Galileon gravity and its relevance to late time cosmic acceleration
aa r X i v : . [ g r- q c ] J un Galileon gravity and its relevance to late time cosmic acceleration
Radouane Gannouji and M. Sami IUCAA, Post Bag 4, Ganeshkhind, Pune 411 007, India Centre of Theoretical Physics, Jamia Millia Islamia, New Delhi-110025, India
We consider the covariant galileon gravity taking into account the third order and fourth orderscalar field Lagrangians L ( π ) and L ( π ) consisting of three and four π ’s with four and five deriva-tives acting on them respectively. The background dynamical equations are set up for the systemunder consideration and the stability of the self accelerating solution is demonstrated in general set-ting. We extended this study to the general case of the fifth order theory. For spherically symmetricstatic background, we spell out conditions for suppression of fifth force effects mediated by thegalileon field π . We study field perturbations in the fixed background and investigate conditions fortheir causal propagation. We also briefly discuss metric fluctuations and derive evolution equationfor matter perturbations in galileon gravity. I. INTRODUCTION
The phenomenon of late time cosmic acceleration [1–4] is as challenging theoretically as was the problem of blackbody radiation whose resolution unveiled many secrets of micro physics. At present, there is no definite clue forthe theoretical understanding of the nature of cosmic repulsion. In recent years, a variety of approaches have beenemployed to attack the problem. According to the standard lore, the late time acceleration can be accounted forby supplementing the energy momentum tensor by an exotic fluid component with large negative pressure dubbed dark energy [5, 6] The simplest candidate of dark energy is provided by cosmological constant Λ. However, itssmall numerical value leads to fine tuning problem and we do not understand why it becomes important today a lacoincidence problem.Scalar fields provide an interesting alternative to cosmological constant though they do not address the cosmologicalconstant problem. To this effect, cosmological dynamics of a variety of scalar fields has been investigated in theliterature(see review [5] for details). They can mimic cosmological constant like behavior at late times and canprovide a viable cosmological dynamics at early epochs. Scalar field models with generic features are capable ofalleviating the fine tuning and coincidence problems. As for the observation, at present, it is absolutely consistentwith Λ but at the same time, a large number of scalar field models are also permitted. Future data should allow tonarrow down the class of permissible models of dark energy.It is quite possible that there is no dark energy and the late cosmic acceleration is an artifact of infrared modificationof gravity. We know that gravity is modified at short distance and there is no guarantee that it would not suffer anycorrection at large scales where it is never verified directly. Large scale modifications might arise from extra dimensionaleffects or can be inspired by fundamental theories. They can also be motivated by phenomenological considerationssuch as f ( R ) theories of gravity [7] or the massive theories of gravity. However, any large scale modification of gravityshould reconcile with local physics constraints and should have potential of being distinguished from cosmologicalconstant.The infrared modified theories of gravity essentially contain additional degrees of freedom. The f(R) theories containa scalar field which mediates fifth force and might contradicts the local gravity constraints such as the solar systemor laboratory tests. Broadly, two mechanisms for hiding the scalar field effects locally have been employed in theliterature. In f ( R ) theories of gravity, the scalar field is screened via the so-called chameleon mechanism [8], bymaking scalar field mass dependent on the local matter density. In generic models of f(R) gravity [9], the chameleonmechanism allows to satisfy the local gravity constraints but at