Conserved Quantities and the Evolution of Perturbations in Lemaître-Tolman-Bondi Cosmology
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1 NAM /17 Conserved Quantities and the Evolution of Perturbations in Lemaître-Tolman-Bondi Cosmology Alexander Leithes From arxiv: (published CQG) by AL and Karim A. Malik
2 NAM /17 Overview Contents Why Perturb LTB Cosmology? The Standard Model of Cosmology - Flat FRW Image: SDSSIII
3 NAM /17 Why Perturb LTB Cosmology? Homogeneous versus Inhomogeneous Cosmologies and Inflation What if the universe is not homogeneous on large scales? What if the universe is inhomogeneous on -some- larger scale? What if the growth of structure in the universe is governed by perturbations around an inhomogeneous background cosmology? Simplest of these to investigate: Lemaître-Tolman-Bondi Cosmology
4 NAM /17 Flat FRW vs LTB Flat FRW vs LTB FRW: Maximally symmetric spatial section - expansion time dependent only ds 2 = dt 2 + a 2 (t)dr 2 + a 2 (t)r 2 ( dθ 2 + sin 2 θdφ 2) LTB: Spherically symmetric spatial section - expansion time and r coordinate dependant (not θ, φ) a a Bondi 1947 ds 2 = dt 2 + X 2 (r, t)dr 2 + Y 2 (r, t) ( dθ 2 + sin 2 θdφ 2)
5 NAM /17 The Standard Model of Cosmology - Flat FRW The Standard Model of Cosmology - Flat FRW Background metric: Perturbed metric: ds 2 = dt 2 + a 2 δ ij dx i dx j ds 2 = (1 + 2Φ)dt 2 + 2aB i dx i dt + a 2 (δ ij + 2C ij )dx i dx j with scalar, vector and tensor perturbations a a e.g. Bardeen 1980
6 NAM /17 The Standard Model of Cosmology - Flat FRW The Standard Model of Cosmology - Flat FRW Further decomposition of 3-spatial perturbations gives curvature perturbation ψ, identified with the intrinsic scalar curvature: * On 3-spatial hypersurfaces C ij = E,ij ψδ ij + vector + tensor quantities
7 NAM /17 The Standard Model of Cosmology - Flat FRW Constructing Gauge Invariant Quantities Splitting quantities into background + perturbation: no longer covariant - gauge dependent; construct gauge invariant quantities General gauge transformations: Tilde denotes new coordinates bar denotes background. δt = δt + δx µ T x µ = x µ + δx µ Key quantities gauge transformations: ψ FRW = ψ FRW + ȧ a δt δρ FRW = δρ FRW + ρδt
8 NAM /17 The Standard Model of Cosmology - Flat FRW Constructing Gauge Invariant Quantities Gauge choice: uniform density hypersurfaces, δρ FRW = 0 δt = δρ FRW ρ Get gauge invariant curvature perturbation ζ ψ FRW + ȧ/a ρ δρ Evolution equations for ζ from time derivative, δρ from energy conservation µ T µν = 0... ζ conserved in large scale limit - conserved perturbed quantities allow easily relate early to late times (e.g. curvature/physics early time relates to density/observables late time)
9 NAM /17 Perturbed LTB Background metric: ds 2 = dt 2 + X 2 (r, t)dr 2 + Y 2 (r, t) ( dθ 2 + sin 2 θdφ 2) Perturbed Metric: ds 2 = (1 + 2Φ)dt 2 + 2B i dx i dt + (δ ij + 2C ij )dx i dx j where dx i = [X(r, t)dr, Y (r, t)dθ, Y (r, t) sin θdφ] and we reserve dx i for [dr, dθ, dφ] X(r, t) = 1 W (r) where W (r) is an arbitrary function of r. Y (r, t), r
10 NAM /17 Perturbed LTB We have performed 1+3 decomposition into time and spatial sections of metric In FRW we have S, V, T decomposition, here it is even more complicated and needs spherical harmonics but... a our undecomposed perturbations give simpler expressions, easing constructing conserved quantities. a e.g. Clarkson, Clifton, February 2009, Gundlach, Martin-Garcia 2000, Gerlach, Sengupta 1979
