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. For barotropic fluids ζ SMTP = 0 δp nad From arxiv:1403.7661 (submitted to CQG) by AL and Karim A. Malik
2/15 Overview ζ SMTP Evolution Equation: ζ SMTP = H X + 2H Y δp nad 3 ρ I will leave this equation on the board from the beginning of this talk, as it is the main result and I d hate to run out of time without getting to it. Contents Why Perturb LTB Cosmology? The Standard Model of Cosmology - Flat FRW
3/15 Why Perturb LTB Cosmology? Why Perturb LTB Cosmology? Recent observations (e.g. SN1a) suggest late time accelerated expansion - driven by Dark Energy Inhomogeneous Cosmologies explain observations through inhomogeneous expansion not acceleration - no Dark Energy Many possible inhomogeneous cosmologies; LTB type models still actively researched (simplest model, toy model) Ongoing work interpreting observations (galaxy surveys, large scale structure surveys, CMB) and other redshift dependent observations in context of LTB background
4/15 The Standard Model of Cosmology - Flat FRW Starting with the standard. FRW: Maximally symmetric spatial section - expansion time dependent only 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
5/15 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
6/15 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
7/15 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)
1 Bondi 1947 8/15 Compare with our inhomogeneous model. LTB: Spherically symmetric spatial section - expansion time and r coordinate dependant (not θ, φ) 1 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 = [Xdr, Y dθ, Y sin θdφ] and we reserve dx i for [dr, dθ, dφ]
9/15 Perturbed LTB We have performed 1+3 decomposition into time and spatial sections of metric Decomposition of perturbations not completely straightforward without use of methods like spherical harmonic decomposition but... a Our undecomposed perturbations give simpler expressions, easing constructing conserved quantities. a e.g. Clarkson, Clifton, February 2009
10/15 Perturbed LTB 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 = 1 2 δgk k = C rr + C θθ + C φφ
11/15 Constructing Gauge Invariant Quantities ψ transformation behaviour: ] [ [Ẋ 3 ψ X ] = 3ψ + X + 2Ẏ δt + Y X + 2Y δr + i δx i + δθ cot θ, Y Gauge choices; uniform density: δt = 1 ρ [δρ + ρ δr]. δ ρ=0 comoving: δx i = v i dt.
12/15 Constructing Gauge Invariant Quantities Gives gauge invariant Spatial Metric Trace Perturbation (SMTP) on comoving, uniform density hypersurfaces: { ( ζ SMTP = ψ + 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 }
13/15 Constructing Gauge Invariant Quantities Get gauge invariant density perturbation on uniform curvature hypersurfaces { ( X ) δ ρ = δρ + ρ 3ψ + ψ=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 ψ=0
14/15 ζ SMTP Evolution Equation: ζ SMTP = H X + 2H Y 3 ρ Valid on all scales. For barotropic fluids ζ SMTP = 0 δp nad
15/15 Conclusion and Further Research ζ SMTP = H X + 2H Y 3 ρ δp nad Research already extended to other spacetimes. i.e. ζ SMTP already extended to Lemaitre and FRW Potential wider use of ζ SMTP in inhomogeneous spacetimes generally versus standard FRW model. arxiv:1403.7661