the same time make these models vulnerable tocurvature singularity whose resolution requires the fine tuning worse than the one encountered in Λ CDM model.The problem can be alleviated by invoking R correction but the scenario becomes problematic if extended to earlyuniverse [9].An alternative possibility of large scale modification of gravity is provided by an effective scalar field π dubbedgalileon [10]. In particular such a field appears in the decoupling limit of DGP. The Lagrangian of the field respectsthe so called shift symmetry in a Minkowskian background: π → π + c and ∂ µ π → ∂ µ π + b µ where c and b µ areconstants. Thank to this symmetry, the equations of motion for the field contain only second derivatives. In four spacetime dimensions, there exist five Lagrangians L i , i = 1 , L is linear in π , L contains normal kinetic term. L involves three π ’s and four derivatives acting on them. This Lagrangian is obtained in the decoupling limit of DGP.The fourth and the fifth order Lagrangians involve four π ’s and six derivatives, five π ’s and seven derivatives actingon the field respectively. A general covariant form of galileon Lagrangian is obtained in Ref.[11](see also Ref.[12] onthe related theme).In DGP or its 4 dimensional generalizations-galileon gravity, the effects of extra degree are suppressed using theVainshtein [13] mechanism which allows us to recover general relativity at small scales due to non-linear interaction.From this point of view, the DGP model is attractive model which has a self-accelerating solution, an asymptoticallyde Sitter solution even in the absence of vacuum energy. Unfortunately this solution suffers from instabilities [14–19].galileon gravity can give rise to late time acceleration and is interesting for the following reasons: (i) It is freefrom negative energy instabilities. (ii) Unlike f ( R ) theories, galileon modified gravity does not suffer from curvaturesingularity. (iii)The chameleon mechanism in f ( R ) might come into conflict with the equivalence principle if the testbodies are considered as extended whereas the Vainshtein mechanism is free from this problem [20].In this paper we study 4th order galileon gravity including L and L terms in the Lagrangian. We set up FRWbackground dynamics and examine the self accelerating solution. We carry out detailed investigations on the stabilityof the solutions and discuss the spherical symmetric solutions to check the local suppression of π effects. We alsoinvestigate matter perturbations in the model under consideration. II. LOWEST ORDER GALILEON GRAVITY AND ITS SELF ACCELERATING FRW BACKGROUND
Recently, an interesting generalization of the DGP action in 4D was proposed in Ref.[10]. The authors considereda consistent general action with a self interacting scalar field ( π ) coupled. It is remarkable that the action can bemotivated by higher dimensional considerations [21]. In what follows we shall consider the action is invariant underGalilean transformation π ( x ) → π ( x ) + b µ x µ + c (1)For the sake of simplicity, we first examine the galileon model in the lowest non-trivial order keeping up to thirdorder term L in the Lagrangian, S = Z d x √− g (cid:18) R c π − c ∇ π ) − c ∇ π ) (cid:3) π (cid:19) + S m [ ψ m , e βπ g µν ] (2)Similar expression occurs in the DGP model. The corresponding Einstein’s equations are G µν = T ( m ) µν + c πg µν + c (cid:18) π ; µ π ; ν − g µν ( ∇ π ) (cid:19) + c (cid:0) π ,µ π ; ν (cid:3) π + g µν π ; λ π ; λρ π ; ρ − π ; ρ [ π ; µ π ; νρ + π ; ν π ; µρ ] (cid:1) (3)0 = βT ( m ) + c + c (cid:3) π + c (cid:0) ( (cid:3) π ) − π ; µν π ; µν − R µν π ; µ π ; ν (cid:1) (4)where T ( m ) is the trace of the matter energy-momentum tensor, T ( m ) µν ≡ − (2 / √− g ) × δ S m /δg µν . In spatially flatFRW background Eq.(4) gives rise to the following Friedmann equation3 H = ρ m − c π + c π − c H ˙ π (5)2 ˙ H + 3 H = − c π − c π − c ˙ π ¨ π (6) βρ m = c − c (3 H ˙ π + ¨ π ) + 3 c ˙ π (cid:16) H ˙ π + ˙ H ˙ π + 2 H ¨ π (cid:17) (7)It is interesting to note that Eq.