Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology

Transcription

1 1/31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology Alex Leithes From arxiv: (published CQG) by AL and Karim A. Malik

2 2/31 Image: SDSSIII Overview Contents Why Conserved Quantities are Important The Standard Model of Cosmology - Flat FRW Conserved Quantities in Perturbed LTB Conserved Quantities in Perturbed Lemaître and Flat FRW Further Questions of Perturbed LTB

3 3/31 Why are Conserved Quantities Important? Why are Gauge Invariant Conserved Quantities Important? Conserved quantities are important, allowing us to link observations at late times to the physics at earlier/all times e.g. in standard cosmology: inflation to CMB Gauge Invariance: i.e. invariant under infinitesimal coordinate shifts x µ = x µ + δx µ to remove gauge artefacts We construct GI conserved quantities in LTB cosmology (simplest inhomogeneous model)

4 4/31 Flat FRW vs LTB Flat FRW vs LTB FRW: Maximally symmetric spatial section - scale factor 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 - scale factors time and r coordinate dependent - Dust dominated a Bondi 1947 ds 2 = dt 2 + X 2 (r, t)dr 2 + Y 2 (r, t) ( dθ 2 + sin 2 θdφ 2) a

5 5/31 The Standard Model of Cosmology - Flat FRW The Standard Model of Cosmology - Flat FRW Background metric: Perturbed metric: ds 2 = dt 2 + a(t) 2 δ ij dx i dx j ds 2 = (1 + 2Φ)dt 2 + 2aB i dx i dt + a(t) 2 (δ ij + 2C ij )dx i dx j with scalar, vector and tensor perturbations a a e.g. Bardeen 1980

6 6/31 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: C ij = E,ij ψδ ij + vector + tensor quantities * On 3-spatial hypersurfaces

7 7/31 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. Key gauge transformations: δt = δt + δx µ T x µ = x µ + δx µ ψ FRW = ψ FRW + ȧ a δt δρ FRW = δρ FRW + ρδt

8 8/31 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 (on constant density hypersurfaces) ψ FRW = ζ ψ FRW + ȧ/a δρ FRW=0 ρ δρ FRW Evolution equations for ζ from energy conservation µ T µν = 0...

9 9/31 The Standard Model of Cosmology - Flat FRW Constructing Gauge Invariant Quantities Evolution equations for ζ from energy conservation µ T µν = 0... Evolution equation for density perturbation on large scales δρ FRW = 3H(δρ FRW + δp ) + 3( ρ + P ) ψ FRW In the large scale limit, ζ = H ρ + P δp nad ( δp = δρc s 2 + δp nad and δρ FRW = 0 selected) ζ conserved for barotropic fluid in large scale limit.

10 10/31 The Standard Model of Cosmology - Flat FRW Constructing Gauge Invariant Quantities Alternatively to ζ we can construct the gauge invariant density perturbation (on flat hypersurfaces) δρ ψ FRW=0 = δρ FRW + ρ H ψ FRW As stated earlier - conserved perturbed quantities allow us to easily relate early to late times (e.g. inflation to CMB) δρ = ρ H ζ ψ FRW=0 (δρ and ζ are GI) ψ FRW=0

11 11/31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology 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

12 12/31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology 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

13 13/31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology 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 = 1 2 δgk k = C rr + C θθ + C φφ

14 14/31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology 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 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

15 15/31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology Constructing Gauge Invariant Quantities Note: working in gauges requiring more than one fixing condition is not unusual - c.f. synchronous gauge in standard cosmology Remember: perturbed flat FRW metric ds 2 = (1 + 2Φ)dt 2 + 2aB i dx i dt + a(t) 2 (δ ij + 2C ij )dx i dx j Gauge choices; Φ = B i = 0, means proper time for comoving observers coincides with coordinate time i.e. dτ = dt s.t. ( ) ψ syn = ψ + H Φdt A(x i ) and ( δρ syn = δρ ρ ) Φdt A(x i ) A(x i ) is an arbitrary fn. Synchronous gauge popular in numerical simulations

16 16/31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology 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 }

17 17/31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology Constructing Gauge Invariant Quantities Alternatively can fix the gauge on uniform curvature hypersurfaces (i.e. ψ 0) [ ( 1 X ) ] δt = 3ψ + H X + 2H Y X + 2Y δr + i δx i + δθ cot θ Y and co-moving as before to fix the δx i terms to give { ( 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 }

18 18/31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology Constructing Gauge Invariant Quantities Note: it is also possible to fix the spatial gauge in a way analogous to longitudindal gauge in flat FRW i.e. set B i = 0 [ ( )] [ ( )] r δr = dt X δρ ρ + Br r δr ρ dt 2 X X 2 ρ [ ( θ δθ = dt Y δρ ρ + B )] [ ( )] θ θ δr ρ dt 2 Y Y 2 ρ [ ( φ δφ = dt Y 2 sin 2 θ δρ ρ + B )] [ ( )] φ φ δr ρ dt Y sin θ Y 2 sin 2 θ ρ Gauge fixing expressions far longer than constant density/constant curvature and comoving and so we did not pursue this gauge choice Back to constant density/constant curvature and comoving hypersurfaces...

