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

Vainshtein mechanism in a cosmological background in the most general second-order scalar-tensor theory Rampei Kimura (Hiroshima Univ.) Asia Pacific School @ YITP Collaborators : Tsutomu Kobayashi (Kyoto Univ.) Kazuhiro Yamamoto (Hiroshima Univ.) Based on : Phys. Rev. D 85, 024023 (2012) [arXiv:1111.6749]

Contents Introduction and Brief review Vainshtein mechanism in the most general scalar-tensor theory Formulation Equations Specific cases (I, II, III) Conclusion

Alternative : Modification of gravity General relativity Modified gravity ?? Solar system scale Horizon scale (Earth, Sun) (Large scale structure) (Cosmic acceleration)

Alternative : Modification of gravity General relativity Modified gravity ?? Solar system scale Horizon scale (Earth, Sun) (Large scale structure) (Cosmic acceleration) Modified gravity must recover “general relativity behavior” at short distance

Alternative : Modification of gravity General relativity Modified gravity ?? Solar system scale Horizon scale (Earth, Sun) (Large scale structure) (Cosmic acceleration) Modified gravity must recover “general relativity behavior” at short distance Screening mechanism

Vainshtein Mechanism ✓ Example (kinetic gravity braiding) (Deffayet et al. ’ 10) L = M 2 r 2 2 R − 1 Pl 2( ∂φ ) 2 − ( ∂φ ) 2 ⇤ φ + L m [ ψ , g µ ν ] c 2 M Pl φ ( t, x ) → φ ( t )[1 + ϕ ( x )] r s = GM : Schwarzshild radius

Vainshtein Mechanism self-accelerating solution ✓ Example (kinetic gravity braiding) (Deffayet et al. ’ 10) r c ∼ O ( H − 1 0 ) L = M 2 r 2 2 R − 1 Pl 2( ∂φ ) 2 − ( ∂φ ) 2 ⇤ φ + L m [ ψ , g µ ν ] c 2 M Pl φ ( t, x ) → φ ( t )[1 + ϕ ( x )] r s = GM : Schwarzshild radius

Vainshtein Mechanism self-accelerating solution ✓ Example (kinetic gravity braiding) (Deffayet et al. ’ 10) r c ∼ O ( H − 1 0 ) L = M 2 r 2 2 R − 1 Pl 2( ∂φ ) 2 − ( ∂φ ) 2 ⇤ φ + L m [ ψ , g µ ν ] c 2 M Pl Solar system scale Horizon scale “Nonlinear” “Linear” φ ( t, x ) → φ ( t )[1 + ϕ ( x )] r s = GM : Schwarzshild radius

Vainshtein Mechanism self-accelerating solution ✓ Example (kinetic gravity braiding) (Deffayet et al. ’ 10) r c ∼ O ( H − 1 0 ) L = M 2 r 2 2 R − 1 Pl 2( ∂φ ) 2 − ( ∂φ ) 2 ⇤ φ + L m [ ψ , g µ ν ] c 2 M Pl c ) 1 / 3 r V ∼ ( r s r 2 Vainshtein radius Solar system scale Horizon scale “Nonlinear” “Linear” φ ( t, x ) → φ ( t )[1 + ϕ ( x )] r s = GM : Schwarzshild radius

Vainshtein Mechanism self-accelerating solution ✓ Example (kinetic gravity braiding) (Deffayet et al. ’ 10) r c ∼ O ( H − 1 0 ) L = M 2 r 2 2 R − 1 Pl 2( ∂φ ) 2 − ( ∂φ ) 2 ⇤ φ + L m [ ψ , g µ ν ] c 2 M Pl c ) 1 / 3 r V ∼ ( r s r 2 Vainshtein radius Solar system scale Horizon scale “Nonlinear” “Linear” d ϕ dr ∼ r s r 2 φ ( t, x ) → φ ( t )[1 + ϕ ( x )] r s = GM : Schwarzshild radius