(7) exhibits a self accelerating solution given by3 H = − c π + c π − c H ˙ π (8)= − c π − c π (9)which means that c = 0 (we assume ˙ π = 0) and H = − c / c . This last condition is impossible to satisfy as c should be positive for stability of the theory.We therefore conclude that a stable self accelerating solution, in general, does not exist in the third order galileongravity with ( ∇ π ) (cid:3) π term in the Einstein frame. It is therefore necessary to invoke the higher order terms L and L . In the discussion, to follow, we shall demonstrate that the desired solution can be obtained by adding the fourthorder term in the action (2). The analysis becomes cumbersome in the presence of 5th order term which completesthe Lagrangian of galileon gravity. We have included the corresponding discussion and results in the Appendix. III. GENERALIZATION TO NEXT HIGHER ORDER
Let us consider the full covariant action of galileon gravity [10, 11]. S = Z d x √− g (cid:18) R c i L ( i ) (cid:19) + S m [ ψ m , e βπ g µν ] (10)where { c i } are constants and the L ′ si are given by L (1) = π (11) L (2) = −
12 ( ∇ π ) ≡ − π ; µ π ; µ (12) L (3) = −
12 ( ∇ π ) (cid:3) π (13) L (4) = −
12 ( ∇ π ) (cid:2) ( (cid:3) π ) − π ; µν π ; µν + π ; µ π ; µ G µν (cid:3) + ( (cid:3) π ) π ; µ π ; ν π ; µν − π ; µ π ; µν π ; νρ π ; ρ (14)Varying the action (10) with respect π and the metric g µν , we obtained the field equation for π and Einsteinequations c i E ( i ) = − βT ( m ) (15) G µν = T ( m ) µν + c i T ( i ) µν (16)where E ( i ) = (1 / √− g ) × δ S ( i ) δπ and T ( i ) µν = − (2 / √− g ) × δ S ( i ) /δg µν with S ( i ) ≡ R d x √− gL ( i ) where E ′ s and T (1) ′ sµν have the following form E (1) = 1 (17) E (2) = (cid:3) π (18) E (3) = ( (cid:3) π ) − π ; µν π ; µν − R µν π ; µ π ; ν (19) E (4) = 2 ( (cid:3) π ) + 4 (cid:0) π ν ; µ π ρ ; ν π µ ; ρ (cid:1) − (cid:3) π ) ( π ; µν π ; µν ) − ( (cid:3) π ) ( π ; µ π ; µ ) R − π ; µ π ; µν π ; ν ) R − (cid:3) π ) ( π ; µ R µν π ; ν ) + 2( ∇ π ) ( π ; µν R µν ) + 8 ( π ; µ π ; µν R νρ π ; ρ ) + 4 ( π ; µ π ; ν π ; ρσ R µρνσ ) (20) T (1) µν = πg µν (21) T (2) µν = π ; µ π ; ν − g µν ( ∇ π ) (22) T (3) µν = π ,µ π ; ν (cid:3) π + g µν π ; λ π ; λρ π ; ρ − π ; ρ [ π ; µ π ; νρ + π ; ν π ; µρ ] (23) T (4) µν = − (cid:3) π ) π ; ρ (cid:2) π ; µ π ; ρν + π ; ν π ; ρµ (cid:3) + 2 ( (cid:3) π ) ( π ; µ π ; ν ) − (cid:3) π ) ( ∇ π ) ( π ; µν ) − (cid:0) π ; λ π ; λρ π ; ρ (cid:1) ( π ; µν )+4 (cid:0) π ; λ π ; λµ (cid:1) ( π ; ρ π ; ρν ) − (cid:0) π ; λρ π ; λρ (cid:1) ( π ; µ π ; ν ) + 2( ∇ π ) (cid:0) π ; ρ ; µ π ; ρν (cid:1) + 4 π ; λ π ; λρ (cid:2) π ; ρµ π ; ν + π ; ρν π ; µ (cid:3) + ( (cid:3) π ) ( ∇ π ) g µν + 4 ( (cid:3) π ) (cid:0) π ; λ π ; λρ π ; ρ (cid:1) g µν − (cid:0) π ; λ π ; λρ π ; ρσ π ; σ (cid:1) g µν − ( ∇ π ) ( π ; ρσ π ; ρσ ) g µν − ( ∇ π ) ( π ; µ π ; ν ) R + 14 ( ∇ π ) g µν R + 2( ∇ π ) π ; ρ (cid:2) R ρµ π ; ν + R ρν π ; µ (cid:3) −
12 ( ∇ π ) R µν − ∇ π ) ( π ; ρ R ρσ π ; σ ) g µν + 2( ∇ π ) ( π ; ρ π ; σ R µρνσ ) (24)It may be instructive to define the effective energy density and pressure for π matter. Indeed, for each ( i ), We have T (4) µν = − T ′ (4) µν and E (4) = − E ′ (4) compared to [11]. ∇ µ T ( i ) µν = π ; ν E ( i ) (25)which allows us to write the equation of conservation ∇ µ T ( m ) µν = βT ( m ) π ; ν (26)For each ( i ), assuming the perfect fluid form, we can express the field energy momentum tensor as, T ( i ) µν = (cid:0) ρ ( i ) + P ( i ) (cid:1) u µ u ν + P ( i ) g µν with u µ ≡ − σ π ; µ √ − ( ∇ π ) and σ = sign ( π ;0 ). The corresponding expressions for ρ i and P i have following form ρ (1) = − π P (1) = πρ (2) = − ( ∇ π ) P (2) = − ( ∇ π ) ρ (3) = π ; λ π ; λρ π ; ρ − ( ∇ π ) (cid:3) π P (3) = π ; λ π ; λρ π ; ρ ρ (4) = 6 (cid:3) ππ ; λ π ; λρ π ; ρ − (cid:3) π ) ( ∇ π ) + 3 ( ∇ π ) P (4) = ( (cid:3) π ) ( ∇ π ) + 4 (cid:3) π ; λ π ; λρ π ; ρ − π ; λ π ; λρ π ; ρσ π ; σ + R ( ∇ π ) − ( ∇ π ) π ; ρ R ρσ π ; σ − ( ∇ π ) π ; ρσ π ; ρσ + R ( ∇ π ) − ∇ π ) π ; ρ R ρσ π ; σ − π ; ρ π ; σ π ; µ π ; ν R µρνσ − π ; λ π ; λρ π ; ρσ π ; σ In the following section, we shall analyze the background solution of the fourth order theory.
