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Cosmology III - Cosmological Perturbation Theory
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1 The Perturbed Universe
1.1 Abstract
1.2 A Cosmological Field
1.3 Remarks
1.4 Homogeneous and Isotropic Universe
1.5 The Background 3-Space Metric
1.6 The Minkowski Metric
1.7 The Background Metric
1.8 Inhomogeneous and Anisotropic Universe
1.9 The First Equation
1.10 The Spatial Average
1.11 More than One Coordinate Systems
1.12 Some Optional Fun
1.13 A Perturbed Gauge
1.14 Two Perturbed Gauges
1.15 The Transformation Vector
1.16 One Set of Coordinates
1.17 The First Equation in Gauge 1 and Gauge 2
1.18 The First Equation for a Scalar Field
1.19 The First Equation for a Vector Field
1.20 The First Equation for a Tensor Field
2 Some Helpful Equations
2.1 Abstract
2.2 The First Theorem
2.3 The Background Vectors
2.4 The Background Tensors
2.5 A Useful Approximation
2.6 Two More Approximantions
2.7 Another Approximation
2.8 Relating the Coordinates of P1 and P2
2.9 The Transformation Matrices
2.10 Linear Perturbation Theory
3 Transformations at One Point
3.1 Abstract
3.2 From General Relativity
3.3 Vector Transformation
3.4 Tensor Transformation
4 Review 1
4.1 Review 1
5 Transformations at One Set of Coordinates
5.1 Abstract
5.2 A Useful Relationship
5.3 Scalar Field
5.4 Vector Field
5.5 Tensor (0,2) Field
5.6 Tensor (1,1) Field
5.7 Tensor (2,0) Field
5.8 Notation Simplification
6 Review 2
6.1 Review 2
7 Introduction to Linear Decomposition
7.1 Abstract
7.2 An Example from High School Physics
7.3 An Example from Undergraduate Physics
8 Linear Decomposition of Scalar Perturbations
8.1 Abstract
8.2 Fourier Decomposition
8.3 Useful Fourier Expressions
8.4 Dropping the Subscript k in the Fourier Space
9 Linear Decomposition of Vector Perturbations
9.1 Abstract
9.2 Helmholtz Decomposition-1 (Introduction)
9.3 Helmholtz Decomposition-2 (Longitudinal Pa...
9.4 Helmholtz Decomposition -3 (Wave Vector al...
10 Linear Decomposition of Tensor(0,2) Perturbations
10.1 Abstract
10.2 3X3 Trace and Traceless Parts
10.3 SVT Decomposition
10.4 Symmetric and Traceless Scalar Component
10.5 Symmetric and Traceless Vector Component
10.6 Symmetric and Traceless Tensor Component
10.7 Fourier Decomposition of 00 (Scalar)
10.8 Fourier Decomposition of 0i and i0 (Scalar...
10.9 Fourier Decomposition ij (Scalar, Vector a...
10.10 The Wave Vector along the z-axis
10.11 4 x 4 SVT Matrices
11 Linear Decompositions and Gauge Transformations
11.1 Introduction
11.2 00 Component
11.3 0i and i0 Components
11.4 ij Components
11.5 Decomposition Strategy 1
11.6 Decomposition Strategy 2
11.7 Gauge Freedom
11.8 Gauge Independent Scalar Potentials
12 Review 3
13 Metric Tensor
13.1 Non-Expanding Background Universe
13.2 Expanding Background Universe
13.3 Exercise
13.4 Perturbed Non-Expanding Universe
13.5 Perturbed Expanding Universe
13.6 Decomposition of B_i
13.7 Decomposition of h_ij
13.8 SVT Decomposition
13.9 Gauge Transformation of Scalars
13.10 Exercise
13.11 Inverse of Perturbed Metric
13.12 Exercise
13.13 Exercise
13.14 Gauge Independent Scalar Potentials
13.15 Exercise
14 Connection Coefficients
14.1 Six Components
15 Fluid Velocity
15.1 Proper Time
15.2 Three-Velocity
15.3 Four-Velocity
15.4 Exercise
15.5 Background Universe
15.6 Exercise
15.7 Perturbed Universe
15.8 Exercise
16 Threads, Slices and Worldlines
16.1 Threads and Slices
16.2 Orthogonal Worldlines
16.3 Shift and Lapse
17 Conformal Newtonian Gauge
18 Perturbed Curvature
19 Perturbed Energy-Momentum Tensor
20 Einstein Equations
21 Review 4
22 xxxThe Background Universe
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Teacher: Simerdeep Singh
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Simerdeep Singh
mycosmicschool@gmail.com
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