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Vainshtein mechanism in modified gravity theories

Rampei Kimura Hiroshima university

TJ2012 : Japan-Thai workshop in cosmology 12/25/2012

Based on : RK, T. Kobayashi, K. Yamamoto PRD 85 024023 (2012) [arXiv : 1111.6749]

+ α

Introduction

(from WMAP website)

Accelerated expansion of the universe

• Implication of cosmological constant?

• Observationally, Λ is fine !

• Cosmological constant problem ~120 orders of magnitude differences

Gµ⌫+⇤gµ⌫ = 8⇡GTµ⌫

Can modification of gravity solve this puzzle ???

Modification of gravity

Gµ⌫ = 8⇡GTµ⌫? ? Gµ⌫ ⇡ 8⇡GTµ⌫

Horizon scaleSolar system scale

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

In order to explain an accelerated expansion of the universe, ONLY long range modification is needed

Modification of gravity Example： Brans-Dicke theory

Inconsistent with long range and short range

S =

Z d

4 x

p �g

'R� !BD

'

(r')2 �

• Self-accelerating solution

2.2 The most general second-order scalar-tensor theory 15

a different form than (2.6), and later it was rediscovered by Deffayet et al. [64] as a generalization of the galileon. The equivalence of the two expressions is shown by the authors of Ref. [65]. In this thesis, we employ the galileon-like expression (2.6) since it is probably more useful than its original form when discussing the Vainshtein mechanism. The gravitational and scalar-field equations can be found in the Appendix of Ref. [65].

2.2.1 Brans-Dicke theory

The prototype Brans-Dicke theory [14] corresponds to the choice of the following arbitrary functions,

K = ωBD φ

X, G4 = φ, G3 = G5 = 0, (2.8)

where ωBD is called the Brans-Dicke parameter, which is dimensionless parameter. In this model, the parametrized post-Newtonian parameter γ is given by

γ(ωBD) = 1 + ωBD 2 + ωBD

. (2.9)

Since solar-system test indicates that |1 − γ| < 2.3 × 10−5 [66], the Brans-Dicke parameter must satisfy ωBD > 4×104. On the other hand, ωBD < −3/2 is required in order to explain the accelerated expansion of the universe [67]. Thus the Brans-Dicke theory is not consistent with the solar system experiment and the accelerated expansion of the universe. However, if we add the potential term V (φ) in K(φ, X) and choose an appropriate potential form, then this local constraints can be avoided through Chameleon mechanism [15–17].

2.2.2 f(R) theories

The Lagrangian of f(R) theories [18–21] is given by

L = M2Pl 2

f(R), (2.10)

which corresponds to the choice,

K = − M2Pl 2

(Rf,R − f), G4 = MPl 2 φ, G3 = G5 = 0, φ = MPlf,R, (2.11)

where φ is a scalar degree of freedom and f,R = ∂f/∂R.

2.2.3 Covariant galileon

The covariant galileon (2.4) corresponds to

K = c1φ− c2X, G3 = c3 M3

X, G4 = M2Pl 2

− c4 M6

X2, G5 = c5 M9

X2, (2.12)

where ci are dimensionless model parameters and M is the constant, which has the dimension of mass.

• Solar system constraint

2.2 The most general second-order scalar-tensor theory 15

a different form than (2.6), and later it was rediscovered by Deffayet et al. [64] as a generalization of the galileon. The equivalence of the two expressions is shown by the authors of Ref. [65]. In this thesis, we employ the galileon-like expression (2.6) since it is probably more useful than its original form when discussing the Vainshtein mechanism. The gravitational and scalar-field equations can be found in the Appendix of Ref. [65].

2.2.1 Brans-Dicke theory

The prototype Brans-Dicke theory [14] corresponds to the choice of the following arbitrary functions,

K = ωBD φ

X, G4 = φ, G3 = G5 = 0, (2.8)

where ωBD is called the Brans-Dicke parameter, which is dimensionless parameter. In this model, the parametrized post-Newtonian parameter γ is given by

γ(ωBD) = 1 + ωBD 2 + ωBD

. (2.9)

Since solar-system test indicates that |1 − γ| < 2.3 × 10−5 [66], the Brans-Dicke parameter must satisfy ωBD > 4×104. On the other hand, ωBD < −3/2 is required in order to explain the accelerated expansion of the universe [67]. Thus the Brans-Dicke theory is not consistent with the solar system experiment and the accelerated expansion of the universe. However, if we add the potential term V (φ) in K(φ, X) and choose an appropriate potential form, then this local constraints can be avoided through Chameleon mechanism [15–17].