IV. BACKGROUND DYNAMICS
Assuming the spatially flat background, we obtain evolution equations of the fourth order galileon cosmology,3 H = ρ m + c π − c H ˙ π + 452 c H ˙ π (27)2 ˙ H + 3 H = − c π − c ˙ π ¨ π + 32 c ˙ π (cid:16) H ˙ π + 2 ˙ H ˙ π + 8 H ¨ π (cid:17) (28) βρ m = − c (3 H ˙ π + ¨ π ) + 3 c ˙ π (cid:16) H ˙ π + ˙ H ˙ π + 2 H ¨ π (cid:17) − c H ˙ π (cid:16) H ˙ π + 2 ˙ H ˙ π + 3 H ¨ π (cid:17) , (29)where we have assumed, c = 0 as we do not want include the cosmological constant explicitly. In this case, theconservation has standard form in presence of coupling β ˙ ρ m + 3 Hρ m = βρ m ˙ π (30)We may also define the total energy density and pressure for the scalar field πρ π = c π − c H ˙ π + 452 c H ˙ π (31) P π = c π + c ˙ π ¨ π − c ˙ π (cid:16) H ˙ π + 2 ˙ H ˙ π + 8 H ¨ π (cid:17) (32)which can be used to check for the total equation of state parameter w π = P π /ρ π . In the next section, we discussthe self accelerating solution of galileon cosmology. V. SELF ACCELERATING SOLUTION
A self acceleration solution is characterized by ρ m = 0 and H ≡ H = C st .In this case, using equation (27), we find that ˙ π ≡ ˙ π = C st and H ˙ π ± = c ± p c − c c c (33)48 H = ( ˙ π ± ) A ± (34)With A ± = c − c c ± c √ c − c c c .The existence of the self accelerating solution then implies the following conditions on constants c , c , c and c , c − c c > A + > A − > H = H + δH, ˙ π = ˙ π + δ ˙ π (37)It can easily be checked that ˙ δH = − H δH , which means that the self-accelerating solutions are stable. VI. SPHERICALLY SYMMETRIC SOLUTION
We shall now be interested in the spherically symmetric static solution . We consider a static point-like source ofmass M , located at the origin: T ( m ) = − M δ ( −→ x ) and look for a spherically symmetric static solution for the field π ( r ) described by the following differential equation c r dd r (cid:2) r π ′ ( r ) (cid:3) + 2 c r dd r (cid:2) rπ ′ ( r ) (cid:3) + 4 c r dd r (cid:2) π ′ ( r ) (cid:3) = βM δ ( −→ x ) (38)Integration of Eq.(38) gives the following relation, c (cid:18) π ′ ( r ) r (cid:19) + 2 c (cid:18) π ′ ( r ) r (cid:19) + 4 c (cid:18) π ′ ( r ) r (cid:19) = β r s r (39)Where r s is the Schwarzschild radius of the source.The conditions of existence of the solution are derived following Ref.[10]:if β > ⇒ sign( c )=sign( c ) and c > −√ c c which means that c > √ c c if we consider the condition (35)if β < ⇒ sign( c )=sign( c ) and c < √ c c which means that c < −√ c c if we consider the condition (35)In case β <
0, at short distances, the solution is not analytic in the neighborhood of r = 0 and we shall not considerthis case any further.Whereas for β > π ′ ( r ) = (cid:0) c r s β (cid:1) / / c (40)Then the galileon-mediated force is suppressed compared to the gravitational force: F π F grav = (cid:18) rr ⋆ (cid:19) ≪ , with r ⋆ = (cid:18) | c | β (cid:19) / r s (41)At large distances, we have F π F grav = 2 βc (42)If β ≃ c , the galileon field can lead to late time acceleration of universe. VII. STABILITY
