Vainshtein mechanism in a cosmological background in the most - - PowerPoint PPT Presentation

Collaborators : Tsutomu Kobayashi (Kyoto Univ.) Kazuhiro Yamamoto (Hiroshima Univ.)

Rampei Kimura (Hiroshima Univ.)

Based on : Phys. Rev. D 85, 024023 (2012) [arXiv:1111.6749]

Vainshtein mechanism in a cosmological background in the most general second-order scalar-tensor theory

Asia Pacific School @ YITP

Contents

Introduction and Brief review Vainshtein mechanism in the most general scalar-tensor theory

Formulation Equations Specific cases (I, II, III)

Conclusion

Horizon scale Solar system scale General relativity Modified gravity ??

(Cosmic acceleration) (Large scale structure) (Earth, Sun)

Alternative : Modification of gravity

Horizon scale Solar system scale General relativity Modified gravity ??

(Cosmic acceleration) (Large scale structure) (Earth, Sun)

Alternative : Modification of gravity

Modified gravity must recover “general relativity behavior” at short distance

Horizon scale Solar system scale General relativity Modified gravity ??

(Cosmic acceleration) (Large scale structure) (Earth, Sun)

Alternative : Modification of gravity

Screening mechanism

Modified gravity must recover “general relativity behavior” at short distance

Vainshtein Mechanism

✓ Example (kinetic gravity braiding)

L = M 2

Pl

2 R − 1 2(∂φ)2 − r2

c

2MPl (∂φ)2⇤φ + Lm[ψ, gµν]

(Deffayet et al. ’10)

rs = GM : Schwarzshild radius

φ(t, x) → φ(t)[1 + ϕ(x)]

Vainshtein Mechanism

✓ Example (kinetic gravity braiding)

L = M 2

Pl

2 R − 1 2(∂φ)2 − r2

c

2MPl (∂φ)2⇤φ + Lm[ψ, gµν]

(Deffayet et al. ’10) self-accelerating solution

rc ∼ O(H−1

0 )

rs = GM : Schwarzshild radius

φ(t, x) → φ(t)[1 + ϕ(x)]

Vainshtein Mechanism

✓ Example (kinetic gravity braiding)

L = M 2

Pl

2 R − 1 2(∂φ)2 − r2

c

2MPl (∂φ)2⇤φ + Lm[ψ, gµν]

(Deffayet et al. ’10) self-accelerating solution

rc ∼ O(H−1

0 )

Horizon scale Solar system scale “Nonlinear” “Linear”

rs = GM : Schwarzshild radius

φ(t, x) → φ(t)[1 + ϕ(x)]

Vainshtein Mechanism

✓ Example (kinetic gravity braiding)

L = M 2

Pl

2 R − 1 2(∂φ)2 − r2

c

2MPl (∂φ)2⇤φ + Lm[ψ, gµν]

(Deffayet et al. ’10) self-accelerating solution

rc ∼ O(H−1

0 )

Horizon scale Solar system scale “Nonlinear” “Linear”

Vainshtein radius

rV ∼ (rsr2

c)1/3

rs = GM : Schwarzshild radius

φ(t, x) → φ(t)[1 + ϕ(x)]

Vainshtein Mechanism

✓ Example (kinetic gravity braiding)

L = M 2

Pl

2 R − 1 2(∂φ)2 − r2

c

2MPl (∂φ)2⇤φ + Lm[ψ, gµν]

(Deffayet et al. ’10) self-accelerating solution

rc ∼ O(H−1

0 )

Horizon scale Solar system scale “Nonlinear” “Linear”

Vainshtein radius

rV ∼ (rsr2

c)1/3

rs = GM : Schwarzshild radius

dϕ dr ∼ rs r2

φ(t, x) → φ(t)[1 + ϕ(x)]

Vainshtein Mechanism

✓ Example (kinetic gravity braiding)

L = M 2

Pl

2 R − 1 2(∂φ)2 − r2

c

2MPl (∂φ)2⇤φ + Lm[ψ, gµν]

(Deffayet et al. ’10) self-accelerating solution

rc ∼ O(H−1

0 )

Horizon scale Solar system scale “Nonlinear” “Linear”

Vainshtein radius

rV ∼ (rsr2

c)1/3

rs = GM : Schwarzshild radius

dϕ dr ∼ rs r2 dϕ dr ⇠ rs r2 ✓ r rV ◆3/2 ⌧ rs r2

φ(t, x) → φ(t)[1 + ϕ(x)]

