Vainshtein mechanism in second-order scalar-tensor theories

Transcription

1 Vainshtein mechanism in second-ode scala-tenso theoies A. De Felice, R. Kase, S. Tsujikawa Phys. Rev. D (202)! Tokyo Univesity of Science Antonio De Felice, Ryotao Kase and Shinji Tsujikawa

2 Motivation } Discovey of late-time cosmic acceleation In 998, the discovey of late-time cosmic acceleation based on Type Ia supenovae is epoted. The souce fo this acceleation is named dak enegy. The equation of state defined below chaacteizes dak enegy. w P/ρ Condition fo acceleation : w < /3 Constaint of the equation of state fom obsevations (SNe+BAO+CMB+ ) H 0 Paameteization w(a) =w 0 + w a ( a) Constaint on w 0.26 <w 0 < Constaint on w a 0.53 <w a < 0.56 σ N. SUZUKI et al. axiv: v [asto-ph.co]

3 } Candidates fo dak enegy Λ w = Cosmological constant ( ) Dak enegy poblem may imply some modification of gavity on lage scales. Modified gavitational theoies w = w(t) These models need to ecove Geneal Relativistic (GR) behavio at shot distances to satisfy sola system constaints.

4 } Recovey of GR behavio at shot distances. Chameleon mechanism Howeve a fine tuning of initial conditions is equied to ealize a viable cosmology. 2. Vainshtein mechanism φ( µ φ µ φ) Howeve the DGP model suffes fom a ghost, in addition to the difficulty of consistency with the combined data analysis.

5 2 Field equation in a spheically symmetic backgound Letʼs study how the Vainshtein mechanism woks in the pesence of! the tem, genealization of φ( µ φ µ φ), in the geneal action.! G(φ, X)φ S = d 4 x g F (φ) R + P (φ, X) G(φ, X)φ 2 ds 2 = e 2Ψ() dt 2 + e 2Φ() d (dθ 2 + sin 2 θ dφ 2 ) + S m We want to check the consistency with sola system! constaints. Then, the line element is! X = g µν µ φ ν φ/2 Fo example, the (0,0) component of the full equation of motion becomes 2F + F 2φ XG,X Φ + F e 2Φ 2 F 2F + φ 2 G,φ +2φ XG,X = e 2Φ (ρ m P ) Fφ F,φφ F,... These quantities should be suppessed inside the Vainshtein adius. i

6 } Equations of gavitational potential and scala field deived fom EOM! Ψ =µ ρ m + µ 2 φ + µ 3 µ i = µ i (F, P, G) φ = µ 4 ρ m + µ 5 { Φ, i } µ 4 (F,φ φ β + φ 2 G,X ) 2Fβ β (P,X +2XP,XX 2G,φ 2XG,φX ) 4(G,X + XG,XX )φ Xφ B() In DGP model, the self inteaction tem, i.e. G(φ, X) =X, gives ise to! the Vainshtein mechanism and local gavitational constaints ae satisfied.! B() 4(G,X + XG,XX )φ In the egime satisfying this condition, the Vainshtein mechanism should be at wok.! In ode to see how the mechanism woks, we need to examine the behavio of the! solutions in the egimes,! whee V () V, (2) V, (3)! is the Vainshtein adius chaacteized by! B( V ) V = 4(G,X + XG,XX )( V )φ ( V ). Ψ

7 3 Vainshtein mechanism } Foms of the functions In what follows, we need to specify the foms of the functions.! F (φ) =φ p φ q Fφ = pφ /φ pq Fφ p, q O() F (φ) =M 2 ple 2Qφ/M pl φ q Fφ = 2Qφ /M pl 2Qq q /M pl Taking account of this condition, we define the foms of functions as the action! coves a wide ange of modified gavitational models.! F (φ) =M 2 ple 2Qφ/M pl,p(φ, X) =f(φ)x, G(φ, X) =M 4n g(φ)x n Extended Galileon! Bans-Dicke theoies with! dilatoinic coupling Q O()

8 } Qualitative behavio of the solution and the Vainshtein adius! In the following, we deive the solution in the egimes () V, (2) V, (3) and study the Vainshtein mechanism. () V B( V ) V 4(G,X + XG,XX )( V )φ ( V ) φ () QM pl B g 2 F,φ φ β B Const. 3 V 4QM pl g B( V ) 2 (G,X + XG,XX )( V ) When G = X/M 3 and! f = Const. V ( Q M pl g ) /3 M ( Q g 2 c) /3 c H 0 M 3 M pl 2 c which ecoves the Vainshtein adius in the DGP model V ( g c) 2 /3,! as long as Q is of the ode of unity.!

