Cosmology in generalized Proca theories

Transcription

1 3-rd Korea-Japan workshop on dark energy, April, 2016 Cosmology in generalized Proca theories Shinji Tsujikawa (Tokyo University of Science) Collaboration with A.De Felice, L.Heisenberg, R.Kase, S.Mukohyama, Y.Zhang, G.Zhao (arxiv: , )

2 There have been many attempts for constructing dark energy models in the framework of scalar-tensor theories. Most of them belong to the so-called Horndeski theories: Most general scalar-tensor theories with second-order equations Horndeski (1973)

3 What happens if the vector degree of freedom is present? (i) Maxwell field (massless) Lagrangian: L F = 1 4 F µ F µ There are two transverse polarizations (electric and magnetic fields). (ii) Proca field (massive) Lagrangian: L F = 1 4 F µ F µ 1 2 m2 A µ A µ Introduction of the mass m of the vector field A µ allows the propagation in the longitudinal direction due to the breaking of U(1) gauge invariance. 2 transverse and 1 longitudinal = 3 DOFs Longitudinal propagation

4 Generalized Proca theories Heisenberg (2014), Tasinato (2014) In a curved background, it is possible to extended the massive Proca theories to those containing three DOFs (besides two tensor polarizations). Heisenberg Lagrangian L (2014) L F = 1 4 F µνf µν, L 2 = G 2 (X), L 3 = G 3 (X) µ A µ, L 4 = G 4 (X)R + G 4,X (X) [ ( µ A µ ) 2 + c 2 ρ A σ ρ A σ (1 + c 2 ) ρ A σ σ A ρ], L 5 = G 5 (X)G µν µ A ν 1 6 G 5,X(X)[( µ A µ ) 3 3d 2 µ A µ ρ A σ ρ A σ 3(1 d 2 ) µ A µ ρ A σ σ A ρ +(2 3d 2 ) ρ A σ γ A ρ σ A γ +3d 2 ρ A σ γ A ρ γ A σ ]. where X = A µ A µ /2 and F µ = r µ A r A µ. The terms proportional to c 2 and d 2 can be expressed in terms of F µ, so they correspond to pure vector modes. The non-minimal derivatives couplings like G 4 (X)R are required to keep the equations of motion up to second order.

5 Cosmology in generalized Proca theories arxiv: Can we realize a viable cosmology with the late-time acceleration? Vector field: A µ =(φ(t), 0, 0, 0) Variation of the Heisenberg action with respect to g µ on the flat FLRW background leads to G 2 G 2,X φ 2 3G 3,X Hφ 3 +6G 4 H 2 6(2G 4,X + G 4,XX φ 2 )H 2 φ 2 + G 5,XX H 3 φ 5 +5G 5,X H 3 φ 3 = ρ M, G 2 φφ 2 G 3,X +2G 4 (3H 2 +2Ḣ) 2G 4,Xφ (3H 2 φ +2H φ +2Ḣφ) 4G 4,XXH φφ 3 + G 5,XX H 2 φφ 4 + G 5,X Hφ 2 (2Ḣφ +2H2 φ +3H φ) = P M. The matter density M and the pressure P M obey the continuity equation ( ) ρ M +3H(ρ M + P M )=0 Variation of the action with respect to A µ leads to (which does not break spatial isotropy) φ ( G 2,X +3G 3,X Hφ +6G 4,X H 2 +6G 4,XX H 2 φ 2 3G 5,X H 3 φ G 5,XX H 3 φ 3) =0. The branch 6= 0 gives the solution where depends on H alone, which allows the existence of de Sitter solutions with constant and H.

6 Vector Galileons ( ) The Lagrangian of vector Galileons which recover the Galilean symmetry in the scalar limit (A µ ) on the flat space-time is given by G 2 (X) =b 2 X, G 3 (X) =b 3 X, G 4 (X) = M 2 pl 2 + b 4X 2, G 5 (X) =b 5 X 2. We substitute these functions into the vector-field equation: G 2,X +3G 3,X Hφ +6G 4,X H 2 +6G 4,XX H 2 φ 2 3G 5,X H 3 φ G 5,XX H 3 φ 3 =0, Taking note that X = 2 /2, the background EOM admits the solution H = constant. The temporal component is small in the early cosmological epoch, but it grows with the decrease of H. The solution finally approaches the de Sitter attractor characterized by = constant, H = constant.