The most general scalar-tensor theory

L2 = K(φ, X) L3 = G3(φ, X)⇤φ L4 = G4(φ, X)R + G4,X[(⇤φ)2 (rµrνφ)(rµrνφ)] L5 = G5(φ, X)Gµν(rµrνφ) 1 6G5,X  (⇤φ)3 3(⇤φ)(rµrνφ) (rµrνφ) + 2(rµrαφ)(rαrβφ)(rβrµφ)

• X = −(∂φ)2/2,

GiX = ∂Gi/∂X

✓ Horndeski found the most general Lagrangian whose EOM is second-order differential

equation for φ and gμν (also known as Generalized galileon)

Deffayet, Gao, Steer (2011) Kobayashi, Yamaguchi, Yokoyama, Prog. Theor. Phys. 126, 511 (2011) Horndeski, Int. J. Theor. Phys. 10,363 (1974)

K-essence term

L2 ⊃ (∂φ)2, V (φ)

The most general scalar-tensor theory

L2 = K(φ, X) L3 = G3(φ, X)⇤φ L4 = G4(φ, X)R + G4,X[(⇤φ)2 (rµrνφ)(rµrνφ)] L5 = G5(φ, X)Gµν(rµrνφ) 1 6G5,X  (⇤φ)3 3(⇤φ)(rµrνφ) (rµrνφ) + 2(rµrαφ)(rαrβφ)(rβrµφ)

• X = −(∂φ)2/2,

GiX = ∂Gi/∂X

✓ Horndeski found the most general Lagrangian whose EOM is second-order differential

equation for φ and gμν (also known as Generalized galileon)

Deffayet, Gao, Steer (2011) Kobayashi, Yamaguchi, Yokoyama, Prog. Theor. Phys. 126, 511 (2011) Horndeski, Int. J. Theor. Phys. 10,363 (1974)

K-essence term

L2 ⊃ (∂φ)2, V (φ)

Cubic galileon term

L3 ⊃ (∂φ)2⇤φ

The most general scalar-tensor theory

L2 = K(φ, X) L3 = G3(φ, X)⇤φ L4 = G4(φ, X)R + G4,X[(⇤φ)2 (rµrνφ)(rµrνφ)] L5 = G5(φ, X)Gµν(rµrνφ) 1 6G5,X  (⇤φ)3 3(⇤φ)(rµrνφ) (rµrνφ) + 2(rµrαφ)(rαrβφ)(rβrµφ)

• X = −(∂φ)2/2,

GiX = ∂Gi/∂X

✓ Horndeski found the most general Lagrangian whose EOM is second-order differential

equation for φ and gμν (also known as Generalized galileon)

Deffayet, Gao, Steer (2011) Kobayashi, Yamaguchi, Yokoyama, Prog. Theor. Phys. 126, 511 (2011) Horndeski, Int. J. Theor. Phys. 10,363 (1974)

Einstein-Hilbert term

L4 ⊃ (M 2

Pl/2)R

K-essence term

L2 ⊃ (∂φ)2, V (φ)

Cubic galileon term

L3 ⊃ (∂φ)2⇤φ

The most general scalar-tensor theory

L2 = K(φ, X) L3 = G3(φ, X)⇤φ L4 = G4(φ, X)R + G4,X[(⇤φ)2 (rµrνφ)(rµrνφ)] L5 = G5(φ, X)Gµν(rµrνφ) 1 6G5,X  (⇤φ)3 3(⇤φ)(rµrνφ) (rµrνφ) + 2(rµrαφ)(rαrβφ)(rβrµφ)

• X = −(∂φ)2/2,

GiX = ∂Gi/∂X

✓ Horndeski found the most general Lagrangian whose EOM is second-order differential

equation for φ and gμν (also known as Generalized galileon)

Deffayet, Gao, Steer (2011) Kobayashi, Yamaguchi, Yokoyama, Prog. Theor. Phys. 126, 511 (2011) Horndeski, Int. J. Theor. Phys. 10,363 (1974)

Einstein-Hilbert term

L4 ⊃ (M 2

Pl/2)R

K-essence term

L2 ⊃ (∂φ)2, V (φ)

Non-minimal derivative coupling

L5 Gµνrµφrνφ

(Germani et al. 2011; Gubitosi, Linder 2011)

