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

Dynamics of Linear Perturbations in Modified Gravity

Syracuse University

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.....

COSMO 2007

University of Sussex, Brighton, August ‘07

f(R) Gravity

{

S = M 2

P

2

• dx4√−g [R + f(R)] +
• d4x√−gLm[χi, gµν]

(1 + fR)Rµν − 1 2gµν(R + f) + (gµν − ∇µ∇ν)fR = Tµν M 2

P

∇µT µν = 0

f(R) Gravity

{

S = M 2

P

2

• dx4√−g [R + f(R)] +
• d4x√−gLm[χi, gµν]

(1 + fR)Rµν − 1 2gµν(R + f) + (gµν − ∇µ∇ν)fR = Tµν M 2

P

∇µT µν = 0 The Einstein equations are fourth order.

f(R) Gravity

{

The Einstein equations are fourth order. S = M 2

P

2

• dx4√−g [R + f(R)] +
• d4x√−gLm[χi, gµν]

(1 + fR)Rµν − 1 2gµν(R + f) + (gµν − ∇µ∇ν)fR = Tµν M 2

P

∇µT µν = 0 The trace-equation becomes: (1 − fR)R + 2f − 3fR = T M 2

P

f(R) Gravity

{

The Einstein equations are fourth order. S = M 2

P

2

• dx4√−g [R + f(R)] +
• d4x√−gLm[χi, gµν]

(1 + fR)Rµν − 1 2gµν(R + f) + (gµν − ∇µ∇ν)fR = Tµν M 2

P

(1 − fR)R + 2f − 3fR = T M 2

P

∇µT µν = 0 The trace-equation becomes: NOT algebraic !

Background Viability

1. 2. 3. 4.

fRR > 0 1 + fR > 0 fR < 0 negative, monotonically increasing function of R that asymptotes to zero from below must be small at recent epochs to pass LGC to have a stable high-curvature regime, to have a non-tachyonic scalar field to prevent the graviton from turning into ghost |f 0

R| ≤ 10−6 (Dolgov & Kawasaki, Phys.Lett.B 573 (2003), Navarro et al. gr-qc/0611127, Sawicki and Hu astro-ph/0702278 Amendola et al. astro-ph/0603703-0612180, Amendola & T sujikawa astro-ph/0705.0396) Starobinsky astro-ph/0706.2041, Chiba, Smith, Erickcek astro-ph/0611867

weff ≃ −1

(Hu and Sawicki astro-ph/0705.1158)

Dynamics of Linear Perturbations in f(R) Gravity

Scalar perturbations in Conformal Newtonian gauge

ds2 = −a2(τ) (1 + 2Ψ) dτ 2 + a2(τ) (1 − 2Φ) d x2

{

T 0

0 = −ρ(1 + δ)

T 0

j = (ρ + P)vj

T i

j = (P + δP)δi j + πi j

Dynamics of Linear Perturbations in f(R) Gravity

Scalar perturbations in Conformal Newtonian gauge

ds2 = −a2(τ) (1 + 2Ψ) dτ 2 + a2(τ) (1 − 2Φ) d x2

Anisotropy eq. (Ψ − Φ) = − fRR 1 + fR δR

T 0

0 = −ρ(1 + δ)

T 0

j = (ρ + P)vj

T i

j = (P + δP)δi j + πi j

{

Dynamics of Linear Perturbations in f(R) Gravity

Scalar perturbations in Conformal Newtonian gauge

ds2 = −a2(τ) (1 + 2Ψ) dτ 2 + a2(τ) (1 − 2Φ) d x2

Anisotropy eq. (Ψ − Φ) = − fRR 1 + fR δR χ ≡ fRRδR Φ+ ≡ Φ + Ψ 2

{

New variables:

T 0

0 = −ρ(1 + δ)

T 0

j = (ρ + P)vj

T i

j = (P + δP)δi j + πi j

{

Dynamics of Linear Perturbations in f(R) Gravity

Scalar perturbations in Conformal Newtonian gauge

ds2 = −a2(τ) (1 + 2Ψ) dτ 2 + a2(τ) (1 − 2Φ) d x2

Anisotropy eq. (Ψ − Φ) = − fRR 1 + fR δR χ ≡ fRRδR Φ+ ≡ Φ + Ψ 2

{

New variables:

Φ′

+ = 3

2 aΩv HkF −

• 1 + 1

2 F ′ F

• Φ+ + 3

4 F ′ F χ F

{

χ′ = −2Ω∆ H2 F F ′ +

• 1 + F ′

F − 2H′ H F F ′

• χ − 2FΦ′

+ − 2F

• 1 + 2

3 k2 a2H2 F F ′

• Φ+

T 0

0 = −ρ(1 + δ)

