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Giang Dan 200911 Phd Thesis.Pdf The Study of Inhomogeneous Cosmologies Through Spacetime Matchings by Dan Giang A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto Copyright c 2009 by Dan Giang Abstract The Study of Inhomogeneous Cosmologies Through Spacetime Matchings Dan Giang Doctor of Philosophy Graduate Department of Physics University of Toronto 2009 Our universe is inherently inhomogeneous yet it is common in the study of cosmol- ogy to model our universe after the homogeneous and isotropic Friedmann-Lemaˆıtre- Roberson-Walker (FLRW) model. In this thesis spacetime matchings are applied to investigate more general inhomogeneous cosmologies. The Cheese Slice universe, constructed from matching together FLRW and Kasner regions satisfying the Darmois matching conditions, is used as a prime example of an inhomogeneous cosmology. Some observational consequences of this model are presented. The lookback time verses redshift relation is calculated using a numerical algorithm and it is shown that the relative thickness of the Kasner regions have the greatest impact on anisotropies an observer would see. The number of layers and distribution of layers play a smaller role in this regard. The relative thickness of the Kasner slice should be on the order of one ten thousandth the thickness of the FLRW regions to have the anisotropies fall within the observed CMB limit. The approach to the singularity of a spacetime matching is examined. A criterion is presented for a matched spacetime to be considered Asymptotically Velocity Term Dominated (AVTD). Both sides of the matching must be AVTD and each leaf of the respective foliations mush match as well. It is demonstrated that the open and flat Cheese Slice universe are both AVTD and the singularity is also of AVTD type. The Cheese Slice model is then examined as a braneworld construction. The possi- ii bility of a Cheese Slice brane satisfying all the energy conditions is shown. However, the embedding of such a brane into a symmetric bulk is non-trivial. The general embedding of a matched spacetime into a bulk is investigated using a Taylor series approximation of the bulk. It is found that the energy-momentum tensor of such a brane cannot have discrete jumps if the embedding does not have a corner. A 3+1+1 decomposition of the bulk spacetime is then carried out. With the spacetime being deconstructed along two preferred timelike hypersurfaces, this becomes a natural environment to discuss the matching of branes. We find that there are conditions on the matter content of the branes to be matched if an observer on the brane is to see the matching surface as a boundary surface with no additional stress energy. Matching more than two bulks is also examined and shown to allow for more general brane configurations. iii Dedication For grandma. iv Acknowledgements I would like to thank my supervisor, Charles C. Dyer, for his guidance and financial support toward the completion of this work and his encouragement along the way. Thanks to my committee members, Michael Luke and Stefan Mochnacki for their helpful feedback. Special thanks to the external examiner, Charles Hellaby, for going through the thesis with a fine toothed comb. To my colleagues in physics Megan McClure, Allen Attard, Johann Bayer, Brian Wilson, Mitch Thomson, Parandis Khavari, and Alex Venditti, thanks for paving the way in front of me and showing me how far I’ve come. I am grateful to my family: my grandma, mom, dad, Amy, Lauren, numerous cousins, uncles, and aunts, for their support even though they had no idea what I was studying. Thanks to my friends from Calgary for being a seamless extension of my family: Bernie, Christine, Duffy, Ellen, Emil, Emily, Hai, Hy, Jeff, Jenny, Joanne, John, Joyce, Justin, Lan, Maelynn, Monica, Paul, Phuoc, Rishi, Rosita, Sam, Susan, Susan, Tri, Vincent, and Vivian. Thanks to my friends in Toronto for being my family away from family: Alex, Athar, Ben, Ben, Beth, Brenda, Chad, Davin, Ela, Elanna, Geoff, George, Heather, House, Irena, Janna, Jean-Sebastien, Jenn, Jenny, Julia, Juliet, Karen, Kari, Karine, Kevin, Kristina, Linda, Lisa, Matt, Moiya, Nisha, Patty, Pascal, Robynne, Sapna, Simon, So, Sola, Staveley, and Stephanie. Thanks to the staff and fellows of Massey College for shaping my first few years of life as a Ph.D. student. A special thanks goes out to Mario Nawrocki and Margaret Huntley for the special roles they played. The financial support was provided by the Natural Sciences and Engineering Research Council of Canada and the Department of Physics at the University of Toronto. v Contents 1 Introduction 1 1.1 Whyassumehomogeneity?.......................... 1 1.2 TheInhomogeneousUniverse . 4 1.3 Aboutsingularities .............................. 