Conserved Quantities in Lemaître-Tolman-Bondi Cosmology

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1 Alex Leithes - Queen Mary, University of London Supervisor - Karim Malik Following work in arxiv: by AL and Karim A. Malik Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 Conserved Quantities in Lemaître-Tolman-Bondi Cosmology

2 Section 1 Section 2 Section 3 Conserved Quantities in Lemaı tre-tolman-bondi Cosmology Figure: Universe timeline - ESA Alex Leithes (QMUL) Conserved Quantities in Lemaı tre-tolman-bondi Cosmology BritGrav /11

3 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 Overview Contents Why Perturb LTB Cosmology? The Standard Model of Cosmology - Flat FLRW

4 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 Why Perturb LTB Cosmology? Why Perturb LTB Cosmology? Locally Rotationally Symmetric Cosmologies appealing, explaining Dark Energy interpreted observations (SN1a observations) with inhomogeneous expansion - no DE Of the many possible LRS cosmologies, LTB type models still lie within the constraints provided by observations Ongoing work interpreting observations (galaxy surveys, large scale structure surveys, CMB) and other redshift dependent observations in context of LTB background Ongoing numerical studies perturbing LTB: analytically derived results useful for comparison Structure formation within a region or scale over which LTB assumed valid requires understanding of how perturbations within region evolve

5 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 The Standard Model of Cosmology - Flat FLRW The Standard Model of Cosmology - Flat FLRW Perturbed metric: ds 2 = (1 + 2Φ)dt 2 + 2aB,i dx i dt + a 2 (δ ij + 2C ij )dx i dx j, (1) Further decomposition, revealing curvature perturbation ψ: C ij = E,ij ψδ ij, (2)

6 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 The Standard Model of Cosmology - Flat FLRW The Standard Model of Cosmology - Flat FLRW Perturbed metric: ds 2 = (1 + 2Φ)dt 2 + 2aB,i dx i dt + a 2 (δ ij + 2C ij )dx i dx j, (1) Further decomposition, revealing curvature perturbation ψ: C ij = E,ij ψδ ij, (2) Gauge Transformations: ψ FRW = ψ FRW + ȧ δt, a (3) δρ FRW = δρ FRW + ρδt, (4)

7 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 The Standard Model of Cosmology - Flat FLRW The Standard Model of Cosmology - Flat FLRW Gauge choice, δρ FRW = 0, δt = δρ FRW ρ Gauge invariant curvature perturbation, ζ,, (5) ζ ψ FRW + ȧ/a δρ, (6) ρ Evolution equations for ζ by taking d.w.t.t, for δρ from energy conservation µ T µν = 0...

8 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 The background metric: ds 2 = dt 2 + X 2 (r, t)dr 2 + Y 2 (r, t) ( dθ 2 + sin 2 θdφ 2), (7) Metric Perturbations: δg µν = 2Φ XB r Y B θ Y sin θb φ XB r 2X 2 C rr XY C rθ XY sin θc rφ Y B θ XY C rθ 2Y 2 C θθ Y 2 sin θc θφ Y sin θb φ XY sin θc rφ Y 2 sin θc θφ 2Y 2 sin 2 θc φφ. (8)

9 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 The background metric: ds 2 = dt 2 + X 2 (r, t)dr 2 + Y 2 (r, t) ( dθ 2 + sin 2 θdφ 2), (7) Metric Perturbations: δg µν = 2Φ XB r Y B θ Y sin θb φ XB r 2X 2 C rr XY C rθ XY sin θc rφ Y B θ XY C rθ 2Y 2 C θθ Y 2 sin θc θφ Y sin θb φ XY sin θc rφ Y 2 sin θc θφ 2Y 2 sin 2 θc φφ. (8) Inhomogeneous spacetime, with 1+3 decomposition does not allow further decomposition (S,V,T), but... gives simpler expressions, easing constructing conserved quantities.

10 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 Background Energy Conservation: ρ + ρ(h X + 2H Y ) = 0. (9) Perturbed Energy Conservation: δ ρ + (δρ + δp ) (H X + 2H Y ) + ρ v r (10) + 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 So, for convenience we define a spatial metric perturbation as: 3ψ = δg k k = C rr + C θθ + C φφ, (11)

11 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 ψ transformation behaviour: ] [ [Ẋ 3 ψ X ] = 3ψ+ X + 2Ẏ δt+ Y X + 2Y δr+ i δx i +δθ cot θ, (12) Y Gauge choices; uniform density: δt = 1 ρ [δρ + ρ δr]. (13) δ ρ=0 and comoving: δx i = v i dt. (14) Gives us gauge invariant Spatial Metric Trace Perturbation (SMTP) on comoving, uniform density hypersurfaces:

12 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 Gives us gauge invariant Spatial Metric Trace Perturbation (SMTP) on comoving, uniform density hypersurfaces: ζ SMTP = ψ + δρ 3 ρ + 1 { ( X ) 3 X + 2Y Y + ρ v r dt (15) ρ + r v r dt + θ v θ dt + φ v φ dt + cot θ v θ dt }

13 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 Gives us gauge invariant Spatial Metric Trace Perturbation (SMTP) on comoving, uniform density hypersurfaces: ζ SMTP = ψ + δρ 3 ρ + 1 { ( X ) 3 X + 2Y Y + ρ v r dt (15) ρ + r v r dt + θ v θ dt + φ v φ dt + cot θ v θ dt } SMTP Evolution Equation: which is valid on all scales. ζ SMTP = H X + 2H Y 3 ρ δp nad, (16)

14 Alex Leithes (QMUL) Conserved Quantities in Lemaître-Tolman-Bondi Cosmology BritGrav /11 Conclusion and Further Research Conclusion and Further Research Research already extended to Lemaitre spacetime, allowing background pressure, ( ζ SMTP = H X +H Y 3( ρ+ P ) δp nad - see arxiv: ). Difference in behaviour for ζ SMTP LTB vs ζ FRW help compare structure formation. Wider potential use of SMTP in comparing inhomogeneous spacetimes generally with standard FRW model.