Galileon and,Modi cations of gravityGalileon and, Modi cations of gravity Antonio De Felice, Tokyo...

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Galileon and,Modifications of gravity

Antonio De Felice,

Tokyo University of Science, 東京理科大学

Summer Institute 2010 – 10 August 2010

with Shinji Mukohyama, Takahiro Tanaka, Shinji Tsujikawa

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Mt. Fuji, 富富富士士士山山山

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• Mazinger-Z was built with 超合金 (chougokin)

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• Mazinger-Z was built with 超合金 (chougokin)

• Only found in Mount Fuji

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Introduction

• Mistery of Dark Energy

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Introduction


• Complex problem

3

Introduction


• Complex problem

• Motivated different approaches

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Three approaches

• Introduction of new matter, e.g. quintessence

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Three approaches


• Introduction of new gravity

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Three approaches



1. DGP

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Three approaches



1. DGP

2. f(R), f(R,G) . . .

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Three approaches



1. DGP

2. f(R), f(R,G) . . .

• New gravity and new matter

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Three approaches



1. DGP

2. f(R), f(R,G) . . .


1. Extended Brans-Dicke

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Three approaches



1. DGP

2. f(R), f(R,G) . . .


1. Extended Brans-Dicke

2. Galileon . . .

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Part I – Modified gravity

• Consider L = f(R) theories

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• Equivalent to ST theory L = f ′(φ) (R− φ) + f(φ)

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• In a consistent way? No extra tensors, f ′ = fR > 0

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• What new degrees of freedom? 1 SF

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• What new degrees of freedom? 1 SF

• Interesting phenomenology [Review by ADF, Tsujikawa ’10]

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General MGM

• Introduce the gravity action L = f(R,G)[ADF, Carroll, Duvvuri, Easson, Trodden, Turner ’05]

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General MGM


• No ghost tensor modes, L = fλ(R−λ) +fσ(G−σ) +f

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General MGM



• Two extra new scalar fields, λ, σ

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General MGM



• Two extra new scalar fields, λ, σ

• Only one on FLRW and ω2 ∝ (k/a)4

[ADF, Suyama ’09; ADF, Tanaka ’10]

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Degrees of freedom [ADF, Tanaka ’10]

• On Kasner background, expand S at 2nd order, 3 flds

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• L ∝ AijΦiΦj − [Cij k2 +Mij] ΦiΦj + . . .

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• On Kasner detA < 0, 1 ghost mode

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• On FLRW, detA = 0, A33 = 0, C33 = 0, M33 6= 0

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• On FLRW, detA = 0, A33 = 0, C33 = 0, M33 6= 0

• Φ3, the ghost, integrated out.

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Galileon [Nicolis et al ’09]

• Keeping EOM 2nd order necessary but not sufficient

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• DGP inspired introducing kinetic non-linearities to SF

to implement Veinshtein mech. (mass, Cham)

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• Able to freeze the field avoiding SSC

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• But need to keep EOM 2nd order

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• But need to keep EOM 2nd order

• Possible example for 4D action coming from ED, ST

[de Rham, Tolley ’10]

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The action [Nicolis et al; Deffayet et al ’09]

• Galileon symm in EOM for SF: ∂µφ→ ∂µφ+ bµ

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• Building blocks

L1 = M 3φ ,

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• Building blocks

L1 = M 3φ ,

L2 = ∂µφ∂µφ ,

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• Building blocks

L1 = M 3φ ,

L2 = ∂µφ∂µφ ,

L3 = (φ)(∇φ)2/M 3 ,

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• Building blocks

L1 = M 3φ ,

L2 = ∂µφ∂µφ ,

L3 = (φ)(∇φ)2/M 3 ,

L4 = (∇φ)2 [2(φ)2 − 2φ;µνφ;µν −R(∇φ)2/2

]/M 6 ,

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• Building blocks

L1 = M 3φ ,

L2 = ∂µφ∂µφ ,

L3 = (φ)(∇φ)2/M 3 ,

L4 = (∇φ)2 [2(φ)2 − 2φ;µνφ;µν −R(∇φ)2/2

]/M 6 ,

L5 = (∇φ)2 [(φ)3 − 3(φ)φ;µνφ;µν

+ 2φ;µνφ;ν

ρφ;ρµ − 6φ;µφ

;µνφ;ρGνρ]/M 9 .

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• Similar to Lovelock scalars [Deffayet et al ’09]

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• The Galileon symmetry is broken in general

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• Is it needed for Veinshtein mech?

