PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

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CHAPTER 6. EFFECTS OF PRIMORDIAL FEATURES ON THE FORMATION OF HALOS 6.1.1 Comparison with the CMB and the LSS data We compare the models with the CMB as well as the LSS data. We have worked with the WMAP-7 data [18] and the halo power spectrum data arrived at from the LRG in SDSS DR7 [4]. We have made use of the LSS data to ensure that the parameter values we eventually work with to obtain the formation rate of the halos are consistent with the observed matter power spectrum. As discussed in Chapters 2 and 3, we couple the code we have developed for computing the inflationary perturbation spectrum to CAMB [44, 43] and COSMOMC [45, 46] to arrive at, not only the CMB angular power spectrum, but also the corresponding matter power spectrum, in order to compare them with the data. We have assumed the background to be the spatially flat ΛCDM model and have worked with the same priors as we have done in our earlier analysis [cf. Chapters 2 and 3]. As far as the priors on the inflationary parameters are concerned, for the case of the quadratic potential with the step, we have worked with the same priors that we had mentioned in Chapter 2. In the case of the two potentials with oscillatory terms, viz. the quadratic potential superimposed with sinusoidal oscillations (3.1) and the axion monodromy model (3.2), we have worked with the same priors on the primary parametersmandλas we had done in Chapter 3. However, as should be evident from Table 6.1, we have widened the priors of the parameters α and β for these potentials, when compared to the values listed in Table 3.1. We should add that we have allowed the phase parameterδ to vary from −π toπ as before. The motivations for choosing these priors are two fold. Firstly, the parameters m and λ determine the amplitude of the scalar power spectrum. We find that choosing them to be close to their COBE normalized values allows Model Potential Lower Upper parameter limit limit Quadratic potential α 0 2×10 −3 with sine modulation ln(β/M Pl ) −3.9 0 Axion monodromy α 0 2×10 −4 model ln(β/M Pl ) −8 0 Table 6.1: The priors that we work with on the parametersαandβ which characterize the potentials that contain oscillatory terms [cf. Eqs. (3.1) and (3.2)]. Note that the priors are wider than the priors we had chosen in Chapter 3 (cf. Table 3.1). Also, to help us cover a larger range, we have worked with the logarithmic value ofβ. 110
6.2. FROM THE PRIMORDIAL SPECTRUM TO THE FORMATION RATE OF HALOS for a faster convergence of the Markov Chains. Secondly, we have chosen the priors on α and β such that the scalar field does not get trapped by the oscillations in the potential. We should also note that unlike in Chapters 2 and 3, we have not included the tensor perturbations, as the corresponding effects are negligible. Moreover, as we have discussed in Chapter 3, in the case of potentials with oscillations, which lead to fine features in the scalar power spectrum, we will have to modify CAMB in order to ensure that the CMB angular power spectrum is evaluated for every multipole, and compare them with the data [98, 101]. But, we have not implemented this point here since we are only interested in the marginalized probabilities of the potential parameters. These probabilities shall indicate the extent to which deviations from a nearly scale invariant spectrum is allowed by the data, which we shall then make use of to study the corresponding effects on the formation of dark matter halos. We should also mention that we not taken the nonlinear [127] effects on the matter power spectrum, but have included the SZ effect and the effects due to gravitational lensing in our analysis. Finally, as in our earlier analysis, we shall set the Gelman and Rubin parameter|R−1| to be0.03 for convergence in all the cases. 6.2 From the primordial spectrum to the formation rate of halos In this section, we shall quickly outline the standard formalism to arrive at the formation rate of halos from the primordial power spectrum. 6.2.1 The matter power spectrum Recall that the matter power spectrum at a given redshiftP M (k,z) is related to the primordial power spectrumP S (k) through the expression (1.30). Evidently, given the primordial spectrum, we require the CDM transfer function T(k) and the growth factor D + (z) to arrive at the matter power spectrum. If we define D + (a) = g(a)/a, then, one finds that, in the spatially flat ΛCDM model, the quantity g satisfies the differential equation [18, 128] d 2 g dlna 2 + 1 2 [5+3Ω Λ(a)] dg dlna +3Ω eff(a)g = 0, (6.1) where Ω eff (a) = Ω Λ H 2 0 /H2 , with Ω Λ denoting the dimensionless density parameter associated with the cosmological constant today. We shall solve this differential equa- 111
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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
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3.2. RESULTS 7000 150 6000 100 50 l
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3.3. DISCUSSION WMAP as well as the
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Chapter 4 Bi-spectra associated wit
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Page 87 and 88: 4.2. THE SCALAR BI-SPECTRUM IN THE
Page 89 and 90: 4.3. THE NUMERICAL COMPUTATION OF T
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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
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Page 131 and 132: Chapter 6 Effects of primordial fea
Page 133: 6.1. MODELS AND COMPARISON WITH THE
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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