ∂ t α. (5.53) 

Relative to an observer with 4-velocity U µ , the null vector K µ determines a redshift 

factor (−K µ U µ ). Hence the observed cosmological redshift z for comoving matter is 

given by Ellis & van Elst (1999) 

1 + z E = (K µU µ ) E 

(K µ U µ ) 0 

(5.54)


where subscript E and O refer to photon emission and reception respectively, U µ 0 

is the 

velocity vector of the observer, U µ E 

is the velocity vector of the emitting surface. We want 

to study the effect of the redshift given in eqn. (5.54) on cosmic microwave background 

radiation which is considered to have a black body spectrum at a temperature T E , 

coming from the surface of last scattering. The observed temperature T 0 is then 

T 0 = 

T E 

1 + z E (θ 0 , φ 0 ) 

(5.55) 

If we define p i to be the direction cosines of the null geodesic, p i = K i /K = (cos θ, sin θ cos φ, sin θ sin φ) 

the above expression to first order, will become 

T 0 = T E Γ −1 [ 1 + (p i U i ) 0 − (p i U i ) E − Σ ] . (5.56) 

where 

Γ = e α E−α 0 

, Σ = 

∫ 0 

E 

p i p k σ ik dt 

The CMB pattern in eqn. (5.56) has an easy interpretation. The first term on the 

right hand side is a Doppler term and exhibits a dipole in the temperature anisotropy 

due to the motion of the observer relative to the hypersurfaces of constant time at the 

time of observation. This causes a net dipole. The second term represents a Doppler 

shift due to the peculiar motion of the source of emitting photon on the surface of last 

scattering. This will cause a local redshift/blueshift but usually not a net dipole. In 

type I and IX, p i E 

= pi 0 , so the second term also gives a dipole variation Collins & 

Hawking (1973b). These two terms are absent in orthogonal models but are present in 

tilted models. The third term, Σ, gives the temperature anisotropy cause by the shear 

of the Universe due to the anisotropic expansion. In types I and IX this causes a pure 

quadrupole pattern equal to p i 0 pj 0 (β 0 − β E ) ij . 

To obtain the signature of Bianchi models on CMB sky, we should solve equation 

(5.71). The time dependence of U i and σ ij are given by Einstein equations. And the 

variation of θ and φ with time obtained from the (null-) geodesic equation which was 

discussed in section 5.6. Following is a very brief review of this matter.


5.7.1 CMB Patterns in Type I Bianchi Models 

This is the simplest type for which the null geodesic direction cosines are all constant, 

π i = constant. α and β are given by 

e α = t 2/3 , β = 0 (5.57) 

The peculiar velocity is zero and the vorticity vanishes. For small perturbations, β ij = 

A ij t −1 and the shear is σ ij = −A ij t −2 where A ij is a constant matrix. Hence the only 

contribution to CMB pattern is a quadrupole which comes from the Σ term and is equal 

to 

p i 0 pj 0 A ij(t −1 

0 − t −1 

E ) (5.58) 

5.7.2 CMB Patterns in Type V Bianchi Models 

This models is the simplest generalization of the k = −1 FLRW models. α and β 

are given by 

e α = 8πM 

3 

sinh 2 τ/2, β = 0 (5.59) 

where τ is a new time variable defined by 6 dt/dτ = e α with τ = 0 when e α = 0, and 

M = ρ 0 (H0 2 − 8πρ 0/3) −3/2 ρ 

= 

0 

. High density type V models behave like type 

H0 3(1−Ω 0) 3/2 

I models, so here we consider low density models. The null geodesics for these models 

are calculated in the previous section. For small anisotropies, σ ij = A ij e −3α where A ij 

is a constant matrix given by A 1j = (8πM/3)U j e α and A 23 = 0 (one can choose the 2-3 

axis such that this happens). The Σ term will then cause an anisotropy in this form 

− 3 2 A 11I 1 + 1 2 (A 22 − A 33 )I 1 cos φ + 2(A 12 cos φ + A 13 sin φ)I 2 (5.60) 

where 

I 1 = 

∫ 0 

E 

e −2α cos 2 θdτ, and I 2 = 

∫ 0 

E 

e −2α sin θ cos θdτ. (5.61) 

6 I will denote ∂ t α 0 which is the present value of the expansion rate as H 0 .


5.7.3 CMB Patterns in Type VII 0 Bianchi Models 

This is the most general family of models which include the k = 0 FLRW solution. 

α and β are given by 

e α = ( 9t2 

4H 0 

) 1/3 x, β = 0 (5.62) 

where x is a constant. The parameter x is equal to the characteristic length scale, e α 0 

, 

over which the invariant vectors change their orientation, multiplied to the expansion 

rate, H 0 . The value of parameter x is does not make any physical difference to the FLRW 

model but it governs the length scale of the anisotropy in an anisotropic model.For 

small anisotropies, β is not zero as given in Collins & Hawking (1973b). Null geodesic 

equations are calculated in the previous section. The Doppler terms (p i U i ) 0 and (p i U i ) E 

arising from the 1-component of the peculiar velocity give rise to a dipole anisotropy 

of magnitude U (1) 

0 (Γ − 1) cos θ 0 . The 2 and 3 components of the peculiar velocity will 

cause the following pattern 

( [ ( 2 

Γ cos 

x 

( 

− Γ sin 

1 − 1 

Γ 1/2 ) 

] 

cos θ 

[ 2 

x 

( 

1 − 1 

Γ 1/2 ) 

cos θ 

]) 

) 

− 1 

(U (2) 

0 cos φ + U (3) 

0 sin φ) sin θ (5.63) 

(U (2) 

0 sin φ − U (3) 

0 cos φ) sin θ 

For x ≪ 1 this is a tightly wound spiral pattern which performs 2/pix(1 − 1/Γ 1/2 ) 

revolutions and has spacing πx/ sin θ 0 [Γ 1/2 /(Γ 1/2 −1)] between successive maxima Collins 

& Hawking (1973b). The Σ term gives a quadrupole pattern through σ 11 which is like 

(β 0 − β E ) 11 cos 2 θ. The effect of the other components of σ ij is complicated and some 

approximate results near the equator has been given in Collins & Hawking (1973b). 

For x ≫ 1 

the second term vanishes and it becomes just a dipole pattern (Γ − 

1)((U (2) 

0 cos φ + U (3) 

0 sin φ) sin θ. Also the p i 0 

gives a quadrupole pattern p i p j (β 0 − β E ) ij . 

are almost constant and hence the Σ term 

5.7.4 CMB Patterns in Type VII h Bianchi Models 

This is the most general family of models which includes the k = 0 FLRW solutions. 

It has some of the features of both types V and type VII 0 . There is an adjustable


parameter h in this model which is given by the square of structure constants (see 

previous section). In an unperturbed FLRW model, the expansion scale factor, α, and 

β are given by 

e α = 

As in type VII 0 , we introduce a factor x = H 0 e α 0 

by 

( ) 

h 1/2 Ω 0 

h 1/2 τ 

H 0 (1 − Ω 0 ) 3/2 sinh2 , β = 0 (5.64) 

2 

which is related to the parameter h 

√ 

h 

x = 

(5.65) 

1 − Ω 0 

As we said before, x has no physical effect on the FLRW models. It can be seen from 

the fact that the present value of the scale factor can be arbitrarily chosen, but from 

(5.64) we have 

e α 0 

= x H 0 

, (5.66) 

And hence we see that parameter x can be scaled out of the solution. However, for large 

values of x (or equally h), the models are similar to those of type V. As the present 

density tends towards the critical density, Ω 0 → 1, and h → 0 in such a way that x 

remains finite, the behavior of the models tend to that of type VII 0 . Behavior of these 

models with different parameters can be seen in figures 5.2 to 5.5. 

Null geodesics have been given in the previous section and have a complicated form. 

They start near the South Pole and spiral in the negative φ-direction up towards the 

equator, and then spiral in the positive φ-direction up towards the North Pole. Small 

anisotropies can be treated as perturbations to FLRW model. β has been calculated 

for them and is given in Collins & Hawking 

(1973b). Here we follow Barrow et al. 

(1985) who used these to calculate the contribution to temperature anisotropies from 

the vorticity alone. They argue that the inclusion of other pure shear distortions which 

are independent of the rotation could only make the temperature anisotropies larger. 

And hence their results would give the maximum level of vorticity consistent with a 

given value of temperature anisotropy. 

In type VII h , there are two independent vorticity components ω 2 and ω 3 . They are 

given in terms of the off-diagonal shear elements as 

ω 2 = (3h − 1)σ 13 − 4h 1/2 σ 12 

3x 2 Ω 0 

, (5.67)


ω 3 = (1 − 3h)σ 12 − 4h 1/2 σ 13 

3x 2 Ω 0 

. 

The observables in this model are two dimensionless amplitudes: ( σ 12 

H 

vorticity is given by 

) 

and ( σ 13 

0 H 

) 

0 . The 

ω = 1 2 e−α (1 + h) 1/2 [ (u 2 ) 2 + (u 3 ) 2] 1/2 , (5.68) 

where u 2 and u 3 are the velocity components and their present values are given by 

(u 2 ) 0 = 1 [ ( 

3h 1/2 σ12 

3xΩ 0 H 

(u 3 ) 0 = 1 [( σ12 

3xΩ 0 H 

Hence the present value of vorticity is 

( ω 

H 

) 

0 

) 

0 

) 

0 

− 

( σ13 

H 

+ 3h 1/2 ( σ13 

H 

= (1 + [ h)1/2 (1 + 9h) 1/2 (σ12 ) 2 

+ 

6x 2 Ω 0 H 0 

To first order, temperature fluctuations are given by 

) ] 

, (5.69) 

0 

) ] 

. 

0 

( σ13 

) 2 

0 

H 

] 1/2 

. (5.70) 

∆T (θ 0 , φ 0 ) 

T 0 

≃ (p i U i ) 0 − (p i U i ) E − 

∫ 0 

E 

p i p k σ ik dt, (5.71) 

where θ 0 and φ 0 are related to the actual observing angles by 

θ = π − θ 0 , φ = π + φ 0 (5.72) 

Substituting for null geodesics from eqn. (5.51) and for velocities from eqn. (5.69) into 

eqn. (5.71), we will obtain 

∆T (θ 0 , φ 0 ) 

T 0 

= 

+ 

[( σ12 

) 

H 

[( σ12 

) 

H 

0 

0 

A(θ 0 ) + 

B(θ 0 ) − 

where coefficients A(θ 0 ) and B(θ 0 ) are defined by 

( σ13 

H 

( σ13 

H 

) 

) 

0 

0 

] 

B(θ 0 ) sin φ 0 (5.73) 

] 

A(θ 0 ) cos φ 0 , 

A(θ 0 ) = C 1 [sin θ 0 − C 2 (cos ψ E − 3h 1/2 sin ψ E )] (5.74) 

∫ τ0 

s(1 − s 2 ) sin φdτ 

+ C 3 

(1 + s 2 ) 2 sinh 4 h 1/2 τ/2 , 

τ E 

B(θ 0 ) = C 1 [3h 1/2 sin θ 0 − C 2 (sin ψ E + 3h 1/2 cos ψ E )] 

∫ τ0 

s(1 − s 2 ) cos φdτ 

− C 3 

(1 + s 2 ) 2 sinh 4 h 1/2 τ/2 . 

τ E


The limits of integration are defined as 

τ 0 = 2h −1/2 sinh −1 (Ω −1 

0 − 1) 1/2 , (5.75) 

τ E = 2h −1/2 sinh −1 ( Ω 

−1 

0 − 1 

1 + z E 

) 1/2 

, 

and constants C 1 , C 2 , C 3 , s and ψ are defined by 

C 1 = (3Ω 0 x) −1 ; (5.76) 

C 2 = 2s E(1 + z E ) 

; 

1 + s 2 E 

C 3 = 4h 1/2 (1 − Ω 0 ) 3/2 Ω 

( ( −2 

0 

) 

; 

θ θ0 

s = tan = tan exp [−(τ − τ 0 ) 

2) 

√ h], 

2 

ψ = φ 0 + (τ − τ 0 ) − √ 1 ( ) ( ) 

ln {sin 2 θ0 

+ cos 2 θ0 

exp [2(τ − τ 0 ) √ h]}. 

h 2 

2 

The expression for ∆T (θ 0,φ 0 ) 

T 0 

where ˜φ is defined as 

and σ is given by 

can be written in a compact form 

∆T 

T 0 

(θ 0 , φ 0 ) = (A 2 + B 2 ) 1/2 ( σ 

H 

cos ˜φ = 

[( σ12 

σ 

) 

B − 

( σ13 

σ 

) 

0 

cos (φ 0 + ˜φ) (5.77) 

) ] 

A (A 2 + B 2 ) −1/2 (5.78) 

σ 2 = σ 2 12 + σ 2 13 (5.79) 

Eqn. (5.77) helps to understand the behavior of the CMB pattern in these models. If 

we look around any circle at a given θ 0 on the sky, the temperature variation will have 

a pure cos φ 0 behavior. ˜φ determines the relative orientation of adjacent θ0 =constant 

rings. In type V models, A(θ 0 ) = 0 and hence we have no spiraling in these models. On 

the other hand, B(θ 0 ) determines the amplitude of ∆T 

T 0 

and the focusing into a hot spot 

[Barrow et al. (1985)]. Some of these maps have been shown in figures 5.2 to 5.5. 

5.7.5 CMB Patterns in Type IX Bianchi Models 

This is a generalization of the k = +1 FLRW models with 

e α = s α 0 

sin τ/4 , β = 0 (5.80)

5.8 UNVEILING HIDDEN PATTERNS OF BIANCHI VII H 96 

Figure 5.2 Bianchi patterns in type VII h models for x = 0.1 and Ω 0 = 0.1 (left),Ω 0 = 0.5 

(middle) and Ω 0 = 0.9 (right). 

For small anisotropies, β ij is given by a perturbative series derived by Collins & Hawking 

(1973b). The null geodesics have p i = constant and hence the Doppler terms (p i U i ) 0 

and (p i U i ) E give rise to a dipole anisotropy equal to (p i U i ) 0 (Γ − 1). The Σ term causes 

a quadrupole anisotropy of the form p i p j (β 0 − β E ) ij . 

5.8 Unveiling Hidden Patterns of Bianchi VII h 

Recently Bianchi VII h models were compared to the first-year WMAP data on large 

scales and it was shown that the best fit Bianchi model corresponds to a highly hyperbolic 

model (Ω 0 = 0.5) with a right-handed vorticity ( ω ) H 0 = 4.3 × 10 −10 [Jaffe et 

al. (2005)]. There are several aspects to this hidden pattern which should be mentioned. 

First of all, the proposed model is an extremely hyperbolic model and does not 

agree with the location of the first peak in the best fit power spectrum of the CMB. In 

the regime of VII h models, one should study the nearly flat models in order to have a 

consistent angular power spectrum. But the problem is that as it has been shown in 

figs 5.2 to 5.5, the spiral patterns in high density models are not concentrated in one 

hemisphere and can’t be used to cure the large-scale power asymmetry in the WMAP 

data reported by Eriksen et al. (2004a). Second is that the Bianchi patterns we studied 

so far, are only valid for a matter dominated universe. Patterns must be recalculated in 

a universe with a dark energy component if one wants to compare them with the real 

observed CMB data which is out there in a Λ dominated universe. 