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• Is it needed for Veinshtein mech?

• Study started with L3

[Silva and Koyama ’09]

[Chow and Khoury ’09]

[Kobayashi, Tashiro, Suzuki ’10]

[Kobayashi, Yamaguchi, Yokoyama ’10]

[ADF and Tsujikawa ’10]

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Generalized Brans-Dicke theory[ADF, Tsujikawa ’10; ADF, Mukohyama, Tsujikawa ’10]

• Lagrangian

L = 12F (φ)R− 1

2B(φ)∂φ2 + ξ(φ)L3 .

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• Lagrangian

L = 12F (φ)R− 1

2B(φ)∂φ2 + ξ(φ)L3 .

• Non-minimal coupling and GL symm broken

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• Lagrangian

L = 12F (φ)R− 1

2B(φ)∂φ2 + ξ(φ)L3 .


• F = M 2p (φ/Mp)3−n, B = ω(φ/Mp)1−n,

ξ = (λ/µ3)(φ/Mp)−n

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• Lagrangian

L = 12F (φ)R− 1

2B(φ)∂φ2 + ξ(φ)L3 .


• F = M 2p (φ/Mp)3−n, B = ω(φ/Mp)1−n,

ξ = (λ/µ3)(φ/Mp)−n

• Existence of late dS era

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Results

• µ = (MpH2dS)1/3

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Results

• µ = (MpH2dS)1/3

• Constraints from BG (n < 3) + Lapl inst. (n > 2) +

no-ghost

[ADF, Mukohyama, Tsujikawa ’10]

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Results

• µ = (MpH2dS)1/3

• Constraints from BG (n < 3) + Lapl inst. (n > 2) +

no-ghost

[ADF, Mukohyama, Tsujikawa ’10]

• 2 ≤ n ≤ 3, φ/(Hφ) > 0→ ω < −n(n− 3)2, λ < 0

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Allowed parameter space

Allowed parameter space in

terms of ω vs n. n < 3 for

DM era, n > 2 for

c2s,dS > 0. φ/φ > 0 for no

scalar ghost. F > 0 for no

tensor ghost.

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Typical background evolution

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Pure Galileon [ADF, Tsujikawa ’10]

• Let us come back to the Galileon Lagrangian

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Pure Galileon [ADF, Tsujikawa ’10]

• Let us come back to the Galileon Lagrangian

• Building blocks

L1 = M 3φ ,

L2 = ∂µφ∂µφ ,

L3 = (φ)(∇φ)2/M 3 ,

L4 = (∇φ)2 [2(φ)2 − 2φ;µνφ;µν −R(∇φ)2/2

]/M 6 ,

L5 = (∇φ)2 [(φ)3 − 3(φ)φ;µνφ;µν

+ 2φ;µνφ;ν

ρφ;ρµ − 6φ;µφ

;µνφ;ρGνρ]/M 9 .

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Total Lagrangian

L = 12M

2pR+ 1

2

∑i

ciLi + Lmat ,

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Total Lagrangian

L = 12M

2pR+ 1

2

∑i

ciLi + Lmat ,

• 2nd order EOM

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Total Lagrangian

L = 12M

2pR+ 1

2

∑i

ciLi + Lmat ,

• 2nd order EOM

• Cosmological background

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Total Lagrangian

L = 12M

2pR+ 1

2

∑i

ciLi + Lmat ,

• 2nd order EOM


• Linear perturbation analysis on FLRW

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Total Lagrangian

L = 12M

2pR+ 1

2

∑i

ciLi + Lmat ,

• 2nd order EOM


• Linear perturbation analysis on FLRW

• Studied up to L4 [Gannouji, Sami ’10]

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Field equations

• On FLRW:

3M 2pH

2 = ρDE + ρm + ρr ,

3M 2pH

2 + 2M 2pH = −pDE − 1

3ρr .

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Field equations

• On FLRW:

3M 2pH

2 = ρDE + ρm + ρr ,

3M 2pH

2 + 2M 2pH = −pDE − 1

3ρr .

• Imply ρDE + 3H(ρDE + pDE) = 0

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Field equations

• On FLRW:

3M 2pH

2 = ρDE + ρm + ρr ,

3M 2pH

2 + 2M 2pH = −pDE − 1

3ρr .