In order to study the stability of the aforesaid static solutions, we perturb the scalar field π : π → π + φ in a fixedmetric g µν . We have neglected the perturbations of the metric induced by the perturbations of the scalar field φ ; themethod is referred to test field approximation.In order to proceed with the test field approximation, let us rewrite the quadratic term in φ in the action S φ = Z √− g d x c i Z µν ( i ) φ ; µ φ ; ν (43)with Z µν (1) = 0 (44) Z µν (2) = − g µν (45) Z µν (3) = π ; µν − g µν (cid:3) π (46) Z µν (4) = − π ; µ R νρ π ; ρ − π ; ν R µρ π ; ρ − R µν ( ∇ π ) + Rπ ; µ π ; ν + 6 (cid:3) ππ ; µν − π ; µρ π ν ; ρ + 2 R µρσν π ; ρ π ; σ (47)+ g µν (cid:18) π ; ρσ π ; ρσ − (cid:3) π ) + 2 R ρσ π ; ρ π ; σ + 12 R ( ∇ π ) (cid:19) (48)The equation of motion for perturbations that follow from action (43) is − c i Z µν ( i ) φ ; µν − c i Z µν ; µ φ ; ν + 8 β φT ( m ) = 0 , (49)which we shall use in the subsequent sections. A. Cauchy-problem
Following the theorem due to Leray [22], the scalar field φ propagates causally in the effective metric G µν eff = − c i Z µν ( i ) if spacetime ( M , G µν eff ) is globally hyperbolic. A necessary condition but not sufficient is the requirement of thehyperbolicity of the equation (49) that is a Lorentzian signature of the effective metric G µν eff .For the static spherical solution, the hyperbolicity is defined by c + 2 c (2 π ′ /r + π ′′ ) + 12 c ( π ′ /r + 2 π ′′ ) π ′ /r > c + 4 c π ′ /r + 12 c ( π ′ /r ) > c + 2 c ( π ′ /r + π ′′ ) + 12 c π ′′ π ′ /r > c − β c c r s r > c + 4 β c c r s r > c − β c c r s r > , (55)which reduce to c > c > c > G = −
14 ( A ± + 4 c ) < , (56) a G = 136 ( A ± − c ) > , (57)which implies that A ± > c .We should however emphasize that this solution is derived when the scalar field is dominant (de Sitter phase),therefore any small perturbation of the scalar field leads to a perturbation of the metric and the test field approximationis then no longer true. B. Hamiltonian approach
An alternative way to study the stability is related to the positive definiteness of Hamiltonian of the underlyingtheory. In a locally inertial frame, the Hamiltonian is H = − G ˙ φ + 12 G kl eff φ ,k φ ,l (58)The condition of hyperbolicity of equation (49) is sufficient for the Hamiltonian to be bounded from below. Thecondition of hyperbolicity imposes an important restriction on sound speed which we consider next. C. Speed of sound
From the equation (49), it is obvious to define the ”sound speed” c s ; the condition of hyperbolicity of the equationrestricts c s to real values c s >
0. It is straightforward to see that the condition of c s to be real, restrict the signatureof the effective metric to ( − , + , + , +) or (+ , − , − , − ).However, if we also impose the positivity of the Hamiltonian, we have to consider the effective metric with thesame signature that as that of the original metric g µν which is ( − , + , + , +), in our case . This condition for nonsuperluminal behavior of the scalar field φ is expressed by c s < c s = A ± − c A ± +4 c ) < c > A ± > G ¨ φ + G ∂ r φ + G r ∂ φ + first derivatives of φ + ... = 0 (59)where ∂ is the angular part of the Laplacian.Therefore we can define the speed of radial and angular excitations as follows, c r = − G G = c + 4 c π ′ /r + 12 c π ′ /r c + 2 c (2 π ′ /r + π ′′ ) + 12 c ( π ′ /r + 2 π ′′ π ′ /r ) (60) c = − r G G = c + 2 c ( π ′ /r + π ′′ ) + 12 c π ′′ π ′ /rc + 2 c (2 π ′ /r + π ′′ ) + 12 c ( π ′ /r + 2 π ′′ π ′ /r ) , (61)which at large distances gives rise to c r ≈ β c c r s r (62) c ≈ − β c c r s r , (63)whereas for small distances, we find c r = 1 (64) c ≈ c c rπ ′ (65)It is clear that at large distances, we have a superluminal behavior ( c r >