2.2.2 f(R) theories

The Lagrangian of f(R) theories [18–21] is given by

L = M2Pl 2

f(R), (2.10)

which corresponds to the choice,

K = − M2Pl 2

(Rf,R − f), G4 = MPl 2 φ, G3 = G5 = 0, φ = MPlf,R, (2.11)

where φ is a scalar degree of freedom and f,R = ∂f/∂R.

2.2.3 Covariant galileon

The covariant galileon (2.4) corresponds to

K = c1φ− c2X, G3 = c3 M3

X, G4 = M2Pl 2

− c4 M6

X2, G5 = c5 M9

X2, (2.12)

where ci are dimensionless model parameters and M is the constant, which has the dimension of mass.

Brans & Dicke (1961)

How can modification of gravity reduce General relativity at short distances??

Galileon theory ✓ Galileon

L � (@')2 ⇤' second derivative with respect to space-time

✓ Galileon term contains the second derivative term, but ...

No higher-order derivative terms in EOM !!

(Nicolis et al. ’09)

coupling between scalar and curvature

EOM � (⇤')2 � (rµr⌫')2 �Rµ⌫rµ'r⌫'

Vainshtein mechanism ✓ Galileon example (Deffayet et al. ’10)

Horizon scaleSolar system scale

“Nonlinear” “Linear”

Vainshtein radius rV ⇠ (rsr2c )1/3

F� ⇠ GM

r2

✓ r

rV

◆3/2 ⌧ GM

r2

φ is effectively weakly coupled to matter

F� ⇠ GM

r2

φ is strongly coupled to matter

Inside the Vainshtein radius, general relativity can be recovered

self-accelerating solution rc ⇠ O(H�10 )

L� = � 1

2 (@�)2 � r

2 c

2MPl (@�)2⇤�

Horndeski theory

K-essence term L2 � (@�)2, V (�)

Non-minimal derivative coupling

L5 � Gµ⌫rµ�r⌫� (Germani et al. 2011; Gubitosi, Linder 2011)

Cubic galileon term

L3 � (@�)2⇤�

L2 = K(�, X)

L3 = �G3(�, X)⇤�

L4 = G4(�, X)R+G4,X [(⇤�)2 � (rµr⌫�)(rµr⌫�)]

L5 = G5(�, X)Gµ⌫(rµr⌫�)

� 1 6 G5,X

(⇤�)3 � 3(⇤�)(rµr⌫�) (rµr⌫�)

+ 2(rµr↵�)(r↵r��)(r�rµ�) �

X = �(@�)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)

Horndeski, Int. J. Theor. Phys. 10,363 (1974) , Deffayet, Gao, Steer (2011)

Einstein-Hilbert term

L4 � (M2Pl/2)R

Question :

Does Vainshtein mechanism work in Horndeski theory ??

Formulation ✓ Newtonian gauge and perturbations on cosmological background

Neglect

EOM � ⇢

”mass terms”, ”time derivative terms”,

✓ L(t)2 @2✏

◆n ,

✓ L(t) @✏

◆m . . .

�

✏ = ,�, and Q ⌧ 1

Picking up the terms like @2✏, (@2✏)2, (@2✏)3, (@2✏)4, �

2

variety of solutions of the key equation (50). In Appendix C, we present the Fourier transform of the perturbation equations.

II. COSMOLOGY IN THE MOST GENERAL SCALAR-TENSOR THEORY

We consider a theory whose action is given by

S =

∫ d4x

√ −g (LGG + Lm) , (1)

where

LGG = K(φ, X)−G3(φ, X)!φ +G4(φ, X)R+G4X

[ (!φ)2 − (∇µ∇νφ)2

]

+G5(φ, X)Gµν∇µ∇νφ− 1

6 G5X

[ (!φ)3

−3!φ(∇µ∇νφ)2 + 2(∇µ∇νφ)3 ] ,(2)

with four arbitrary functions, K,G3, G4, and G5, of φ and X := −(∂φ)2/2. Here GiX stands for ∂Gi/∂X, and hereafter we will use such a notation without stating so. The Lagrangian LGG is a mixture of the gravitational and scalar-field portions, as the Ricci scalar R and the Einstein tensor Gµν are included. Note in particular that if a constant piece is present in G4 then it gives rise to the Einstein-Hilbert term. We assume that matter, described by Lm, is minimally coupled to gravity.

The Lagrangian (2) gives the most general scalar- tensor theory with second-order field equations in four dimensions. The most general theory was constructed for the first time by Horndeski [7] in a different form than (2), and later it was rediscovered by Deffayet et al. [9] as a generalization of the Galileon. The equivalence of the two expressions is shown by the authors of Ref. [13]. In this paper, we employ the Galileon-like expression (2) since it is probably more useful than its original form when discussing the Vainshtein mechanism. The gravitational and scalar-field equations can be found in the Appendix of Ref. [13].

We now replicate the cosmological background equa- tions in