7 Phase-space trajectories for vector Galileons (c) The de Sitter fixed point (c) is always stable against homogeneous perturbations, so it corresponds to the late-time attractor. The dark energy equation of state w DE is 2 during the matter era. (b) (a) This behavior is the same as a tracker solution of scalar Galileons, which is in tension with the observational data (Nesseris, De Felice, ST, 2010). (a) Radiation point (b) Matter point (c) De Sitter point

8 Generalizations of vector Galileons Let us consider the case in which p / H 1 This solution can be realized for is related with H according to (p>0) G 2 (X) =b 2 X p 2, G 3 (X) =b 3 X p 3, G 4 (X) = M 2 pl 2 + b 4X p 4, G 5 (X) =b 5 X p 5, where p 3 = 1 2 (p +2p 2 1), p 4 = p + p 2, p 5 = 1 2 (3p +2p 2 1). The dark energy and radiation density parameters obey Ω DE = (1 + s)ω DE(3 + Ω r 3Ω DE ) 1+s Ω DE, Ω r = Ω r[1 Ω r +(3+4s)Ω DE ] 1+s Ω DE, where s p 2 p. The vector Galileon corresponds to p 2 = p = 1. There are 3 fixed points: (a) (Ω DE, Ω r )=(0, 1) (b) (Ω DE, Ω r )=(0, 0) (c) (Ω DE, Ω r )=(1, 0)

9 The dark energy equation of state w DE = 3(1 + s)+sω (a) w DE = 1 4s/3 in the radiation era, r 3(1 + s Ω DE ). (b) w DE = 1 s in the matter era, (c) w DE = 1 in the de Sitter era For smaller s = p 2 /p close to 0, w DE = 1 s approaches 1. The joint data analysis of SNIa, CMB, and BAO give the bound 0 apple s apple 0.36 (95 %CL) Smaller s (De Felice and ST, 2012) For larger p the field evolves more slowly as / H 1/p,sow DE approaches 1. Vector Galileons

10 Cosmological perturbations in generalized Proca theories We need to study perturbations on the flat FLRW background to study (i) Conditions for avoiding ghosts and instabilities, (ii) Observational signatures for the matter distribution in the Universe. In doing so, let us consider the perturbed metric in flat gauge: ds 2 = (1 + 2α) dt 2 +2( i χ + V i ) dt dx i + a 2 (t)(δ ij + h ij ) dx i dx j, where, are scalar perturbations, V i and h ij are the vector and tensor perturbations, respectively, obeying i V i =0, i h ij =0, h i i =0. We also consider the perturbations of the vector field, as A 0 = φ(t)+δφ, A i = 1 a 2 δij ( j χ V + E j ) where and V are scalar perturbations, while E j is the vector perturbation j E j = 0.

11 Tensor perturbations : 2 Dofs There are two polarization modes h + and h for the tensor perturbation: h ij = h + e + ij + h e ij Expanding the Heisenberg action up to second order in tensor [ perturbations, the resulting second-order action is given by [ S (2) T = λ=+, dt d 3 xa 3 q T 8 [ ḣ 2 λ c2 T a 2 ( h λ) 2 ], where q T =2G 4 2φ 2 G 4,X + Hφ 3 G 5,X, c 2 T = 2G 4 + φ 2 φ G5,X q T. The tensor perturbation obeys (in Fourier space) ḧ λ + ( 3H + q T q T ) ḣ λ + c 2 k 2 T a 2 h λ =0 The ghost and instability can be avoided for q T > 0, c 2 T > 0

12 Vector perturbations : 2 Dofs Besides the vector field, we take into account a single perfect fluid described by the Schutz-Sorkin action: S M = d 4 x [ g ρ M (n)+j µ ( µ l + A 1 µ B 1 + A 2 µ B 2 ) ] ( ) Related with the number density, as n = q J J g /g Scalar part Vector part After integrating out the matter action, introducing the combination Z i = E i + (t)v i, and finally taking the small-scale limit, the resulting vector action (for two dofs Z 1,Z 2 ) reads where S (2) V 2 i=1 dt d 3 x aq V 2 (Ż 2i + k2 a 2 c2 V Z 2 i q V =1 2c 2 G 4,X 2d 2 Hφ G 5,X, ), c 2 V =1+ φ2 (2G 4,X G 5,X Hφ) 2 2q T q V + d 2G 5,X (Hφ φ) q V.

