Late time cosmological evolution in DHOST models
LLate time cosmological evolution in DHOST models
Hamza Boumaza, David Langlois, and Karim Noui
3, 2 Laboratory of Theoretical Physics and Department of Physics,Faculty of Exact and Computer Sciences, University Mohamed Seddik Ben Yahia,BP 98, Ouled Aissa, Jijel 18000, Algeria Astroparticule et Cosmologie (APC),CNRS, Universit´e de Paris, F-75013 Paris, France Institut Denis Poisson, CNRS, Universit´e de Tours, 37200 Tours, France (Dated: April 23, 2020)We study the late cosmological evolution, from the nonrelativistic matter dominated erato the dark energy era, in modified gravity models described by Degenerate Higher-OrderScalar-Tensor (DHOST) theories. They represent the most general scalar-tensor theoriespropagating a single scalar degree of freedom and include Horndeski and Beyond Horndeskitheories. We provide the homogeneous evolution equations for any quadratic DHOST theory,without restricting ourselves to theories where the speed of gravitational waves coincides withthat of light since the present constraints apply to wavelengths much smaller than cosmo-logical scales. To illustrate the potential richness of the cosmological background evolutionin these theories, we consider a simple family of shift-symmetric models, characterized bythree parameters and compute the evolution of dark energy and of its equation of state. Wealso identify the regions in parameter space where the models are perturbatively stable.
I. INTRODUCTION
One possible explanation for the observed acceleration of the cosmological expansion is thatgravity is modified on cosmological scales. Concrete realisations of this idea often rely on scalar-tensor theories, which represent the simplest extension of general relativity since a scalar degree offreedom is added to the usual tensor modes of general relativity. The most general family of scalar-tensor theories that has been developed so far is that of Degenerate Higher-Order Scalar-Tensor(DHOST) theories [1], which encompass Horndeski theories [2], Beyond Horndeski (or GLPV)theories [3] which are earlier extensions of Horndeski, as well as disformal transformations of theEinstein-Hilbert action [4]. In the present work, we consider the whole family of quadratic DHOSTtheories, introduced in [1] (see also [5, 6] for further details and [7] for a review), but for simplicitly,we do not include DHOST theories with cubic terms (in second derivatives of the scalar field) whichhave been fully classified in [8].Most of the literature has recently concentrated on DHOST theories where the speed of gravita-tional waves coincides with that of light, following the observation of a neutron star binary mergerthat has set an impressively stringent constraint on the difference between these two velocities[9]. Moreover, it has been pointed out that subsets of DHOST theories can lead to the decay ofgravitational waves, yielding a further tight constraint on DHOST theories [10, 11]. The cosmologyof DHOST theories satisfying either the first or both of the above constraints has been studied in[12–17].However, it should be stressed that the LIGO-Virgo measurements probe wavelengths of order10 km, which are many orders of magnitude smaller than cosmological scales, and an effectivetheory describing cosmological scales might not be adequate to describe physics on much smallerlength scales, as those probed by LIGO-Virgo (see [18] for a discussion on this point). In thepresent work, we adopt the point of view that DHOST theories apply only to cosmological scales a r X i v : . [ a s t r o - ph . C O ] A p r and cannot be extrapolated down to astrophysical scales within the same framework . In thisperspective, all the constraints derived from GW170817 mentioned above are not directly relevantand it thus makes sense to study models that can lead to distinct propagation velocities for lightand gravitational waves on cosmological scales.The outline of the paper is the following. In section II, starting from the most general actionfor quadratic DHOST theories, we derive the Friedmann equations and the scalar field equation.These results extend those obtained recently in [12] and [13]. As in [13] we introduce an auxiliaryscale factor that makes the equations manifestly second-order. The second part of the paper isdevoted to the study of a simple subfamiliy of quadratic DHOST theories characterized by a fewparameters. In section III, we write the equations of motion in the form of a dynamical systemand identify the fixed points and their nature. In section IV, we turn to the linear perturbationsin order to study the perturbative stability of the model. We conclude in the last section. II. GENERAL COSMOLOGICAL EQUATIONS
In this section, we briefly recall the basic properties of DHOST theories and introduce thenotations used throughout this paper. Then, we focus on the case of a homogeneous and isotropicuniverse and provide the cosmological evolution equations for the whole family of quadratic DHOSTtheories.