For these reasons, we study the type VII h Bianchi models only as hidden patterns 

in the CMB anisotropy maps. We pay our attention to the specific model proposed by


Jaffe et al. (2005) and in the next two sections, we address the question whether and 

to what extent one would be able to discover this pattern (Fig. 5.7) and other hidden 

anisotropic pattern in the CMB.


Figure 5.3 Bianchi patterns in type VII h models for x = 0.55 and Ω 0 = 0.1 (left),Ω 0 = 0.5 


Figure 5.4 Bianchi patterns in type VII h models for x = 1 and Ω 0 = 0.1 (left),Ω 0 = 0.5 


Figure 5.5 Bianchi patterns in type VII h models for x = 5 and Ω 0 = 0.1 (left),Ω 0 = 0.5 

(middle) and Ω 0 = 0.9 (right).


1500 

C l l (l+1) / 2 π (µK) 2 

1000 

500 

0 

0 10 20 30 

Multipole l 

Figure 5.6 Top panel: WMAP Internal Linear Combination map. Middle panel: Best-fit 

Bianchi VII h template (enhanced by a factor of four to show the structure). Bottom 

panel: Difference between ILC and best-fit Bianchi template; the “Bianchi-corrected” 

ILC map. Figure and caption courtesy of Jaffe et al. (2005). 

Figure 5.7 Comparison of power spectra. The gray and black solid lines show the power 

spectrum estimated from the co-added V+W map before and after correcting for the 

Bianchi template, respectively. The dotted gray and black lines shows the theoretical 

best-fit power-spectra from the WMAP-team analysis and Hansen et al. (2004) respectively. 

The latter is a fit to northern hemisphere data alone. The light gray dashed line 

corresponds to the full sky power spectrum of the best-fit Bianchi template. Figure and 

caption from Jaffe et al. (2005).


Bipolar Power Spectrum Analysis 

To do a statistical study of these patterns, we generate the pattern for a given model. 

This pattern is given by the shear components, x parameter and the Ω 0 . We will then 

simulate random CMB maps from the best fit C l of the WMAP data. We add these 

two maps with a strength factor α which will let us control the relative strength of the 

pattern and the random map (see Fig. 5.8). The resultant map is then given by 

Figure 5.8 Adding a pattern template with a strength α to a random realization of the 

CMB anisotropy map. Maps are rotated to the Galactic center for illustration 

∆T (ˆn) = ∆T CMB (ˆn) + α∆T Bianchi (ˆn), (5.81) 

and hence the power spectrum of this map will be given by 

C l = C CMB 

l 

+ α 2 C Bianchi 

l (5.82) 

α is related to ( σ ) H 0 through ( σ ) H 0 = α T 0 

where T 0 = 2.73 × 10 6 µK. 

We analysis the above Bianchi added CMB maps using BiPS method. Since BiPS is 

orientation independent, we don’t need to worry about the relative orientation between 

the background map and the Bianchi template. The procedure is then as follows. For 

different strength factors we simulate 100 Bianchi added CMB maps 7 

at HEALPix 

resolution of N side = 32 corresponding to l max = 95. Some of these maps are shown 

in Figure 5.9. 

eqn. (5.81) are a lm = a CMB 

lm 

BiPS of each map is obtained from the total a lm which according to 

+ α a Bianchi 

lm . We compute the BiPS for each map using the 

estimator given in eqn. (2.37). We average these 100 BiPS and compute the dispersion 

in them. The dispersion is an estimate of 1σ error bars. At last we use the total 

angular power spectrum given in eqn. (5.82) to estimate the bias. We use the best fit 

7 Simulation of background maps and the spherical harmonic expansion of the resultant map is done 

by HEALPix.


Figure 5.9 Four Bianchi added CMB maps with different strength factors, rotated to 

the Galactic center for illustration. The Bianchi template in the above maps has been 

computed with x = 0.55, Ω 0 = 0.5 and strength factors ( σ ) H 0 = 1.83 × 10 −9 (top left), 

( σ ) H 0 = 1.09 × 10 −9 (top right), ( σ ) H 0 = 7.3 × 10 −10 (bottom left), ( σ ) H 0 = 3.66 × 10 −10 

(bottom right). 

theoretical power spectrum from the WMAP analysis, Spergel et al. 

(2003), as C CMB 

l 

in computing the bias by the estimator of eqn. (2.40). Some results of BiPS for different 

strength factors are shown in Figure 5.10. As it has been discussed in Chapter 2, we 

can use different window functions in harmonic space, W l , in order to concentrate on 

a particular l-range. This is proved to be a useful and strong tool to detect deviations 

from SI using BiPS method. In other words, multipole space windows that weigh down 

the contribution from the SI region of multipole space will enhance the signal relative 

to cosmic error. We use simple filter functions in l space to isolate different ranges of 

angular scales; a low pass, Gaussian filter, Wl 

G (l s ) and a band pass filter, Wl S (l t , l s ). 

These filters are plotted in the lower panel of Figure 7.1. 

We use a simple χ 2 statistics to compare our BiPS results with zero, 

l∑ 

max 

χ 2 = 

l=0 

( 

κl 

σ κl 

) 2 

. (5.83)


0 

0 

5 10 15 

5 10 15 

0 

0 

5 10 15 

5 10 15 

Figure 5.10 Bipolar power spectrum of Bianchi added CMB maps with strength factors 

of ( σ H ) 0 = 4.76 × 10 −10 , 3.66 × 10 −10 , 2.56 × 10 −10 , 1.25 × 10 −10 . We have used W S 

l (4, 13) 

to filter the maps. 

The probability of detecting a map with a given BiPS is then given by the probability 

distribution of the above χ 2 . Probability of χ 2 versus the strength factor is plotted in 

Figure 5.11. Being on the conservative side we can constrain the strength factor to be 

( σ ) H 0 = 3.15 × 10 −10 at a 99.9% confidence level, Ghosh et al. (2006). 

We can repeat the above exercise for nearly flat (Ω 0 = 0.99) Bianchi VII h templates. 

Some of these Bianchi added CMB maps are shown in Figure 5.12. As we expect, the 

spiral patterns of these models are wider than the hyperbolic models and is not focused 

in one hemisphere. As it can be seen in Figure 5.13 they have a weaker BiPS signal. 

And hence we get a weaker limit of ( σ ) H 0 = 4.7 × 10 −10 at a 99.9% confidence level on 

them (see Figure 5.14).


5 

1 

0.32 

0.1 

P(χ 2 ) 

0.01 

0.001 

0.0001 

2.52 

2.83 

3.15 

3.48 

4.3 

4.7 

(σ/H) 0 in (10 -10 ) 

Figure 5.11 Probability distribution of χ 2 versus the strength factor for Ω 0 = 0.5 Bianchi 

template. Results of two different window functions, Wl 

G (25) (blue line) and Wl S (4, 13) 

(green line) are shown. 

Figure 5.12 Two Bianchi added CMB maps, Ω 0 = 0.99, with different strength factors, 

( σ H ) 0 = 1.09 × 10 −9 and 3.66 × 10 −10 , rotated to the Galactic center for illustration.


0 

0 

5 10 15 

5 10 15 

Figure 5.13 Bipolar power spectrum of Bianchi added CMB maps with strength factors 

of ( σ H ) 0 = 7.33 × 10 −10 (left) and ( σ H ) 0 = 4.76 × 10 −10 (right). 

5 

1 

0.32 

0.1 

P(χ 2 ) 

0.01 

0.001 

0.0001 

3.96 4.4 4.7 

(σ/H) 0 in (10 -10 ) 

5 5.55.8 

Figure 5.14 Probability distribution of χ 2 versus the strength factor for Ω 0 = 0.99 

Bianchi template. Results of two different window functions, Wl 

G (25) (red line) and 

W S 

l (4, 13) (cyan line) are shown.

Chapter 6 

Observational Artifacts 

Foregrounds and observational artifacts (such as non-circular beam, incomplete/nonuniform 

sky coverage and anisotropic noise) would also manifest themselves as violations 

of SI. 

6.1 Anisotropic noise 

The CMB temperature measured by an instrument is a linear sum of the cosmological 

signal as well as instrumental noise. The two point correlation function then has two 

parts, one part comes from the signal and the other one comes from the noise 

C(ˆn 1 , ˆn 2 ) = C S (ˆn 1 , ˆn 2 ) + C N (ˆn 1 , ˆn 2 ). (6.1) 

Both signal and noise should be statistically isotropic to have a statistically isotropic 

CMB map. So even for a statistically isotropic signal, if the noise fails to be statistically 

isotropic the resultant map will turn out to be anisotropic. The noise matrix can fail 

to be statistically isotropic due to non-uniform coverage. Also if the noise is correlated 

between different pixels the noise matrix could be statistically anisotropic. A simple 

example of this is the diagonal (but anisotropic) noise given by the following correlation 

C N (ˆn, ˆn ′ ) = σ 2 (ˆn)δˆnˆn ′. (6.2) 

105

6.1 ANISOTROPIC NOISE 106 

This noise clearly violates the SI and will result a non-zero BiPS given by 

κ l = 

l∑ 

m=−l 

|f lm | 2 , (6.3) 

where f lm are spherical harmonic transform of the noise, f lm = ∫ dΩˆn Y ∗ 

lm (ˆn)σ2 (ˆn). 

As an example we study the BiPS of the coadded noise covariance matrix of WMAP 

which is a simple diagonal matrix. We have plotted this matrix in figure 6.1 where the 

value on each pixel represents the 1/σ 2 i 

on that pixel. We compute the BiPS prediction 

Figure 6.1 The coadded noise covariance matrix of WMAP. This is in fact 1/σi 2 . The 

two spots represent cleaner regions with less noise. 

for this case using eqn. (6.3). The result is shown in figure 6.2. The characteristic 

component of this covariance matrix is the l = 2 component. This is very well expected 

and can be explained. This is a two point correlation of noise with a preferred direction. 

As it is shown in Chapter 3, the signature of such a preferred axis is a non-zero l = 2 

component in BiPS. It is always very important to know how well a statistical measure 

can be estimated from a map. We would like to know what would we see if we estimate 

BiPS from a CMB map. To answer that one should make realizations of the above 

covariance matrix and add them to simulated CMB maps and the estimate the BiPS 

from them and study the average BiPS obtained from many realizations. However at

6.2 ANISOTROPIC NOISE 107 

4e-05 

"bips_cttp_coadd_noise_3.5deg_32.dat" 

3.5e-05 

3e-05 

2.5e-05 

2e-05 

1.5e-05 

1e-05 

5e-06 

0 

0 2 4 6 8 10 12 14 

Figure 6.2 BiPS prediction for the diagonal noise covariance matrix of Figure 6.1. The 

characteristic component is the l = 2. 

this point we are only interested in the behavior of the noise patterns. So, we will stick 

to estimating BiPS from noise maps solely. We simulate noise maps from the noise 

covariance matrix of Fig. 6.1. To do that we use our fast methods of generating CMB 

maps from anisotropic correlation functions explained in Appendix F. The result is 

shown in Fig. F.1 and we see that the noise dispersion is smaller on the place of those 

two clean spots of Fig. 6.1. We make 100 realizations of noise maps and compute the 

BiPS for them. The average BiPS has a clear non-zero component at l = 2. Fig. 6.3 

shows this for simulated noise maps.

6.2 THE EFFECT OF NON-CIRCULAR BEAM 108 

0 

0 

2 4 6 8 

Figure 6.3 Estimating BiPS from 100 simulations of noise maps with a sharp cut off in 

l-space (top) and with a Gaussian filter, G(120), (bottom). 

6.2 The effect of non-circular beam 

as 

The temperature of the CMB anisotropy measured by an instrument can be written 

∫ 

∆ ˜T (ˆn) = ∆T (ˆn ′ )B(ˆn, ˆn ′ )dΩˆn ′. (6.4) 

Here B(ˆn, ˆn ′ ) is the beam function that encodes the instrumental response, ˆn denotes the 

direction to the center of the beam and ˆn ′ denotes the direction of the incoming photon. 

Using this relation to calculate the correlation function ˜C(ˆn 1 , ˆn 2 ) = 〈∆ ˜T (ˆn 1 )∆ ˜T (ˆn 2 )〉 

one would get 

∫ ∫ 

˜C(ˆn 1 , ˆn 2 ) = dΩˆn ′ dΩˆn ′′〈∆T (ˆn ′ )∆T (ˆn ′′ )〉B(ˆn 1 , ˆn ′ )B(ˆn 2 , ˆn ′′ ) (6.5)

6.3 MASK EFFECTS 109 

= 

∫ 

dΩˆn ′ 

∫ 

dΩˆn ′′C(ˆn ′ , ˆn ′′ )B(ˆn 1 , ˆn ′ )B(ˆn 2 , ˆn ′′ ). 

Only for a circular beam where B(ˆn, ˆn ′ ) ≡ B(ˆn · ˆn ′ ), the correlation function is statistically 

isotropic, ˜C(ˆn1 , ˆn 2 ) ≡ ˜C(ˆn 1 · ˆn 2 ). Breakdown of SI is obvious since even C l get 

mixed for a non-circular beam, ˜Cl = ∑ l 

A ′ ll ′C l ′ Mitra et al. (2004). Non-circularity 

of the beam in CMB anisotropy experiments is becoming increasingly important as 

experiments go for higher resolution measurements at higher sensitivity. 

6.3 Mask effects 

Most CMB experiments map only a part of the sky. Even in the best case, contamination 

by galactic foreground residuals make parts of the sky unusable. The incomplete 

sky or mask effect is another source of breakdown of SI. But, this effect can be readily 

modeled out. The effect of a general mask on the temperature field is as follows 

∆T masked (ˆn) = ∆T (ˆn)W (ˆn), (6.6) 

where W (ˆn) is the mask function. One can cut different parts of the sky by choosing 

appropriate mask functions. Masked a lm coefficients can be computed from the masked 

temperature field, 

a masked 

lm = 

∫ 

∆T masked (ˆn)Y ∗ 

lm (ˆn)dΩˆn (6.7) 

= ∑ l 1 m 1 

a l1 m 1 

∫ 

Y l1 m 1 

(ˆn)Y ∗ 

lm(ˆn)W (ˆn)dΩˆn . 

Where a l1 m 1 

are spherical harmonic transforms of the original temperature field. We 

can expand W (ˆn) in spherical harmonics as well 

W (ˆn) = ∑ lm 

w lm Y lm (ˆn), (6.8) 

and after substituting this into eqn. (6.7) it is seen that the masked a lm is given by the 

effect of a kernel K l 1m 1 

lm on original a lm Prunet et al. (2004) 

a masked 

lm = ∑ l 1 m 1 

a l1 m 1 

K l 1m 1 

lm . (6.9)

6.4 RESIDUALS FROM FOREGROUND REMOVAL 110 

The kernel contains the information of our mask function and is defined by 

K l 1m 1 

lm 

= ∑ ∫ 

w l2 m 2 

Y l1 m 1 

(ˆn)Y l2 m 2 

(ˆn)Ylm ∗ (ˆn)dΩˆn (6.10) 

l 2 m 2 

= ∑ l 2 m 2 

w l2 m 2 

√ 

(2l 1 + 1)(2l 2 + 1) 

Cl l0 

4π(2l + 1) 1 0l 2 0Cl lm 

1 m 1 l 2 m 2 

. 