Vainshtein Mechanism self-accelerating solution ✓ Example (kinetic gravity braiding) (Deffayet et al. ’ 10) r c ∼ O ( H − 1 0 ) L = M 2 r 2 2 R − 1 Pl 2( ∂φ ) 2 − ( ∂φ ) 2 ⇤ φ + L m [ ψ , g µ ν ] c 2 M Pl c ) 1 / 3 r V ∼ ( r s r 2 Vainshtein radius Solar system scale Horizon scale “Nonlinear” “Linear” ✓ r ◆ 3 / 2 d ϕ dr ∼ r s d ϕ dr ⇠ r s ⌧ r s r 2 r 2 r 2 r V φ ( t, x ) → φ ( t )[1 + ϕ ( x )] r s = GM : Schwarzshild radius

The most general scalar-tensor theory ✓ Horndeski found the most general Lagrangian whose EOM is second-order differential equation for φ and g μν (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) L 2 = K ( φ , X ) L 3 = � G 3 ( φ , X ) ⇤ φ L 4 = G 4 ( φ , X ) R + G 4 ,X [( ⇤ φ ) 2 � ( r µ r ν φ )( r µ r ν φ )] L 5 = G 5 ( φ , X ) G µ ν ( r µ r ν φ ) � 1  ( ⇤ φ ) 3 � 3( ⇤ φ )( r µ r ν φ ) ( r µ r ν φ ) 6 G 5 ,X � + 2( r µ r α φ )( r α r β φ )( r β r µ φ ) X = − ( ∂φ ) 2 / 2 , G iX = ∂ G i / ∂ X

The most general scalar-tensor theory ✓ Horndeski found the most general Lagrangian whose EOM is second-order differential equation for φ and g μν (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) L 2 ⊃ ( ∂φ ) 2 , V ( φ ) K-essence term L 2 = K ( φ , X ) L 3 = � G 3 ( φ , X ) ⇤ φ L 4 = G 4 ( φ , X ) R + G 4 ,X [( ⇤ φ ) 2 � ( r µ r ν φ )( r µ r ν φ )] L 5 = G 5 ( φ , X ) G µ ν ( r µ r ν φ ) � 1  ( ⇤ φ ) 3 � 3( ⇤ φ )( r µ r ν φ ) ( r µ r ν φ ) 6 G 5 ,X � + 2( r µ r α φ )( r α r β φ )( r β r µ φ ) X = − ( ∂φ ) 2 / 2 , G iX = ∂ G i / ∂ X

The most general scalar-tensor theory ✓ Horndeski found the most general Lagrangian whose EOM is second-order differential equation for φ and g μν (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) L 2 ⊃ ( ∂φ ) 2 , V ( φ ) K-essence term L 2 = K ( φ , X ) Cubic galileon term L 3 = � G 3 ( φ , X ) ⇤ φ L 3 ⊃ ( ∂φ ) 2 ⇤ φ L 4 = G 4 ( φ , X ) R + G 4 ,X [( ⇤ φ ) 2 � ( r µ r ν φ )( r µ r ν φ )] L 5 = G 5 ( φ , X ) G µ ν ( r µ r ν φ ) � 1  ( ⇤ φ ) 3 � 3( ⇤ φ )( r µ r ν φ ) ( r µ r ν φ ) 6 G 5 ,X � + 2( r µ r α φ )( r α r β φ )( r β r µ φ ) X = − ( ∂φ ) 2 / 2 , G iX = ∂ G i / ∂ X

The most general scalar-tensor theory ✓ Horndeski found the most general Lagrangian whose EOM is second-order differential equation for φ and g μν (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) L 2 ⊃ ( ∂φ ) 2 , V ( φ ) K-essence term L 2 = K ( φ , X ) Cubic galileon term L 3 = � G 3 ( φ , X ) ⇤ φ L 3 ⊃ ( ∂φ ) 2 ⇤ φ Einstein-Hilbert term L 4 = G 4 ( φ , X ) R + G 4 ,X [( ⇤ φ ) 2 � ( r µ r ν φ )( r µ r ν φ )] L 4 ⊃ ( M 2 Pl / 2) R L 5 = G 5 ( φ , X ) G µ ν ( r µ r ν φ ) � 1  ( ⇤ φ ) 3 � 3( ⇤ φ )( r µ r ν φ ) ( r µ r ν φ ) 6 G 5 ,X � + 2( r µ r α φ )( r α r β φ )( r β r µ φ ) X = − ( ∂φ ) 2 / 2 , G iX = ∂ G i / ∂ X