19 19/31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology Constructing Gauge Invariant Quantities May be related to ζ SMTP as δ ρ = 3 ρζ SMTP ψ=0 c.f. ζ and δ ρ ψ FRW=0 in flat FRW

20 20/31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology Conserved Quantities in Perturbed LTB ζ SMTP Evolution Equation: Valid on all scales. ζ SMTP = H X + 2H Y 3 ρ For barotropic fluids ζ SMTP = 0 δp nad

21 21/31 Perturbed The Elephant in the Room Pressure perturbation around zero background?

22 22/31 Perturbations and Conservation in Lemaître Cosmology Perturbed Lemaître Allows for background pressure (additional scale factor on the t coordinate, reduces to LTB or FRW in limits) Background metric: ds 2 = f 2 (r, t)dt 2 + X 2 (r, t)dr 2 + Y 2 (r, t) ( dθ 2 + sin 2 θdφ 2), Perturbed Metric: ds 2 = f 2 (r, t)(1+2φ)dt 2 +2f(r, t)b 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]

23 23/31 Perturbations and Conservation in Lemaître Cosmology Constructing Gauge Invariant Quantities Similar procedure to LTB: the perturbed energy conservation equation is ( ) )(Ẋ δ ρ + δρ + δp X + 2Ẏ + ( ρ + Y P )v r + fb r X P + + ( B θ θ Y + B φ φ [ Y sin ] θ f f + X X + 2Y Y ) f P + ( ρ + P ) v r + B rf X + cot θvθ) = 0. and we construct the gauge invariant quantity ζ SMTP = ψ + + r δρ 3( ρ + P ) v r dt + θ { ( X X + 2 Y Y + v θ dt + φ ( ψ + v r + θ v θ + φ v φ ) ρ ρ + P v φ dt + cot θ v r dt v θ dt }

24 24/31 Perturbations and Conservation in Lemaître Cosmology Constructing Gauge Invariant Quantities The evolution equation for ζ SMTP ζ SMTP = + ρ ( ρ + P ) 2 δp nad ( B θ θ Y B φ + φ Y sin θ ( X [ t X + Y Y + P ( ρ + P ) v r ) f P ( ρ + P ) ρ ρ + P )] fb r X ( ρ + P ) P v r dt f f vr + Brf X which becomes in the large scale limit (like ζ in FRW) ζ SMTP = H X + 2H Y 3( ρ + P ) δp nad c.f. LTB - all scales

25 25/31 Perturbations and Conservation in Flat FRW Cosmology Constructing Gauge Invariant Quantities The evolution equation for ζ SMTP FRW in flat FRW ζ SMTP FRW = H ( ρ + P ) δp nad valid on all scales and ζ SMTP FRW conserved for barotropic fluid (like LTB)( δρ FRW = 0 and ṽ = 0 selected). c.f. ζ FRW ζ = H ρ + P δp nad valid only in large scale limit ( δρ FRW = 0 selected)

26 Further Questions of Perturbed LTB Further Questions of Perturbed LTB (Beyond scope of paper) At first glance LTB seems incompatible with inflation (inhomogeneous) Without inflation then what seeds the primordial cosmological perturbations? Without inflation how are problems such as relics explained? These arguments could apply to any inhomogeneous cosmology 26/31

27 27/31 Further Questions of Perturbed LTB Further Questions of Perturbed LTB Some attempts made to reconcile LTB with inflation - e.g. Multistream inflation - Wang, Li 2010, more generally - Linde, Linde, Mezhlumian 1994 Wang, Li 2010 Would allow scalar field to generate perturbations once again. Allow dilution of relic particle densities

28 28/31 Further Questions of Perturbed LTB Further Questions of Perturbed LTB LTB not applicable at early times when background pressure important Other inhomogeneous cosmologies may provide an answer but... Potentially still needs an answer of how the universe came to be well described by an inhomogeneous model in the first place... (although Linde, Linde, Mezhlumian 1994 provide interesting argument) image: CTC -

29 29/31 Further Questions of Perturbed LTB Final Word on Further Questions Should we discard the Copernican principle? If observations say we should, then yes. Maybe we are only throwing it away a bit - an LTB patch among many, in a larger homogeneous whole.

30 arxiv: /31 Perturbations and Conservation in Lemaître-Tolman-Bondi Cosmology 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.

31 31/31 Additional Notes Additional notes - time permitting Comparison with 2+2 Spherical Harmonic Decomposition ζ SMTP 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,