PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

More documents

Recommendations

Info

CHAPTER 1. INTRODUCTION say, λ P = λa, associated with any mode of a given wavelength λ, always grows proportional to the scale factor. In contrast, the Hubble radius, say, d H = H −1 , which reflects the size of the horizon in such cases, behaves as a 1/q . Therefore, λ P /d H ∝ a (q−1)/q , which, for q < 1, turns larger than unity at a sufficiently early time. In other words, the modes associated with the perturbations leave the Hubble radius as we go back in time. The fact that, at early times, the modes of cosmological interest (say, wavelengths larger than the scale of homogeneity today) are outside the Hubble radius in the hot big bang model implies that the model cannot provide a causal mechanism for seeding the perturbations. Inflation, a period of rapid expansion during the stages of the radiation dominated epoch, solves the horizon problem (and, in fact, other issues too, such as, for example, the flatness problem) of the standard big bang cosmology in an elegant fashion [6, 7]. Evidently, the horizon problem can be resolved if we can have a phase during the early stages of radiation domination wherein the modes of cosmological interest are inside the Hubble radius. It is clear from the above discussion that this can be achieved, provided, during this period, the physical wavelengths of the modes decrease faster than the Hubble radius as we go back in time, i.e. if −d(λ P /d H )/dt < 0. This condition corresponds to ä > 0, and it is such an era of accelerated expansion that is referred to as inflation. In most models of inflation, such as the slow roll scenarios that have drawn constant attention, the Hubble radius remains approximately constant. As illustrated in Figure 1.2, this property allows us to bring the modes of cosmological interest inside the Hubble radius at early times. It can be shown that, in order to ensure that the forward light cone at decoupling is at least as large as the backward light cone, one requires the universe to expand by a factor of about 10 28 during inflation 1 . For convenience, the extent of inflation is often measured in terms of the number of e-folds N, which is defined as the logarithmic ratio of the scale factor at any given instant to its value at another fixed time. It can, in fact, be expressed as N = ∫ t t ∗ dtH = [ ] da a(t) t ∗ a = ln , (1.4) a(t ∗ ) ∫ t where t ∗ denotes some fixed time. Typically, one requires about 60–70 e-folds of inflation to overcome the horizon problem [6, 7]. As we shall soon discuss, the fact that the modes are inside the Hubble radius during the early phase of inflation allows us to impose well motivated initial conditions on the perturbations. 1 It should be clarified that the factor of10 28 that we have quoted is an approximate upper limit, and the actual number depends on the energy scale at which inflation takes place (in this context, see, for instance, Refs. [32]). 8
1.2. THE INFLATIONARY PARADIGM log (length) 1 λ 2 2 λ ( > λ ) ~ a ~ a ~ a d 2 H d H ~ constant Behaviour of Hubble radius in non−inflationary cosmology INFLATION log a(t) Radiation domination Figure 1.2: Evolution of the physical wavelength λ P (in green) and the Hubble radius d H (in blue) has been plotted as a function of the scale factor a on a logarithmic plot during the inflationary and the radiation dominated epochs. It is clear from the figure that a nearly constant Hubble radius (as is encountered in slow roll inflation) ensures that the modes emerge from inside the Hubble radius at a sufficiently early epoch, thereby resolving the horizon problem. 1.2.2 Driving inflation with scalar fields It is evident from the second of the Friedmann equations, viz. Eq. (1.2b), that one requires (ρ + 3p) < 0 for inflation, i.e. a period of accelerated expansion, to occur. Since neither matter corresponding to p m = 0 and radiation with p r = ρ r /3 (and, needless to add, a positive energy density) do not satisfy the condition, they cannot drive inflation. As we have alluded to before, it is scalar fields that are often invoked to achieve inflation [30]. Consider a single, canonical scalar field, say, φ, which is the dominant source for the expansion of the universe at a particular epoch. The energy density and pressure associated 9