Cubic galileon term

L3 ⊃ (∂φ)2⇤φ

The most general scalar-tensor theory

L2 = K(φ, X) L3 = G3(φ, X)⇤φ L4 = G4(φ, X)R + G4,X[(⇤φ)2 (rµrνφ)(rµrνφ)] L5 = G5(φ, X)Gµν(rµrνφ) 1 6G5,X  (⇤φ)3 3(⇤φ)(rµrνφ) (rµrνφ) + 2(rµrαφ)(rαrβφ)(rβrµφ)

• X = −(∂φ)2/2,

GiX = ∂Gi/∂X

✓ Horndeski found the most general Lagrangian whose EOM is second-order differential

equation for φ and gμν (also known as Generalized galileon)

Deffayet, Gao, Steer (2011) Kobayashi, Yamaguchi, Yokoyama, Prog. Theor. Phys. 126, 511 (2011) Horndeski, Int. J. Theor. Phys. 10,363 (1974)

QUESTION :

Does Vainshtein mechanism work in the most general second-order scalar-tensor theory in a cosmological background??? QUESTION :

✏ = Ψ, Φ, and Q

EOM ⊃ ⇢ ”mass terms”, ”time derivative terms”, ✓ L(t)2 @2✏ ◆n , ✓ L(t) @✏ ◆m . . .

• Formulation

✓ In field equations,

⌧ 1

Q ≡ H δφ ˙ φ

Neglect

✏ = Ψ, Φ, and Q

EOM ⊃ ⇢ ”mass terms”, ”time derivative terms”, ✓ L(t)2 @2✏ ◆n , ✓ L(t) @✏ ◆m . . .

• Formulation

✓ In field equations,

⌧ 1

Q ≡ H δφ ˙ φ

Neglect Quasi-static approximation

∂t ⌧ ∂x

✏ = Ψ, Φ, and Q

EOM ⊃ ⇢ ”mass terms”, ”time derivative terms”, ✓ L(t)2 @2✏ ◆n , ✓ L(t) @✏ ◆m . . .

• Formulation

✓ In field equations,

⌧ 1

Q ≡ H δφ ˙ φ

Neglect Quasi-static approximation

∂t ⌧ ∂x

✏ = Ψ, Φ, and Q

L(t) ∼ O(H−1)

higher-order terms

EOM ⊃ ⇢ ”mass terms”, ”time derivative terms”, ✓ L(t)2 @2✏ ◆n , ✓ L(t) @✏ ◆m . . .

• Formulation

✓ In field equations,

⌧ 1

Q ≡ H δφ ˙ φ

Neglect Quasi-static approximation

∂t ⌧ ∂x

✏ = Ψ, Φ, and Q

L(t) ∼ O(H−1)

higher-order terms Picking up the terms like

@2✏, (@2✏)2, (@2✏)3, (@2✏)4,

EOM ⊃ ⇢ ”mass terms”, ”time derivative terms”, ✓ L(t)2 @2✏ ◆n , ✓ L(t) @✏ ◆m . . .

• Formulation

✓ In field equations,

⌧ 1

Q ≡ H δφ ˙ φ

Traceless part of the Einstein Equations

r2 (FT Ψ GT Φ A1Q) = B1 2a2H2 Q(2) + B3 a2H2

• r2Φr2Q ∂i∂jΦ∂i∂jQ
• Q(2) ⌘
• r2Q

2 (∂i∂jQ)2

Non-linear terms

Traceless part of the Einstein Equations

r2 (FT Ψ GT Φ A1Q) = B1 2a2H2 Q(2) + B3 a2H2

• r2Φr2Q ∂i∂jΦ∂i∂jQ
• Q(2) ⌘
• r2Q

2 (∂i∂jQ)2

Non-linear terms

Traceless part of the Einstein Equations

r2 (FT Ψ GT Φ A1Q) = B1 2a2H2 Q(2) + B3 a2H2

• r2Φr2Q ∂i∂jΦ∂i∂jQ
• Q(2) ⌘
• r2Q

2 (∂i∂jQ)2 ✓ The propagation speed of the gravitational waves

c2

h ≡ FT

GT

Kobayashi, Yamaguchi, Yokoyama,

• Prog. Theor. Phys. 126, 511 (2011)