1) of the scalar field φ for the staticspherically solution, but this behavior is physically possible if the theory does not have Closed Causal Curves (CCCs)which leads to paradoxes [22, 23]. It is known that if a spacetime is stably causal, it does not possesses CCCs whichmeans that a global time can be defined. This is the case if we can define a global time for the two metrics g µν and G µν .For the static spherically symmetric solution, we will consider the Minkowsky time η µν ∇ µ t ∇ ν t = − G µνeff ∇ µ t ∇ ν t = − c − c (2 π ′ /r + π ′′ ) − c ( π ′ /r + 2 π ′′ ) π ′ /r (66)Eq.(66), at large distances, reduces to G µνeff ∇ µ t ∇ ν t = − c + 36 β c c r s r (67)which is negative iff r > β r s c /c .If this condition is satisfied then the space time ( M, G eff µν ) is stably causal which means that no closed timelikecurves exist. We should emphasize that this condition is satisfied if the equation (49) is hyperbolic. VIII. METRIC PERTURBATIONS
Let us consider the perturbed FLRW spacetime with scalar metric perturbations in the longitudinal gauged s = − (1 + 2 φ )d t + a (1 − ψ ) d x (68)The linear matter perturbations δ m on super horizon scales satisfy the evolution equation similar to the one inEinstein gravity ¨ δ m + 2 H ˙ δ m − G eff ρ m δ m = 0 (69)with the modified Newtonian constant, G eff = 1 + 2 (cid:0) c ˙ π + 2 β (cid:1) + c N c − c ˙ π − c H ˙ π − c ¨ π + c D (70)where N and D are given by N = 14 c ˙ π + c ˙ π − c H ˙ π + 4 c β ˙ π + 20 c ˙ π ¨ π − Hβ ˙ π − β ˙ π + 96 β ˙ π ¨ π + c (cid:16) − c ˙ π − c H ˙ π − c ˙ π ¨ π + 492 H ˙ π − Hβ ˙ π − H ˙ π ¨ π + 168 ˙ H ˙ π + 18 β ˙ π − β ˙ π ¨ π + 288 ˙ π ¨ π (cid:17) + c (cid:16) (cid:16) H − H (cid:17) ˙ π + 648 H ˙ π ¨ π (cid:17) (71) D = − c ˙ π − c ˙ π + 80 c H ˙ π − c ˙ π ¨ π + 8 (cid:16) H + 6 ˙ H (cid:17) ˙ π + 96 H ˙ π ¨ π + c (cid:16) c ˙ π + 12 c H ˙ π + 54 c ˙ π ¨ π − (cid:16) H + 6 ˙ H (cid:17) ˙ π + 288 H ˙ π ¨ π (cid:17) + c (cid:16) (cid:16) H − H (cid:17) ˙ π − H ˙ π ¨ π (cid:17) (72)The study of generic models of modified gravity shows that there is a characteristic signature in the growth function f = d ln δ m d ln a which can allow us to distinguish these models from Λ CDM and other dynamical dark energy modelswithin the frame work of Einstein gravity. We expect similar features in galileon gravity. We shall address thisimportant issue in our future work.
IX. CONCLUSION
In this paper, we have investigated galileon gravity in its general form. The model consists of an effective field π Lagrangian consisting of five terms P c i L i added to Einstein-Hilbert action such that the field equation are ofsecond order. In spatially flat FRW background, we set up the evolutions equations in the model and examine theexistence and stability of self accelerating solutions. We point out that these solutions, in general ( ˙ π = 0), are notstable in the third order galileon theory. We extend the analysis to the fourth and fifth order theory. In fourth ordertheory, self accelerating solutions exist provided that c − c c > A + > A − >