11 NAM /17 LTB Governing Equations Background Energy Conservation: ρ + ρ(h X + 2H Y ) = 0, H X = Ẋ X, Perturbed Energy Conservation: δ ρ + (δρ + δp ) (H X + 2H Y ) + ρ v r H Y = Ẏ Y + ρ ( Ċ rr + Ċ θθ + Ċ φφ + r v r + θ v θ + φ v φ [ X ] + X + 2Y v r + cot θv θ) = 0 Y Convenient to define spatial metric perturbation: 3ψ SMTP = 1 2 δgk k = C rr + C θθ + C φφ
12 NAM /17 Constructing Gauge Invariant Quantities ψ SMTP transformation behaviour: [ 3 ψ Ẋ SMTP = 3ψ SMTP + X + 2 Ẏ ] [ X δt+ Y X + 2 Y ] δr + iδx i +δθ cot θ, Y δρ and 3-velocity transformation behaviour: δ ρ = δρ + ρδt + ρ δr ṽ i = v i δx i Gauge choices; uniform density (partially fixes the t coordinate): δt = 1 ρ [δρ + ρ δr] δ ρ=0 comoving (completes the gauge fixing in the spatial coordinates): δx i = v i dt
13 NAM /17 Constructing Gauge Invariant Quantities Gives gauge invariant Spatial Metric Trace Perturbation (SMTP) on comoving, uniform density hypersurfaces: { ( ζ SMTP = ψ SMTP + δρ 3 ρ + 1 X 3 X + 2 Y ) Y + ρ v r dt ρ } + r v r dt + θ v θ dt + φ v φ dt + cot θ v θ dt
14 NAM /17 Constructing Gauge Invariant Quantities Gauge invariant density perturbation on uniform curvature hypersurfaces also constructed [ (i.e. ( ψsmtp 0) 1 X ) ] δt = 3ψ SMTP + H X + 2H Y X + 2Y δr + i δx i + δθ cot θ Y { ( X ) δ ρ = δρ + ρ 3ψ + ψsmtp=0 X + 2Y Y + ρ v r dt ρ } + r v r dt + θ v θ dt + φ v φ dt + cot θ v θ dt May be related to ζ SMTP as δ ρ = 3 ρζ SMTP ψsmtp=0 c.f. ζ and δ ρ ψ FRW=0 in flat FRW
15 NAM /17 ζ SMTP Evolution Equation: ζ SMTP = H X + 2H Y 3 ρ Valid on all scales. For barotropic fluids ζ SMTP = 0 δp nad
16 arxiv: NAM /17 Conclusion and Further Research ζ SMTP = H X + 2H Y 3 ρ δp nad Research already extended to other spacetimes i.e. ζsmtp already extended to Lemaitre and FRW ζ SMTP can provide a useful analytical check for numerical codes in perturbed LTB/inhomogeneous spacetimes... as described in Bartelmann et al. e.g. potentially useful in study of structure formation in large voids (see e.g Das et al.)
17 NAM /17 Additional Notes Additional notes - time permitting Comparison with 2+2 Spherical Harmonic Decomposition ζ in Clifton, Clarkson, February formalism: ζ SMTP = ( 1 (1 κr 2 ) 6 a 2 h rry + h Ȳθθ a 2 r 2 + h Ȳφφ a 2 r 2 sin 2 θ + 2KY + GY :θθ + GY ) :φφ sin 2 θ + δρ 3 ρ ( { θ 3 a 2 r 2 v Ȳθ + ṽ Y θ h axial t Ȳ θ h polar ) t Y θ dt 1 ( + φ a 2 r 2 sin 2 v θ + ṽ Y φ h axial t Ȳ φ h polar ) t Y φ dt + 1 ( cot θ a 2 r 2 v Ȳθ + ṽ Y θ h axial t Ȳ θ h polar ) t Y θ dt r ( ) Y(1 κr 2 ) 1 a 2 2 h a tr + (1 κr 2 ) w dt + [ ( ) a (a r) a (1 κr ) a r (1 κr 2 ) ρ a ] ( ) } (1 κr 2 ) 1 ρ (1 κr 2 ) a 2 Y 2 h a tr + (1 κr 2 ) w dt,
Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology
1/28 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology Alex Leithes From arxiv:1403.7661 (submitted to CQG) by AL and Karim A. Malik 2/28 Image: SDSSIII Overview Contents Why Conserved
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1/15 Section 1 Section 2 Section 3 Conserved Quantities in Lemaître-Tolman-Bondi Cosmology Alex Leithes - Blackboard Talk Outline ζ SMTP Evolution Equation: ζ SMTP = H X + 2H Y 3 ρ Valid on all scales.
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