FT ≡ 2

• G4 − X
• ¨

φG5X + G5φ

• ,

GT ≡ 2

• G4 − 2XG4X − X
• H ˙

φG5X − G5φ

• Coefficients

00 component of the Einstein equation

Q(3) ⌘

• r2Q

3 3r2Q (∂i∂jQ)2 + 2 (∂i∂jQ)3 U(3) ⌘ Q(2)r2Φ 2r2Q∂i∂jQ∂i∂jΦ + 2∂i∂jQ∂j∂kQ∂k∂iΦ

A0r2Q A1r2Ψ A2r2Φ + B0 a2H2 Q(2) B1 a2H2

• r2Ψr2Q ∂i∂jΨ∂i∂jQ
• B2

a2H2

• r2Φr2Q ∂i∂jΦ∂i∂jQ
• B3

a2H2

• r2Φr2Ψ ∂i∂jΦ∂i∂jΨ
• C0

a4H4 Q(3) C1 a4H4 U(3) = 0

GT r2Ψ = a2 2 ρmδ A2r2Q

• B2

2a2H2 Q(2) B3 a2H2

• r2Ψr2Q ∂i∂jΨ∂i∂jQ
• C1

3a4H4 Q(3)

Scalar field equation

Non-linear terms Non-linear terms

00 component of the Einstein equation

Q(3) ⌘

• r2Q

3 3r2Q (∂i∂jQ)2 + 2 (∂i∂jQ)3 U(3) ⌘ Q(2)r2Φ 2r2Q∂i∂jQ∂i∂jΦ + 2∂i∂jQ∂j∂kQ∂k∂iΦ

A0r2Q A1r2Ψ A2r2Φ + B0 a2H2 Q(2) B1 a2H2

• r2Ψr2Q ∂i∂jΨ∂i∂jQ
• B2

a2H2

• r2Φr2Q ∂i∂jΦ∂i∂jQ
• B3

a2H2

• r2Φr2Ψ ∂i∂jΦ∂i∂jΨ
• C0

a4H4 Q(3) C1 a4H4 U(3) = 0

GT r2Ψ = a2 2 ρmδ A2r2Q

• B2

2a2H2 Q(2) B3 a2H2

• r2Ψr2Q ∂i∂jΨ∂i∂jQ
• C1

3a4H4 Q(3)

Scalar field equation

Spherically Symmetric Case

where

c2

h

Ψ0 r − Φ0 r − α1 Q0 r = β1 H2 ✓Q0 r ◆2 + 2 β3 H2 Φ0 r Q0 r Ψ0 r + α2 Q0 r = 1 8πGT δM(t, r) r3 − β2 H2 ✓Q0 r ◆2 − 2 β3 H2 Ψ0 r Q0 r − 2 3 γ1 H4 ✓Q0 r ◆3 α0 Q0 r − α1 Ψ0 r − α2 Φ0 r = 2  − β0 H2 ✓Q0 r ◆2 + β1 H2 Ψ0 r Q0 r + β2 H2 Φ0 r Q0 r + β3 H2 Φ0 r Ψ0 r + γ0 H4 ✓Q0 r ◆3 + γ1 H4 Φ0 r ✓Q0 r ◆2

c2

h ≡ FT

GT , αi(t) ≡ Ai GT , βi(t) ≡ Bi GT , γi(t) ≡ Ci GT .

✓ EOM for gravity and scalar field can be integrated once,

Spherically Symmetric Case

where

c2

h

Ψ0 r − Φ0 r − α1 Q0 r = β1 H2 ✓Q0 r ◆2 + 2 β3 H2 Φ0 r Q0 r Ψ0 r + α2 Q0 r = 1 8πGT δM(t, r) r3 − β2 H2 ✓Q0 r ◆2 − 2 β3 H2 Ψ0 r Q0 r − 2 3 γ1 H4 ✓Q0 r ◆3 α0 Q0 r − α1 Ψ0 r − α2 Φ0 r = 2  − β0 H2 ✓Q0 r ◆2 + β1 H2 Ψ0 r Q0 r + β2 H2 Φ0 r Q0 r + β3 H2 Φ0 r Ψ0 r + γ0 H4 ✓Q0 r ◆3 + γ1 H4 Φ0 r ✓Q0 r ◆2

c2

h ≡ FT

GT , αi(t) ≡ Ai GT , βi(t) ≡ Bi GT , γi(t) ≡ Ci GT .