In the last step of the above expression we used rule G.3 of spherical harmonics to 

simplify the expression. The covariance matrix of a masked sky will no longer have the 

diagonal form of eqn.(2.17) because of the action of the kernel 

〈a masked 

lm a masked l ∗ 

′ m 〉 = 〈a ′ l 1 m 1 

a ∗ l 1 ′ m′ 〉Kl 1m 1 

1 lm Kl′ 1 m′ 1 

l ′ m 

(6.11) 

′ 

= C l1 δ l1 l 1 ′ δ m 1 m ′ Kl 1m 1 

1 lm Kl′ 1 m′ 1 

l ′ m ′ 

= ∑ 

C l1 K l 1m 1 

lm Kl 1m 1 

l ′ m 

. ′ 

l 1 ,m 1 

This clearly violates the SI and results a non-zero BiPS for masked CMB skies. In the 

next section we apply a galactic mask to ILC map and show that signature of this mask 

on BiPS is a rising tail at low l, (l < 20). 

6.4 Residuals from foreground removal 

Besides the cosmological signal and instrumental noise, a CMB map also contains 

foreground emission such as galactic emission, etc. The foreground is usually modeled 

out using spectral information. However, residuals from foreground subtractions in the 

CMB map will violate SI. Interestingly, BiPS does sense the difference between maps 

with grossly different emphasis on the galactic foreground. As an example we compute 

BiPS of a Wiener filtered map in the next section to make this point clear (see Fig. 7.4). 

It shows a signal very similar to that of a galactic cut sky. This can be understood if 

one writes the effect of the Wiener filter as a weight on some regions of the map 

∆T W (ˆn) = ∆T (ˆn)(1 + W (ˆn)). (6.12) 

This explains the similarity between a cut sky and a Wiener filtered map. The effect of 

foregrounds on BiPS has recently been studied in great details by Saha et al. (2006).

Chapter 7 

Statistical isotropy of CMB anisotropy from 

WMAP 

After the first year of WMAP data, the SI of the CMB anisotropy (i.e. rotational 

invariance of n-point correlations) has attracted considerable attention. Tantalizing 

evidence of SI breakdown (albeit, in very different guises) has mounted in the WMAP 

first year sky maps, using a variety of different statistics. It was pointed out that the 

suppression of power in the quadrupole and octopole are aligned Tegmark et al. (2004). 

Further “multipole-vector” directions associated with these multipoles (and some other 

low multipoles as well) appear to be anomalously correlated Copi et al. (2004); Schwarz 

et al. (2004). There are indications of asymmetry in the power spectrum at low 

multipoles in opposite hemispheres Eriksen et al. (2004a); Hansen et al. (2004); 

Naselsky et al. (2004). Possibly related, are the results of tests of Gaussianity that 

show asymmetry in the amplitude of the measured genus amplitude (at about 2 to 3σ 

significance) between the north and south galactic hemispheres Park (2004); Eriksen 

et al. (2004b,c). Analysis of the distribution of extrema in WMAP sky maps has 

indicated non-Gaussianity, and to some extent, violation of SI Larson & Wandelt (2004). 

However, what is missing is a common, well defined, mathematical language to quantify 

SI (as distinct from non Gaussianity) and the ability to ascribe statistical significance 

to the anomalies unambiguously. 

As a statistical tool of searching for departures from SI, we use the bipolar power 

spectrum (BiPS) which is sensitive to structures and patterns in the underlying to- 

111

7.0 112 

tal two-point correlation function Hajian & Souradeep 

(2003b); Souradeep & Hajian 

(2003). The BiPS is particularly sensitive to real space correlation patterns (preferred 

directions, etc.) on characteristic angular scales. In harmonic space, the BiPS at multipole 

l sums power in off-diagonal elements of the covariance matrix, 〈a lm a l ′ m ′〉, in the 

same way that the ‘angular momentum’ addition of states lm, l ′ m ′ have non-zero overlap 

with a state with angular momentum |l − l ′ | < l < l + l ′ . Signatures, like a lm and 

a l+nm being correlated over a significant range l are ideal targets for BiPS. These are 

typical of SI violation due to cosmic topology and the predicted BiPS in these models 

have a strong spectral signature in the bipolar multipole l space Hajian & Souradeep 

(2003a). The orientation independence of BiPS is an advantage since one can obtain 

constraints on cosmic topology that do not depend on the unknown specific orientation 

of the pattern (e.g., preferred directions). 

We carry out measurement of the BiPS, on the following CMB anisotropy maps 

A) a foreground cleaned map (denoted as ‘TOH’) Tegmark et al. (2004), 

B) the Internal Linear Combination map (denoted as ‘ILC’ in the figures) Bennett et 

al. 

(2003), and 

C) a customized linear combination of the QVW maps of WMAP with a galactic cut 

(denoted as ‘CSSK’). 

Also for comparison, we measure the BiPS of 

D) a Wiener filtered map of WMAP data (denoted as ‘Wiener’) Tegmark et al. 

(2004), and 

E) the ILC map with a 10 ◦ cut around the equator (denoted as ‘Gal. cut.’). 

Angular power spectra of these maps are shown in Fig. 7.1. The best fit theoretical 

power spectrum from the WMAP analysis 1 

Spergel et al. (2003) is plotted on the same 

figure. C l from observed maps are consistent with the theoretical curve, C T l 

, (except for 

low multipoles. The bias and cosmic variance of BiPS depend on the total SI angular 

1 Based on an LCDM model with a scale-dependent (running) spectral index which best fits the 

dataset comprised of WMAP, CBI and ACBAR CMB data combined with 2dF and Ly-α data

C T l . We use the estimator given in eqn.(2.37) to measure BiPS for the given CMB maps. 

7.0 113 

power spectrum of the signal and noise C l = C S l + C N l . However, we have restricted 

our analysis to l < 60 where the errors in the WMAP power spectrum is dominated by 

cosmic variance. It is conceivable that the SI violation is limited to particular range of 

angular scales. Hence, multipole space windows that weigh down the contribution from 

the SI region of multipole space will enhance the signal relative to cosmic error, σ SI 

. We 

use simple filter functions in l space to isolate different ranges of angular scales; a low 

pass, Gaussian filter 

W G 

l (l s ) = exp(−(l + 1/2) 2 /(l s + 1/2) 2 ) (7.1) 

that cuts off power on small angular scales (< 1/l s ) and a band pass filter, 

W S 

l (l t , l s ) = [2(1 − J 0 ((l + 1/2)/(l t + 1/2)))] exp(−(l + 1/2) 2 /(l s + 1/2) 2 ) (7.2) 

that retains power within a range of multipoles set by l t and l s . The windows are 

normalized such that ∑ l (l + 1/2)/(l(l + 1))W l = 1, i.e., unit rms for unit flat band 

power C l = 1/(l(l+1)). The window functions used in our work are plotted in figure 7.1. 

We use the C T l 

to generate 1000 simulations of the SI CMB maps. a lm ’s are generated 

up to an l max of 1024 (corresponding to HEALPix resolution N side = 512). These 

are then multiplied by the window functions Wl 

G (l s ) and Wl S (l t , l s ). We compute the 

BiPS for each realization. 

Fig.7.1 shows that the average power spectrum obtained 

from the simulation matches the theoretical power spectrum, C T l 

realizations. We use C T l 

, used to generate the 

to analytically compute bias and cosmic variance estimation for 

˜κ l . This allows us to rapidly compute BiPS with 1σ error bars for different theoretical 

We compute the BiPS for all window functions shown in Fig 7.1. Results for three of 

these windows are plotted in Figs. 7.2, 7.3, and 7.4. In the low-l regime, where we have 

kept the low multipoles, BiPS for all three given maps are consistent with zero (Fig. 

7.2). But in the intermediate-l regime (Fig. 7.3), although BiPS of ILC and TOH maps 

are well consistent with zero, the CSSK map shows a rising tail in BiPS due to the 

galactic mask. To confirm it, we compute the BiPS for the ILC map with a 10-degree 

cut around the galactic plane (filtered with the same window function). The result is

7.0 114 

shown on the top panel of Fig. 7.3. Another interesting effect is seen when we apply a 

W S 

l (20, 45) filter, where Wiener filtered map has a non zero BiPS very similar to that 

of CSSK but weaker (Fig. 7.4). The reason is that Wiener filter takes out more modes 

from regions with more foregrounds since these are inconsistent with the theoretical 

model. As a result, a Wiener filtered map at Wl s (20, 45) filter has a BiPS similar to a 

cut sky map. The fact that Wiener map has less power at the Galactic plane can even 

be seen by eye! Comparing Fig. 7.3 to Fig.7.4 we see that using different filters allows 

us to uncover different types of violation of SI in a CMB map. In our analysis we have 

used a set of filters which enables us to probe SI breakdown on angular scales l < 60. 

The BiPS measured from 1000 simulated SI realizations of C T l 

is used to estimate the 

probability distribution functions (PDF), p(˜κ l ). A sample of the PDF for two windows 

is shown in Fig. 7.5. Measured values of BiPS for ILC, TOH and CSSK maps are plotted 

on the same plot. BiPS for ILC and TOH maps are located very close to the peak of 

the PDF. We compute the individual probabilities of the map being SI for each of the 

measured ˜κ l . This probability is obtained by integrating the PDF beyond the measured 

˜κ l . To be precise, we compute 

P (˜κ l |C T l ) = P (κ l > ˜κ l ) = 

= P (κ l < ˜κ l ) = 

∫ ∞ 

˜κ l 

dκ l p(κ l ), ˜κ l > 0, (7.3) 

∫ ˜κl 

−∞ 

dκ l p(κ l ), ˜κ l < 0. 

The probabilities obtained are shown in Figs. 7.6 and 7.8 for W S (20, 30), Wl 

G (40) and 

W G 

l (4). The probabilities for the W S 

l (20, 30) window function are greater than 0.25 

and the minimum probability at ∼ 0.05 occurs at κ 4 for W G (40). The reason for 

systematically lower SI probabilities for W l S (20, 30) as compared to W l G (40) is simply 

due to lower cosmic variance of the former. The contribution to the cosmic variance of 

BiPS is dominated by the low spherical harmonic multipoles. Filters that suppress the 

a lm at low multipoles have a lower cosmic variance. 

It is important to note that the above probability is a conditional probability of 

measured ˜κ l being SI given the theoretical spectrum C T l 

(used to estimate the bias). 

A final probability emerges as the Bayesian chain product with the probability of the 

theoretical C T l 

used given data. Hence, small difference in these conditional probabilities 

for the two maps are perhaps not necessarily significant. 

Since the BiPS is close to

7.0 115 

zero, the computation of a probability marginalized over the Cl 

T may be possible using 

Gaussian (or, improved) approximation to the PDF of κ l . The important role played 

by the choice of the theoretical model for the BiPS measurement is shown for a W l that 

retains power in the lowest multipoles, l = 2 and l = 3. Assuming Cl T , there are hints 

of non-SI detections in the low l’s (left panel of Fig. 7.7). We also compute the BiPS 

using a Cl 

T for a model that accounts for suppressed quadrupole and octopole in the 

WMAP data Shafieloo & Souradeep (2004). The mild detections of a non zero BiPS 

vanish for this case (right panel of Fig. 7.7). 

In summary, We find no strong evidence for SI violation in the WMAP CMB 

anisotropy maps considered here. We have verified that our null results are consistent 

with measurements on simulated SI maps. The BiPS measurement reported here 

is a Bayesian estimate of the conditional probability of SI (for each κ l of the BiPS) 

given an underlying theoretical spectrum Cl T . We point out that the excess power in 

the Cl 

T with respect to the measured C l from WMAP at the lowest multipoles tends to 

indicate mild deviations from SI. BiPS measurements are shown to be consistent with 

SI assuming an alternate model Cl 

T that is consistent with suppressed power on low 

multipoles.

7.0 116 

Figure 7.1 Top: C l of the two WMAP CMB anisotropy maps. The red, magenta and 

green curves correspond to map A, B and C, respectively. 

The black line is a ‘best 

fit’ WMAP theoretical C l used for simulating SI maps. Blue dots are the average 

C l recovered from 1000 realizations. Bottom: These plots show the window functions 

used. The dashed curves with increasing l coverage are ‘low-pass’ filter, Wl 

G (l s ), with 

l s = 4, 18, 40, respectively. The solid lines are ‘band-pass’ filter W S 

l (l t , l s ) with (l s , l t ) = 

(13, 2), (30, 5), (30, 20), (45, 20), respectively.

7.0 117 

100 

0 

-100 

-200 

CSSK 

100 

0 

-100 

-200 

ILC 

100 

0 

-100 

-200 

TOH 

5 10 15 20 

Figure 7.2 Measured values of κ l for maps A, B and C filtered with a Gaussian window 

with l s = 40. Each map appears to show the violation of statistical isotropy for some 

l at the level of ∼ 1σ. However, this is because the PDF for these models are skewed 

with the maximum probability shifted away from the mean to negative values (see Fig. 

7.5).

7.0 118 

20 

10 

0 

-10 

Gal. cut 

20 

10 

0 

-10 

Wiener 

20 

10 

0 

-10 

CSSK 

20 

10 

0 

-10 

ILC 

20 

10 

0 

-10 

TOH 

5 10 15 20 

Figure 7.3 Measured BiPS for maps A, B and C filtered with a window with l s = 30, 

l t = 20. This is to check the statistical isotropy of the WMAP in the modest 20 < l < 40 

in the multipole space where certain anomalies have been reported. ILC with a 10- 

degree-cut (top) has the same BiPS as map C (l s = 30, l t = 20) which explains that the 

raising tail of CSSK map is because of the mask.

7.0 119 

15 

10 

5 

Wiener 

0 

-5 

15 

10 

5 

CSSK 

0 

-5 

15 

10 

5 

ILC 

0 

-5 

15 

10 

5 

TOH 

0 

-5 

5 10 15 20 

Figure 7.4 BiPS measured from 4 different WMAP maps filtered by W S 

l (20, 45). From 

bottom to up: TOH map, ILC map, CSSK foreground free map with a galactic cut, 

Wiener filtered map. We see that Wiener filtering has the similar effect as galactic cut. 

(But not exactly the same)

7.0 120 

0.8 

0.6 

0.4 

0.2 

0 

0.8 

0.6 

0.4 

0.2 

0 

0.8 

0.6 

0.4 

0.2 

0 

0.8 

0.6 

0.4 

0.2 

0 

0.8 

0.6 

0.4 

0.2 

0 

l=5 

l=4 

l=3 

l=2 

l=1 

-2 -1 0 1 2 

l=5 

l=4 

l=3 

l=2 

l=1 

-2 -1 0 1 2 

Figure 7.5 Probability distribution function for κ 1 to κ 5 constructed from 1000 realizations. 

The left panel shows the PDF for the maps filtered with W S 

l (20, 30) (left panel) 

and Wl 

G (l s = 40) (right panel). The latter is more skewed, which explains the ∼ 1σ 

shift in κ l values shown in Fig. (7.2). The green, magenta and red (circular, pentagonal 

and rectangular) points represent ILC, CSSK and TOH maps, respectively. The smooth 

solid curves are Gaussian approximations.

