PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

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CHAPTER 5. THE SCALAR BI-SPECTRUM DURING PREHEATING This expression can also be rewritten in terms of the parameters describing the postinflationary evolution. We obtain that [117] f eq (k) = 115ǫ 1 (γ NL 288π Γ2 + 1 ) 2 2γ+1 (2γ +1) 2 sin(2πγ) |γ +1| −2(γ+1) [ 1−e ] −3(N f−N e)/2 2 [ (π ) ] 2 −1/4 g ∗ × (1+z eq ) 1/4 ρ 1/4 −(2γ+1) ( ) −(2β+1) c k , (5.41) 30 T rh a 0 H 0 where g ∗ denotes the effective number of relativistic degrees of freedom at reheating, T rh the reheating temperature andz eq the redshift at the epoch of equality. Also, recall that,ρ c , a 0 andH 0 represent the critical energy density, the scale factor and the Hubble parameter today, respectively. The above expression is mainly determined by the ratio ρ 1/4 c /T rh . For a model with γ ≃ −2 and a reheating temperature of T rh ≃ 10 10 GeV, one obtains that f NL ∼ 10 −60 for the modes of cosmological interest (i.e. for k such that k/a 0 ≃ H 0 ), a value which is completely unobservable. This confirms and quantifies our result that, in the case of single field inflation, the epoch of preheating does not alter the amplitude of the scalar bi-spectrum generated during inflation [118]. However, it is worthwhile to add that, while the amplitude of the above non-Gaussianity parameter f NL is small, it seems to be strongly scale dependent. We believe that a couple of points require further emphasis at this stage of our discussion. Recall that, to fall within the first instability band during preheating, the modes need to satisfy the condition (5.15). But, in order to neglect the term involving k 2 in the differential equation (5.11), the modes of interest are actually required to satisfy the condition (5.16). As we have emphasized earlier, evidently, the condition (5.16) will be easily satisfied by the large scale modes that already lie within the instability band and thereby satisfying the condition (5.17). Therefore, it is important to appreciate the fact that the conclusions we have arrived at above apply to all cosmologically relevant scales. 5.3 Discussion In this chapter, we have analyzed the effects of preheating on the primordial bi-spectrum in inflationary models involving a single canonical scalar field. We have illustrated that, certain contributions to the bi-spectrum, such as those due to the combination of the fourth and the seventh terms and that due to the second term, vanish identically at late times. Further, assuming the inflationary potential to be quadratic around its minimum, we have shown that the remaining contributions to the bi-spectrum are completely in- 104
5.3. DISCUSSION significant during the epoch of preheating when the scalar field is oscillating at the bottom of the potential immediately after inflation. It is important to appreciate the fact that the results we have arrived at apply to any single field inflationary potential that has a parabolic shape near the minimum. A couple of other points also need to be stressed regarding the conclusions we have arrived at. The results we have obtained supplement the earlier results wherein it has been shown that the power spectrum generated during inflation remains unaffected during the epoch of preheating (see, Ref. [62]; in this context, also see Ref. [119]). Moreover, our results are in support of earlier discussions which had pointed to the fact that the contributions to the correlation functions at late times will be insignificant if the interaction terms in the actions at the cubic and the higher orders depend on either a time or a spatial derivative of the curvature perturbation [120]. Broadly, our effort needs to be extended in two different directions. Firstly, it is important to confirm that the conclusions we have arrived at hold true for potentials which behave differently, say, quartically, near the minima. Further, the exercise needs to be repeated for models involving non-canonical scalar fields [121, 122]. In this context, it is worth mentioning that the generalization of the conserved quantityR k in the Dirac-Born- Infeld case has been shown to stay constant in amplitude on scales larger than the sonic horizon, a property which allows us to propagate the spectrum from horizon exit till the beginning of the radiation dominated era [121]. Secondly, as we had mentioned, preheating is followed by an epoch of reheating when the energy from the inflaton is expected to be transferred to radiation. It will be interesting to examine the evolution of the bispectrum during reheating. However, in order to achieve reheating, the scalar field needs to be coupled to radiation. It is clear that the formalism for evaluating the bi-spectrum involving just the inflaton is required to be extended to a situation wherein radiation too is present and is also coupled to the scalar field. However, possibly, the most interesting direction opened up by our work concerns multi-field inflation and associated non-Gaussianities [123]. Unlike single field models wherein the curvature perturbation associated with the large scale modes is conserved at late times, such a behavior is not necessarily true in multi-field inflation. When many fields are present, the entropy (i.e. the iso-curvature) fluctuations can cause the evolution of curvature perturbations even on super-Hubble scales. Further, in the case of multi-field inflation, it is known that the two-point correlation function can be affected by metric preheating [124]. In other words, the power spectrum calculated at the end of multi-field inflation is not necessarily the power spectrum observed in, say, the CMB data because 105
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Primordial features and non-Gaussia
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Declaration This thesis is a presen
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Acknowledgments According to a popu
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Abstract Currently, inflation is th
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ABSTRACT tensors prove to be small
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ABSTRACT the best fit values arrive
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Contents 1 Introduction 1 1.1 The c
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CONTENTS 6 Effects of primordial fe
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List of Figures 1.1 A schematic tim
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Chapter 1 Introduction At a first g
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2.725K. It is also found to be high
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1.1. THE CONCORDANT, BACKGROUND COS
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1.2. THE INFLATIONARY PARADIGM 1.2
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1.2. THE INFLATIONARY PARADIGM log
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1.2. THE INFLATIONARY PARADIGM wher
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1.2. THE INFLATIONARY PARADIGM wher
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1.2. THE INFLATIONARY PARADIGM The
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1.2. THE INFLATIONARY PARADIGM yet
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1.3. THE GENERATION AND IMPRINTS OF
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1.5. THE FORMATION OF THE LARGE SCA
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1.7. NOTATIONS AND CONVENTIONS resu
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Chapter 2 Generation of localized f
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2.1. THE INFLATIONARY MODELS OF INT
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2.2. METHODOLOGY, DATASETS, AND PRI
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2.2. METHODOLOGY, DATASETS, AND PRI
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2.3. BOUNDS ON THE PARAMETERS 2.3 B
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2.3. BOUNDS ON THE PARAMETERS Model
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2.4. THE SCALAR AND THE CMB ANGULAR
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2.4. THE SCALAR AND THE CMB ANGULAR
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2.5. DISCUSSION by the canonical sc
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Chapter 3 Non-local features in the
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3.1. MODELS AND METHODOLOGY on the
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3.1. MODELS AND METHODOLOGY values
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3.2. RESULTS Datasets WMAP-7 WMAP-7
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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 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 123 and 124: 5.2. THE CONTRIBUTIONS TO THE BI-SP
Page 125 and 126: 5.2. THE CONTRIBUTIONS TO THE BI-SP
Page 127: 5.2. THE CONTRIBUTIONS TO THE BI-SP
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.
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PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

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