Dynamics of Linear Perturbations in Modified Gravity COSMO 2007 - PowerPoint PPT Presentation

Dynamics of Linear Perturbations in Modified Gravity COSMO 2007 University of Sussex, Brighton, August ‘07 Alessandra Silvestri collaborators: M.T rodden, L.Pogosian, R.Bean, S.M.Carroll, I.Sawicki and D.Bernat references : astro-ph/0611321, PRD’07 astro-ph/0607458, NJP’06, astro-ph/07.08..... Syracuse University

f(R) Gravity S = M 2 � � dx 4 √− g [ R + f ( R )] + d 4 x √− g L m [ χ i , g µ ν ] P 2 (1 + f R ) R µ ν − 1 2 g µ ν ( R + f ) + ( g µ ν � − ∇ µ ∇ ν ) f R = T µ ν { M 2 P ∇ µ T µ ν = 0

f(R) Gravity S = M 2 � � dx 4 √− g [ R + f ( R )] + d 4 x √− g L m [ χ i , g µ ν ] P 2 (1 + f R ) R µ ν − 1 2 g µ ν ( R + f ) + ( g µ ν � − ∇ µ ∇ ν ) f R = T µ ν { M 2 P ∇ µ T µ ν = 0 The Einstein equations are fourth order.

f(R) Gravity S = M 2 � � dx 4 √− g [ R + f ( R )] + d 4 x √− g L m [ χ i , g µ ν ] P 2 (1 + f R ) R µ ν − 1 2 g µ ν ( R + f ) + ( g µ ν � − ∇ µ ∇ ν ) f R = T µ ν { M 2 P ∇ µ T µ ν = 0 The Einstein equations are fourth order. The trace-equation becomes: T (1 − f R ) R + 2 f − 3 � f R = M 2 P

f(R) Gravity S = M 2 � � dx 4 √− g [ R + f ( R )] + d 4 x √− g L m [ χ i , g µ ν ] P 2 (1 + f R ) R µ ν − 1 2 g µ ν ( R + f ) + ( g µ ν � − ∇ µ ∇ ν ) f R = T µ ν { M 2 P ∇ µ T µ ν = 0 The Einstein equations are fourth order. The trace-equation becomes: T NOT algebraic ! (1 − f R ) R + 2 f − 3 � f R = M 2 P

Background Viability 1. to have a stable high-curvature regime, to have a f RR > 0 non-tachyonic scalar field 2. to prevent the graviton from turning into ghost 1 + f R > 0 3. negative, monotonically increasing function of R that f R < 0 asymptotes to zero from below 4. | f 0 R | ≤ 10 − 6 must be small at recent epochs to pass LGC (Hu and Sawicki astro-ph/0705.1158) w eff ≃ − 1 (Dolgov & Kawasaki, Phys.Lett.B 573 (2003), Navarro et al. gr-qc/0611127, Sawicki and Hu astro-ph/0702278 Starobinsky astro-ph/0706.2041, Chiba, Smith, Erickcek astro-ph/0611867 Amendola et al. astro-ph/0603703-0612180, Amendola & T sujikawa astro-ph/0705.0396)

Dynamics of Linear Perturbations in f(R) Gravity Scalar perturbations in Conformal Newtonian gauge T 0 0 = − ρ (1 + δ ) { ds 2 = − a 2 ( τ ) (1 + 2 Ψ ) d τ 2 + a 2 ( τ ) (1 − 2 Φ ) d � x 2 T 0 j = ( ρ + P ) v j T i j = ( P + δ P ) δ i j + π i j

Dynamics of Linear Perturbations in f(R) Gravity Scalar perturbations in Conformal Newtonian gauge T 0 0 = − ρ (1 + δ ) { ds 2 = − a 2 ( τ ) (1 + 2 Ψ ) d τ 2 + a 2 ( τ ) (1 − 2 Φ ) d � x 2 T 0 j = ( ρ + P ) v j T i j = ( P + δ P ) δ i j + π i j Anisotropy eq. ( Ψ − Φ ) = − f RR δ R 1 + f R

Dynamics of Linear Perturbations in f(R) Gravity Scalar perturbations in Conformal Newtonian gauge T 0 0 = − ρ (1 + δ ) { ds 2 = − a 2 ( τ ) (1 + 2 Ψ ) d τ 2 + a 2 ( τ ) (1 − 2 Φ ) d � x 2 T 0 j = ( ρ + P ) v j T i j = ( P + δ P ) δ i j + π i j Anisotropy eq. ( Ψ − Φ ) = − f RR δ R 1 + f R Φ + ≡ Φ + Ψ New variables: { 2 χ ≡ f RR δ R

Dynamics of Linear Perturbations in f(R) Gravity Scalar perturbations in Conformal Newtonian gauge T 0 0 = − ρ (1 + δ ) { ds 2 = − a 2 ( τ ) (1 + 2 Ψ ) d τ 2 + a 2 ( τ ) (1 − 2 Φ ) d � x 2 T 0 j = ( ρ + P ) v j T i j = ( P + δ P ) δ i j + π i j Anisotropy eq. ( Ψ − Φ ) = − f RR δ R 1 + f R Φ + ≡ Φ + Ψ New variables: { 2 χ ≡ f RR δ R ( F ≡ 1 + f R ) � � F ′ F ′ + = 3 a Ω v 1 + 1 Φ + + 3 χ Φ ′ { HkF − 2 2 F 4 F F � � � k 2 � χ ′ = − 2 Ω∆ 1 + F ′ F − 2 H ′ F F 1 + 2 F χ − 2 F Φ ′ Φ + F ′ + + − 2 F H 2 a 2 H 2 H F ′ 3 F ′