7 1.4 AboutSpacetimeMatchings ......................... 9 1.5 AboutBraneworlds .............................. 10 1.6 TheStorytoCome .............................. 10 2 Matching Regions of Spacetimes 12 2.1 ReviewofMatchingConditions . 13 2.1.1 Matchings Across a Boundary Surface . 15 2.1.2 MatchingataCorner.. .. .. ... .. .. .. .. .. ... .. 18 2.1.3 Matchings Across Thin Shells . 22 2.1.4 NullMatchings ............................ 24 2.2 TheCheeseSliceUniverse .......................... 25 3 Lookback Time and Observational Consequences 30 3.1 Preliminaries ................................. 31 3.1.1 NullVectors.............................. 31 3.1.2 BendingAngles ............................ 33 3.1.3 Calculating the Redshift and Lookback Time . 35 vi 3.1.4 NumericalAlgorithm . 37 3.2 Results..................................... 38 3.2.1 Lookback Time and Redshift Relations . 38 3.2.2 PossibleCMBData.......................... 42 3.3 SummaryandDiscussion........................... 47 4 The Structure of the Singularity 48 4.1 DefinitionofaSingularity . 49 4.2 Classification of Singularities . 50 4.2.1 IsotropicSingularities. 50 4.2.2 ClassificationScheme. 51 4.2.3 StrengthofaSingularity . 52 4.3 MoreGeneralSingularityStructures . 52 4.3.1 PropertiesoftheMatching. 52 4.3.2 BKL Picture of Cosmological Singularities . 53 4.3.3 Cauchy Horizon Singularities . 55 4.4 TheAVTDSingularity ............................ 55 4.4.1 Definitions............................... 56 4.4.2 AVTD Property of Matched Spacetimes . 59 4.5 Singularities in the Cheese Slice Universe . 60 4.5.1 Case (i) Flat FLRW, k =0...................... 61 4.5.2 Case (ii) Open FLRW, k = 1.................... 62 − 4.5.3 SummaryandDiscussion. 65 5 Cheese Slice Braneworlds 68 5.1 BraneworldCosmologies ........................... 69 5.1.1 Randall-SundrumBraneworlds. 69 5.1.2 Cosmological Braneworlds . 70 vii 5.1.3 AnisotropicBraneworlds . 75 5.1.4 BraneCollisions............................ 77 5.2 ConstructinganInhomogeneousBrane . 78 5.2.1 TheCheeseSliceBrane. 79 5.2.2 EnergyConditions .......................... 81 5.3 ExtendingtheMatchingintotheBulk . 86 5.3.1 The Bulk of the Cheese Slice Brane . 88 5.4 GeneralEmbeddingofMatchedBranes . 92 5.4.1 Set-up ................................. 92 5.4.2 AnEmbeddingWithnoCorners . 93 5.4.3 TheBulkMatchingSurface . 94 5.4.4 ApproximationoftheBulk. 96 5.4.5 MatchingtheBulk .......................... 97 5.4.6 Consequences of Assuming No Corner . 100 5.5 The3+1+1Decomposition. 101 5.5.1 Defining the Normals, Bases and Metrics . 102 5.5.2 FixingtheCoordinates . 108 5.5.3 FindingtheMetrics. 110 5.5.4 TheBulkMetric ........................... 112 5.6 TheMatchingoftheBulk .......................... 115 5.6.1 TheMatchingConditions . 115 5.6.2 The Second Fundamental Form and Matter Content . 118 5.6.3 MatchingFourBulks . 122 5.6.4 Special Cases: Breaking the Angle Condition . 124 5.7 SummaryandDiscussion. 126 6 Summary and Conclusions 128 viii A Taylor Expansion of a Tensor Field 132 Bibliography 135 ix List of Tables 5.1 Positivity of matter density, ρ, as a function of cosmological time, t. 83 x List of Figures 1.1 Thecubiclatticeuniverse. 6 2.1 The construction of a matched spacetime. 14 2.2 Matchingacrossacorner. .......................... 19 2.3 Findingthecornerconditions. 20 2.4 Illustrations of the Cheese Slice universe with (a) flat FLRW slices and (b) openFLRWslices............................... 28 3.1 Light ray propagating through different regions. ....... 35 3.2 Lookback time and redshift relation for an Einstein de Sitter model. 38 3.3 Lookback time and redshift relation for a large Kasner region. ...... 39 3.4 Lookback time and redshift relation for a three slice model ........ 40 3.5 Average lookback time and redshift relation for different models. 41 3.6 RedshiftoftheCMBfordifferentmodels. 43 3.7 Changing the position of a thin Kasner slice in a predominantly FLRW model...................................... 44 3.8 Changing the thickness of a thin Kasner slice in a predominantly FLRW model...................................... 45 3.9 Changing the number of Kasner slices while keeping the total ratio of KasnertoFLRWconstant.. 46 4.1 Spacelike foliation of a spacetime. 57 xi 4.2 Matching two leaves of the foliations across Σ. ...... 60 4.3 The Singularities of the Cheese Slice Model. ..... 66 5.1 Regions in which ρ ispositive......................... 83 5.2 RegioninwhichtheSECissatisfied. 85 5.3 RegioninwhichtheDECissatisfied. 87 5.4 Matching of two branes extended into the bulk. ..... 98 5.5 DefinitionoftheZ-Wplane. 103 5.6 Illustration of the matching conventions that are being used. ....... 104 5.7 Matchingfourdifferentbulks. 123 5.8 Assumeonesideofthebraneisavacuum. 125 xii Chapter 1 Introduction
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