Cosmic Microwave Background Anisotropies and Theories of the Early Universe A dissertation submitted in satisfaction of the final requirement for the degree of Doctor Philosophiae SISSA – International School for Advanced Studies Astrophysics Sector arXiv:astro-ph/9512161v1 26 Dec 1995 Candidate: Supervisor: Alejandro Gangui Dennis W. Sciama October 1995 TO DENISE Abstract In this thesis I present recent work aimed at showing how currently competing theories of the early universe leave their imprint on the temperature anisotropies of the cosmic microwave background (CMB) radiation. After some preliminaries, where we review the current status of the field, we con- sider the three–point correlation function of the temperature anisotropies, as well as the inherent theoretical uncertainties associated with it, for which we derive explicit analytic formulae. These tools are of general validity and we apply them in the study of possible non–Gaussian features that may arise on large angular scales in the framework of both inflationary and topological defects models. In the case where we consider possible deviations of the CMB from Gaussian statis- tics within inflation, we develop a perturbative analysis for the study of spatial corre- lations in the inflaton field in the context of the stochastic approach to inflation. We also include an analysis of a particular geometry of the CMB three–point func- tion (the so–called ‘collapsed’ three–point function) in the case of post–recombination integrated effects, which arises generically whenever the mildly non–linear growth of perturbations is taken into account. We also devote a part of the thesis to the study of recently proposed analytic models for topological defects, and implement them in the analysis of both the CMB excess kurtosis (in the case of cosmic strings) and the CMB collapsed three–point function and skewness (in the case of textures). Lastly, we present a study of the CMB anisotropies on the degree angular scale in the framework of the global texture scenario, and show the relevant features that arise; among these, the Doppler peaks. i ii Contents Abstract i Acknowledgements vi Units and Conventions vii Introduction 1 1 Standard cosmology: Overview 5 1.1 Theexpandinguniverse .......................... 5 1.2 The Einstein field equations . 6 1.3 Thermal evolution and nucleosynthesis . .... 9 1.4 The CMB radiation and gravitational instability . ....... 11 1.5 The perturbation spectrum at horizon crossing . ...... 13 1.6 Drawbacksofthestandardmodel . 15 2 Theories of the early universe 19 2.1 Inflation ................................... 20 2.1.1 Theparadigm: somehistory . 21 2.1.2 Scalarfielddynamics . 22 2.1.3 Problemsinflationcansolve . 25 2.1.4 Generation of density perturbations . .. 27 2.1.5 Evolutionoffluctuations . 29 2.1.6 Anoverviewofmodels . 32 2.1.7 Stochasticapproachtoinflation . 38 2.2 Topologicaldefects ............................. 42 iii 2.2.1 Phase transitions and finite temperature field theory . ..... 45 2.2.2 TheKibblemechanism . 46 2.2.3 Asurveyoftopologicaldefects. 48 2.2.4 Local and global monopoles and domain walls . 50 2.2.5 Aredefectsinflatedaway? . 52 2.2.6 Cosmicstrings ........................... 53 2.2.7 Observational features from cosmic strings . .... 54 2.2.8 Globaltextures ........................... 55 2.2.9 Evolutionofglobaltextures . 57 3 CMB anisotropies 61 ∆T 3.1 Contributions to T ............................ 62 3.2 Suppressionofanisotropies. .. 67 ∆T 3.3 Adiabatic fluctuations as source of T .................. 68 3.4 Non–lineareffects.............................. 70 3.5 Overview................................... 70 4 CMB statistics 75 4.1 Dealing with spherical harmonics . .. 75 4.2 The CMB two–point correlation function . ... 77 4.3 The CMB three–point correlation function . .... 79 4.4 Predictions of a theory: ensemble averages . ..... 81 5 Primordial non–Gaussian features 83 5.1 Stochastic inflation and the statistics of the gravitational potential . 84 5.1.1 Theinflatonbispectrum . 85 5.1.2 The three–point function of the gravitational potential ..... 88 5.2 TheCMBangularbispectrum . 90 5.3 TheCMBskewness............................. 92 5.3.1 Worked examples: inflationary models . 93 5.3.2 On the correlation between n and the sign of the skewness . 95 5.4 The r.m.s. skewness of a Gaussian field . 105 5.5 Discussion .................................. 106 iv 6 Integrated effects 109 6.1 Introduction................................. 109 6.2 Contribution to the CMB three–point correlation function from the Rees–Sciamaeffect ............................. 111 6.3 The r.m.s. collapsed three–point function . ...... 115 6.4 Results for the COBE–DMRexperiment . 115 7 On the CMB kurtosis from cosmic strings 119 7.1 Introduction................................. 119 7.2 The four–point temperature correlation function . ........ 121 7.3 The r.m.s. excess kurtosis of a Gaussian field . ..... 127 7.4 Discussion .................................. 131 8 Analytic modeling of textures 133 8.1 Introduction................................. 