7.0 121 

0.7 

0.6 

ILC, W G 

TOH, W G 

ILC, W S 

TOH, W S 

0.5 

0.4 

P(κ l 

) 

0.3 

0.2 

0.1 

0 

0 2 4 6 8 10 12 14 16 18 20 

l 

Figure 7.6 The probability of the two WMAP based CMB maps being SI when filtered 

by W S 

l (20, 30) and a Gaussian filter W G 

l (40). 

200 

200 

100 

100 

0 

0 

-100 

ILC 

-100 

ILC 

-200 

-200 

2 4 6 8 10 

2 4 6 8 10 

200 

200 

100 

100 

0 

0 

-100 

Tegmark 

-100 

Tegmark 

-200 

-200 

2 4 6 8 10 

2 4 6 8 10 

Figure 7.7 Comparing the measured values of κ l for maps A and B filtered to retain 

power only on the lowest multipoles, l = 2 and l = 3 assuming the WMAP theoretical 

spectrum WMAPbf (left) and a model spectrum that matches the suppressed power 

at the lowest multipoles Shafieloo & Souradeep (2004). The non zero κ l ‘detections’ 

assuming the WMAP theoretical spectrum become consistent with zero for a Cl 

T that 

has power suppressed at low multipoles.

7.0 122 

0.7 

ILC 

TOH 

0.7 

ILC 

TOH 

0.6 

0.6 

0.5 

0.5 

0.4 

0.4 

P(κ l 

) 

P(κ l 

) 

0.3 

0.3 

0.2 

0.2 

0.1 

0.1 

0 

0 2 4 6 8 10 12 14 16 18 20 

l 

0 

0 2 4 6 8 10 12 14 16 18 20 

l 

Figure 7.8 probability of the two WMAP based CMB maps being SI when filtered 

by Wl 

G (4) to only keep the lowest multipoles when assuming the WMAP theoretical 

spectrum, C T l 

lowest multipoles (right). 

(left) and a model spectrum that matches the suppressed power at the

7.0 123 

100 

0 

Spergel 

50 

Spergel 

-100 

0 

-200 

0 5 10 15 20 

0 5 10 15 20 

100 

0 

ILC 

50 

ILC 

-100 

0 

-200 

0 5 10 15 20 

0 5 10 15 20 

100 

0 

Tegmark 

50 

Tegmark 

-100 

0 

-200 

0 5 10 15 20 

0 5 10 15 20 

20 

Spergel 

10 

Spergel 

0 

0 

0 5 10 15 20 

0 5 10 15 20 

20 

ILC 

10 

ILC 

0 

0 

0 5 10 15 20 

0 5 10 15 20 

20 

Tegmark 

10 

Tegmark 

0 

0 

0 5 10 15 20 

0 5 10 15 20 

Figure 7.9 Measured BiPS for maps A, B and C filtered by bandpass filters W S 

l (2, 13) 

(top left panel), W S 

l (5, 30) (top right), W S 

l (20, 30) (bottom left) and W S 

l (20, 45) (bottom 

right).

7.0 124 

100 

0 

-100 

-200 

Spergel 

0 5 10 15 20 

100 

0 

-100 

-200 

Spergel 

0 5 10 15 20 

100 

0 

-100 

-200 

0 5 10 15 20 

ILC 

100 

0 

-100 

-200 

ILC 

0 5 10 15 20 

100 

0 

-100 

-200 

Tegmark 

0 5 10 15 20 

100 

0 

-100 

-200 

Tegmark 

0 5 10 15 20 

0 

Spergel 

-200 

0 5 10 15 20 

0 

ILC 

-200 

0 5 10 15 20 

0 

Tegmark 

-200 

0 5 10 15 20 

Figure 7.10 Measured BiPS for maps A, B and C filtered by Gaussian filters W G 

l (40) 

(top left panel), W G 

l 

(18) (top right) and W G (4) (bottom). 

l

7.0 125 

1 

l=5+5 

1 

l=15+5 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=5+4 

1 

l=15+4 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=5+3 

1 

l=15+3 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=5+2 

1 

l=15+2 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=5+1 

1 

l=15+1 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=5 

1 

l=10+5 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=4 

1 

l=10+4 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=3 

1 

l=10+3 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=2 

1 

l=10+2 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=1 

1 

l=10+1 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

Figure 7.11 PDF of κ l for a Gaussian filter Wl 

G (40) for 1000 realizations of a SI CMB 

map and the WMAP results. Red point corresponds to the TOH map, green point to 

ILC and magenta to the CSSK map.

7.0 126 

1 

l=5+5 

1 

l=15+5 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=5+4 

1 

l=15+4 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=5+3 

1 

l=15+3 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=5+2 

1 

l=15+2 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=5+1 

1 

l=15+1 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=5 

1 

l=10+5 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=4 

1 

l=10+4 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=3 

1 

l=10+3 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=2 

1 

l=10+2 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

1 

l=1 

1 

l=10+1 

0 

-4 -2 0 2 4 

0 

-4 -2 0 2 4 

Figure 7.12 PDF of κ l for a band pass filter W S 

l (20, 30) for 1000 realizations of a SI CMB 

map and the WMAP results. Red point corresponds to the TOH map, green point to 

ILC and magenta to the CSSK map.

Chapter 8 

Summary and future directions 

We have studied statistics of the CMB anisotropy as a Gaussian random field on a 

sphere. We studied statistical isotropy (SI) of CMB temperature anisotropy and used 

that to draw interesting conclusions on the physics, topology and large scale structure 

of the universe as well as properties of the systematics in CMB experiments. Statistical 

isotropy is usually assumed in almost all CMB studies. But there are certain mechanisms 

that violate this condition. This thesis is dedicated to the study of this fundamental 

assumption in CMB. We have formulated the problem mathematically, constructed a 

quantitative method to describe SI violation, studied possible ways of violation of SI 

condition and finally examined this fundamental assumption using the most recent CMB 

anisotropy data. 

The SI of the CMB anisotropy has been under scrutiny after the release of the first 

year of WMAP data. We use the BiPS which is sensitive to structures and patterns 

in the underlying two-point correlation function as a statistical tool of searching for 

departures from SI. We carry out a BiPS analysis of WMAP full sky maps. We find no 

strong evidence for SI violation in the WMAP CMB anisotropy maps considered here. 

We have verified that our null results are consistent with measurements on simulated SI 

maps. The BiPS measurement reported here is a Bayesian estimate of the conditional 

probability of SI (for each κ l of the BiPS) given an underlying theoretical spectrum 

Cl T . We point out that the excess power in the C l 

T with respect to the measured C l 

from WMAP at the lowest multipoles tends to indicate mild deviations from SI. BiPS 

measurements are shown to be consistent with SI assuming an alternate model Cl 

T that 

127

8.0 128 

is consistent with suppressed power on low multipoles. The full sky maps and the 

restriction to low l < 60 (where instrumental noise is sub-dominant) permits the use of 

our analytical bias subtraction and error estimates. 

There are strong theoretical motivations for hunting for SI violation in the CMB 

anisotropy. The possibility of compact spaces in cosmic topology is a theoretically well 

motivated possibility that has also been observationally targeted Ellis (1971); Lachieze- 

Rey & Luminet (1995); Levin (2002); Linde (2004). The breakdown of statistical 

homogeneity and isotropy of cosmic perturbations is a generic feature of ultra large scale 

structure of the cosmos, in particular, of non trivial cosmic topology Bond, Pogosyan & 

Souradeep (1998, 2000). The underlying correlation patterns in the CMB anisotropy in 

a multiply connected universe is related to the symmetry of the Dirichlet domain. The 

BiPS expected in flat, toroidal models of the universe has been computed and shown 

to be related to the principle directions in the Dirichlet domain Hajian & Souradeep 

(2003a). As a tool for constraining cosmic topology, the BiPS has the advantage of being 

independent of the overall orientation of the Dirichlet domain with respect to the sky. 

Hence, the null result of BiPS can have important implication for cosmic topology. This 

approach complements direct search for signature of cosmic topology Cornish, Spergel 

& Starkman (1998); de Oliveira-Costa et al. (1996) and our results are consistent 

with the absence of the matched circles and the null S-map test of the WMAP CMB 

maps Cornish et al. (2003); de Oliveira-Costa et al. (2003). Full Bayesian likelihood 

comparison to the data of specific cosmic topology models is another approach that has 

applied to COBE-DMR data Bond, Pogosyan & Souradeep (1998, 2000). We also study 

Poincare Dodecahedral spaces suggested as a possible explanation of low quadrupole of 

the WMAP data by Luminet et al. (2003). We show that the Poincaré Dodecahedron 

with Ω k = 0.013 breaks down the statistical isotropy at an observable level which was 

not observed in the WMAP data. Our results are consistent with the complete Bayesian 

likelihood analysis of Poincaré Dodecahedral spaces [Hajian et al. (2006)]. 

Discovery of a hidden pattern in the WMAP data was recently reported by Jaffe et 

al. (2005) which according to the authors was due to a rotating universe template (a 

Bianchi pattern). The model studied by Jaffe et al. (2005) was examined in this thesis. 

We show that hiding a Bianchi template in a ΛCDM CMB anisotropy temperature map,

8.0 129 

will lead to measurable non-zero bipolar power spectra. The null BiPS of WMAP can 

be used to put tight constraints on the existence of such hidden patterns in the WMAP 

data. 

The null BiPS results also has implications for the observation and data analysis 

techniques used to create the CMB anisotropy maps. Observational artifacts such as 

non-circular beam, inhomogeneous noise correlation, residual striping patterns, etc. are 

potential sources of SI breakdown. Our null BiPS results confirm that these artifacts 

do not significantly contribute to the maps studied here. Foreground residuals can also 

be sources of SI breakdown. The extent to which BiPS probes foreground residuals is 

yet to be fully studied and explored. We do not see any significant effect of the residual 

foregrounds in ILC and the TOH maps as it was mentioned by Eriksen et al. (2004c). 

This can not be necessarily called a discrepancy between the two results unless we have 

a model that can predict what should have been seen in the BiPS. However the question 

which remains is if the signal is strong enough and whether the effect smeared out in 

bipolar multipole space within our angular l-space window. On the other hand, we have 

verified that BiPS does show a strong signal for the residuals from foregrounds in the 

Wiener filtered map. 

In the very near future we are going to have CMB polarization sky maps. The wealth 

of information in the CMB polarization field will enable us to test and characterize the 

initial perturbations and inflationary mechanisms with great precision. The cosmological 

polarized microwave radiation in a simply connected universe is expected to be 

statistically isotropic. This is a very important feature which allows us to fully describe 

the field by its power spectrum. Violation of SI in CMB polarization maps is an interesting 

issue which is going to be very important soon. Importance of having statistical 

tests of departures from SI for CMB polarization maps lies not only in the theoretical 

motivations discussed above but also in testing the cleaned CMB polarization maps for 

residuals from polarized foreground emission. Unlike the foregrounds in temperature 

anisotropies, polarized foreground emissions which are mostly from our galaxy and also 

from extra-galactic sources on large scales (where we may see the primordial B-mode 

due to inflationary gravitational waves) is likely to dominate the signal at all frequencies. 

A strong tool to distinguish between the primordial polarized radiation and polarized

8.0 130 

foreground emissions is the test of SI. BiPS, by construction is applicable to polarized 

maps. We can check for departures from SI in E and B polarization modes as well as 

the cross terms such as TE by computing BiPS for them.

AppendixA 

Detailed Steps of Calculations 

Gaussianity of CMB anisotropy 

The temperature anisotropy at a given space-time point (x, τ) can be written as a 

superposition of plane waves in k-space 

∫ 

∆T (x, ˆn, τ) = d 3 k e ik·x ∆T (k, ˆn, τ). 

(A.1) 

As discussed in Ma & Bertschinger (1995), the dependence on ˆn arises only through 

k · ˆn. Therefore ∆T (k, ˆn, τ) may be represented as 

∞∑ 

∆T (k, ˆn, τ) = (−i) l (2l + 1)∆T l (k, τ)P l (k · ˆn). 

(A.2) 

l=0 

The anisotropy coefficients ∆T l (k, τ) are random variables whose amplitudes and phases 

depend on the initial perturbations and can be written as 

∆T l (k, τ) = φ 0 (k)∆T l (k, τ), 

(A.3) 

where φ 0 (k) is the initial potential perturbation and ∆T l (k, τ) is the photon transfer 

function, i.e. solution of Boltzmann equation with φ 0 (k) = 1. In general, ∆T l (k, τ) is 

given by the integral solution 

∆T l (β) = 

∫ τ0 

0 

dτΦ l β(τ 0 − τ)S(β, τ), 

(A.4) 

in which S(β, τ) is the source function, Φ l β is the ultra-spherical Bessel function and 

β = k − K, where K = H0(Ω 2 0 − 1) is the curvature Zaldarriaga et al. (1998). In the 

131

A.0 132 

flat case, the above expression has a very simple form 

∆T (k) = 

∫ τ0 

0 

dτ e i k·ˆn 

k (τ−τ 0) S(k, τ), (A.5) 

and hence, the temperature anisotropies are given by 

∫ 

∫ τ0 

∆T (x, ˆn) = d 3 k e ik·x φ 0 (k)[ dτ e i k·ˆn 

k (τ−τ0) S(k, τ)]. (A.6) 

Equation (A.6) clearly shows that Gaussian perturbations in primordial gravitational 

potential result a Gaussian CMB temperature anisotropy field. 

0 

SI condition in harmonic space 

It was shown in eqn. 

(2.17) that statistical isotropy as invariance of two point 

correlation under rotation, translates to a diagonal covariance matrix in harmonic space. 

To prove this we substitute the spherical harmonic coefficients, a lm , from eqn. (2.8) in 

the covariance matrix to express it in terms of the two point correlation 

∫ ∫ 

〈a lm a ∗ l ′ m ′〉 = dΩˆn dΩˆn ′Ylm ∗ (ˆn)Y l ′ m ′(ˆn′ )〈∆T (ˆn)∆T (ˆn ′ )〉 

∫ ∫ 

= dΩˆn dΩˆn ′Ylm(ˆn)Y ∗ 

l ′ m ′(ˆn′ )C(ˆn, ˆn ′ ). 

(A.7) 

Under the condition of SI, eqn. (2.13), and after making use of the spherical harminc 

expansion of Legendre polynomials, eqn. (G.6), in eqn. (2.14), the above expression 

will become 

∫ ∫ 

〈a lm a ∗ l ′ m ′〉 = dΩˆn dΩˆn ′Ylm(ˆn)Y ∗ 

lm (ˆn ′ ) ∑ l 1 

= ∑ ∑ 

C l1 

l 1 

m 1 

δ ll1 δ l ′ l 1 

δ mm1 δ m ′ m 1 

= C l δ ll ′δ mm ′, 

which is the SI condition in harmonic space, eqn. 

C l1 

∑ 

m 1 

Y l1 m 1 

(ˆn)Y l1 m 1 

(ˆn ′ ) (A.8) 

(2.17), and in the last line, the 

orthonormality relation of spherical harmonics, eqn. (G.2) has been used.