✓ EOM for gravity and scalar field can be integrated once,

enclosed mass

Functions of K, G3, G4, G5

Spherically Symmetric Case

where

c2

h

Ψ0 r − Φ0 r − α1 Q0 r = β1 H2 ✓Q0 r ◆2 + 2 β3 H2 Φ0 r Q0 r Ψ0 r + α2 Q0 r = 1 8πGT δM(t, r) r3 − β2 H2 ✓Q0 r ◆2 − 2 β3 H2 Ψ0 r Q0 r − 2 3 γ1 H4 ✓Q0 r ◆3 α0 Q0 r − α1 Ψ0 r − α2 Φ0 r = 2  − β0 H2 ✓Q0 r ◆2 + β1 H2 Ψ0 r Q0 r + β2 H2 Φ0 r Q0 r + β3 H2 Φ0 r Ψ0 r + γ0 H4 ✓Q0 r ◆3 + γ1 H4 Φ0 r ✓Q0 r ◆2

c2

h ≡ FT

GT , αi(t) ≡ Ai GT , βi(t) ≡ Bi GT , γi(t) ≡ Ci GT .

✓ EOM for gravity and scalar field can be integrated once,

enclosed mass

Linear Solution

✓ For sufficiently large r,

Φ0 = 1 8πGT c2

hα0 − α2 1

α0 + (2α1 + c2

hα2)α2

δM r2 Ψ0 = 1 8πGT α0 + α1α2 α0 + (2α1 + c2

hα2)α2

δM r2 Q0 = 1 8πGT α1 + c2

hα2

α0 + (2α1 + c2

hα2)α2

δM r2

(time-dependent) effective gravitational coupling

G(Linear)

eff

(6= GN)

Linear Solution

✓ For sufficiently large r,

Φ0 = 1 8πGT c2

hα0 − α2 1

α0 + (2α1 + c2

hα2)α2

δM r2 Ψ0 = 1 8πGT α0 + α1α2 α0 + (2α1 + c2

hα2)α2

δM r2 Q0 = 1 8πGT α1 + c2

hα2

α0 + (2α1 + c2

hα2)α2

δM r2

Case 1 : G4X = 0, G5 = 0

Non-linear Solution

Case 2 : Case 3 : B. G5X = 0

G5X 6= 0

Case 1 : G4X = 0, G5 = 0

Non-linear Solution

Case 2 : Case 3 : B. G5X = 0

G5X 6= 0

Quadratic equation for Q’

a Q02 + b Q0 + c = 0

Case 1 : G4X = 0, G5 = 0

Non-linear Solution

Case 2 : Case 3 : B. G5X = 0

G5X 6= 0

Quadratic equation for Q’

a Q02 + b Q0 + c = 0

Cubic equation for Q’

a Q03 + b Q02 + c Q0 + d = 0

Case 1 : G4X = 0, G5 = 0

Non-linear Solution

Case 2 : Case 3 : B. G5X = 0

G5X 6= 0

Quadratic equation for Q’

a Q02 + b Q0 + c = 0

Cubic equation for Q’

a Q03 + b Q02 + c Q0 + d = 0

Very complicated ... (sextic equation for Q’)

a Q06 + b Q05 + c Q04 + d Q03 + e Q02 + f Q0 + g = 0

Case 1 :

G4X = 0, G5 = 0

✓ Kinetic gravity braiding (with non-minimal coupling)

L = G4(φ)R + K(φ, X) − G3(φ, X)✷φ.

✓In this case, the propagation speed of the gravitational waves is

(Deffayet et al. 2010)

Generalization of the cubic galileon theory

⊃ (∂φ)2⇤φ

In this c2

h = 1.

Case 1 :

✓ In this case, the equation for Q reduces to a quadratic equation

G4X = 0, G5 = 0

µ ≡ δM 8πGT

B(t) H2 ✓Q0 r ◆2 + 2Q0 r = 2 C(t) µ r3

Case 1 :

✓ In this case, the equation for Q reduces to a quadratic equation

G4X = 0, G5 = 0

Q′ r = H2 B

• 1 + 2BCµ

H2r3 − 1

• ✓ Solving for Q’

µ ≡ δM 8πGT

B(t) H2 ✓Q0 r ◆2 + 2Q0 r = 2 C(t) µ r3

Case 1 :

✓ In this case, the equation for Q reduces to a quadratic equation

G4X = 0, G5 = 0

Q′ r = H2 B

• 1 + 2BCµ

H2r3 − 1

• ✓ Solving for Q’

The sign is chosen to be connected to the linear solution

µ ≡ δM 8πGT

B(t) H2 ✓Q0 r ◆2 + 2Q0 r = 2 C(t) µ r3

Case 1 :