A. Quadratic DHOST theories
The most general theory of quadratic DHOST theory is described by the action S = (cid:90) d x √− g (cid:32) P ( X, ϕ ) + Q ( X, ϕ ) (cid:50) ϕ + F ( X, ϕ ) R + (cid:88) i =1 A i ( X, ϕ ) L i (cid:33) (2.1)where the functions A i , F, Q and P depend on the scalar field ϕ and its kinetic term X ≡ ∇ µ ϕ ∇ µ ϕ , R is the Ricci scalar. The five elementary Lagrangians L i quadratic in second derivatives of ϕ aredefined by L ≡ ϕ µν ϕ µν , L ≡ ( (cid:50) ϕ ) , L ≡ ϕ µ ϕ µν ϕ ν (cid:50) ϕ ,L ≡ ϕ µ ϕ µν ϕ νρ ϕ ρ , L ≡ ( ϕ µ ϕ µν ϕ ν ) , (2.2)where we are using the standard notations ϕ µ ≡ ∇ µ ϕ and ϕ µν ≡ ∇ ν ∇ µ ϕ for the first and second(covariant) derivatives of ϕ . For the theory to be degenerate and thus propagate only one extrascalar degree of freedom in addition to the usual tensor modes of gravity, the functions F and A i have to satisfy some conditions [1, 6] whereas P and Q are totally free.It has been established in [6] that these DHOST theories can be classified into three classeswhich are stable under general disformal transformations, i.e. transformations of the metric of theform g µν −→ ˜ g µν = C ( X, ϕ ) g µν + D ( X, ϕ ) ϕ µ ϕ ν , (2.3)where C and D are arbitrary functions (provided that the metric ˜ g µν remains regular). Our motivation here is that dark energy can be described by a DHOST model. DHOST theories with a very differentset of parameters could still be used to describe modified gravity in astrophysical systems, but without being ableto account for dark energy because of the LIGO-Virgo constraints.
The theories belonging to the first class, named class Ia in [6], can be mapped into a Horndeskiform by applying a disformal transformation. The other two classes are not physically viable [19]and will not be considered in the present work. Theories in class Ia are labelled by the three freefunctions
F, A and A (in addition to P and Q ) and the three remaining functions are given bythe relations [1] A = − A , (2.4) A = 18 ( F + XA ) (cid:16) A A (cid:0) X F X − XF (cid:1) + 4 A (16 XF X + 3 F ) + 16 A (4 XF X + 3 F ) F X +16 XA + 8 A F ( XF X − F ) − X A F + 48 F F X (cid:17) , (2.5) A = 18 ( F + XA ) (cid:16) A + XA − F X (cid:17)(cid:16) XA A − A F X − A + 4 A F (cid:17) , (2.6)where F X denotes the derivative of F ( X, ϕ ) with respect to X . Similarly F ϕ will denote the partialderivative of F with respect to ϕ and the same notations will be used for all functions.The above relations (2.4-2.6) are a direct consequence of the three degenerate conditions thatguarantee only one scalar degree of freedom is present [1, 20]. In conclusion, this means that all theDHOST theories we study here are characterized by five free functions of X and ϕ , which are P , Q , F , A and A . Notice that we have implicitly supposed the condition F + XA (cid:54) = 0. Theorieswhere F + XA = 0 belong to the sub-class Ib which is not physically relevant [6]. B. Homogeneous and isotropic cosmology
We now wish to study the behaviour of these theories in a homogeneous and isotropic spacetime,endowed with the metric ds = − N ( t ) dt + a ( t ) δ ij dx j dx i , (2.7)where the lapse function N ( t ) and the scale factor a ( t ) depend on time only. As a consequence ofthe spacetime symmetries, the scalar field must also be homogeneous and therefore depends onlyon time.Substituting the above metric (2.7) into the action (2.1), and taking into account the degeneracyconditions (2.4-2.6), one finds that the corresponding homogeneous action can be written as afunctional of N ( t ), a ( t ) and of the homogeneous scalar field ϕ ( t ). It reads S hom [ N, a, ϕ ] = (cid:90) dt a N (cid:40) P + Q (cid:32) ˙ NN ˙ ϕ − aa N ˙ ϕ − ¨ ϕN (cid:33) − F ϕ aa N ˙ ϕ − f N (cid:32) ˙ aa + f f (cid:32) ˙ N ˙ ϕ N − ¨ ϕ ˙ ϕN (cid:33)(cid:33) (cid:41) , (2.8)where we have introduced the new functions, f ≡ F − XA , f ≡ F X − A + XA , (2.9)and, everywhere, the expression of X is explicitly given by X = − ˙ ϕ N . (2.10)The Euler-Lagrange equations derived from the above action (2.8) lead to equations of motionthat appear higher than second order. However, due to the degeneracy of the theory, these equationscan be recast into a second order system. As done in [13], this can be demonstrated explicitly byintroducing an auxiliary scale factor b , defined by the relation a ≡ Λ (
