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Nonlinear perturbations in Relativistic cosmology
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.advisor | 이수종 | - |
| dc.contributor.author | 변상규 | - |
| dc.date.accessioned | 2017-07-19T06:07:44Z | - |
| dc.date.available | 2017-07-19T06:07:44Z | - |
| dc.date.issued | 2015-02 | - |
| dc.identifier.other | 000000026803 | - |
| dc.identifier.uri | http://dcollection.snu.ac.kr:80/jsp/common/DcLoOrgPer.jsp?sItemId=000000026803 | - |
| dc.description | 학위논문(박사)--서울대학교 대학원 :자연과학대학 물리·천문학부,2015. 2. 이수종. | - |
| dc.description.abstract | In this dissertation, we discuss the inhomogeneity growth in the universe.
The inhomogeneity growth is originated from the gravitational instability and we consider the Newtonian and the Einstein gravity in this dissertation. In the Newtonian cosmology based on the Newtonian gravity, we derive the nonlinear order solutions with the nonlinear perturbation theory, and show the next-leading-order power spectrum. The next-leading-order Newtonain power spectrum is effective on the small scale. However, the Newtonian cosmology is only appropriate to study subhorion scale. To overcome the limitation of the Newtonian cosmology, we need to study the Relativistic cosmology based on the Einstein gravity. In the Relativistic, we discuss the gauge issue and show that the proper time gauge is relevant to study the universe. We also derive the higher solutions in the proper time gauge and show the next-leading-order power spectra. From the synchronous gauge nonlinear order study, we conclude that the synchronous gauge is improper and the comoving gauge is proper to study nonlinear perturbation. Furthermore, we show the next-leading-order bispectrum in the comoving guage and the relativistic effect is much smaller than the Newtonian results. At the end or this dissertation, we study the Dark energy effect on the (nonlinear)inhomogeneity growth, and the Dark energy affects significantly the Relativistic next-leading-order power spectrum rather than the Newtonian power spectrum. | - |
| dc.description.tableofcontents | 1 Introduction 1
2 Homogeneous Universe 7 2.1 Cosmological Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Friedmann-Robertson-Walker Metric . . . . . . . . . . . . . . . . . . . . . 9 2.3 Behavior of neighboring geodesics . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Friedmann equation : Dynamics of the Universe . . . . . . . . . . . . . . . 12 2.5 Application Tool, Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5.1 Deceleration Parameter . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.2 Look-back time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5.3 Comoving Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5.4 Luminosity Distance . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.5 Angular Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Standard Perturbation theory in the Newtonian Cosmology 23 3.1 Basic hydrodynamical equations . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Main perturbation equations in the Newtonian cosmology . . . . . . . . . . 25 3.3 Linear Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Nonlinear Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4.1 Second Order solution . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4.2 Third Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.5 Cosmological applications of the nonlinear solutions . . . . . . . . . . . . 31 3.5.1 From Linear to Nonlinear . . . . . . . . . . . . . . . . . . . . . . . 32 3.5.2 Next-leading-order matter Power spectrum . . . . . . . . . . . . . . 35 3.6 Disscusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 Perturbation Theory on the Relativistic cosmology 39 4.1 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1.1 Metric perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1.2 Energy momentum tensor perturbation . . . . . . . . . . . . . . . . 40 i Contents ii 4.1.3 The 4-Velocity perturbation . . . . . . . . . . . . . . . . . . . . . . 41 4.1.4 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Linear Perturbation Equations . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2.1 Scalar perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.2 Vector Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2.3 Tensor Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3 Strategy of solving the perturbation equations . . . . . . . . . . . . . . . . 45 4.4 Gauge Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.4.1 Category of gauge transformation . . . . . . . . . . . . . . . . . . . 47 4.4.2 Gauge-Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 Proper time gauge Vs Non-proper time gauge . . . . . . . . . . . . . . . . 49 4.5.1 Proper time gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.2 Non-Proper time gauge . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.5.3 Leading order matter Power Spectra in the Relativistic cosmology . 57 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5 Nonlinear perturbation of the proper time Gauge 61 5.1 Fully non-linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2 Third-order perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2.1 Comoving gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2.2 Synchronous gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3 Power spectra in the synchronous gauge . . . . . . . . . . . . . . . . . . . 