Averaging Robertson-Walker Cosmologies Iain A. Brown Institut f ur - PowerPoint PPT Presentation

Averaging Robertson-Walker Cosmologies Iain A. Brown Institut f¨ ur Theoretische Physik, Universit¨ at Heidelberg “Backreaction from Perturbations”, J. Behrend, IB and G. Robbers, JCAP 0801 013 ‘Averaging Robertson-Walker Cosmologies”, IB, G. Robbers and J. Behrend, in preperation Cosmo 08, Madison, 25th August 2008 1 / 23

Motivation Standard Cosmology Averaging in Cosmology Backreaction Numerical Study Summary Motivation 2 / 23

Standard Cosmology Copernican Principle + CMB observations ⇒ Universe homogeneous ■ Motivation Standard Cosmology and isotropic. Averaging in Cosmology Backreaction Numerical Study Summary 3 / 23

Standard Cosmology Copernican Principle + CMB observations ⇒ Universe homogeneous ■ Motivation Standard Cosmology and isotropic. Averaging in Cosmology ■ Robertson-Walker cosmology: foliate spacetime with Backreaction maximally-symmetric three-spaces Numerical Study Line element: ds 2 = − dt 2 + a 2 ( t ) δ ij dx i dx j Summary — a/a ) 2 = (8 πG/ 3) ρ + Λ / 3 Friedmann equation: (˙ — Raychaudhuri equation: ¨ a/a = − (4 πG/ 3)( ρ + p ) + Λ / 3 — Perturb metric with O ( ǫ ) ≈ 10 − 5 — 3 / 23

Standard Cosmology Copernican Principle + CMB observations ⇒ Universe homogeneous ■ Motivation Standard Cosmology and isotropic. Averaging in Cosmology ■ Robertson-Walker cosmology: foliate spacetime with Backreaction maximally-symmetric three-spaces Numerical Study Line element: ds 2 = − dt 2 + a 2 ( t ) δ ij dx i dx j Summary — a/a ) 2 = (8 πG/ 3) ρ + Λ / 3 Friedmann equation: (˙ — Raychaudhuri equation: ¨ a/a = − (4 πG/ 3)( ρ + p ) + Λ / 3 — Perturb metric with O ( ǫ ) ≈ 10 − 5 — ■ We have assumed the existence of an average and added perturbations 3 / 23

Averaging in Cosmology ■ Motivation An implicit averaging in cosmology transfers local equations to global Standard Cosmology cosmology; should be made explicit Averaging in Cosmology � ∂ t ρ � � = ∂ t � ρ � ⇒ Na¨ ■ ıve EFE for assumed averages does not reflect a Backreaction true average of small-scale physics. Numerical Study Summary 4 / 23

Averaging in Cosmology ■ Motivation An implicit averaging in cosmology transfers local equations to global Standard Cosmology cosmology; should be made explicit Averaging in Cosmology � ∂ t ρ � � = ∂ t � ρ � ⇒ Na¨ ■ ıve EFE for assumed averages does not reflect a Backreaction true average of small-scale physics. Numerical Study ■ We should be using Summary � G µν ( g µν ) � = 8 πG � T µν � + Λ � g µν � instead of G µν ( � g µν � ) = 8 πG � T µν � + Λ � g µν � . “Backreaction” may not be dark energy, but all cosmological models ■ should be properly averaged ■ Aim: Express Buchert equations in general form, apply to range of perturbed Robertson-Walker models from radiation domination to present day. 4 / 23

Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Backreaction Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary 5 / 23

Formalism: 3+1 Split Motivation Backreaction Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary Line-element: ds 2 = − α 2 dt 2 + h ij dx i dx j ■ Fluids ( Λ , φ , b, CDM, γ , ν ): ̺ = n µ n ν T µν , j i = − n µ T iµ , S ij = T ij ■ Perfect fluids, T µν = ( ρ + p ) u µ u ν + pg µν : ■ ̺ = n µ n ν T µν = ρ ( n µ u µ ) 2 + p ( n µ u µ ) 2 − 1 � � , S = T i i = 3 p + ( ρ + p ) u i u i . 6 / 23

Formalism: Buchert Averaging ■ Motivation Select average √ � A � = 1 � Backreaction hd 3 x , A Formalism: 3+1 Split V Formalism: Buchert D Averaging Define averaged “scale factor” and Hubble rate by Formalism: Modifications to Standard Cosmology ˙ √ Link to Perturbation 3 H D = 3 ˙ a D V V = − 1 � hd 3 x = − � αK � = �H� , Theory = αK Link to Perturbation a D V Theory: Backreaction D Terms Numerical Study ■ Buchert equations: Summary � ˙ � 2 a D 8 πG + Λ − 1 α 2 ̺ α 2 � � � � = 6 ( Q D + R D ) a D 3 3 a D ¨ − 4 πG + Λ + 1 α 2 ( ̺ + S ) α 2 � � � � = 3 ( Q D + P D ) a D 3 3 7 / 23

Formalism: Modifications to Standard Cosmology ■ Motivation Kinematical “backreaction”: Backreaction − 2 � α 2 � K 2 − K i �� Formalism: 3+1 Split j K j 3 � αK � 2 Q D = Formalism: Buchert i Averaging Formalism: Modifications to Standard Cosmology � αD i D i α � Dynamical “backreaction”: P D = � ˙ αK � + Link to Perturbation ■ Theory α 2 R � � Curvature contribution: R D = ■ Link to Perturbation Theory: Backreaction ■ Deviation from average density and pressure: Terms Numerical Study 3 T ( a ) 3 S ( a ) Summary α 2 ̺ ( a ) α 2 S ( a ) D D � � � � 8 πG = − ρ ( a ) , 4 πG = − S ( a ) 8 / 23