X, ϕ ) b ≡ e λ ( X,ϕ ) b , (2.11)where λ satisfies the condition λ X = − f f , (2.12)so that the terms quadratic in ¨ ϕ in the action (2.8) are reabsorbed in the derivatives of the newscale factor.It is also convenient to use a Hubble parameter associated with this auxiliary scale factor,defined by H b ≡ ˙ bN b = H − λ X ˙ XN − λ ϕ ˙ ϕN , ˙ X = 2 N (cid:32) ˙ N ˙ ϕ N − ˙ ϕ ¨ ϕ (cid:33) . (2.13)In fact, the auxiliary variable b corresponds to the scale factor of the disformally transformed metric˜ g µν in (2.3) when the DHOST theory coincides with a Horndeski theory. The drawback of usingthis “Horndeski frame” is that matter is no longer minimally coupled, as it was assumed in theinital frame, which we will call here the “DHOST frame”. The Horndeski and DHOST frames are,respectively, the analogs of the Einstein and Jordan frames for traditional scalar-tensor theories.When expressed in terms of b instead of a , the Lagrangian L hom in the action (2.8) becomes L hom = Λ b N (cid:40) − λ ϕ ˙ ϕ N (2 f λ ϕ + Q + 2 F ϕ ) + P − f H b − ϕN (4 λ ϕ f + 2 F ϕ + Q ) H b + (cid:18) Q − λ X (2 F ϕ + Q ) ˙ ϕ N (cid:19) (cid:32) ˙ N ˙ ϕN − ¨ ϕN (cid:33)(cid:41) . (2.14)The coupling to matter is described by adding to L hom a matter Lagrangian L m , and the totalLagrangian is denoted L ≡ L hom + L m .We get the equations of motion by writing the Euler-Lagrange equations for N , b and ϕ . Thefirst two equations provide the generalizations of the Friedmann equations. The last equation,corresponding to the scalar field equation of motion, is obtained from an Euler-Lagrange equationof the form − d dt ∂L∂ ¨ ϕ + ddt ∂L∂ ˙ ϕ − ∂L∂ϕ = 0 . (2.15)Once we have derived the equations of motion, we fix the time coordinate such that N = 1 (andthus ˙ N = 0) in order to simplify the equations.As for matter, we assume that it is described by a perfect fluid whose equation of state is P = wρ , where w is constant. The variation of the matter action S m gives the energy-momentumtensor of the fluid, defined as usual by T µν = 2 √− g δS m δg µν , S m = (cid:90) d x √− g L m . (2.16)As a consequence, in the DHOST frame, where matter is minimally coupled, the variation of thematter Lagrangian is immediately given by δL m = − a ρ m δN + 3 N a P m δa , (2.17)where ρ m and P m are the fluid energy density and pressure, respectively. Using δaa = − Xλ X δNN + δbb (2.18)which follows from the definition (2.11) of b , one finds that the variation of the matter Lagrangianin the Horndeski frame is given by δL m = N b Λ (cid:20) − ( ρ m + 6 Xλ X P m ) δNN + 3 P m δbb (cid:21) . (2.19)Inserting the above variation of the matter Lagrangian into the Euler-Lagrange equations, weobtain the set of equations of motion for the cosmological dynamics. The analogs of the twoFriedmann equations (with N = 1) take the form g + g H b ˙ ϕ + g H b = (cid:0) − wλ X ˙ ϕ (cid:1) ρ m , (2.20) g + g (2 ˙ H b + 3 H b ) + g H b ˙ ϕ + g ¨ ϕ + g H b ˙ ϕ ¨ ϕ = − wρ m . (2.21)where all the coefficients g i can be written explicitly in terms of the functions that appear in theLagrangian and of λ . They are given in Appendix A.Finally the scalar field equation can be written as ddt (cid:0) b Λ J (cid:1) + b Λ U = 0 , (2.22)where we have defined J = 1 b Λ (cid:20) ddt (cid:18) ∂L hom ∂ ¨ ϕ (cid:19) − ∂L hom ∂ ˙ ϕ (cid:21) , U = 1 b Λ ∂L hom ∂ϕ . (2.23)The two functions U and J are of the form U = g + g H b ˙ ϕ + g H b + g ¨ ϕ , J = g ˙ ϕ + g H b + g H b ˙ ϕ , (2.24)where the corresponding coefficients g i are also given explicitly in Appendix A. As usual, theequation for the scalar field is not independent and can be obtained from the two Friedmannequations.One recovers the equations of motion given in [13] when the theory is shift symmetric, i.e. whenthe functions in the Lagrangian are invariant under the transformation ϕ → ϕ + c and thus dependonly on X . In particular, the quantity U defined above vanishes and J is conserved. III. AN ILLUSTRATIVE TOY MODEL
