Constraints on general second-order scalar-tensor models from - PowerPoint PPT Presentation

Constraints on general second-order scalar-tensor models from gravitational Cherenkov radiation Rampei Kimura Hiroshima University JGRG @Tokyo University 11/14/2012 Based on RK and Kazuhiro Yamamoto, JCAP 07 (2012) 050

Introduction Accelerating universe • Implication of cosmological constant? G µ ν + Λ g µ ν = 8 π GT µ ν • Observationally, fine !! • Cosmological constant problem 121 orders of magnitude differences Can modification of gravity solve this puzzle ??? (from WMAP website)

Galileon theory ✓ Galileon (Nicolis et al. ʼ 09) ⇤ = r µ r µ L ⊃ ( ∂ϕ ) 2 ⇤ ϕ second derivative with respect to space-time ✓ Galileon term contains the second derivative term, but ... coupling between scalar and curvature EOM � ( ⇤ ϕ ) 2 � ( r µ r ν ϕ ) 2 � R µ ν r µ ϕ r ν ϕ No higher-order derivative terms in EOM !!

Most general second-order scalar-tensor theory (MGST) ✓ Horndeski found the most general Lagrangian whose EOM is second-order differential equation for φ and g μν (also known as Generalized galileon ) Horndeski, Int. J. Theor. Phys. 10,363 (1974) , Deffayet, Gao, Steer (2011) L 2 ⊃ ( ∂φ ) 2 , V ( φ ) L 2 = K ( φ , X ) K-essence term 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

Why galileon?? Self-accelerating solution Free of ghost-instabilities Vainshtein mechanism (Vainshtein 1972) • Scalar field is effectively weakly coupled to matter in a high density region • Reduce general relativity at small scales Relation with decoupling limit in massive gravity (de Rham, Gabadadze, Tolley, 2010)

Cosmological observations RK, Kazuhiro Yamamoto, JCAP 04 (2011) 025 RK, Tsutomu Kobayashi, Kazuhiro Yamamoto, Physical Review D 85 (2012) 123503 Standard rulers (supernovae + CMB shift parameter) • Not powerful tools to constrain model parameters in modified gravity theories, but useful tools to determine cosmological parameters Galaxy distribution (SDSS LRG sample) • The error bar is still large to constrain model parameters Cross correlation between LSS and ISW • Excellent tool to constrain modified gravity • Indicates that the effective gravitational coupling G eff has to be smaller than ~1.2 G N , otherwise CCF becomes negative which contradicts with observations

Other signatures ??

Sound speed of graviton in MGST Quadratic action for a tensor mode in the most general scalar-tensor theory Kobayashi, Yamaguchi, Yokoyama, Prog. Theor. Phys. 126, 511 (2011), � � = 1 � ij − F T S (2) G T ˙ dtd 3 xa 3 h 2 a 2 ( � ∇ h ij ) 2 T 8 Sound speed of graviton T ≡ F T c 2 . G T h ⇣ ⌘i ¨ F T ≡ 2 G 4 − X φ G 5 X + G 5 φ h ⇣ ⌘i H ˙ G T ≡ 2 G 4 − 2 XG 4 X − X φ G 5 X − G 5 φ Sound speed of graviton could be different from speed of light !!!

Gravitational Cherenkov radiation If the sound speed of graviton is smaller than the speed of light, particle should emit graviton through the similar process to Cherenkov radiation Moore and Nelson (2001) graviton particle particle Highest energy cosmic ray (p ~ 3 × 10 11 GeV) can provide us the lower bound on the sound speed of graviton

Gravitational Cherenkov radiation Consider the complex scalar in a FRW background d 4 x √− g � − g µ ν ∂ µ Ψ ∗ ∂ ν Ψ − m 2 Ψ ∗ Ψ − ξ R Ψ ∗ Ψ � � S m = Quantize the complex scalar and tensor field as d 3 p Ψ ( η , x ) = 1 � � b p ψ p ( η ) e i p · x + ˆ p ( η ) e − i p · x � ˆ ˆ c † p ψ ∗ [ˆ b p , ˆ b † (2 π ) 3 / 2 a p 0 ] = δ ( p − p 0 ) d 3 k � c † h µ ν = 1 2 � � � p 0 ] = δ ( p − p 0 ) [ˆ c p , ˆ a k h k ( η ) e i k · x + ε ( λ ) ˆ a † � ε ( λ ) k ( η ) e − i k · x k h ∗ µ ν ˆ µ ν ˆ a G T (2 π ) 3 / 2 λ Mode functions satisfy ✓ d 2 ◆ d η 2 + p 2 + m 2 a 2 ψ p ( η ) = 0 ✓ d 2 T k 2 − a 00 ◆ d η 2 + c 2 h k ( η ) = 0 a