A.0 133 

Cosmic Variance Surplus 

The angular power spectrum, C l , is an “isotropized” measure. The isolation of SI 

implies that the C l do not contain the entire information contents of the anisotropy 

field. This leads to the surplus of cosmic variance of C l in cases where SI is violated. 

Here is a proof of this. The estimator of angular power spectrum is given by 

˜C l = 1 

2l + 1 

l∑ 

m=−l 

a lm a ∗ lm, 

(A.9) 

for a Gaussian isotropic theory, this has the cosmic variance 

σ 2 SI( ˜C l ) = 

2C2 l 

2l + 1 . 

For a general theory, the cosmic variance of the above estimator is given by 

σ 2 ( ˜C l ) = 

2 

(2l + 1) [∑ ∑ 

〈a 2 lm a ∗ lm ′〉〈a lma ∗ lm ′〉∗ δ mm ′ 

m m ′ 

+ ∑ ∑ 

〈a lm a ∗ lm ′〉〈a lma ∗ lm ′〉∗ ] 

m≠m ′ m ′ 

= 2C2 l 

2l + 1 + 2 

(2l + 1) 2 ∑ 

m≠m ′ ∑ 

= σ 2 SI ( ˜C l ) + σ 2 SA . 

m ′ 〈a lm a ∗ lm ′〉〈a lma ∗ lm ′〉∗ 

(A.10) 

(A.11) 

The second term (σSA 2 ) is the sum of non-negative numbers and hence is non-negative. 

Therefore the cosmic variance is always greater than or equal to that of SI models 

σ 2 ( ˜C l ) ≥ σ 2 SI ( ˜C l ). 

(A.12) 

The fact that cosmic variance of angular power spectrum is minimized for SI theories, 

was shown in Ferreira & Magueijo (1997). Also Bond, Pogosyan & Souradeep 

(2000) in a detailed calculations for statistically anisotropic theories showed that the incompleteness 

of the information contained in the C l is reflected in the enhanced cosmic 

variance. This can be seen by splitting the two point correlation into an isotropic part 

determined by C l and an anisotropic term containing the remainder 

C(ˆn 1 , ˆn 2 ) = C SI (ˆn 1 , ˆn 2 ) + C SA (ˆn 1 , ˆn 2 ). 

(A.13)

A.0 134 

It is then straightfoward to calculate the cosmic variace due to each part and the final 

expression as it is given in Bond, Pogosyan & Souradeep (2000) is 

σ 2 ( ˜C l ) = 

2C2 l 

2l + 1 + l2 (l + 1) 2 ∫ 

32π 4 

dΩˆn1 

∫ 

∫ 

dΩˆn2 [ 

dΩˆn3 C SA (ˆn 1 , ˆn 2 )P l (ˆn 1 · ˆn 2 )] 2 . 

(A.14) 

In the above expression, the second term is a non-negative term and therefore we can 

again see that in isotropic theories, the cosmic variance of the angular power spectrum 

has its smallest value. 

A lM 

l 1 l 2 

are linear combinations of 〈a lm a ∗ l ′ m ′ 〉 

The BiPoSH coefficients, A lM 

l 1 l 2 

, are linear combinations a lm . 

To see that, in the 

definition of two point correlation we can substitute for ∆T/T from eqn. (2.7), which 

will lead to the expansion of C(ˆn 1 , ˆn 2 ) in terms of spherical harmonics 

C(ˆn 1 , ˆn 2 ) = 〈∆T (ˆn 1 )∆T (ˆn 2 )〉 (A.15) 

= ∑ ∑ 

〈a lm a ∗ l ′ m ′〉Y lm(ˆn 1 )Yl ∗ ′ m ′(ˆn 2). 

lm l ′ m ′ 

This can be substituted in the definition of A lM 

l 1 l 2 

and eqn. (2.20) can be written as 

∫ ∫ 

A lM 

l 1 l 2 

= dΩ 1 dΩ 2 C(ˆn 1 , ˆn 2 ){Y l1 (ˆn 1 ) × Y l2 (ˆn 2 )} ∗ lM 

(A.16) 

∫ ∫ 

∑ ∑ 

= dΩ 1 dΩ 2 〈a lm a ∗ l ′ m ′〉Y lm(ˆn 1 )Yl ∗ ′ m ′(ˆn 2) ∑ Cl lM 

1 m 1 l 2 m 2 

Yl ∗ 

1 m 1 

(ˆn 1 )Yl ∗ 

2 m 2 

(ˆn 2 ) 

lm l ′ m ′ m 1 m 

∫ ∫ 

2 

∑ ∑ 

= dΩ 1 dΩ 2 〈a lm a ∗ l ′ m ′〉Y lm(ˆn 1 )Yl ∗ ′ m ′(ˆn 2) 

lm l ′ m ′ 

× 

∑ 

m 1 m 2 

C lM 

l 1 m 1 l 2 m 2 

Y ∗ 

l 1 m 1 

(ˆn 1 )(−1) m 2 

Y l2 −m 2 

(ˆn 2 ) 

= ∑ m 1 m 2 

(−1) m 2 

〈a l1 m 1 

a ∗ l 2 −m 2 

〉C lM 

l 1 m 1 l 2 m 2 

By changing the dummy variable m 2 → −m 2 we will get 

A lM 

l 1 l 2 

= ∑ m 1 m 2 

〈a l1 m 1 

a ∗ l 2 m 2 

〉(−1) m 2 

C lM 

l 1 m 1 l 2 −m 2 

. 

(A.17)

A.0 135 

SI condition in bipolar harmonic space 

If we substitute the covariance matrix of a SI theory, eqn. (2.17), into the expression 

for A lM 

ll , eqn. (A.17), we would get 

′ 

A lM 

ll ′ = ∑ mm ′ C l δ ll ′δ m−m ′(−1) m′ C lM 

lml ′ m ′ 

∑ 

= C l δ ll ′δ M0 (−1) m C lM 

m 

lml−m. 

(A.18) 

Using summation rules of Clebsch-Gordan equations, eqn. (G.11), the above expression 

will become 

A lM 

ll ′ = (−1)l C l (2l + 1) 1/2 δ ll ′ δ l0 δ M0 . (A.19) 

This is the SI condition in bipolar harmonic space and it arises as we carry out the sum 

over m. 

Simplifying the real-space expression for BiPS 

The BiPS in real-space is given by the following expression. 

κ l = ( 2l + 1 ∫ ∫ ∫ 

∫ 

8π 2 )2 dΩ 1 dΩ 2 dR 1 χ l (R 1 ) dR 2 χ l (R 2 )C(R 1ˆn 1 , R 1ˆn 2 )C(R 2ˆn 1 , R 2ˆn 2 ) 

(A.20) 

Now if we change the variables in the following way 

R 1ˆn 1 = ˆn 1 

(A.21) 

R 1ˆn 2 = ˆn 2 

We will have 

κ l = ( 2l + 1 ∫ 

) 2 

8π 2 

dΩ 1 

∫ 

dΩ 2 

∫ 

∫ 

dR 1 χ l (R 1 ) 

dR 2 χ l (R 2 )C(ˆn 1 , ˆn 2 )C(R 2 R −1 

1 ˆn 1, R 2 R −1 

1 ˆn 2), 

(A.22) 

and with 

R = R 2 R −1 

1 (A.23) 

we will get 

κ l = ( 2l + 1 ∫ 

) 2 

8π 2 

dΩ 1 

∫ 

dΩ 2 

∫ 

∫ 

dR 1 χ l (R 1 ) 

dRχ l (RR 1 )C(ˆn 1 , ˆn 2 )C(Rˆn 1 , Rˆn 2 ). 

(A.24)

A.0 136 

Using eqn. (G.9) we can write 

∫ 

dR 1 χ l (R 1 )χ l (RR 1 ) = ∑ m 1 

∑ 

mm ′ ∫ 

= ∑ m 1 

∑ 

= 

= 

8π 2 

2l + 1 

mm ′ D l mm 

dR 1 D ∗l 

m 1 m 1 

(R 1 )D l mm ′(R)Dl m ′ m(R 1 ) (A.25) 

8π2 

′(R) 

2l + 1 δ m 1 m ′δ m 1 m 

∑ 

Dm l 1 m 1 

(R) 

m 1 

8π 2 

2l + 1 χ l(R). 

So, the κ l can be written in this form: 

κ l = 2l + 1 ∫ ∫ 

∫ 

dΩ 

8π 2 1 dΩ 2 C(ˆn 1 , ˆn 2 ) 

dRχ l (R)C(Rˆn 1 , Rˆn 2 ). 

(A.26) 

Computing the Cosmic Variance of BiPS 

In section 2.6 we presented the analytical expression for the cosmic variance of BiPS. 

Here we show the details of our calculation. The cosmic variance, 〈˜κ 2 l 〉 − 〈˜κ l〉 2 , has 96 

terms similar to the one shown in eqn. (2.44) but with different combinations of R’s and 

ˆn’s. Among them, 40 terms contain at least one term like C(Rˆn 1 , Rˆn 2 ). These terms 

only contribute to κ 0 because when SI holds, C(Rˆn 1 , Rˆn 2 ) = C(ˆn 1 , ˆn 2 ) and hence the 

integral in eqn. (A.26) will be proportional to δ l0 , which is not in the interest of us. 

So, we can ignore them and we will be left with 56 terms (more details on calculation 

of these terms are given in Appendix B). These terms, depending on the number of 

connected correlations in them, can be classified into 4 major groups: 

1. One-loop terms, such as 

C(ˆn 1 , R ′ˆn 3 )C(ˆn 3 , Rˆn 2 )C(ˆn 2 , R ′ˆn 4 )C(ˆn 4 , Rˆn 1 ). 

(A.27) 

These terms get reduced to expressions with a C 4 l 

coefficients (summation symbols are omitted for brevity), 

factor and 4 Clebsch-Gordan 

C 4 l 1 

C lM 

l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 7 l 1 m 5 

C lM ′ 

l 1 −m 1 l 1 −m 3 

. 

(A.28) 

There are 24 terms like this but with different combinations of indices.

A.0 137 

2. Two-loop terms – type I, such as 

C(ˆn 1 , ˆn 3 )C(R ′ˆn 3 , Rˆn 2 )C(ˆn 2 , Rˆn 1 )C(ˆn 4 , R ′ ˆn 4 ). 

(A.29) 

These terms get reduced to expressions with a C 3 l 1 

C l2 factor and 4 Clebsch-Gordan 

coefficients 

C 3 l 1 

C l7 C lM 

l 1 −m 1 l 1 −m 3 

C lM 

l 1 −m 3 l 1 m 5 

C lM ′ 

l 1 m 1 l 7 −m 7 

C lM ′ 

l 1 −m 5 l 7 −m 7 

. 

There are 16 terms like this but with different combinations of indices. 

3. Two-loop terms – type II, such as 

(A.30) 

C(ˆn 1 , R ′ˆn 4 )C(ˆn 4 , Rˆn 1 )C(ˆn 2 , R ′ˆn 3 )C(Rˆn 2 , ˆn 3 ). 

(A.31) 


C 2 l 2 

factor and 4 Clebsch-Gordan 

coefficients 

C 2 l 1 

C 2 l 3 

C lM 

l 1 −m 1 l 3 −m 3 

C lM 

l 1 m 5 l 3 m 7 

C lM ′ 

l 1 m 5 l 3 m 7 

C lM ′ 

l 1 −m 1 l 3 −m 3 

. 


4. Three-loop terms, such as 

(A.32) 

C(ˆn 1 , ˆn 3 )C(R ′ˆn 3 , Rˆn 1 )C(ˆn 2 , Rˆn 2 )C(ˆn 4 , R ′ˆn 4 ). 

(A.33) 


C l2 C l3 

Gordan coefficients 

factor and 4 Clebsch- 

C 2 l 1 

C l3 C l7 C lM 

l 3 −m 3 l 1 −m 1 

C lM 

l 3 −m 3 l 1 m 5 

C lM ′ 

l 1 m 1 l 7 −m 7 

C lM ′ 

l 1 −m 5 l 7 −m 7 

. 

(A.34) 


Using symmetry properties of Clebsch-Gordan coefficients, eqn. (G.10), and their 

summation rules, eqn. (G.11), it is possible to simplify these 56 terms immensely. After 

tedious algebra we would obtain the final expression for the kernel of cosmic variance of 

the bipolar power spectrum: 

4 Cl 4 (2l + 1)2 

1 

[2 

2l 1 + 1 

+ (−1)l (2l + 1) + (1 + 2(−1) l )F l 

ll ] {l 1l 1 l} + 

4 C 2 l 1 

C 2 l 2 

[(2l + 1) + F l 

l 1 l 2 

] {l 1 l 2 l} + 

8 C 2 l 1 

C l2 C l3 

(2l + 1) 2 

2l 1 + 1 {l 1l 2 l} {l 1 l 3 l} + 

16 (−1) l Cl 3 (2l + 1) 2 

1 

C l3 

2l 1 + 1 {l 1l 1 l} {l 1 l 3 l} 

(A.35)

A.0 138 

where the 3j-symbol {l 1 l 2 l} is given by 

⎧ 

⎨ 1 if l 1 + l 2 + l is integer and |l 1 − l 2 | ≤ l ≤ (l 1 + l 2 ), 

{l 1 l 2 l} = 

⎩ 0 otherwise, 

and F l 

l 1 l 3 

is defined as 

F l 

l 1 l 3 

= 

l 1 

∑ 

m 1 m 2 =−l 1 

l 3 

∑ 

l∑ 

m 3 m 4 =−l 3 M,M ′ =−l 

C lM 

l 1 −m 1 l 3 −m 3 

C lM 

l 1 m 2 l 3 m 4 

C lM ′ 

l 3 m 4 l 1 m 1 

C lM ′ 

l 3 −m 3 l 1 −m 2 

. 

(A.36) 

This should be summed over all indices except l to result the analytical expression for 

the cosmic variance given in eqn. (2.45).

AppendixB 

Cosmic Variance of BiPS, Handling 

Gaussian 8 Point Functions 

As it is shown in eqn. (2.42), to compute the cosmic variance (CV) of BiPS, we 

must deal with an eight point correlation. Assuming Gaussiantiy of the field we can 

rewrite the eight point correlation in terms of two point correlations. This will give us 

(8 − 1)!! = 7 × 5 × 3 = 105 terms. These 105 terms are given below: 

x1x2 x3x4 x5x6 x7x8 + x1x2 x3x4 x5x7 x6x8 + x1x2 x3x4 x5x8 x6x7 + 




















139

B.0 140 















x1x8 x2x7 x3x4 x5x6 + x1x8 x2x7 x3x5 x4x6 + x1x8 x2x7 x3x6 x4x5 

Where by x1x2 I mean the two point correlation between 1 and 2, i.e. 〈∆T (R 1ˆq 1 )∆T (R 1ˆq 2 )〉. 

