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Topics in Cosmology: Island Universes, Cosmological Perturbations and Dark Energy

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Topics in Cosmology: Island Universes, Cosmological Perturbations and Dark Energy TOPICS IN COSMOLOGY: ISLAND UNIVERSES, COSMOLOGICAL PERTURBATIONS AND DARK ENERGY by SOURISH DUTTA Submitted in partial fulfillment of the requirements for the degree Doctor of Philosophy Department of Physics CASE WESTERN RESERVE UNIVERSITY August 2007 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of ______________________________________________________ candidate for the Ph.D. degree *. (signed)_______________________________________________ (chair of the committee) ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ (date) _______________________ *We also certify that written approval has been obtained for any proprietary material contained therein. To the people who have believed in me. Contents Dedication iv List of Tables viii List of Figures ix Abstract xiv 1 The Standard Cosmology 1 1.1 Observational Motivations for the Hot Big Bang Model . 1 1.1.1 Homogeneity and Isotropy . 1 1.1.2 Cosmic Expansion . 2 1.1.3 Cosmic Microwave Background . 3 1.2 The Robertson-Walker Metric and Comoving Co-ordinates . 6 1.3 Distance Measures in an FRW Universe . 11 1.3.1 Proper Distance . 12 1.3.2 Luminosity Distance . 14 1.3.3 Angular Diameter Distance . 16 1.4 The Friedmann Equation . 18 1.5 Model Universes . 21 1.5.1 An Empty Universe . 22 1.5.2 Generalized Flat One-Component Models . 22 1.5.3 A Cosmological Constant Dominated Universe . 24 1.5.4 de Sitter space . 26 1.5.5 Flat Matter Dominated Universe . 27 1.5.6 Curved Matter Dominated Universe . 28 1.5.7 Flat Radiation Dominated Universe . 30 1.5.8 Matter Radiation Equality . 32 1.6 Gravitational Lensing . 34 1.7 The Composition of the Universe . 39 v 1.7.1 Measuring the Total Matter Content . 39 1.7.2 Ordinary and Dark Matter . 42 1.7.3 Supernovae and Cosmic Acceleration . 45 1.7.4 Cosmic Microwave Background Anisotropies . 49 1.8 Shortcomings of the Standard Cosmology . 58 1.8.1 The Flatness Problem . 58 1.8.2 The Entropy Problem . 59 1.8.3 The Horizon Problem . 59 1.8.4 The Monopole Problem . 60 1.9 Inflation . 62 1.9.1 Inflation and the Problems of the Standard Cosmology . 62 1.9.2 The Dynamics of Inflation . 64 1.9.3 The Slow Roll Parameters . 66 1.9.4 Models of Inflation . 67 1.9.5 Issues with Inflation . 69 2 Island Cosmology 76 2.1 Introduction . 76 2.2 The Model . 77 2.3 NEC violations in de Sitter space . 80 2.4 Extent and duration of NEC violation . 82 2.5 Likelihood – the role of the observer . 87 2.6 The NEC violating field . 89 2.7 Assumptions . 91 2.8 Conclusions . 94 3 Perturbation Spectra 98 3.1 Introduction . 98 3.2 The Theory of Cosmological Perturbations . 98 3.2.1 The metric perturbations . 99 3.2.2 Gauge Issues in Cosmology . 101 3.2.3 The Comoving Curvature Perturbation . 103 3.2.4 The Power Spectrum . 104 3.2.5 The Equations of Motion . 106 3.2.6 Quantum Theory of Cosmological Perturbations . 108 3.3 Difficulties in computing perturbations from Island Cosmology . 112 3.3.1 Classical vs Quantum Fields . 112 3.3.2 Backreaction on spacetime . 113 3.4 Perturbations in a Spectator Field . 117 3.5 A Classical Treatment of Island Cosmology . 120 vi 3.5.1 Introduction . 120 3.5.2 The de Sitter phase . 121 3.5.3 The Phantom Phase . 122 3.5.4 The Radiation Dominated FRW Phase . 125 3.5.5 Calculational Strategy . 125 3.6 Spectrum from a Classical Treatment . 127 3.6.1 vk in the de Sitter and FRW phases . 127 3.6.2 vk in the Phantom Phase . 128 3.6.3 Calculation of Unknown Constants . 135 3.6.4 Determination of the Scalar Power Spectrum . 138 3.6.5 Determination of the Tensor Power Spectrum . 138 3.7 Conclusion . 140 4 Dark Energy Voids 144 4.1 Introduction . 144 4.2 What is causing the acceleration? . 145 4.3 The model . 147 4.3.1 Linearization . 150 4.3.2 Potential . 153 4.3.3 Initial conditions . 153 4.4 Results . 155 4.4.1 Density contrasts . 155 4.4.2 Equation of state . 160 4.5 Discussion . 163 4.5.1 Void formation . 163 4.5.2 Generality . 165 4.6 Conclusions . 166 Bibliography 172 vii List of Tables 3.1 Scalar-Vector-Tensor decomposition of metric perturbations . 100 viii List of Figures 1.1 The spectrum of the CMB, as seen by COBE. http://lambda.gsfc. nasa.gov/product/cobe/firas_image.cfm .............. 4 1.2 Gravitational lensing geometry. The lens distorts the ”true” angles β (that would have been seen without any lensing effects) into the angles θ. Source [15] . 