68 5.4 Convective derivative interpretation of the synchronous gauge time derivative 73 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6 Non-linear matter bispectrum in general relativity 79 6.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2 One-loop bispectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2.1 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2.2 Tree bispectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.2.3 One-loop bispectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2.3.1 B(1) 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2.3.2 B(1) 114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2.3.3 B(1) 123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.4 Tree bispectrum in other gauges . . . . . . . . . . . . . . . . . . . . . . . . 89 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7 1-loop Powerspectrum in the General Dark energy Model 95 7.1 Perturbation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.2 Second Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.3 Third Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.4 Evolutions of Coe cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.5 1-loop Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Contents iii 7.5.1 The Newtonian 1-loop power spectrum . . . . . . . . . . . . . . . . 106 7.5.2 Relativistic 1-loop powerspectrum . . . . . . . . . . . . . . . . . . . 108 7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 8 Summary and Outlook 113 A ADM Vs Covariant Formalism 117 A.1 Irrotational Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 A.2 Covariant Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 A.2.1 The Gauss Relation(Energy Constraint) . . . . . . . . . . . . . . . 121 A.2.2 The Codazzi Relation(Momentum Constraint) . . . . . . . . . . . . 123 A.2.3 The Raychaudhuri Equation . . . . . . . . . . . . . . . . . . . . . . 123 A.2.4 The Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . 124 A.2.5 The Momentum Conservation . . . . . . . . . . . . . . . . . . . . . 125 A.3 ADM Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.3.0.1 The Energy constraint . . . . . . . . . . . . . . . . . . . . 125 A.3.1 The Momentum Constraint . . . . . . . . . . . . . . . . . . . . . . 126 A.3.2 The ADM Propagation Equation . . . . . . . . . . . . . . . . . . . 126 A.3.3 The Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . 127 A.3.4 The Momentum Conservation . . . . . . . . . . . . . . . . . . . . . 127 A.3.5 The Tracefree ADM propagation equation . . . . . . . . . . . . . . 128 A.4 Linear Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 B Non-Linear Gauge Transformation in General Relativity 131 B.1 Di eomorphism Vs Coordinate Transformation . . . . . . . . . . . . . . . 131 B.2 Pull-Back( ) and Push-Forward( ) . . . . . . . . . . . . . . . . . . . . . 132 B.3 De nition of Perturbation in the Relativistic cosmology . . . . . . . . . . . 133 B.4 The Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 B.5 Tensor gauge transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 134 C Stochastic Properties 137 C.1 Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 C.2 Fourier Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 C.3 Correlation Function and Averages . . . . . . . . . . . . . . . . . . . . . . 138 C.4 Ergodic Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 D Gauge transformation 141 D.1 Gauge transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 D.2 Comoving gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 D.3 Synchronous gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 D.4 From CG to SG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 D.5 From SG to CG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 E Mode analysis in the synchronous gauge 145 Contents iv E.1 Linear order solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 E.2 Higher order solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 E.3 Second order solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 E.4 Third order solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 F Newtonian non-linear power spectrum in the Eulerian and Lagrangian frames 151 F.0.1 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 G Relativistic Nonlinear Solutions in the Comoving Gauge 155 G.1 EdS universe Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 G.1.1 Third Order Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 155 G.1.2 Fourth Order Solution . . . . . . . . . . . . . . . . . . . . . . . . . 156 G.2 General Dark energy Universe Solution . . . . . . . . . . . . . . . . . . . . 158 Bibliography 161 | - |
| dc.format.extent | viii,166 | - |
| dc.language.iso | kor | - |
| dc.publisher | 서울대학교 대학원 | - |
| dc.subject | Relativistic cosmology, Nonlinear perturbation theory, Dark | - |
| dc.subject.ddc | 523 | - |
| dc.title | Nonlinear perturbations in Relativistic cosmology | - |
| dc.type | Thesis | - |
| dc.type | Dissertation | - |
| dc.contributor.department | 자연과학대학 물리·천문학부 | - |
| dc.description.degree | Doctor | - |
| dc.date.awarded | 2015-02 | - |
| dc.identifier.holdings | 000000000021▲000000000023▲000000026803▲ | - |
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