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10 1 1e+12 1e+10 1e+08 1e+06 10000 100 τ·(Ha) Figure4.3: Ha≈τ −1 approximation Haτisplotted over theredshift z,taken fromacmbeasyrun. Theapproximationisexactaslong asthe plotted line equals one. When matter begins to dominate, the exactness breaks down. It further deviates during the stageofvacuum domination. The scaleλ=τ exceeds thehorizon three times then. 28 4.9 TheHorizon Scale z 1 0.01
5 Cosmological Perturbation Theory Attention! The dot denotes no longer a deviation with respect to the physical time ˙a= da , but a deviation with respect to the dt conformal time: !!! ˙a= da dτ !!! From now on. (5.1) The physics, described in this chapter are be closely related to [4], [25] and [26], becausecmbeasy is used for the investigation of the perturbation evolution. In the most times, the space-like curvature will be neglected. 5.1 Perturbationsofthebackgroundquantities 5.1.1 Metric perturbations The unperturbed metric ¯g µν of eq. 4.37 can be perturbed by adding a small deviation a 2 h µν. Usually the general ansatz for the perturbed metric is: h µν d x µ d x ν = −2Adτ 2 − 2V idτd x i + 2H i jd x i d x j (5.2) g µν = ¯g µν+ a 2 h µν , (5.3) where A is a scalar field, V i =φ |i+ B i is a vector field, decomposed into a gradient of a scalarφ |i and a transverse, rotational vector field B i, B |i i = 0. The tensor field � Hi j= HLγ i j+ ∇i∇ j− 1 3 γ � i j∆ HT+ 1� 2 H (V) i|j + H(V) j|i � + H (T) i j can be decomposed into a trace generating scalar HL, a ’double gradient’ scalar HT, a transverse vector H (V) i , H (V)|i i = 0 and a transverse traceless tensor H (T) i j , H (T)|j i j = 0, H(T)i i = 0. The scalars contribute to one degree of freedom, the vectors to 3−1 constraint = 2 degrees of freedom and the tensor to 6−(3+1) constraints = 2, yielding a total number of 1·4 scalar + 2·2 vector + 2·1 tensor = 10 degrees of freedom. This corresponds to n(n+1) = 10 independent parameters in 2 the 4-metric of the physical space-time. A harmonic decomposition can be made by using the laplacian eigenfunctions: ∆Y=−k 2 Y , ∆Y (V) j for each eigenvalue / wave vector k, and defining: With this, the decomposition is of the form: =−k2 Y (V) j , ∆Y (T) ji =−k2 Y (T) ji Yj :=−k −1 Y |j , Yi j :=−k −2 Y |i j+ 1 3 γi jY , Y (V) i j :=− 1� 2k Y (V) i|j + Y(V) j|i (5.4) (5.5) � . (5.6) A = AY (5.7) V i H i j Altogether the scalar perturbed metric reads: g µν= a 2 = BY i+ B (V) Y (V) i = HLYγ i j+ HT Yi j+ H (V) Y (V) i j + H(T) Y (T) i j � −1−2AY −BYi −BY j (1+2H LY)γ i j− 2H T Y i j � (5.8) (5.9) (5.10) 5 Cosmological Perturbation Theory 29
Page 1: Structure Formation in the Universe
Page 5 and 6: Contents 1 Introduction 5 2 Equival
Page 7: 1 Introduction At high energies the
Page 10 and 11: 1. Spacetime is endowed with a symm
Page 12 and 13: One can see, that the differentials
Page 14 and 15: 3.5 The Connection 3.5.1 Geometric
Page 16 and 17: 3.7 Torsion andCurvature The Torsio
Page 18 and 19: It is an important property that th
Page 20 and 21: One concludes that the space of con
Page 22 and 23: 4.4 ScalefactorandRedshift For simp
Page 24 and 25: 4.6 The FriedmannEquations Summing
Page 26 and 27: 4.7.1 Values ofthedensityparameters
Page 28 and 29: 4.8 Inflation Inflation, first intr
Page 32 and 33: With this metric, the velocity pert
Page 34 and 35: 5.2 InitialConditions According to
Page 36 and 37: CDM density perturbation today. Unf
Page 38 and 39: 100000 10000 1000 100 10 1 Dgγ Dg
Page 40 and 41: 10000 1000 100 10 1 0.001 0.01 k/h
Page 42 and 43: timesteps 1e+09 1e+08 1e+07 1e+06 1
Page 44 and 45: Performing on the other hand a Four
Page 46 and 47: α(Ω b,Ω c)=0.8873 andβ(Ω b,
Page 49 and 50: 8 The search for a cutoff Now the q
Page 51 and 52: 8 The searchfor a cutoff 49 ∆ 2 (
Page 53 and 54: Figure8.4: Luminosityfunction ofgal
Page 55: λ0/m Inflation aRH Radiation domin
Page 59 and 60: Descriptions Acbar: Arcminute Cosmo
Page 61 and 62: Bibliography [1] Clifford M. Will,
Page 63: [48] Simon P. Driver, Paul D. Allen

Master Thesis

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