Sub-Horizon CDM equation: � � k 2 1 + H ′ Φ + − χ � � δ ′′ + δ ′ + = 0 a 2 H 2 H 2 F

Sub-Horizon CDM equation: � � k 2 1 + H ′ Φ + − χ � � δ ′′ + δ ′ + = 0 a 2 H 2 H 2 F Einstein equations: Φ + ≃ 3 Ω H 2 a 2 δ k 2 F 2 k 2 Ω δ a 2 χ ≃ k 2 + 3 a 2 H 2 F ′ /F k 2

Sub-Horizon CDM equation: � � k 2 1 + H ′ Φ + − χ � � δ ′′ + δ ′ + = 0 a 2 H 2 H 2 F Einstein equations: Φ + ≃ 3 Ω H 2 a 2 δ k 2 F 2 k 2 Ω δ a 2 χ ≃ k 2 + 3 a 2 H 2 F ′ /F k 2 1 + 4 k 2 f RR − 3 1 Ω δ a 2 F 1 + 3 k 2 f RR F 2 a 2 F { time and scale dependent ≡ G eff rescaling of Newton constant

Sub-Horizon CDM equation: � � k 2 1 + H ′ Φ + − χ � � δ ′′ + δ ′ + = 0 a 2 H 2 H 2 F Einstein equations: k 2 f RR Φ + ≃ 3 Ω H 2 a 2 δ a 2 F k 2 F 2 k 2 Ω δ a 2 There is a scale associated χ ≃ k 2 + 3 a 2 H 2 F ′ /F k 2 with the extra d.o.f. : � f RR 1 λ C ≡ = F m f R 1 + 4 k 2 f RR − 3 1 Ω δ a 2 F 1 + 3 k 2 f RR F 2 a 2 F { time and scale dependent ≡ G eff rescaling of Newton constant

Sub-Horizon Below this scale there is a significant departure from std GR and a scale dependence in the behavior of perturbations λ ≫ λ C χ ≃ 0 , G eff ≃ G F Ψ ≃ Φ λ ≪ λ C G χ ≃ − 2 3 F Φ + , G eff ≃ 4 F 3 Ψ ≃ 2 Φ

w eff = − 1 d Φ + f 0 R = − 10 − 4 dz

w eff = − 1 · ∆ ( k, z ) − d Φ + f 0 R = − 10 − 4 ∆ ( k, z i ) dz

Dynamics of Linear Perturbations in Modified Source Gravity � M 2 dx 4 √− g � � 2 e 2 ψ R + 3 e 2 ψ ( ∇ ψ ) 2 − U ( ψ ) P S = + s m [ g, χ i ] Carroll, Sawicki, Silvestri, T rodden astro-ph/0607458, NJP’06 e 2 ψ G µ ν = κ 2 � � T µ ν + T ( ψ ) µ ν G µ ν = ˜ ψ = ψ ( T ) = ψ ( ρ m ) T µ ν ( ρ )

Dynamics of Linear Perturbations in Modified Source Gravity � M 2 dx 4 √− g � � 2 e 2 ψ R + 3 e 2 ψ ( ∇ ψ ) 2 − U ( ψ ) P S = + s m [ g, χ i ] Carroll, Sawicki, Silvestri, T rodden astro-ph/0607458, NJP’06 e 2 ψ G µ ν = κ 2 � � T µ ν + T ( ψ ) µ ν G µ ν = ˜ ψ = ψ ( T ) = ψ ( ρ m ) T µ ν ( ρ ) Linear perturbations d ψ Φ − Ψ = − 2 dlna δ 3 � 3 Ω e − 2 ψ k 2 � � � k 2 d ψ d ψ 1 + 2 − 2 δ a 2 Ψ = − a 2 2 3 dlna 3 dlna

6 MSG, k = 0.003 Mpc − 1 MSG, k = 0.001 Mpc − 1 5 MSG, k = 0.0003 Mpc − 1 LCDM scale-dependent runaway growth 4 d ln a d∆ 3 rapid structure formation drives the growth of 2 gravitational potentials 1 0 5 4 3 2 1 0 z the ISW effect is enhanced at the lowest 7 multipoles MSG, k = 0.003 Mpc − 1 MSG, k = 0.001 Mpc − 1 6 MSG, k = 0.0003 Mpc − 1 LCDM 5 negative LSS-ISW correlation 4 Φ − Ψ 3 2 1 0 3 2.5 2 1.5 1 0.5 0 z

CONCLUSIONS For f(R) and MSG models that reproduce the desired background evolution, we investigated the dynamics of linear perturbations, finding: transition scale related to new d.o.f. mass scale and effective shear slip between metric potentials Ψ Φ modified ISW signal modified, scale-dependent evolution of the metric potentials modified, scale-dependent evolution of matter perturbations The ISW, its correlation with LSS and Weak Lensing might be very useful probes of modifications of gravity

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w eff = − 1 w eff = − 0 . 99 w eff = − 1 . 01 w eff = − 0 . 99 → − 1 . 01