The most general scalar-tensor theory ✓ Horndeski found the most general Lagrangian whose EOM is second-order differential equation for φ and g μν (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) L 2 ⊃ ( ∂φ ) 2 , V ( φ ) K-essence term L 2 = K ( φ , X ) Cubic galileon term L 3 = � G 3 ( φ , X ) ⇤ φ L 3 ⊃ ( ∂φ ) 2 ⇤ φ Einstein-Hilbert term L 4 = G 4 ( φ , X ) R + G 4 ,X [( ⇤ φ ) 2 � ( r µ r ν φ )( r µ r ν φ )] L 4 ⊃ ( M 2 Pl / 2) R L 5 = G 5 ( φ , X ) G µ ν ( r µ r ν φ ) Non-minimal derivative coupling � 1  L 5 � G µ ν r µ φ r ν φ ( ⇤ φ ) 3 � 3( ⇤ φ )( r µ r ν φ ) ( r µ r ν φ ) 6 G 5 ,X (Germani et al. 2011; Gubitosi, Linder 2011) � + 2( r µ r α φ )( r α r β φ )( r β r µ φ ) X = − ( ∂φ ) 2 / 2 , G iX = ∂ G i / ∂ X

QUESTION :

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

Formulation Q ≡ H δφ ˙ φ ✓ In field equations, ✏ = Ψ , Φ , and Q ⌧ 1 ⇢ EOM ⊃ ” mass terms ” , ” time derivative terms ” , ◆ n ◆ m ✓ ✓ � L ( t ) 2 @ 2 ✏ L ( t ) @✏ , . . .

Formulation Q ≡ H δφ ˙ φ ✓ In field equations, ✏ = Ψ , Φ , and Q ⌧ 1 Neglect ⇢ EOM ⊃ ” mass terms ” , ” time derivative terms ” , ◆ n ◆ m ✓ ✓ � L ( t ) 2 @ 2 ✏ L ( t ) @✏ , . . .

Formulation Q ≡ H δφ ˙ φ ✓ In field equations, ✏ = Ψ , Φ , and Q ⌧ 1 Quasi-static approximation Neglect ∂ t ⌧ ∂ x ⇢ EOM ⊃ ” mass terms ” , ” time derivative terms ” , ◆ n ◆ m ✓ ✓ � L ( t ) 2 @ 2 ✏ L ( t ) @✏ , . . .

Formulation Q ≡ H δφ ˙ φ ✓ In field equations, ✏ = Ψ , Φ , and Q ⌧ 1 Quasi-static approximation Neglect ∂ t ⌧ ∂ x ⇢ EOM ⊃ ” mass terms ” , ” time derivative terms ” , ◆ n ◆ m ✓ ✓ � L ( t ) 2 @ 2 ✏ L ( t ) @✏ , . . . L ( t ) ∼ O ( H − 1 ) higher-order terms

Formulation Q ≡ H δφ ˙ φ ✓ In field equations, ✏ = Ψ , Φ , and Q ⌧ 1 Quasi-static approximation Neglect ∂ t ⌧ ∂ x ⇢ EOM ⊃ ” mass terms ” , ” time derivative terms ” , ◆ n ◆ m ✓ ✓ � L ( t ) 2 @ 2 ✏ L ( t ) @✏ , . . . L ( t ) ∼ O ( H − 1 ) higher-order terms Picking up the terms like @ 2 ✏ , ( @ 2 ✏ ) 2 , ( @ 2 ✏ ) 3 , ( @ 2 ✏ ) 4 , �

Traceless part of the Einstein Equations r 2 ( F T Ψ � G T Φ � A 1 Q ) B 1 B 3 2 a 2 H 2 Q (2) + r 2 Φ r 2 Q � ∂ i ∂ j Φ ∂ i ∂ j Q � � = a 2 H 2 � 2 � ( ∂ i ∂ j Q ) 2 Q (2) ⌘ r 2 Q �

Traceless part of the Einstein Equations r 2 ( F T Ψ � G T Φ � A 1 Q ) B 1 B 3 2 a 2 H 2 Q (2) + r 2 Φ r 2 Q � ∂ i ∂ j Φ ∂ i ∂ j Q � � = a 2 H 2 Non-linear terms � 2 � ( ∂ i ∂ j Q ) 2 Q (2) ⌘ r 2 Q �