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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