Gravitational Cherenkov radiation The total radiation energy from the complex scalar field a † ( λ ) a ( λ ) X X ⌦ ↵ E = ( ω k /a ) ˆ ˆ k k λ k where � t � t 2 � � � � a † ( λ ) a ( λ ) a † ( λ ) a ( λ ) ˆ ˆ = 2 ℜ dt 2 dt 1 H I ( t 1 )ˆ ˆ k H I ( t 2 ) k k k t in t in Z d 3 xh ij ∂ i Ψ ∂ j Ψ ∗ H I = a Graviton emission rate (using sub-horizon approximation) dt ' G N p 4 4(1 � c T ) 2 dE in a 4 3(1 + c T ) 2 A particle with momentum p cannot possibly have been traveling for longer than t ∼ a 4 (1 + c T ) 2 1 4(1 − c T ) 2 p 3 G N

＞ Gravitational Cherenkov radiation Observations from cosmic rays tells Time scale that cosmic Time scale that ray turn into radiation cosmic ray travels energy of graviton from origin to us The highest energy cosmic ray E highest ∼ 10 11 GeV • Energy • Distance c t ∼ 10 kpc Constraint on the sound speed of graviton ∼ 2 × 10 − 15 1 − c T <

Toy Model 1 K = X Gubitosi and Linder model (Gubitosi and Linder 2011) G 3 = 0 G 4 = M 2 Pl / 2  M 2 Z d 4 x p� g � λ Pl S = 2 R + X + G µ ν r µ φ r µ φ + L m [ g µ ν , ψ ] G 5 = − λφ /M 2 M 2 Pl Pl Condition for existence of self-accelerating solution and avoiding the ghost-instability M 2 M 2 − 1 < λ < − 1 λ is always negative Pl Pl H 2 H 2 18 30 0 0 Sound speed of graviton Pl + 2 λ ˙ T = M 2 φ 2 /M 2 c 2 Pl < 1 Pl − 2 λ ˙ M 2 φ 2 /M 2 Pl Inconsistent with the constraint from the gravitational Cherenkov radiation...

Toy Model 2 c T > 1 − ✏ ✏ = 2 × 10 − 15 Extended galileon model (De Felice and Tsujikawa 2011) p = 1 and q = 1/2 K = − c 2 M 4(1 − p 2 ) X p 2 , 2 G 3 = c 3 M 1 − 4 p 3 X p 3 , 1.0 3 c T > 1 - e G 4 = 1 pl − c 4 M 2 − 4 p 4 2 M 2 X p 4 , 4 0.5 G 5 = 3 c 5 M − (1+4 p 5 ) X p 5 , 5 0.0 b p 2 = p - 0.5 p 3 = p + (2 q − 1 / 2) 4 model parameters p, q, α (c No ghost–instabilities - 1.0 ), β (c p 4 = p + 2 q ) 3 4 p 5 = p + (6 q − 1) / 2 - 1.5 - 2 - 1 0 1 2 a Strong constraints for the model parameters α and β

Summary • In the most general scalar-tensor theory, sound speed of graviton could be different from speed of light • The constraints from gravitational Cherenkov radiation would be a powerful probe. • Gravitational Cherenkov radiation could be a criteria for the construction of modification of gravity

Gravitational Cherenkov radiation for massive graviton Quadratic action for a tensor mode for massive graviton Gumrukcuoglu, Kuroyanagi, Lin, Mukohyama, Tanahashi (2012) � � γ ij + c 2 dtd 3 x Na 3 √ tensor = M 2 g ( t ) � 1 γ ij ˙ a 2 γ ij ( △ − 2 K ) γ ij − M 2 I (2) P l GW ( t ) γ ij γ ij N 2 ˙ Ω 8 Dispersion relation g k 2 + a 2 M 2 ω 2 k = c 2 GW For c g =c, there is no gravitational Cherenkov radiation even if m ≠ 0 Currently, checking the case c g ≠ c and m ≠ 0...