T 0

j = (ρ + P)vj

T i

j = (P + δP)δi j + πi j

{

(F ≡ 1 + fR)

Sub-Horizon

δ′′ +

• 1 + H′

H

• δ′ +

k2 a2H2

• Φ+ − χ

2F

• = 0

CDM equation:

Sub-Horizon

Einstein equations:

χ ≃ k2 k2 + 3a2H2F ′/F Ωδa2 k2 δ′′ +

• 1 + H′

H

• δ′ +

k2 a2H2

• Φ+ − χ

2F

• = 0

CDM equation:

Φ+ ≃ 3ΩH2 2 a2 k2 δ F

Sub-Horizon

CDM equation:

δ′′ +

• 1 + H′

H

• δ′ +

k2 a2H2

• Φ+ − χ

2F

• = 0

Einstein equations: time and scale dependent rescaling of Newton constant −3 2 Ωδ

{

≡ Geff

χ ≃ k2 k2 + 3a2H2F ′/F Ωδa2 k2

1 F 1 + 4 k2

a2 fRR F

1 + 3 k2

a2 fRR F

Φ+ ≃ 3ΩH2 2 a2 k2 δ F

Sub-Horizon

CDM equation:

δ′′ +

• 1 + H′

H

• δ′ +

k2 a2H2

• Φ+ − χ

2F

• = 0

Einstein equations: time and scale dependent rescaling of Newton constant −3 2 Ωδ

{

≡ Geff

χ ≃ k2 k2 + 3a2H2F ′/F Ωδa2 k2

1 F 1 + 4 k2

a2 fRR F

1 + 3 k2

a2 fRR F

λC ≡

• fRR

F = 1 mfR

There is a scale associated with the extra d.o.f. :

k2 a2 fRR F

Φ+ ≃ 3ΩH2 2 a2 k2 δ F

Sub-Horizon

Ψ ≃ Φ

λ ≫ λC

Below this scale there is a significant departure from std GR and a scale dependence in the behavior of perturbations

Ψ ≃ 2Φ

λ ≪ λC

χ ≃ 0, Geff ≃ G F χ ≃ −2 3FΦ+, Geff ≃ 4 3 G F

dΦ+ dz

weff = −1 f 0

R = −10−4

−dΦ+ dz · ∆(k, z) ∆(k, zi)

weff = −1 f 0

R = −10−4

Dynamics of Linear Perturbations in Modified Source Gravity

e2ψGµν = κ2 Tµν + T (ψ)

µν

• ψ = ψ(T) = ψ(ρm)

Gµν = ˜ Tµν(ρ) S =

• dx4√−g

M 2

P

2 e2ψR + 3e2ψ(∇ψ)2 − U(ψ)

• + sm[g, χi]

Carroll, Sawicki, Silvestri, T rodden astro-ph/0607458, NJP’06

Dynamics of Linear Perturbations in Modified Source Gravity

e2ψGµν = κ2 Tµν + T (ψ)

µν

• ψ = ψ(T) = ψ(ρm)

Gµν = ˜ Tµν(ρ) S =

• dx4√−g

M 2

P

2 e2ψR + 3e2ψ(∇ψ)2 − U(ψ)

• + sm[g, χi]

Carroll, Sawicki, Silvestri, T rodden astro-ph/0607458, NJP’06

Φ − Ψ = −2 3 dψ dlnaδ 3Ωe−2ψ 2

• 1 + 2

3 dψ dlna

• − 2

3 k2 a2 dψ dlna

• k2

a2 Ψ = − δ

Linear perturbations

0.5 1 1.5 2 2.5 3 1 2 3 4 5 6 7 z Φ − Ψ MSG, k = 0.003 Mpc−1 MSG, k = 0.001 Mpc−1 MSG, k = 0.0003 Mpc−1 LCDM

scale-dependent runaway growth rapid structure formation drives the growth of gravitational potentials the ISW effect is enhanced at the lowest multipoles negative LSS-ISW correlation

1 2 3 4 5 1 2 3 4 5 6 z

d∆ d ln a

MSG, k = 0.003 Mpc−1 MSG, k = 0.001 Mpc−1 MSG, k = 0.0003 Mpc−1 LCDM

CONCLUSIONS

modified, scale-dependent evolution of matter perturbations

For f(R) and MSG models that reproduce the desired background evolution, we investigated the dynamics of linear perturbations, finding:

modified, scale-dependent evolution of the metric potentials effective shear slip between metric potentials Ψ

Φ

and modified ISW signal The ISW, its correlation with LSS and Weak Lensing might be very useful probes

• f modifications of gravity

transition scale related to new d.o.f. mass scale

THANK YOU!

weff = −1 weff = −1.01

weff = −0.99

weff = −0.99 → −1.01