On the other hand < κ l > has a 4 point correlation in it 

< ∆T (R 1ˆq 1 )∆T (R 1ˆq 2 )∆T (R 2ˆq 1 )∆T (R 2ˆq 2 ) > (B.1) 

which can be written as, say, x1x2 x3x4 + x1x3 x2x4 + x1x4 x2x3; so < κ l > 2 will be 

(x1x2 x3x4 + x1x3 x2x4 + x1x4 x2x3) (x5x6 x7x8 + x5x7 x6x8 + x5x8 x6x7) = 




If we subtract these two, we will get: 

< κ 2 l > − < κ l > 2 = 









x1x8 x2x7 x3x4 x5x6 + x1x7 x2x8 x3x4 x5x6 + x1x8 x2x4 x3x7 x5x6 +

B.0 141 
























Which are 96 terms. Now if we notice that every term which has a x(2k + 1)x2k will 

only contribute to κ 0 then we can get rid of many more terms. And these will be the 

terms we are left with: 
















x1x3 x2x8 x4x5 x6x7 + x1x5 x2x3 x4x8 x6x7 + x1x3 x2x5 x4x8 x6x7 +

B.0 142 



x1x5 x2x3 x4x7 x6x8 + x1x3 x2x5 x4x7 x6x8 

Which are 56 terms. 

Now if we remember that 

x1 = R1q1 (B.2) 

x2 = R1q2 

x3 = R2q1 

x4 = R2q2 

x5 = R3q3 

x6 = R3q4 

x7 = R4q3 

x8 = R4q4 

The above terms can be written in the following way. 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+

B.0 143 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

 

It is easier to use the simpler expression for BiPS which we derived in A. Using this 

simplified form of BiPS, we can rewrite the kernel of CV in the following form: 

+ 

+ 

+ 

+ 

+

B.0 144 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+

B.0 145 

+ 

+ 

+ 

+ 

+ 

+ 

 

All of these temrs are somehow similar. And all of them have the same integrals. So if we 

do some of them we can find a shortcut to the answer of integration of each term. This 

was explained in A and can be seen that having known that, the rest of our calculation 

is more like a book keeping task. We should very carefully keep track of indices. I wrote 

a FORTRAN90 program (!) to do that. It reads the above expression and outputs the 

expression below in LATEX format, 

C l1 C l3 C l1 C l3 C lM 

l 1 −m 1 l 3 −m 3 

C lM 

l 1 m 5 l 3 m 7 

C lM ′ 

l 1 m 5 l 3 m 7 

C lM ′ 

l 1 −m 1 l 3 −m 3 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 7 l 1 m 5 

C lM ′ 

l 1 −m 1 l 1 −m 3 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 3 l 1 m 7 

C lM ′ 

l 1 −m 1 l 1 −m 5 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 1 l 1 m 7 

C lM ′ 

l 1 −m 3 l 1 −m 5 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 7 l 1 m 3 

C lM ′ 

l 1 −m 1 l 1 −m 5 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 m 5 l 3 m 7 

C lM ′ 

l 3 m 7 l 1 m 1 

C lM ′ 

l 3 −m 3 l 1 −m 5 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 7 l 1 m 5 

C lM ′ 

l 1 −m 1 l 1 −m 3 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 m 5 l 3 m 7 

C lM ′ 

l 1 m 5 l 3 m 7 

C lM ′ 

l 1 −m 1 l 3 −m 3 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 7 l 1 m 3 

C lM ′ 

l 1 −m 1 l 1 −m 5 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 m 5 l 3 m 7 

C lM ′ 

l 3 m 7 l 1 m 1 

C lM ′ 

l 3 −m 3 l 1 −m 5 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 3 l 1 m 7 

C lM ′ 

l 1 −m 1 l 1 −m 5 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 1 l 1 m 7 

C lM ′ 

l 1 −m 3 l 1 −m 5 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 3 l 1 m 5 

C lM ′ 

l 1 −m 1 l 1 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 1 l 1 m 5 

C lM ′ 

l 1 −m 3 l 1 −m 7 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 m 5 l 3 m 7 

C lM ′ 

l 1 m 5 l 3 m 3 

C lM ′ 

l 1 −m 1 l 3 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 5 l 1 m 1 

C lM ′ 

l 1 −m 3 l 1 −m 7 

+

B.0 146 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 m 5 l 3 m 7 

C lM ′ 

l 1 m 1 l 3 m 3 

C lM ′ 

l 1 −m 5 l 3 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 3 l 1 m 1 

C lM ′ 

l 1 −m 5 l 1 −m 7 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 m 5 l 3 m 7 

C lM ′ 

l 1 m 5 l 3 m 3 

C lM ′ 

l 1 −m 1 l 3 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 5 l 1 m 1 

C lM ′ 

l 1 −m 3 l 1 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 3 l 1 m 5 

C lM ′ 

l 1 −m 1 l 1 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 1 l 1 m 5 

C lM ′ 

l 1 −m 3 l 1 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 m 5 l 1 m 7 

C lM ′ 

l 1 m 3 l 1 m 1 

C lM ′ 

l 1 −m 5 l 1 −m 7 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 m 5 l 3 m 7 

C lM ′ 

l 1 m 1 l 3 m 3 

C lM ′ 

l 1 −m 5 l 3 −m 7 

+ 


l 3 −m 3 l 1 −m 1 

C lM 

l 3 −m 3 l 1 m 5 

C lM ′ 

l 1 m 5 l 7 −m 7 

C lM ′ 

l 1 −m 1 l 7 −m 7 

+ 


l 1 −m 3 l 1 −m 1 

C lM 

l 1 −m 1 l 1 m 5 

C lM ′ 

l 1 m 5 l 7 −m 7 

C lM ′ 

l 1 −m 3 l 7 −m 7 

+ 


l 3 −m 3 l 1 −m 1 

C lM 

l 3 −m 3 l 1 m 5 

C lM ′ 

l 1 m 1 l 7 −m 7 

C lM ′ 

l 1 −m 5 l 7 −m 7 

+ 


l 1 m 10 l 1 −m 1 

C lM 

l 1 −m 1 l 1 m 5 

C lM ′ 

l 1 −m 2 l 7 −m 7 

C lM ′ 

l 1 −m 5 l 7 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 −m 3 l 1 m 5 

C lM ′ 

l 1 m 5 l 7 −m 7 

C lM ′ 

l 1 −m 1 l 7 −m 7 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 −m 1 l 3 m 5 

C lM ′ 

l 3 m 5 l 7 −m 7 

C lM ′ 

l 3 −m 3 l 7 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 −m 3 l 1 m 5 

C lM ′ 

l 1 m 1 l 7 −m 7 

C lM ′ 

l 1 −m 5 l 7 −m 7 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 −m 1 l 3 m 5 

C lM ′ 

l 3 m 3 l 7 −m 7 

C lM ′ 

l 3 −m 5 l 7 −m 7 

+ 


l 3 −m 3 l 1 −m 1 

C lM 

l 3 −m 3 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 m 5 

C lM ′ 

l 1 −m 1 l 1 −m 7 

+ 


l 1 m 10 l 1 −m 1 

C lM 

l 1 −m 1 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 m 5 

C lM ′ 

l 1 m 2 l 1 −m 7 

+ 


l 3 −m 3 l 1 −m 1 

C lM 

l 3 −m 3 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 m 1 

C lM ′ 

l 1 −m 5 l 1 −m 7 

+ 


l 1 m 10 l 1 −m 1 

C lM 

l 1 −m 1 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 −m 2 

C lM ′ 

l 1 −m 5 l 1 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 −m 3 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 m 5 

C lM ′ 

l 1 −m 1 l 1 −m 7 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 −m 1 l 3 m 5 

C lM ′ 

l 3 −m 7 l 3 m 5 

C lM ′ 

l 3 −m 3 l 3 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 −m 3 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 m 1 

C lM ′ 

l 1 −m 5 l 1 −m 7 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 −m 1 l 3 m 5 

C lM ′ 

l 3 −m 7 l 3 m 3 

C lM ′ 

l 3 −m 5 l 3 −m 7 

+ 


l 3 −m 3 l 1 −m 1 

C lM 

l 3 −m 3 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 m 5 

C lM ′ 

l 1 −m 1 l 1 −m 7 

+ 


l 1 −m 3 l 1 −m 1 

C lM 

l 1 −m 1 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 m 5 

C lM ′ 

l 1 −m 3 l 1 −m 7 

+ 


l 3 −m 3 l 1 −m 1 

C lM 

l 3 −m 3 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 m 1 

C lM ′ 

l 1 −m 5 l 1 −m 7 

+

B.0 147 


l 1 −m 3 l 1 −m 1 

C lM 

l 1 −m 1 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 m 3 

C lM ′ 

l 1 −m 5 l 1 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 −m 3 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 m 5 

C lM ′ 

l 1 −m 1 l 1 −m 7 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 −m 1 l 3 m 5 

C lM ′ 

l 3 −m 7 l 3 m 5 

C lM ′ 

l 3 −m 3 l 3 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 −m 3 l 1 m 5 

C lM ′ 

l 1 −m 7 l 1 m 1 

C lM ′ 

l 1 −m 5 l 1 −m 7 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 −m 1 l 3 m 5 

C lM ′ 

l 3 −m 7 l 3 m 3 

C lM ′ 

l 3 −m 5 l 3 −m 7 

+ 


l 3 −m 3 l 1 −m 1 

C lM 

l 3 −m 3 l 1 m 5 

C lM ′ 

l 1 m 5 l 7 −m 7 

C lM ′ 

l 1 −m 1 l 7 −m 7 

+ 


l 1 m 10 l 1 −m 1 

C lM 

l 1 −m 1 l 1 m 5 

C lM ′ 

l 1 m 5 l 7 −m 7 

C lM ′ 

l 1 m 2 l 7 −m 7 

+ 


l 3 −m 3 l 1 −m 1 

C lM 

l 3 −m 3 l 1 m 5 

C lM ′ 

l 1 m 1 l 7 −m 7 

C lM ′ 

l 1 −m 5 l 7 −m 7 

+ 


l 1 m 10 l 1 −m 1 

C lM 

l 1 −m 1 l 1 m 5 

C lM ′ 

l 1 −m 2 l 7 −m 7 

C lM ′ 

l 1 −m 5 l 7 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 −m 3 l 1 m 5 

C lM ′ 

l 1 m 5 l 7 −m 7 

C lM ′ 

l 1 −m 1 l 7 −m 7 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 −m 1 l 3 m 5 

C lM ′ 

l 3 m 5 l 7 −m 7 

C lM ′ 

l 3 −m 3 l 7 −m 7 

+ 


l 1 −m 1 l 1 −m 3 

C lM 

l 1 −m 3 l 1 m 5 

C lM ′ 

l 1 m 1 l 7 −m 7 

C lM ′ 

l 1 −m 5 l 7 −m 7 

+ 


l 1 −m 1 l 3 −m 3 

C lM 

l 1 −m 1 l 3 m 5 

C lM ′ 

l 3 m 3 l 7 −m 7 

C lM ′ 

l 3 −m 5 l 7 −m 7 

(B.3) 

This can be then simplified much further to get the final expression for cosmic 

variance of BiPS as it is explained in Appendix A.

AppendixC 

Search for Planar Symmetries in CMB 

Ansiotropy Maps 

Flat spaces have planar symmetries which can be used to detect them. These symmetries 

as pointed out by Starobinsky 

(1993), emerge if the size of the fundamental 

domain is smaller than the diameter of the surface of last scattering. 

For example, 

imagine a case where the fundamental domain is a cuboid and only one of its cell sizes 

is smaller than R SLS . In this case the large scale temperature fluctuations have a symmetry 

plane which is perpendicular to the compact direction. One can then look for 

these planar symmetries as signatures of compact flat spaces. We perform this search 

for symmetries using the statistics originally proposed in de Oliveira-Costa & Smoot 

(1995) and later made easier by de Oliveira-Costa et al. (2003). This is known as 

S-statistics and is done by computing the function S(ˆn i ) defined by 

S(ˆn i ) = 1 

N 

∑ pix ( ∆T 

N pix T (ˆn j) − ∆T ) 2 

T (ˆn ij) 

(C.1) 

j=1 

where N pix is the number of pixels in the CMB map and ˆn ij denotes the reflection of ˆn j 

in the plane whose normal is ˆn i , i.e., 

ˆn ij = ˆn j − 2(ˆn i · ˆn j )ˆn i 

(C.2) 

The original S-statistics proposed by de Oliveira-Costa & Smoot (1995) was slightly 

different and had a division by the r.m.s. errors associated with the pixels in the form 

148

C.0 149 

of 

S(ˆn i ) = 1 

N 

∑ pix 

N pix 

j=1 

( ∆T 

(ˆn T j) − ∆T 

σ 2 (ˆn j ) + σ 2 (ˆn ij ) 

(ˆn T ij) ) 2 

. (C.3) 

But we used the one given in eqn. (C.1) because at that time we did not know the r.m.s. 

pixel noise of the WMAP data. 

S(ˆn i ) itself can be viewed as a map, and is a measure of how much reflection symmetry 

there is in the mirror plane perpendicular to S(ˆn i ). This resultant map is referred 

to as an S-map. This is a very useful visualization tool to search for symmetries and 

works in the following way. 

(a) 

(b) 

As it is seen in eqn. C.1, the S(ˆn i ) is a sum over non-negative quantities. 

These quantities are the difference between ∆T/T at every pixel and it’s 

reflection with respect to an axis. At the axis of symmetry, ∆T/T at each 

pixel is similar to the ∆T/T at the position of reflection of this point with 

respect to that axis. And hence the minimum of the S-map, denoted as S 0 , 

which is visualized as the coldest spot of the map is pointing to the axis of 

symmetry. 

The distribution of S 0 values changes with the size of the fundamental domain. 

The smaller the fundamental domain, the smaller the value of S 0 

would be. 

(c) 

The value of S 0 depends on the size of the fundamental domain and is independent 

of its orientation. 

The S-statistics will then be best done by obtaining S-map and looking for the coldest 

spots on it. We computed the S-map for ILC map. This map as it is shown in Fig. C.1 

has some cold spots, major ones are on the poles and two long ones (spots appear in 

pairs because the S-map has been built symmetrically). In Fig. C.1 we show the S-map 

obtained from a random simulation of the CMB sky with the best fit C l of WMAP. The 

patterns on these two S-maps are clearly different. But what is important here, is how 

cold the coldest spots, S 0 , in the ILC S-map are. To answer this question, we degrade 

the ILC map to the HEALPix resolution of N side = 32 and compute the S 0 in that. We 

will then make 500 random realizations of the CMB sky from the best fit C l of WMAP

C.0 150 

and compute the S 0 in each of them. The question of how cold is the coldest spot in 

ILC S-map would then be equivalent to in the random realizations of a normal CMB 

sky, how many times do I get S 0 values bigger than that of S 0 of ILC map. If ILC 

represents the CMB in a small compact flat space, its S 0 should be smaller than most 

of the S 0 values obtained from the random realizations. The result of this test is shown 

in the left panel of Fig. C.2. Which means that S 0 of ILC is not low at all. And this 

rules out the possibility of small spaces. The above test was done by de Oliveira-Costa 

et al. (1996) for the 4 year COBE/DMR data and by de Oliveira-Costa et al. (2003) 

for the WMAP data. However, the quadrupole in the first year WMAP data has been 

the source of many debates. To make sure that we have not been misled by that, we 

do the above test once again, this time by excluding the quadrupole from ILC map as 

well as the simulated maps. The result is shown in right panel of Fig. C.2 and is still 

consistent with a not-so-low S 0 of the ILC map.