Page 1: Primordial features and non-Gaussia
Page 5: Declaration This thesis is a presen
Page 9 and 10: Acknowledgments According to a popu
Page 11 and 12: Abstract Currently, inflation is th
Page 13 and 14: ABSTRACT tensors prove to be small
Page 15 and 16: ABSTRACT the best fit values arrive
Page 17 and 18: Contents 1 Introduction 1 1.1 The c
Page 19: CONTENTS 6 Effects of primordial fe
Page 23 and 24: List of Figures 1.1 A schematic tim
Page 25 and 26: Chapter 1 Introduction At a first g
Page 27 and 28: 2.725K. It is also found to be high
Page 29 and 30: 1.1. THE CONCORDANT, BACKGROUND COS
Page 31: 1.2. THE INFLATIONARY PARADIGM 1.2
Page 35 and 36: 1.2. THE INFLATIONARY PARADIGM wher
Page 37 and 38: 1.2. THE INFLATIONARY PARADIGM wher
Page 39 and 40: 1.2. THE INFLATIONARY PARADIGM The
Page 41 and 42: 1.2. THE INFLATIONARY PARADIGM yet
Page 43 and 44: 1.3. THE GENERATION AND IMPRINTS OF
Page 45 and 46: 1.5. THE FORMATION OF THE LARGE SCA
Page 47 and 48: 1.7. NOTATIONS AND CONVENTIONS resu
Page 49 and 50: Chapter 2 Generation of localized f
Page 51 and 52: 2.1. THE INFLATIONARY MODELS OF INT
Page 53 and 54: 2.2. METHODOLOGY, DATASETS, AND PRI
Page 55 and 56: 2.2. METHODOLOGY, DATASETS, AND PRI
Page 57 and 58: 2.3. BOUNDS ON THE PARAMETERS 2.3 B
Page 59 and 60: 2.3. BOUNDS ON THE PARAMETERS Model
Page 61 and 62: 2.4. THE SCALAR AND THE CMB ANGULAR
Page 63 and 64: 2.4. THE SCALAR AND THE CMB ANGULAR
Page 65 and 66: 2.5. DISCUSSION by the canonical sc
Page 67 and 68: Chapter 3 Non-local features in the
Page 69 and 70: 3.1. MODELS AND METHODOLOGY on the
Page 71 and 72: 3.1. MODELS AND METHODOLOGY values
Page 73 and 74: 3.2. RESULTS Datasets WMAP-7 WMAP-7
Page 75 and 76: 3.2. RESULTS 0.2 0.15 0.1 ∆ χ 2
Page 77 and 78: 3.2. RESULTS 7000 150 6000 100 50 l
Page 79 and 80: 3.3. DISCUSSION WMAP as well as the
Page 81 and 82: Chapter 4 Bi-spectra associated wit
Page 83 and 84:
4.1. MODELS OF INTEREST AND POWER S
Page 85 and 86:
4.1. MODELS OF INTEREST AND POWER S
Page 87 and 88:
4.2. THE SCALAR BI-SPECTRUM IN THE
Page 89 and 90:
4.3. THE NUMERICAL COMPUTATION OF T
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
4.4. RESULTS IN THE EQUILATERAL LIM
Page 107 and 108:
4.4. RESULTS IN THE EQUILATERAL LIM
Page 109 and 110:
Chapter 5 The scalar bi-spectrum du
Page 111 and 112:
5.1. BEHAVIOR OF BACKGROUND AND PER
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
5.2. THE CONTRIBUTIONS TO THE BI-SP
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
5.3. DISCUSSION significant during
Page 131 and 132:
Chapter 6 Effects of primordial fea
Page 133 and 134:
6.1. MODELS AND COMPARISON WITH THE
Page 135 and 136:
6.2. FROM THE PRIMORDIAL SPECTRUM T
Page 137 and 138:
6.3. THE NUMERICAL METHODS: ESSENTI
Page 139 and 140:
6.4. RESULTS Model Quadratic Quadra
Page 141 and 142:
6.4. RESULTS 6e-09 Quadratic potent
Page 143 and 144:
6.4. RESULTS 30 4 20 Change in R Ch
Page 145 and 146:
6.5. DISCUSSION wherein one works w
Page 147 and 148:
Chapter 7 Imprints of primordial no
Page 149 and 150:
7.1. FORMALISM and the LSS studies.
Page 151 and 152:
7.1. FORMALISM only mildly non-line
Page 153 and 154:
7.1. FORMALISM If the bi-spectrum c
Page 155 and 156:
7.2. RESULTS 0.12 0.13 2e3 , 2.2e3
Page 157 and 158:
7.3. CONCLUSIONS 7.3 Conclusions To
Page 159 and 160:
Chapter 8 Summary and outlook In th
Page 161 and 162:
8.2. OUTLOOK factorily [69, 70]. Ra
Page 163 and 164:
Bibliography [1] See,http://magnum.
Page 165 and 166:
BIBLIOGRAPHY [27] See,http://cfht.h
Page 167 and 168:
BIBLIOGRAPHY [55] F. Arroja, A. E.
Page 169 and 170:
BIBLIOGRAPHY [77] R. K. Jain, P. Ch
Page 171 and 172:
BIBLIOGRAPHY [103] R. Allahverdi, J
Page 173 and 174:
BIBLIOGRAPHY [132] S. Sasaki, Publ.
show all

PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

Create successful ePaper yourself

Delete template?

Save as template?