13 ) Scalar perturbations : 2 Dofs (1 scalar +1 matter) The second-order Lagrangian for scalar perturbations is given by L (2) S = a3 ( X t K X + k 2 a 2 X t G X X t M X X t B X where = V + (t) and M is the matter perturbation. ), X X t =(ψ, δρ M ). If the two eigenvalues of the 2 2 matrix K are positive, the ghosts are absent. One of them is M + P M > 0, and another is Q S = a3 H 2 q T (3w q T w 4 ) 2 (w 1 2w 2 ) 2 In the small-scale limit, the dispersion relation is given by det (ω 2 K k2 a 2 G ) =0, One of the solutions is the matter propagation speed squared, while another one is c 2 S = 1 { ( ) 2 w 2 ( 2w 3 (ρ M + P M ) w 3 (w 1 2w 2 )[w 1 w 2 + φ(w 1 2w 2 )w 6 ] φ/φ H w 3 2w 2 2 ẇ 1 w1ẇ 2 ) 2 + φ (w 1 2w 2 ) 2 [ w 3 ẇ 6 + φ(2w 3 w 7 + w6) 2 ] [ ]} + w 1 w 2 w 1 w 2 +(w 1 2w 2 )(2φ w 6 w 3 φ/φ), where =8H 2 2 q T q V q S, and w 1 etc are the known from the background. > 0

14 A model consistent with no-ghost and stability conditions G 2 (X) =b 2 X, G 3 (X) =b 3 X p 3, G 4 (X) = M 2 pl 2 + b 4X p 4, G 5 (X) =0. Provided that 0 < 4 < 1/[6(2p + 1)], there exists the parameter space in which all the theoretically consistent conditions are satisfied. q T > 0,q V > 0,Q S > 0 c 2 T > 0,c 2 V > 0,c 2 S > 0 c 2 T is sub-luminal. (right) versus 1 + z for p 2 =1,p =5,β 4 =0.01, β 5 =0,c 2 = 1, and l and m 2, respectively. The background initial conditions are chosen to 6. er the inequality (5.50). On the other hand the condition (5.52) ), so the allowed range of β 4 shrinks to

15 A model with a super-luminal tensor propagation speed From the Cherenkov radiation, the sub-luminal tensor propagation speed is constrained to be 1 c T < In the previous model, this puts tight constraints on b 4. In the presence of the Lagrangian L 5, it is possible to avoid the Cherenkov radiation constraint with c 2 T > 1. p 2 =1/2, p =5/2, 4 =0.01, 5 =0.05, c 2 = 1, d 2 =0 It remains to see to find viable parameter spaces in which all the theoretical and observational constraints are satisfied.

16 Effective gravitational couplings associated with the cosmic growth Under the quasi-static approximation on sub-horizon scales, the matter perturbation obeys M +2H M 4 G e M M ' 0 where the effective gravitational coupling is ξ 1 =4πφ 2 (w 2 +2Hq T ) 2, G e = ξ 2 =[H (w 2 +2Hq T ) ẇ 1 +2ẇ 2 + ρ M ] φ 2 w2 2, q [ V 1 ξ 3 = 8H 2 φ 2 qs 3 q T c 2 2φ 2 {q S [w 2 ẇ 1 (w 2 2Hq T )ẇ 2 ]+ρ M w 2 [3w 2 (w 2 +2Hq T ) q S ]} S q ] 2 S w 2 {w 6 φ(w 2 +2Hq T ) w 2 (w 2 2Hq T )}. q V 3 is positive under the no-ghost and stability conditions (which enhances the gravitational attraction). For smaller q V close to 0, there is a tendency that G e decreases.

17 [ ] Additional contribution to the Heisenberg Lagrangian L 6 = G 6 (X)L µναβ µ A ν α A β G 6,X(X) F αβ F µν α A µ β A ν Horndeski, 1976 Jimenez and Heisenberg, 2016 G e is modified only through the change of q V : q V =1 2c 2 G 4,X 2d 2 H G 5,X +2G 6 H 2 +2G 6,X H 2 2 The term G 6 can decrease q V. For smaller q V approaching 0, the e ect of the vector field tends to reduce the gravitational attraction. It is possible to distinguish the models with G 6 = 0 and G 6 6= 0 observationally. Smaller q V p 2 =1/2, p =5/2, 4 = 10 4, 5 =0.052, c 2 = 0, d 2 =0

18 Conclusions and outlook 1. Generalized Proca theories give rise to interesting cosmological solutions with a late-time de Sitter attractor. 2. We derived 6 no-ghost and stability conditions associated with tensor, vector, and scalar perturbations for the consistency of the theory. 3. We constructed a class of models in which all the theoretically consistent conditions are satisfied during the cosmic expansion history. 4. We also derived the e ective gravitational coupling that can be used to put observational constraints on the models. It will be of interest to put observational constraints on the models in the framework of generalized Proca theories.