Formalism: Modifications to Standard Cosmology ■ Motivation The Buchert equations can then be written as Backreaction � ˙ � 2 Formalism: 3+1 Split a D 8 πG ρ ( a ) + Λ 3 + 8 πG � Formalism: Buchert = ρ eff , Averaging a D 3 3 Formalism: a Modifications to a D ¨ − 4 πG + Λ 3 − 4 πG Standard Cosmology � � � � � = ρ ( a ) + S ( a ) ρ eff + S eff Link to Perturbation a D 3 3 Theory Link to Perturbation a Theory: Backreaction Terms with effective correction fluid Numerical Study � Λ 8 πG 3 − 1 Summary α 2 − 1 T ( a ) � � ρ eff = + 6 ( Q D + R D ) , D 3 a + 1 α 2 − 1 S ( a ) � � � 16 πGp eff = 4 D − 2Λ 3 ( R D − 3 Q D − 4 P D ) , a α 2 − 1 a S ( a ) � � R D − 3 Q D − 4 P D + 12 � D − 6Λ − 1 w eff = . a T ( a ) 3 − 2Λ � α 2 − 1 � R D + Q D − 6 � D 9 / 23

Link to Perturbation Theory ■ Motivation Identify ADM and Newtonian co-ordinates (c.f. Mukhanov et. al.) Backreaction ds 2 = − (1+2Ψ) dt 2 + a 2 ( t )(1 − 2Φ) δ ij dx i dx j = − α 2 dt 2 + h ij dx i dx j Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology a D ( t ) is “observational”, a ( t ) is “physical” – drawback of re-averaging ■ Link to Perturbation Theory assumed average (Kolb, Marra, Matarrese 08; IB, Behrend, Robbers Link to Perturbation Theory: Backreaction 08) Terms Numerical Study Summary 10 / 23

Link to Perturbation Theory ■ Motivation Identify ADM and Newtonian co-ordinates (c.f. Mukhanov et. al.) Backreaction ds 2 = − (1+2Ψ) dt 2 + a 2 ( t )(1 − 2Φ) δ ij dx i dx j = − α 2 dt 2 + h ij dx i dx j Formalism: 3+1 Split Formalism: Buchert Averaging Formalism: Modifications to Standard Cosmology a D ( t ) is “observational”, a ( t ) is “physical” – drawback of re-averaging ■ Link to Perturbation Theory assumed average (Kolb, Marra, Matarrese 08; IB, Behrend, Robbers Link to Perturbation Theory: Backreaction 08) Terms ■ Quickly find Numerical Study a D ˙ = ˙ a � � Summary ˙ a − Φ (1 + 2Φ) a D 10 / 23

Link to Perturbation Theory: Backreaction Terms ■ Motivation Kinematical and dynamical backreactions: Backreaction �� � 2 � Formalism: 3+1 Split Φ 2 � � ˙ ˙ Formalism: Buchert Q D = 6 − Φ , Averaging Formalism: Modifications to 1 ∇ 2 Ψ − ( ∇ Ψ) 2 + 2Φ ∇ 2 Ψ − ( ∇ Φ) · ( ∇ Ψ) � � Standard Cosmology P D = a 2 Link to Perturbation Theory +3 ˙ a Link to Perturbation � � � � Ψ − 2Ψ ˙ ˙ Ψ ˙ ˙ Ψ − 3 Φ Theory: Backreaction a Terms Numerical Study Summary ■ Curvature correction: R D = 2 2 ∇ 2 Φ + 3( ∇ Φ) 2 + 4(2Φ + Ψ) ∇ 2 Φ � � . a 2 11 / 23

Link to Perturbation Theory: Backreaction Terms ■ Motivation Fluid corrections: Backreaction 8 πG Formalism: 3+1 Split δ + 2Ψ + (1 + w ) a 2 v 2 + 2Ψ δ � � T D = ρ , Formalism: Buchert 3 Averaging Formalism: 4 πG s δ + 6 w Ψ + (1 + w ) a 2 v 2 + 6 c 2 Modifications to 3 c 2 � � S D = ρ s Ψ δ Standard Cosmology 3 Link to Perturbation Theory Link to Perturbation Theory: Backreaction Terms Numerical Study Summary 12 / 23

Link to Perturbation Theory: Backreaction Terms ■ Motivation Fluid corrections: Backreaction 8 πG Formalism: 3+1 Split δ + 2Ψ + (1 + w ) a 2 v 2 + 2Ψ δ � � T D = ρ , Formalism: Buchert 3 Averaging Formalism: 4 πG s δ + 6 w Ψ + (1 + w ) a 2 v 2 + 6 c 2 Modifications to 3 c 2 � � S D = ρ s Ψ δ Standard Cosmology 3 Link to Perturbation Theory Link to Perturbation Theory: Backreaction Note: alternative gauges – uniform density to simplify T D and S D , ■ Terms uniform curvature to remove R D , synchronous gauge to remove P D . Numerical Study Summary Q D cannot be entirely removed except in EdS matter domination. 12 / 23

Motivation Backreaction Numerical Study Ergodic Averaging Quintessence Cosmology Early Dark Energy Inverse Power Law Numerical Study Potential Exponential Potential Equations of State Summary 13 / 23