✓ In this case, the equation for Q reduces to a quadratic equation

G4X = 0, G5 = 0

Q′ r = H2 B

• 1 + 2BCµ

H2r3 − 1

• ✓ Solving for Q’

The sign is chosen to be connected to the linear solution

µ ≡ δM 8πGT

B(t) H2 ✓Q0 r ◆2 + 2Q0 r = 2 C(t) µ r3

Vainshtein radius rV (~ 100 pc for sun)

Case 1 :

✓ In this case, the equation for Q reduces to a quadratic equation

G4X = 0, G5 = 0

Q′ r = H2 B

• 1 + 2BCµ

H2r3 − 1

• ✓ Solving for Q’

The sign is chosen to be connected to the linear solution

µ ≡ δM 8πGT

B(t) H2 ✓Q0 r ◆2 + 2Q0 r = 2 C(t) µ r3

Vainshtein radius rV (~ 100 pc for sun)

• ∗

BC Q′ ≃ H B

• 2BCµ

r . ⌧ GM

r2

✓ Inside the Vainshtein radius rV

Case 1 :

✓ In this case, the equation for Q reduces to a quadratic equation

G4X = 0, G5 = 0

Q′ r = H2 B

• 1 + 2BCµ

H2r3 − 1

• ✓ Solving for Q’

The sign is chosen to be connected to the linear solution

µ ≡ δM 8πGT

B(t) H2 ✓Q0 r ◆2 + 2Q0 r = 2 C(t) µ r3

Vainshtein radius rV (~ 100 pc for sun)

Scalar field is screened !!

• ∗

BC Q′ ≃ H B

• 2BCµ

r . ⌧ GM

r2

✓ Inside the Vainshtein radius rV

Case 1 :

G4X = 0, G5 = 0

✓ Gravitational Coupling GN

8πGN ≡ 1 2G4(t)

time-dependent

✓ Friedmann equation

3H2 = 8πGcos (ρm + ρφ)

Case 1 :

G4X = 0, G5 = 0

✓ Gravitational Coupling GN

8πGN ≡ 1 2G4(t)

time-dependent

✓ Friedmann equation

3H2 = 8πGcos (ρm + ρφ)

Case 1 :

G4X = 0, G5 = 0

✓ Gravitational Coupling GN

8πGN ≡ 1 2G4(t)

time-dependent

(Uzan 2011)

Time-dependence can be tested by BBN !!

where Gcos = 1/16πG4 = GN ˙

• 1 − GN|BBN

GN|now

• 0.1

✓ Friedmann equation

3H2 = 8πGcos (ρm + ρφ)

Case 1 :

G4X = 0, G5 = 0

✓ Gravitational Coupling GN

8πGN ≡ 1 2G4(t)

time-dependent

✓ PPN parameter

γ ≡ Ψ0 Φ0 = 1

(Uzan 2011)

Time-dependence can be tested by BBN !!

where Gcos = 1/16πG4 = GN ˙

• 1 − GN|BBN

GN|now

• 0.1

✓ Lagrangian

B. G5X = 0

L = K(φ, X) G3(φ, X)⇤φ + G4(φ, X)R + G4X ⇥ (⇤φ)2 (rµrνφ)2⇤ + G5(φ)Gµνrµrνφ

Case 2 :

c2

h = 1 + 2β1 6= 1 ✓In this case, the propagation speed of the gravitational waves is

does not depend on the kinetic term X

✓ Lagrangian

B. G5X = 0

L = K(φ, X) G3(φ, X)⇤φ + G4(φ, X)R + G4X ⇥ (⇤φ)2 (rµrνφ)2⇤ + G5(φ)Gµνrµrνφ

Case 2 :

c2

h = 1 + 2β1 6= 1 ✓In this case, the propagation speed of the gravitational waves is

does not depend on the kinetic term X

✓ Cubic eqn

(Q0)3 + C2H2r(Q0)2 + ✓C1 2 H4r2 − H2Cβ µ r ◆ Q0 − H4Cαµ 2 = 0

B. G5X = 0

Case 2 :

✓ Cubic eqn

(Q0)3 + C2H2r(Q0)2 + ✓C1 2 H4r2 − H2Cβ µ r ◆ Q0 − H4Cαµ 2 = 0

B. G5X = 0

Case 2 :

Functions of K, G3, G4, G5

✓ Cubic eqn

(Q0)3 + C2H2r(Q0)2 + ✓C1 2 H4r2 − H2Cβ µ r ◆ Q0 − H4Cαµ 2 = 0

B. G5X = 0

Case 2 :

Functions of K, G3, G4, G5

✓ 3 possible solutions at short distance can be matched to the linear solution

Q′ ≃ +H

• Cβ

µ r , −H

• Cβ

µ r , −Cα Cβ H2r 2 .