0. We show that there isat least one stable self-accelerating solution in this case. The analysis is cumbersome in case of 5th order theory andwe have included the corresponding results in the appendix. The conclusions reached in fourth order galileon theoryare shown to hold in general. In case of the spherically symmetric static solution, we find that the solution existsprovided that c > √ c c . The solution is stable and the fifth force can lead the acceleration of the universe if weassume β ≃ c and and c >
0. We find as expected that the galileon force mediated by the scalar field π is negligiblysmall at small scales, because of the non-linear terms in the Lagrangian. However, the fifth force is of the order ofthe gravitational force at large scales in case, β ≃ c .Subsequently, we investigated the stability issues associated with the spherically symmetric solution. Using the fixedbackground method, we found superluminal behavior of perturbations as was noticed in [10]. It is really interestingthat despite the superluminal behavior, there exist static solutions which do not possess any Closed Causal Curveallowing to avoid paradoxes related to micro-causality and making the solution physically acceptable. The modelhas a well posed Cauchy problem and no Closed Causal Curves exists in this model even if we have a superluminalbehavior of the perturbation of the scalar field in the static spherically symmetric situation at large distances.We have included brief discussion on the metric perturbations and have set up the evolution equation for linearmatter perturbation in the galileon gravity. In our opinion, it is important to study the growth function f = d ln δ m d ln a which can provide a discriminating signature of galileon gravity; we defer this analysis to our future work. ACKNOWLEDGEMENTS
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Appendix A: The full Lagrangian of galileon gravity: Extension of the model to the fifth order term, L (5) We consider the term L (5) derived in [11] L (5) = −
12 ( ∇ π ) h ( (cid:3) π ) − (cid:3) π ) ( π ; µν π ; µν ) + 2 (cid:0) π ν ; µ π ρ ; ν π µ ; ρ (cid:1) − π ; µ π ; ν π ; ρσ R µρνσ ) −
18 ( π ; ν π ; νρ R ρσ π ; σ ) + 3 ( (cid:3) π ) ( π ; ν R νρ π ; ρ ) + 152 ( ∇ π ) ( π ; ν π ; νρ π ; ρ ) R (cid:21) +3 (cid:2) π ; µ π ; µν π ; νρ π ; ρλ π ; λ − ( (cid:3) π ) ( π ; µ π ; µν π ; νρ π ; ρ ) (cid:3) + 32 h ( (cid:3) π ) ( π ; µ π ; µν π ; ν ) − ( π ; µν π ; µν ) (cid:0) π ; ρ π ; ρλ π ; λ (cid:1)i (A1)then the equations (15,16) are modified by E (5) = 52 ( (cid:3) π ) −
15 ( (cid:3) π ) ( π ; µν π ; µν ) −
154 ( (cid:3) π ) ( ∇ π ) R −
152 ( (cid:3) π ) ( π ; µ R µν π ; ν )+20 ( (cid:3) π ) (cid:0) π ν ; µ π ρ ; ν π µ ; ρ (cid:1) −
152 ( (cid:3) π ) ( π ; µ π ; µν π ; ν ) R + 15 ( (cid:3) π ) ( ∇ π ) ( π ; νρ R νρ )+30 ( (cid:3) π ) ( π ; µ π ; µν R νρ π ; ρ ) + 15 ( (cid:3) π ) ( π ; µ π ; ν π ; ρσ R µρνσ ) + 152 ( π ; µν π ; µν ) −
15 ( π ; µν π ; νρ π ; ρσ π ; σµ )+ 154 ( ∇ π ) ( π ; νρ π ; νρ ) R + 152 ( π ; µ π ; µν π ; νρ π ; ρ ) R + 152 ( π ; µν π ; µν ) ( π ; ρ R ρσ π ; σ )+15 ( π ; µ π ; µν π ; ν ) ( π ; ρσ R ρσ ) − ∇ π ) (cid:0) π ρ ; ν R σρ π ν ; σ (cid:1) −
30 ( π ; µ π ; µν π ; νρ R ρσ π ; σ ) −
15 ( π ; µ π ; µν R νρ π ; ρσ π ; σ ) −
152 ( ∇ π ) (cid:0) π ; νρ π ; σλ R νσρλ (cid:1) − (cid:0) π ; µ π ; ν π ; ρσ π ; σλ R µρνλ (cid:1) +30 (cid:0) π ; λ π ; λµ π ; νρ π ; σ R µνρσ (cid:1) + 154 ( ∇ π ) ( π ; ν R νρ π ; ρ ) R −
152 ( ∇ π ) ( π ; ν R νρ R ρσ π ; σ ) −