C.0 151 

Figure C.1 SMAP of ILC map (top) and of a random realization of the CMB sky obtained 

from the best fit C l of the WMAP data (bottom). Units are arbitrary and figures are 

presented for a shape comparison of the two maps.

C.0 152

AppendixD 

Symmetries of the Space 

By symmetries of the space, we mean those transformations of the space into itself 

under which metric and every other physical and geometrical properties of the space 

remain invariant. Here we only consider continuous symmetries. These symmetries as we 

shal see, form continuous groups of symmetries and are generated by vector fields which 

are generators of these groups. In the following section we will study the mathematical 

preliminaries of the notions used in Chapter 5. 

Mathematical preliminaries 

Let M be an m-dimensional differentiable manifold and X be a vector field in M. 

An integral curve x(t) of X is a curve in M, whose tangent vector at x(t) is X| x 

dx µ (t) 

dt 

= X µ (x(t)) (D.1) 

Let σ(t, x 0 ) be an integral curve of X which passes a point x 0 at t = 0 and denote the 

coordinate by σ µ (t, x 0 ). Eqn. (D.1) then becomes 

with the initial condition 

d 

dt σµ (t, x 0 ) = X µ (σ(t, x 0 )) 

σ µ (0, x 0 ) = x µ 0 . 

(D.2) 

(D.3) 

The map σ : R × M ↦−→ M is called a flow generated by X ∈ X(M) 1 . For fixed t ∈ R 

a flow is a diffeomorphism from M to M, denoted by σ t : M ↦−→ M and is made into a 

1 X(M) denotes the set of vector fields on M. 

153

D.0 154 

commutative group by the folloing rules 

1. σ t (σ s (x)) = σ t+s (x), 

2. σ 0 = the identity map, 

3. σ −t = (σ t ) −1 . This group is called the one-parameter group of transformations. 

Diffeomorphism: A C ∞ map is called a diffeomorphism if it is one-to-one and onto 

(and hence invertible) and its inverse is C ∞ too. Let (M, g) be a (pseudo-) Riemannian 

manifold. A diffeomorphism f : M → M is an isometry if it preserves the metric 

f ∗ g f(p) = g p . 

(D.4) 

This means that f is an isometry if for each X, Y ∈ T p M we have g f(p) (f ∗ X, f ∗ Y ) = 

g p (X, Y ). In components, this condition becomes 

∂y α 

∂x µ ∂y β 

∂x ν g αβ(f(p)) = g µν (p), 

(D.5) 

where x and y are coordinates of p and f(p) respectively. The isometries of a space of 

dimension n form a group, because the identity map, the composition of two isometries 

and the inverse of an isometry are all isometries too. For example in R n , the Euclidean 

group E n is the isometry group, that is f : x → Ax + T , where A ∈ SO(n) are rotations 

and T ∈ R n causes the translations. 

Let (M, g) be a Riemannian manifold and X ∈ X(M). If an infinitesimal displacement 

given by ɛX generates an isometry (ɛ is an infinitesimal parameter), the vector 

field X is called a Killing vector fields. The coordinates x µ of a point p ∈ M change 

to x µ + ɛX µ (p) under this displacement 2 . If f : x µ ↦−→ x µ + ɛX µ is an isometry, it 

satisfies (D.5), 

∂(x α + ɛX α ) 

∂x µ ∂(x β + ɛX β ) 

∂x ν g αβ (x + ɛX) = g µν (x) (D.6) 

The above expression tells us that g µν and X µ satisfy the Killing equation 

X η ∂ η g µν + ∂ µ X η g ην + ∂ ν X η g µη = 0. 

(D.7) 

2 σ µ ɛ (x) = σ µ (ɛ, x) = x µ + ɛX µ (x) is a one-parameter group of transformations and the vector field 

X is called the infinitesimal generator of this transformation.

D.0 155 

And from the definition of the Lie derivative, this is written as 

(L X g) µν = 0. (D.8) 

This has a nice interpretation. Let φ t : M ↦−→ M be a one-parameter group of transformations 

which generate the Killing vector field X. Eqn. (D.8 then shows that the local 

geometry does not change as we move along φ t . In this sense, the Killing vector fields 

represent the direction of the symmetry of a manifold. 

A set of Killing vector fields is said to be linearly dependent if one of them is expressed 

as a linear combination of others with constant coefficients 3 . The maximum number of 

symmetries is related to dimM and in an m-dimensional manifold is m(m+1)/2. Spaces 

which admit m(m+1)/2 Killing vector fields are called maximally symmetric spaces. 

Let X a and X b be two Killing vector fields. It can be seen that 

(a) 

a linear combination αX a + βX b is a killing vector field (α, β ∈ R), 

(b) the commutator [X a , X b ] is a Killing vector field 4 . 

Thus all Killing vector fields form a Lie algebra of the symmetric operations on the 

manifold M, with structure constants Cab 

c 

[X a , X b ] = Cab c X c (D.9) 

where a, b, c = 1, 2, ..., r and r ≤ m(m + 1)/2. Structure constants satisfy 

(a) 

(b) 

Skew-symmetry 

Jacobi identity 

C c ab = −C c ba 

C a e[b Ce cd] = 0 

(D.10) 

(D.11) 

These are the integrability conditions that must be satisfied in order that the Lie algebra 

exist in a consistent way. The transformations generated by the Lie algebra form a Lie 

group of the same dimension. 

3 There may be more Killing vector fields than the dimension of the manifold. The number of 

independent symmetries has no direct connection with the dimension of M. 

4 Because [X a , X b ] g = L Xa (L Xb g) − L Xb (L Xa g) = 0

D.0 156 

In the case of our interest, a Lie group often appears as a set of transformations 

acting on a manifold. We need to study the action of a Lie group G on a manifold M 

in more details. Let G be a Lie group and M a manifold. The action of G on M is a 

differentiable map σ : G × M ↦−→ M which satisfies the conditions 

(i) σ(e, p) = p for any p ∈ M 

(ii) 

σ(g 1 , σ(g 2 , p)) = σ(g 1 g 2 , p). 

The action σ is said to be transitive if, for any p 1 , p 2 ∈ M, there exists an element 

g ∈ G such that σ(g, p 1 ) = p 2 . Which means it can move any point of M into any other 

point of M. Given a point p ∈ M, the action of G on p takes p to various points in M. 

The orbit of p is the set of all points into which p can be moved by the action of the 

isometries of a space. This is 

Gp = {σ(g, p)|g ∈ G}. 

(D.12) 

Any orbit Gp is clearly a subset of M and obviously the action of G on any orbit Gp is 

transitive. Orbits are in fact the largest surfaces through each point on which the group 

is transitive. If the action of G on M is transitive, the orbit of any p ∈ M is M itself. 

So, the maximum dimension of orbits is dimM. 

dim orbits = s, where in a space of dimension n (dimM = n), s ≤ n. 

A subgroup H(p) of a Lie group G that acts on M is called an isotropy group if 

it leaves the point p ∈ M fixed, 

H(p) = {g ∈ G|σ(g, p) = p} 

(D.13) 

H(p) is also called the little group or stabilizer. We have 

dim of isotropy group = q, where q ≤ 1 n(n − 1). 

2 

Interestingly, the cosets of the isotropy group correspond to the elements in the orbit, 

G p ∼ G/H(p) 

(D.14) 

G/H has a very interesting property indeed. There is a theorem which says Nakahara 

(1984): for any subgroup H of a Lie group G, the coset space G/H admits a differentiable

D.0 157 

structure and becomes a manifold called homogeneous space. Dimension of this coset 

space is given by 

dimG/H = dimG − dimH 

(D.15) 

In our case, G is a Lie group which acts on a manifold M transitively, and H(p) is an 

isotropy group of p ∈ M. This theorem tells us that H(p) is a Lie group and the coset 

G/H(p) is a homogeneous space. 

The dimension r of the group of symmetries of the space (group of isometries), is 

then given by 

r = s + q =⇒ 0 ≤ r ≤ n + n 2 (n − 1) = 1 n(n + 1) 

2 (D.16) 

which is what we had seen before. The above relation can be viewed as 

dim group of isometries = dim group of rotational symmetries + dim group of translational 

symmetries. 

If q = 0 then r = s which means that dimension of group of isometries is just enough 

to move each point in an orbit into any other point. This is called a simply transitive 

group. There is no continuous isotropy group in this case.

AppendixE 

Bianchi Models: Bianchi’s Original 

Classification 

In his historical paper On the 3-dimensional spaces which admit a continous group 

of motions 1 , Luigi Bianchi begins with finite-dimensional continuous Lie group G r generated 

by r infinitesimal transformations X 1 , X 2 , ..., X r of an n-dimensional Riemannian 

manifold. Then his problem of determining which spaces possess a continuous group of 

motions gets reduced to the classification of all possible forms of metrics which possess 

a Lie group G r = {X 1 , · · · , X n } which transforms the metric into itself. Lie’s classification 

of equivalence classes of 3-dimensional Lie algebras over the complex numbers was 

then refined by him to the real case by adding several types, giving Bianchi’s canonical 

form for the generating Lie algebra commutation relations for each type designated by 

consecutive Roman numerals I through IX, now known as the Bianchi classification. 

Below is Bianchi’s frist classification of all possible spaces which admit continous 

groups of motions. They are divided into six categories according to the type of their 

complete group G of motions. Spaces whose group of motions is a G 1 are class (A), classe 

(B) are spaces with group of motions G 2 , they are class (C) when it is an intransitive 

G 3 . The other two categories (D) , (E) contain the spaces whose group of motions is 

1 Original title: Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti, 

Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, Serie Terza, Tomo XI, pp. 267– 

352 (1898). I found an english translation of this paper Translated by Robert Jantzen. This paper 

was also reprinted in: Opere [The Collected Works of Luigi Bianchi], Rome, Edizione Cremonese, 1952, 

vol. 9, pp. 17-109. 

158

E.0 159 

transitive, (D) those with a G 3 , and (E) those with a G 4 . The sixth category (F) includes 

the spaces of constant curvature which are maximally symmetric spaces which admit a 

group G 6 of motions. He proves the impossibility of other spaces with continuous groups 

of motions. For example that there does not exist any 3-dimensional space whose group 

of motions contains a real 5-parameter subgroup. 

Category A 

Groups G 1 

Type I 

ds 2 = Σ a ik dx i dx k 

with coefficients a ik independent of x 1 

group: 

X 1 f = ∂f 

∂x 1 

Or, the group G 3 is abelian and offers the first and simplest composition 

[X 1 , X 2 ]f = [X 1 , X 3 ]f = [X 2 , X 3 ]f = 0 . 

Category B 

Groups G 2 

ds 2 = dx 2 1 + α dx2 2 + 2β dx 2dx 3 + γ dx 2 3 

with α, β, γ functions only of x 1 

group: 

composition: 

X 1 f = ∂f 

∂x 2 

, X 2 f = ∂f 

∂x 3 

[X 1 , X 2 ] = 0 

ds 2 = dx 2 1 + α dx 2 2 + 2(β − αx 2 ) dx 2 dx 3 + (αx 2 2 − 2βx 2 + γ) dx 2 3

E.0 160 

with α, β, γ functions of x 1 

group: 

composition: 

X 1 f = ∂f 

∂x 3 

, X 2 f = e x 3 ∂f 

∂x 2 

[X 1 , X 2 ]f = X 2 f 

Category C 

Groups G 3 intransitive 

α) ds 2 = dx 2 1 + ϕ2 (x 1 )(dx 2 2 + dx2 3 ) 

with ϕ(x 1 ) an arbitrary function of x 1 

group: 

composition: 

X 1 f = ∂f 

∂x 2 

, X 2 f = ∂f 

∂x 3 

, X 3 f = x 3 

∂f 

∂x 2 

− x 2 

∂f 

∂x 3 

[X 1 , X 2 ]f = 0 , [X 1 , X 3 ]f = −X 2 f , [X 2 , X 3 ]f = X 1 f 

β) ds 2 = dx 2 1 + ϕ2 (x 1 )(dx 2 2 + sin2 x 2 dx 3 ) 2 

group: 

composition: 

X 1 f = ∂f 

∂x 3 

, X 2 f = sin x 3 

∂f 

∂x 2 

+ cot x 2 cos x 3 

∂f 

∂x 3 

, 

X 3 f = cos x 3 

∂f 

∂x 2 

− cot x 2 sin x 3 

∂f 

∂x 3 

[X 1 , X 2 ]f = X 3 f , [X 2 , X 3 ]f = X 1 f , [X 3 , X 1 ]f = X 2 f 

γ) ds 2 = dx 2 1 + ϕ2 (x 1 ) (dx 2 2 + e2x 2 

dx 3 ) 2

E.0 161 

group: 

composition: 

X 1 f = ∂f 

∂x 3 

, X 2 f = ∂f 

∂x 2 

− x 3 

∂f 

∂x 3 

, 

X 3 f = x 3 

∂f 

∂x 2 

+ 1 2 (e−2x 2 

− x 2 3) ∂f 

∂x 3 

[X 1 , X 2 ]f = −X 1 f , [X 1 , X 3 ]f = X 2 f , [X 2 , X 3 ]f = −X 3 f 

Category D 

Groups G 3 transitive 

Type IV 

ds 2 = dx 2 1 + ex 1 

[dx 2 2 + 2x 1 dx 2 dx 3 + (x 2 1 + n2 ) dx 3 ) 2 ] 

group: 

composition: 

X 1 f = 2 ∂f 

∂x 2 

, X 2 f = ∂f 

∂x 3 

, 

X 3 f = −2 ∂f 

∂x 1 

+ (x 2 + 2x 3 ) ∂f 

∂x 2 

+ x 3 

∂f 

∂x 3 

[X 1 , X 2 ] = 0 , [X 1 , X 3 ]f = X 1 f , [X 2 , X 3 ]f = X 1 f + X 2 f 

Type VI 

ds 2 = dx 2 1 + e 2x 1 

dx 2 2 + 2ne (h+1)x 1 

dx 2 dx 3 + e 2hx 1 

dx 2 3 

group: 

composition: 

X 1 f = ∂f 

∂x 2 

, X 2 f = ∂f 

∂x 3 

, 

X 3 f = − ∂f 

∂x 1 

+ x 2 

∂f 

∂x 2 

+ hx 3 

∂f 

∂x 3 

[X 1 , X 2 ] = 0 , [X 1 , X 3 ]f = X 1 f , [X 2 , X 3 ]f = hX 2 f

E.0 162 

Type V II 1 

ds 2 = dx 2 1 + (n + cos x 1) dx 2 2 + 2 sin x 1 dx 2 dx 3 + (n − cos x 1 ) dx 2 3 

group: 

with the composition 

X 1 f = ∂f 

∂x 2 

, X 2 f = ∂f 

∂x 3 

, X 3 f = 2 ∂f 

∂x 1 

− x 3 

∂f 

∂x 2 

+ x 2 

∂f 

∂x 3 

, 

[X 1 , X 2 ]f = 0 , [X 1 , X 3 ]f = X 2 f , [X 2 , X 3 ]f = −X 1 f . 