✓ 3 solutions at short distances

B. G5X = 0

Case 2 :

Q0 ' ±H r Cβ µ r , Cα Cβ H2r 2

✓ 3 solutions at short distances

B. G5X = 0

Case 2 :

Q0 ' ±H r Cβ µ r , Cα Cβ H2r 2

✓ Metric perturbations

Ψ0 = Φ0 ' CΨ(t) 8πGT (t) δM r2

Newton’ s constant GN(t)

✓ 3 solutions at short distances

B. G5X = 0

Case 2 :

Q0 ' ±H r Cβ µ r , Cα Cβ H2r 2

✓ Metric perturbations

Ψ0 = Φ0 ' CΨ(t) 8πGT (t) δM r2

Newton’ s constant GN(t)

✓ 3 solutions at short distances

B. G5X = 0

Case 2 :

Q0 ' ±H r Cβ µ r , Cα Cβ H2r 2

✓ Metric perturbations

Ψ0 = Φ0 ' CΨ(t) 8πGT (t) δM r2

Gcos = GN

Time-dependence of GN can be tested by BBN

Friedmann equation

3H2 = 8πGcos (ρm + ρφ)

Newton’ s constant GN(t)

✓ 3 solutions at short distances

B. G5X = 0

Case 2 :

Q0 ' ±H r Cβ µ r , Cα Cβ H2r 2

✓ PPN parameter

γ ≡ Ψ0 Φ0 = 1

✓ Metric perturbations

Ψ0 = Φ0 ' CΨ(t) 8πGT (t) δM r2

Gcos = GN

Time-dependence of GN can be tested by BBN

Friedmann equation

3H2 = 8πGcos (ρm + ρφ)

B. G5X = 0

Case 2 :

Q0 ' ±H r Cβ µ r , Cα Cβ H2r 2

✓ 3 solutions at short distance

B. G5X = 0

Case 2 :

Q0 ' ±H r Cβ µ r , Cα Cβ H2r 2

✓ 3 solutions at short distance ✓ Metric perturbations

Φ0 ' c2

hΨ0 '

c2

h

8πGT δM r2

B. G5X = 0

Case 2 :

Q0 ' ±H r Cβ µ r , Cα Cβ H2r 2

✓ 3 solutions at short distance ✓ Metric perturbations

Φ0 ' c2

hΨ0 '

c2

h

8πGT δM r2

Newton’ s constant GN(t)

B. G5X = 0

Case 2 :

✓ PPN parameter

γ ⌘ Ψ0 Φ0 = 1 c2

h

(6= 1)

propagation speed of gravitational waves

Q0 ' ±H r Cβ µ r , Cα Cβ H2r 2

✓ 3 solutions at short distance ✓ Metric perturbations

Φ0 ' c2

hΨ0 '

c2

h

8πGT δM r2

Newton’ s constant GN(t)

B. G5X = 0

Case 2 :

✓ PPN parameter

γ ⌘ Ψ0 Φ0 = 1 c2

h

(6= 1)

propagation speed of gravitational waves

Q0 ' ±H r Cβ µ r , Cα Cβ H2r 2

Solar-system tests

constrained from solar j1 j < 2:3 105

(Will 2005)

✓ 3 solutions at short distance ✓ Metric perturbations

Φ0 ' c2

hΨ0 '

c2

h

8πGT δM r2

Newton’ s constant GN(t)

✓ At sufficiently small scales,

Case 3 :

G5X 6= 0

Ψ0(r), Φ0(r) ∝ 1 r2

✓ At sufficiently small scales,

Case 3 :

G5X 6= 0

The Vainshtein mechanism no longer works in the presence of G5X !!

Ψ0(r), Φ0(r) ∝ 1 r2

Summary

Vainshtein screening successfully operates in the most general second-order scalar-tensor theory, but

• Newton’

s constant G=G(t)

• constrained from PPN and BBN
• inverse-square law can not be reproduced at small

scales if G5X≠0