9 (2) V φ () QM pl g B( V ) 2 V g V 2n f, g Φ g 2 Ψ g 2 2Q2 B( V ) 4n 2n V 2 2n Q 2 B( V ) V 2 2n Howeve this solution diveges at the oigin. One needs anothe solution inside =.! (3) g c > 0 n = Q M φ 3 /2 ρ c () ± 6M pl g c n> φ () Qρ c 3M pl f c Q<0 B( V ) > 0! (2) V! B( V ) < 0 (2) V B( V ) > 0 : :+

10 } Vainshtein adius and post-newtonian paamete! γ The post-newtonian paamete is stictly constained by obsevations.! Using the solution in the egime, this tight constaint is tanslated as! V γ =, γ Φ Ψ < 2Q2 2n B( V ) n 2n/(4n ) 2Q 2 B( V ) V V 2 /(2n) < PN Consistent with local gavity tests! The stat of ecoveing GR! Stong modification of gavity! PN 0 V

11 4 Application to concete models! The Vainshtein adius in the models esponsible fo dak enegy aound the sun is...! F (φ) =M 2 ple 2Qφ/M pl, P(φ, X) = ( 6Q 2 ) F (φ) Mpl 2 X, G(φ, X) = λf (φ) M 3 Mpl 2 X Vainshtein adius! V /3 4Qλ M pl g ( 6Q 2 ) 2 M 3 Q, λ V ( g 2 c) /3 M 3 M pl c 2 V 0 20 cm PN 0 7 cm c H 0 aphelion distance of Pluto :! 0 4 cm

12 5 Conclusion! } } } } In second-ode scala-tenso theoies we have studied how the Vainshtein mechanism woks in a spheically symmetic backgound with a matte souce. We deived the full equation of motion, and geneal fomula fo the Vainshtein adius V in the pesence of non-minimal coupling F (φ) =Mple 2 2Qφ/M pl. } In the case of Q = 0, we deived analytic solutions in the egimes! () V, (2) V (3) and the constaint of the! post-newtonian paamete.! We applied ou geneal esults to concete models that ae esponsible fo the late-time cosmic acceleation.! 6 Futue wok! It will be of inteest to see how the Vainshtein mechanism woks in Hondeskiʼs most geneal scala-tenso theoies having the tem as well as the X-dependence in.! G 4 G 5

13 } Hondeski L 2 = P (φ, X), L 3 = G(φ, X)φ L 4 = G 4 (φ, X)R + G 4,X (field deivatives) L 5 = G 5 (φ, X)G µν ( µ ν φ)+(g 5,X /6) (field deivatives) DGP G. W. Hondeski, Int. J. Theo. Phys. 0, (974). C. Deffayet, X. Gao, D. A. Stee and G. Zahaiade, Phys. Rev. D84, (20). φ( µ φ µ φ) G(φ, X)φ? L 3 G 4 = F (φ)/2, G 5 =0 F = M 2 pl

14 } S = d 4 x g 2 χr ω BD 2χ ( χ)2 + S = d 4 x g 2 F (φ)r + ( 6Q2 ) F (φ) M 2 pl X + χ = M 2 ple 2Qφ/M pl F (φ) =M 2 ple 2Qφ/M pl,x= 2 ( φ)2,q 2 = 2(3 + 2ω BD ) ω BD = (Q 2 =/2) S = d 4 x g Xφ 2 F (φ)r + ( 6Q2 ) F (φ) Mpl 2 X F (φ) =M 2 ple 2Qφ/M pl λf (φ) M 3 Mpl 2 Xφ

15 } Q =0 n = () V φ () f c 2g c M 3 c! 2 V 2 V n> φ () V 2f c g V f V g c 2/3 (2) V φ () f c M 3 /2 2g c (3) B( V ) V 4(G,X + XG,XX )( V )φ ( V ) = V V! i { f, g } O() V (2) V φ () f c 2g c M 3 F,φ + 2F XG,X (F,φ φ β + φ 2 G,X ) F β G eff 8πF < V + O ()! GX