133 8.2 Texture–spotstatistics . 136 8.3 Spotcorrelations .............................. 139 8.4 Textures and the three–point function . 139 8.5 Discussion .................................. 142 9 Global textures and the Doppler peaks 145 ∆T 9.1 Acoustic oscillations and T ........................ 146 9.2 Lineartheorypowerspectra . 147 9.3 Choosinginitialconditions . 150 9.4 Angular power spectrum from global textures . 152 9.5 Discussion .................................. 153 References 157 v Acknowledgements First and foremost I would like to thank my supervisor Dennis Sciama for his constant support and encouragement. He gave me complete freedom to chose the topics of my research as well as my collaborators, and he always followed my work with enthusiasm. I would also like to thank the whole group of astrophysics for creating a nice envi- ronment suited for pursuing my studies; among them I warmly thank Antonio Lanza for being always so kind and ready–to–help us in any respect. For the many interactions we had and specially for the fact that working with them was lots of fun, I want to express my acknowledgement to Ruth Durrer, Francesco Lucchin, Sabino Matarrese, Silvia Mollerach, Leandros Perivolaropoulos, and Mairi Sakellariadou. Our fruitful collaboration form the basis of much of the work presented here. My officemates, Mar Bastero, Marco Cavagli`a, Luciano Rezzolla, and Luca Zampieri + ‘Pigi’ Monaco also deserve some credits for helping me to improve on my italian and better appreciate the local caff`e, . and specially for putting up with me during our many hours together. I am also grateful to my colleagues in Buenos Aires for their encouragement to pursue graduate studies abroad, and for their being always in contact with me. Richard Stark patiently devoted many 5–minuteses to listen to my often direction- less questions, and also helped me sometimes as an online English grammar. I also thank the computer staff@sissa: Marina Picek, Luisa Urgias and Roberto Innocente for all their help during these years, and for making computers friendly to me. Finally, I want to express my gratitude to my wife, for all her love and patience; I am happy to dedicate this thesis to her. And last but not least, I would like to say thanks to our families and friends; they always kept close to us and helped us better enjoy our stay in Trieste. vi Units and Conventions In this thesis we are concerned with the study of specific predictions for the cosmic microwave background (CMB) anisotropies as given by different early universe models. Thus, discussions will span a wide variety of length scales, ranging from the Planck length, passing through the grand unification scale (e.g., when inflation is supposed to have occurred and cosmological phase transitions might have led to the existence of topological defects), up to the size of the universe as a whole (as is the case when probing horizon–size perturbation scales, e.g., from the COBE–DMR maps).1 As in any branch of physics, it so happens that the chosen units and conventions are almost always dictated by the problem at hand (whether we are considering the microphysics of a cosmic string or the characteristic thickness of the wake of accreted matter that the string leaves behind due to its motion). It is not hard to imagine then that also here disparate units come into the game. Astronomers will not always feel comfortable with those units employed by particle physicists, and vice versa, and so it is worthwhile to spend some words in order to fix notation. The so–called natural units, namelyh ¯ = c = kB = 1, will be employed, unless otherwise indicated.h ¯ is Planck’s constant divided by 2π, c is the speed of light, and kB is Boltzmann’s constant. Thus, all dimensions can be expressed in terms of one energy–unit which is usually chosen as GeV = 109eV, and so [Energy] = [Mass] = [Temperature] = [Length]−1 = [Time]−1 . Some conversion factors that will be useful in what follows are 1GeV =1.60 10−3 erg (Energy) × 1GeV =1.16 1013 K (Temperature) × 1GeV =1.78 10−24 g (Mass) × 1 GeV−1 = 1.97 10−14 cm (Length) × 1 GeV−1 = 6.58 10−25 sec (Time) × mP stands for the Planck mass, and associated quantities are given (both in natural 1COBE stands for COsmic Background Explorer satellite, and DMR is the Differential Microwave Radiometer on board. vii and cgs units) by m = 1.22 1019GeV = 2.18 10−5g, l = 8.2 10−20GeV−1 P × × P × =1.62 10−33cm, and t =5.39 10−44sec. In these units Newton’s constant is given × P × by G =6.67 10−8cm3g−1sec−2 = m−2. × P These fundamental units will be replaced by other, more suitable, ones when study- ing issues on large–scale structure formation. In that case it is better to employ astro- nomical units, which for historical reasons are based on solar system quantities. Thus, the parsec (the distance at which the Earth–Sun mean distance (1 a.u.) subtends one second of arc) is given by 1 pc = 3.26 light years. Of course, even this is much too small for describing the typical distances an astronomer has to deal with and thus many times distances are quoted in megaparsecs, 1 Mpc = 3.1 1024cm.