• Imply ρDE + 3H(ρDE + pDE) = 0

• Admits de Sitter solutions

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• Define for convenience

r1 ≡φdSHdS

φH, r2 ≡

φ4

r1 φ4dS

, xdS ≡φdS

HdSMp

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r1 ≡φdSHdS

φH, r2 ≡

φ4

r1 φ4dS

, xdS ≡φdS

HdSMp

• From FRD eq

ΩDE ≡ρDE

3M 2pH

2 = −16c2x

2dSr

31r2 + c3x

3dSr

21r2

− 152 c4x

4dSr1r2 + 7c5x

5dSr2

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r1 ≡φdSHdS

φH, r2 ≡

φ4

r1 φ4dS

, xdS ≡φdS

HdSMp

• From FRD eq

ΩDE ≡ρDE

3M 2pH

2 = −16c2x

2dSr

31r2 + c3x

3dSr

21r2

− 152 c4x

4dSr1r2 + 7c5x

5dSr2

• Cannot forget L5

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• Define

α ≡ c4x4dS , β ≡ c5x

5dS

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• Define


5dS

• EOM for de Sitter impose

c2x2dS = 6 + 9α− 12β , c3x

3dS = 2 + 9α− 9β

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• Define


5dS

• EOM for de Sitter impose

c2x2dS = 6 + 9α− 12β , c3x

3dS = 2 + 9α− 9β

• EOM for r′1, r′2, and Ω′r, Ωm from FRD

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きききれれれいいいなななGalileon

• EOM for r1:

r′1 = (r1 − 1) f(r1, r2,Ωr) , r′2 = . . . , Ω′r = . . .

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• EOM for r1:

r′1 = (r1 − 1) f(r1, r2,Ωr) , r′2 = . . . , Ω′r = . . .

• Existence of equilibrium point

r1 = 1

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• EOM for r1:

r′1 = (r1 − 1) f(r1, r2,Ωr) , r′2 = . . . , Ω′r = . . .


r1 = 1 , ΩDE = r2 , φ ∝ H−1

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• EOM for r1:

r′1 = (r1 − 1) f(r1, r2,Ωr) , r′2 = . . . , Ω′r = . . .


r1 = 1 , ΩDE = r2 , φ ∝ H−1

• Small BG perturbations

δr′1 = −9 + Ωr + 3r2

2(1 + r2)δr1

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Other variables

• EOM

r′2 =2r2 (3− 3r2 + Ωr)

1 + r2, Ω′r =

Ωr (Ωr − 1− 7r2)1 + r2

,

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Other variables

• EOM

r′2 =2r2 (3− 3r2 + Ωr)

1 + r2, Ω′r =

Ωr (Ωr − 1− 7r2)1 + r2

,

• Universality: no α and β dependence

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Other variables

• EOM

r′2 =2r2 (3− 3r2 + Ωr)

1 + r2, Ω′r =

Ωr (Ωr − 1− 7r2)1 + r2

,


• Three fixed points: RAD = (r1 = 1, r2 = 0,Ωr = 1),

DM = (1, 0, 0), dS = (1, 1, 0)

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Other variables

• EOM

r′2 =2r2 (3− 3r2 + Ωr)

1 + r2, Ω′r =

Ωr (Ωr − 1− 7r2)1 + r2

,


• Three fixed points: RAD = (r1 = 1, r2 = 0,Ωr = 1),

DM = (1, 0, 0), dS = (1, 1, 0)

• Last is stable, others are saddle as r2 unstable

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Features

• BG Signature

wDE ≡pDE

ρDE= − Ωr + 6

3 (r2 + 1), weff ≡ −1−2H ′

3H=

Ωr − 6 r2

3 (r2 + 1).

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Features

• BG Signature

wDE ≡pDE

ρDE= − Ωr + 6

3 (r2 + 1), weff ≡ −1−2H ′

3H=

Ωr − 6 r2

3 (r2 + 1).

• Evolution

wDE = −73 → −2→ −1 , weff = 1

3 → 0→ −1

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Features

• BG Signature

wDE ≡pDE

ρDE= − Ωr + 6

3 (r2 + 1), weff ≡ −1−2H ′

3H=

Ωr − 6 r2

3 (r2 + 1).

• Evolution

wDE = −73 → −2→ −1 , weff = 1

3 → 0→ −1

• To be tested against observations

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Background

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Removing degeneracy

• Universality leads to BG degeneracy

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Removing degeneracy


• LPT removes degeneracy

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Removing degeneracy


• LPT removes degeneracy

• Use no-ghost and c2S > 0, c2

T > 0 conditions

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LPT results

Galileon and,Modi cations of gravityGalileon and, Modi cations of gravity Antonio De Felice, Tokyo...

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