34 1.3 A flow chart showing the classifications of Supernovae http://rsd-www. nrl.navy.mil/7212/montes/snetax.html. 46 1.4 top panel: A Hubble diagram made from data from both the Supernova Cosmology Project and the High-z Supernova Search Team taken from [13].bottom panel The residual of the distances relative to a ΩM = 0.3, ΩΛ = 0.7 Universe. 50 1.5 Best fit regions in the (ΩM ,ΩΛ) plane for data from both the Super- nova Cosmology Project and the High-z Supernova Search Team. The agreement of the two experiments is remarkable. Source [13] . 51 1.6 The Dipole Anisotropy in the CMB as seen by COBE http://map. gsfc.nasa.gov/m_uni/uni_101Flucts.html. 52 1.7 The CMB spectrum once the dipole is subtracted. http://map.gsfc. nasa.gov/m_uni/uni_101Flucts.html. 53 1.8 The WMAP three-year power spectrum (in black) together with data from other recent experiments measuring the CMB angular power spec- trum. Taken from [44] . 56 1.9 A plot of length scale vs (logarithmic) scale factor in an FRW cosmol- ogy. The blue line shows the evolution of the Hubble scale. The red lines show the evolution of physical scales. Since the Hubble length evolves faster than the physical scales, sub-Horizon modes have never been in causal contact prior to Horizon entry. 61 1.10 Physical scales entering the horizon at the time of last scattering have been in causal contact before. 63 ix 2.1 Sketch of the behavior of the Hubble length scale with conformal time, η, in the Island model, and the evolution of fluctuation modes. At early times, inflation is driven by the presently observed dark energy, assumed to be a cosmological constant. As the cosmological constant is very small, the Hubble length scale is very large – of order the present horizon size. Exponential inflation in some horizon volume ends not due to the decay of the vacuum energy as in inflationary scenarios but due to a quantum fluctuation in the time interval (ηi, ηf ) that violates the null energy condition (NEC). The NEC violating quantum fluctu- ation causes the Hubble length scale to decrease. After the fluctuation is over, the universe enters radiation dominated FRW expansion, and the Hubble length scale grows with time. The physical wavelength of a −1 quantum fluctuation mode starts out less than HΛ at some early time ηi. The mode exits the cosmological horizon during the NEC violating fluctuation (ηexit) and then re-enters the horizon at some later epoch (ηentry) during the FRW epoch (The modes are drawn as straight lines for illustrative purposes only, they actually grow in proportion to the scale factor). 78 2.2 We show a classical de Sitter spacetime for conformal time η < ηP , that transitions to a faster expanding classical de Sitter spacetime for η > ηQ. The inverse Hubble size is shown by the white region. A bundle of ingoing null rays originating at point a is convergent initially but becomes divergent in the superhorizon region at point b. This can only occur if the NEC is violated in the region η ∈ (ηP , ηQ). In the quantum domain, a classical picture of spacetime may not be valid and this is made explicit by the question marks. 83 2.3 A spacetime diagram similar to that in Fig. 2.2 but one in which the NEC violation occurs over a sub-horizon region (shaded region in the diagram). Now the null ray bundle from a to b goes from being converg- ing (within the horizon) to diverging (outside the horizon). However, it does not encounter any NEC violation along its path, and this is not possible as can be seen from the Raychaudhuri equation. Since the ingoing null rays are convergent as far out as the point P , the size of the quantum domain has to extend out to at least the inverse Hubble size of the initial de Sitter space. Therefore the NEC violating patch has to extend beyond the initial horizon. 85 x 4.1 The evolution of the DDE overdensity δφ at the center of the matter perturbation, r = 0, with redshift (1+z).The scale of the perturbation is σHi = 0.01, and the mass is m/H0 = 1. Initially homogenous, the DDE develops an underdensity at late times in response to the matter perturbation. 156 4.2 Same as Figure 4.1, with the y-axis on a logarithmic scale. The DDE tends to cluster initially, but eventually forms a void. The kink in the plot signifies the change-over from positive to negative perturbation. 156 4.3 Logarithmic profiles of the matter density contrast log10 |δm| and the dark energy density contrast log10 |δφ|, at three different redshifts. Solid lines denote the DDE profiles, and dotted lines denote the matter profiles.158 4.4 DDE density contrast δφ at the center of the matter perturbation r = 0 as a function of the redshift (1 + z) for fixed mass m/H0 = 1 and different initial matter perturbations’ widths.
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