152 ( ∇ π ) (cid:0) π ; ν π ; ρ R σλ R νσρλ (cid:1) + 154 ( ∇ π ) (cid:0) π ; ν π ; ρ R νσκλ R ρσκλ (cid:1) (A2) T (5) µν = 52 ( (cid:3) π ) ( π ; µ π ; ν ) + 52 ( (cid:3) π ) ( ∇ π ) g µν −
152 ( (cid:3) π ) ( ∇ π ) ( π ; µν ) −
152 ( (cid:3) π ) π ; ρ (cid:2) π ; ρµ π ; ν + π ; ρν π ; µ (cid:3) + 152 ( (cid:3) π ) ( π ; ρ π ; ρσ π ; σ ) g µν + 15 ( (cid:3) π ) ( ∇ π ) ( π ; µσ π ; σν ) −
15 ( (cid:3) π ) ( π ; ρ π ; ρσ π ; σ ) ( π ; µν ) −
152 ( (cid:3) π ) ( π ; ρσ π ; ρσ ) ( π ; µ π ; ν ) + 15 ( (cid:3) π ) ( π ; ρ π ; ρµ ) ( π ; σ π ; σν ) + 15 ( (cid:3) π ) π ; ρ π ; ρσ (cid:2) π ; σµ π ; ν + π ; σν π ; µ (cid:3) −
152 ( (cid:3) π ) ( ∇ π ) (cid:0) π ; σλ π ; σλ (cid:1) g µν −
15 ( (cid:3) π ) (cid:0) π ; ρ π ; ρσ π ; σλ π ; λ (cid:1) g µν −
154 ( (cid:3) π ) ( ∇ π ) ( π ; µ π ; ν ) R + 152 ( (cid:3) π ) ( ∇ π ) π ; σ (cid:2) R σµ π ; ν + R σν π ; µ (cid:3) −
152 ( (cid:3) π ) ( ∇ π ) (cid:0) π ; σ R σλ π ; λ (cid:1) g µν
1+ 152 ( (cid:3) π ) ( ∇ π ) (cid:0) π ; σ π ; λ R µσνλ (cid:1) + 152 ( ∇ π ) (cid:0) π ; σλ π ; σλ (cid:1) ( π ; µν ) − ∇ π ) (cid:0) π ; µσ π ; σλ π ; λν (cid:1) +15 ( π ; ρ π ; ρσ π ; σ ) (cid:0) π ; µλ π ; λν (cid:1) + 15 (cid:0) π ; ρ π ; ρσ π ; σλ π ; λ (cid:1) ( π ; µν ) + 5 (cid:16) π σ ; ρ π λ ; σ π ρ ; λ (cid:17) ( π ; µ π ; ν )+ 152 (cid:0) π ; σλ π ; σλ (cid:1) π ; ρ (cid:2) π ; ρµ π ; ν + π ; ρν π ; µ (cid:3) − π ; ρ π ; ρσ π ; σλ (cid:2) π ; λµ π ; ν + π ; λν π ; µ (cid:3) − π ; ρ π ; ρλ π ; σ (cid:2) π ; λµ π ; σν + π ; λν π ; σµ (cid:3) + 5( ∇ π ) (cid:0) π λ ; σ π κ ; λ π σ ; κ (cid:1) g µν −
152 ( π ; ρ π ; ρσ π ; σ ) (cid:0) π ; λκ π ; λκ (cid:1) g µν +15 (cid:0) π ; ρ π ; ρσ π ; σλ π ; λκ π ; κ (cid:1) g µν + 154 ( ∇ π ) π ; σ (cid:2) π ; σµ π ; ν + π ; σν π ; µ (cid:3) R −
154 ( ∇ π ) (cid:0) π ; σ π ; σλ π ; λ (cid:1) R g µν + 152 ( ∇ π ) (cid:0) π ; σ π ; σλ π ; λ (cid:1) R µν + 152 ( ∇ π ) (cid:0) π ; σ R σλ π ; λ (cid:1) ( π ; µν ) + 152 ( ∇ π ) (cid:0) π ; σλ R σλ (cid:1) ( π ; µ π ; ν ) −
152 ( ∇ π ) π ; σ π ; σλ (cid:2) R λµ π ; ν + R λν π ; µ (cid:3) −
152 ( ∇ π ) π ; λ π ; σ (cid:2) R λµ π ; σν + R λν π ; σµ (cid:3) −
152 ( ∇ π ) π ; σ R σλ (cid:2) π ; λµ π ; ν + π ; λν π ; µ (cid:3) + 15( ∇ π ) (cid:0) π ; σ π ; σλ R λκ π ; κ (cid:1) g µν −
152 ( ∇ π ) π ; σ π ; λκ (cid:2) R µλσκ π ; ν + R νλσκ π ; µ (cid:3) + 152 ( ∇ π ) π ; σ π ; λ (cid:2) R µσλκ π ; κν + R νσλκ π ; κµ (cid:3) −
152 ( ∇ π ) π ; σ π ; σλ π ; κ (cid:2) R µλνκ + R νλµκ (cid:3) + 152 ( ∇ π ) (cid:0) π ; σ π ; λ π ; κτ R σκλτ (cid:1) g µν . (A3)The Friedmann equations for this model are3 H = ρ m + c π − c H ˙ π + 452 c H ˙ π − c H ˙ π (A4)2 ˙ H + 3 H = − c π − c ˙ π ¨ π + 32 c ˙ π (cid:16) H ˙ π + 2 ˙ H ˙ π + 8 H ¨ π (cid:17) − c H ˙ π (cid:16) H ˙ π + 2 ˙ H ˙ π + 5 H ¨ π (cid:17) (A5) βρ m = − c (3 H ˙ π + ¨ π ) + 3 c ˙ π (cid:16) H ˙ π + ˙ H ˙ π + 2 H ¨ π (cid:17) − c H ˙ π (cid:16) H ˙ π + 2 ˙ H ˙ π + 3 H ¨ π (cid:17) + 752 c H ˙ π (cid:16) H ˙ π + 3 ˙ H ˙ π + 4 H ¨ π (cid:17) (A6)Therefore the self accelerating solution exist if there is a real solution of the equation c − c X + 18 c X − c X = 0 (A7) c − c X + 30 c X < , with X = H ˙ π (A8)If we have a solution of this system therefore the self accelerating solution is stable. In fact if we consider aperturbation of the self accelerating solution H = H + δH and ˙ π = ˙ π + δ ˙ π , it is straightforward to see that,˙ δH = − H δHδH