Type V II 2 

ds 2 = dx 2 1 + e−hx 1 

{(n + cos vx 1 ) dx 2 2 + (h cos vx 1 + v sin x 1 + hn) dx 2 dx 3 

( 

) } 

2−v 

+ 

2 

cos vx 

2 1 + hv sin vx 2 1 + n dx 2 3 , h ≠ 0 (0 < h < 2) 

where v = √ 4 − h 2 

group: 

composition: 

X 1 f = ∂f 

∂x 2 

, X 2 f = ∂f 

∂x 3 

, 

X 3 f = ∂f 

∂x 1 

− x 3 

∂f 

∂x 2 

+ (x 2 + hx 3 ) ∂f 

∂x 3 

[X 1 , X 2 ] = 0 , [X 1 , X 3 ]f = X 2 f , [X 2 , X 3 ]f = −X 1 f + hX 2 f 

Type VIII 

( 

) 

ds 2 = Q(4) (x 1 ) 

dx 2 24 1 + Q(x 1 ) dx 2 2 + Q(x 1 )x 2 2 − Q′ (x 1 ) 

x 

2 2 + Q′′ (x 1 ) 

− h 2 2 

( ) 

{ ( ) } 

+2 Q ′′ (x 1 ) 

+ h dx 

12 1 dx 2 + 2 Q ′′′ (x 1 ) 

− Q ′′ (x 1 ) 

+ h x 

24 12 2 dx 1 dx 3 

( ) 

+2 Q ′ (x 1 ) 

− Q(x 

4 1 )x 2 dx 2 dx 3 , 

with Q(x 1 ) a fourth degree polynomial in x 1 with its first 

coefficient positive (or zero), and h a constant 

group: 

X 1 f = e −x 3 ∂f 

∂x 1 

− x 2 2 e−x 3 ∂f 

∂x 2 

− 2x 2 e −x 3 ∂f 

∂x 3 

; 

X 2 f = ∂f 

∂x 3 

, X 3 f = ∂f 

∂x 2 

dx 2 3

E.0 163 

composition: 

[X 1 , X 2 ]f = X 1 f , [X 1 , X 3 ]f = 2X 2 f , [X 2 , X 3 ]f = X 3 f 

ds 2 = Σ i,k a ik dx i dx k 

group: 

composition: 

a 11 = 2e cos 2x 3 + 2f sin 2x 3 + (a 2 + d 2 )/2 , 

a 22 = 2 sin x 1 cos x 1 (b sin x 3 − c cos x 3 ) − a 11 sin 2 x 1 + a 2 + d sin 2 x 1 , 

a 33 = a 2 , a 13 = b cos x 3 + c sin x 3 , 

a 12 = cos x 1 (b cos x 3 + c sin x 3 ) + 2 sin x 1 (e sin 2x 3 − f cos 2x 3 ) , 

a 23 = a 2 cos x 1 + sin x 1 (b sin x 3 − c cos x 3 ) 

X 1 f = ∂f 

∂x 2 

, X 2 f = cos x 2 

∂f 

∂x 1 

− cot x 1 sin x 2 

∂f 

∂x 2 

+ sin x 2 

sin x 1 

∂f 

∂x 3 

, 

X 3 f = − sin x 2 

∂f 

∂x 1 

− cot x 1 cos x 2 

∂f 

∂x 2 

+ cos x 2 

sin x 1 

∂f 

∂x 3 

[X 1 , X 2 ]f = X 3 f , [X 2 , X 3 ]f = X 1 f , [X 3 , X 1 ]f = X 2 f 

Category E 

Groups G 4 

a) Type II 

ds 2 = dx 2 1 + dx 2 2 + 2x 1 dx 2 dx 3 + (x 2 1 + 1) dx 2 3 

group: 

composition: 

X 1 f = ∂f 

∂x 2 

, X 2 f = ∂f 

∂x 3 

, X 3 f = − ∂f 

∂x 1 

+ x 3 

∂f 

∂x 2 

, 

X 4 f = x 3 

∂f 

∂x 1 

+ 1 2 (x2 1 − x2 3 ) ∂f 

∂x 2 

− x 1 

∂f 

∂x 3 

[X 1 , X 2 ] = [X 1 , X 3 ] = [X 1 , X 4 ] = 0 , 

[X 2 , X 3 ]f = X 1 f , [X 2 , X 4 ]f = −X 3 f , [X 3 , X 4 ]f = X 2 f

E.0 164 

b) Types III, VIII 

ds 2 = dx 2 1 + e2x 1 

dx 2 2 + 2nex 1 

dx 2 dx 3 + dx 2 3 

group: 

composition: 

X 1 f = ∂f , X 2 f = ∂f , X 3 f = ∂f ∂f 

− x 2 , 

∂x 2 ∂x 3 ∂x 1 ∂x 2 

∂f 

X 4 f = x 2 + 1 ( e 

−2x 1 

) ∂f 

∂x 1 2 1 − n − 2 x2 2 − ne−x 1 

∂f 

∂x 2 1 − n 2 ∂x 3 

[X 1 , X 2 ] = 0 , [X 1 , X 3 ]f = −X 1 f , [X 1 , X 4 ]f = X 3 f , 

[X 2 , X 3 ] = 0 , [X 2 , X 4 ] = 0 , [X 3 , X 4 ]f = −X 4 f 

c) Type IX 

ds 2 = dx 2 1 + (sin2 x 1 + n 2 cos 2 x 1 ) dx 2 2 + 2n cos x 1 dx 2 dx 3 + dx 2 3 

group: 

composition: 

X 1 f = ∂f 

∂f 

∂f 

, X 2 f = cos x 2 − cot x 1 sin x 2 + n sin x 2 

∂x 2 ∂x 1 ∂x 2 sin x 1 

X 3 f = − sin x 2 

∂f 

∂x 1 

− cot x 1 cos x 2 

∂f 

∂x 2 

+ n cos x 2 

sin x 1 

∂f 

∂x 3 

, 

∂f 

, X 4 f = ∂f 

∂x 3 ∂x 3 

[X 1 , X 2 ]f = X 3 f , [X 2 , X 3 ]f = X 1 f , [X 3 , X 1 ]f = X 2 f , 

[X 1 , X 4 ] = [X 2 , X 4 ] = [X 3 , X 4 ] = 0 

Category F 

Groups G 6 — spaces of constant curvature

AppendixF 

Simulating CMB maps for non-SI 

correlation functions 

We define a zero-mean Gaussian random field T over a grid S with covariance function 

C : R 2 → R 

(F.1) 

Having such a field means that for any finite set of n points 

x 1 , x 2 , · · · , x n ∈ S 

(F.2) 

we have random variables 

T (x 1 ), T (x 2 ), · · · , T (x n ); 

(F.3) 

which are jointly normal with mean zero and covariances 

Cov(T (x i ), T (x j )) = C(T (x i ), T (x j )). 

(F.4) 

Lets denote T (x i ) by T i for simplicity and also the covariance matrix C(T (x i ), T (x j )) 

by C ij . The distribution function for this field is given by 

P = 

1 

(2π) n/2√ ‖C‖ exp (−1 2 T iC −1 

ij T j). 

(F.5) 

In the above expression we have made use of Einstein summation convention to drop 

the summation sign, hereafter we will stick to this notation. And by ‖C‖ we mean the 

determinant of C ij . 

165

F.0 166 

In eqn. (F.5) we can not write the distribution function as a product of n Gaussian 

distributions because the covariance matrix in general may not be diagonal and it will 

have cross correlation terms. 

There are two ways to proceed further. The diagonalization method and the Cholesky 

decomposition. 

Diagonalization method 

The first way is the following: suppose we can diagonalize C −1 which means we 

can find a unitary matrix U such that U −1 C −1 U is diagonal. In that case we can 

immediately write the exponent in (F.5) as a sum of independent quantities and the 

distribution function will break into a product of n Gaussian distributions. 

If we use some matrix D to change C ij into diagonal form say, diag(σ 1 , σ 2 , ..., σ N ) 

then the power of e would change as: 

T T 

i U T UC −1 

ij U T UT i = ψ 2 i /σ2 i 

(F.6) 

and ψ i are nothing but elements of rotated T vector under the effect of U ( −→ ψ = U −→ T ). 

So P (T i ) can be written as: 

P (T i ) = Π i 

1 

(2πσ i ) 1/2 e−ψ2 i /σ2 i 

P i (ψ i ) = 

= Π i P i (ψ i ) , 

1 

√ 

2πσi 

e − ψ2 i 

2σ 2 i 

(F.7) 

So now everything is clear: it is enough to generate a vector −→ ψ whose elements are 

gaussian random numbers, then multiply each element by the corresponding eigenvalue 

of C ij say σ i to find a gaussian variable with variance σ i . Then apply an inverse rotation 

on vector −→ ψ to find T = U −1−→ ψ which is a realization of the temperature fluctuations 

on the sky. 

This has been done using the method of Householder reduction of a real, symmetric, 

N × N matrix into a tridiagonal matrix. And then the method of QL to diagonalize 

that tridiagonal matrix. An example of the use of this method is simulation of noise

F.0 167 

maps using the noise covariance matrix given in Fig. 6.1 which is ofcourse diagonal by 

itself and hence is a trivial example! The resultant map is shown in Fig. F.1.

F.0 168 

Figure F.1 A simulated map from noise covariance matrix using inversion method.

F.0 169 

Cholesky decomposition 

There is another way of proceeding further with the equation(F.5). In matrix notation 

we can write it as 

P (T ) = 

1 

(2π) n/2√ ‖C‖ exp (−1 2 T t C −1 T ) 

(F.8) 

where T t means T transposed. Now if there exists a matrix L such C can be decomposed 

into L and L t in the following manner 

C = L L t . 

(F.9) 

Then we can substitute this for C in the exponent of (F.8) to get 

T t C −1 T = T t (LL t ) −1 T (F.10) 

= (L −1 T ) t L −1 T 

= Y t Y 

Where 

Y = L −1 T or T = L Y. (F.11) 

The distribution function can be written for Y 

P (Y ) = 

= 

= 

1 

(2π) n/2√ ‖C‖ exp (−1 2 Y t Y ) ‖∂T/∂Y ‖ 

1 

(2π) exp (−1 ∑ 

y n/2 i 2 2 

) 

i 

n∏ 

{ 1 

(2π) exp (−1 2 y2 i )} 

i=1 

(F.12) 

This is nothing but the product of n independent Gaussian distributions for a single 

variable. Based on the above procedure we can prescribe a way to make realizations of 

a Gaussian random field with a given covariance matrix: 

Recipe: Map Making 

• Decompose the covariance matrix into L and L t . In mathematics this 

is called the Cholesky decomposition of a positive definite matrix. In 

Cholesky decomposition one decomposes a positive definite matrix into

F.0 170 

a triangular matrix and its transposed. There exists a subroutine 

choldc.f in Numerical Recipes which takes the matrix as an input and 

in output it returns the triangular matrix and in the diagonal it returns 

the square root of the eigenvalues of the matrix. 

• Generate n independent Gaussian random variables(Y ). 

• Use L to rotate back the vector Y and get the joint Gaussian random 

variables T = LY . 

• The computational effort in the one-time Cholesky decomposition scales 

as n 3 . However, Cholesky decomposition is significantly faster than the 

other alternative of diagonalization. 

This method has some advantages to the previous methods of generating CMB sky 

maps which use the angular power spectrum C l to generate CMB sky maps. Below we 

list a number of such advantages: 

1. The first advantage is that in this method we are using the covariance matrix 

to generate the CMB map. This is very important when we deal with statistically 

anisotropic models such as models with non-trivial topologies, or when we 

want to account effects of non-circular beam, foreground residuals and statistically 

anisotropic noise. In these models we can not use the angular power spectrum because 

we don’t have statistical isotropy to let us expand the covariance in terms 

of power spectrum. In the statistically non-isotropic models, power spectrum does 

not carry all the information of the random field and can not be used to generate 

a valid realization of the random field. 

2. The other advantage in this method is that we can generate a map for a small 

part of the sky with the high resolution at a proportional small fraction of the 

cost of generating the whole sky map. It is just enough to have the covariance 

matrix of the region of our interest, and making map without generating the full 

sky map is as we described in the recipe. This is in stark contrast to generating 

maps using the power spectrum. In that case the computational expense is high 

for high resolution even if we are simulating a small fraction of the sky.

F.0 171 

Figure F.2 A simulated CMB map in a toroidal universe using Cholesky decomposition 

method. Only naive Sachs-Wolfe effect has been considered and R SLS = 0.65L.

AppendixG 

Useful Standard Integrals and Summation 

Rules 

For completeness we list a set of standard integrals and summation rules which are 

needed in BiPS calculations. Orthonormality of spherical harmonics 

∫ 

∫ 

dΩˆn Y lm (ˆn) = √ 4πδ l0 δ m0 , (G.1) 

dΩˆn Y l1 m 1 

(ˆn)Y ∗ 

l 2 m 2 

(ˆn) = δ l1 l 2 

δ m1 m 2 

, (G.2) 

∫ 

dΩˆn Y l1 m 1 

(ˆn)Y l2 m 2 

(ˆn)Y ∗ 

l 3 m 3 

(ˆn) = 

√ 

(2l 1 + 1)(2l 2 + 1) 

C l 30 

l 

4π(2l 3 + 1) 1 0l 2 0 Cl 3m 3 

l 1 m 1 l 2 m 2 

. 

(G.3) 

(G.4) 

Symmetry property of spherical harmonics 

Y ∗ 

lm(ˆn) = (−1) m Y l−m (ˆn). 

(G.5) 

Spherical harmonic expansion of Legendre polynomials 

P l (ˆn · ˆn ′ ) = 

4π 

2l + 1 

l∑ 

m=−l 

Y ∗ 

lm (ˆn)Y lm(ˆn ′ ). 

(G.6) 

Orthonormality relation of χ l (ω) 

∫ π 

0 

χ l 1 

(ω)χ l 2 

(ω) sin 2 ω 2 dω = π 2 δ l 1 l 2 

. 

(G.7) 

172

G.0 173 

Orthonormality of D l MM (R) 

∫ 

dRD ∗l 

MM ′(R)Dl mm ′(R) = 

∑ 

8π2 

2l + 1 δ llδ mM δ m ′ M ′, 

mm ′ D l mm ′(R 1)D l m ′ m (R 2) = χ l (R 1 R 2 ) 

Symmetry properties of Clebsch-Gordan coefficients 

C cγ 

aαbβ 

= (−1) a+b−c C cγ 

bβaα , 

C cγ 

aαbβ 

= (−1) a+b−c C c−γ 

a−αb−β . 

(G.8) 

(G.9) 

(G.10) 

Summation rules of Clebsch-Gordan coefficients 

∑ 

αβ 

∑ 

aγ 

∑ 

cγ 

C cγ 

aαbβ Cc′ γ ′ 

aαbβ 

= δ cc ′δ γγ ′{abc}{abc ′ } 

C cγ 

aαbβ Ccγ aαb ′ β ′ = 2c + 1 

2b + 1 δ bb ′δ ββ ′{abc}{ab′ c} 

C cγ 

aαbβ Ccγ aα ′ bβ ′ = δ αα ′δ ββ ′{abc}, (G.11) 

where the 3j-symbol {abc} is defined by 

⎧ 

⎨ 1 if a + b + c is integer and |a − b| ≤ c ≤ (a + b), 

{abc} = 

⎩ 0 otherwise,

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