We now restrict our study to a class of models described by Lagrangians that depend on threeconstant parameters only. These Lagrangians are shift symmetric and characterized by the simplepolynomial functions P = αX, Q = 0 , F = 12 , A = − A = − βX, λ = 12 µX , (3.1)where α, β and µ are arbitrary constants. Hence, from (2.12) and (2.9), we deduce A = − β + 2 µ ) − βµX . (3.2)The expressions of A and A are easily obtained from the degeneracy conditions (2.5) and (2.6).One thus gets A = 2( β + 2 µ − µ X ) , A = 8 µX ( β + 2 µ + 3 βµX ) . (3.3)Note that the particular choice µ = 0 corresponds to a subset of Horndeski theories (in this case,the DHOST and Horndeski frames coincide, i.e. λ = 0 and thus a = b ). A. Horndeski frame: dynamical system analysis
Since we are interested in the transition between the matter and dark energy dominated eras,we also assume that matter is non-relativisitc and thus take w = 0. The Friedmann-like equations(2.20) and (2.21) then reduce to3 H b (cid:0) µ + 5 β ) X + 12 µβ X (cid:1) − α X − αµ X − ρ m = 0 , (3.4)3 H b (cid:0) β X (cid:1) + 2 (cid:0) β X (cid:1) ˙ H b + α X − H b (cid:0) (3 µ + 4 β ) X + 6 βµ X (cid:1) ˙ ϕ ¨ ϕ = 0 . (3.5)Furthermore, the equation of motion for the scalar field becomes (cid:2) α (cid:0) µ X + 18 µ X (cid:1) − X (cid:0) β + 3 µ + 2 µ (11 β + 3 µ ) X + 12 βµ X (cid:1) H b (cid:3) ¨ ϕ +3 (cid:104) α (1 + 3 µX ) − X (4 β + 3 µ + 6 βµX )(3 H b + 2 ˙ H b ) (cid:105) H b ˙ ϕ = 0 . (3.6)To study these cosmological equations, it is convenient to rewrite them as a dynamical system(and analyse the fixed points and their stability) with the new variables x ≡ α XH b , x ≡ βX , s ≡ µβ , (cid:15) h ≡ ˙ H b H b , (cid:15) ϕ ≡ ¨ ϕH b ˙ ϕ , (3.7)following similar treatments for dark energy models (see e.g. [21–23] in the context of Galileonsand [24] for a recent review). These variables are not independent and one can easily see that dx d ln b = 2 ( (cid:15) ϕ − (cid:15) h ) x , (3.8) dx d ln b = 4 (cid:15) ϕ x , (3.9)where ln b plays the role of time. The two equations above can also be formulated in terms of thetime ln a instead of ln b by using the relation between the two Hubble constants, H = (1 + 2 sx (cid:15) ϕ ) H b , (3.10)which follows from (2.13). The two previous relations (3.8) and (3.9) then become x (cid:48) = 2( (cid:15) ϕ − (cid:15) h ) x s(cid:15) ϕ x , (3.11) x (cid:48) = 4 (cid:15) ϕ x s(cid:15) ϕ x , (3.12)where a prime denotes a derivative with respect to N ≡ ln a .So far, we have not yet used the equations of motion, namely the Friedmann-like equations,Eqs. (3.4) and (3.5), and the scalar equation (3.6). They can be reformulated, respectively, as x − sx − sx + 2 sx x − x + Ω m = 1 , (3.13)2(1 + 2 x ) (cid:15) h + 4 x (4 + 3 s + 6 sx ) (cid:15) ϕ + (3 + x + 6 x ) = 0 , (3.14) − (cid:0) x (cid:0) s x + 21 sx + 1 (cid:1) − x (cid:0) s x (2 x + 1) + s (22 x + 3) + 4 (cid:1)(cid:1) (cid:15) ϕ +4 x (cid:15) h ( s (6 x + 3) + 4) + 36 sx + 18 sx − sx x + 24 x − x = 0 . (3.15)These equations can be seen as constraints for the dynamical system (3.11-3.12). The first con-straint, Eq. (3.13), involves the matter density, whereas the last two equations (3.14) and (3.15)can be used to determine (cid:15) ϕ and (cid:15) h in terms of x and x . After a straightforward calculation, onegets (cid:15) h = (cid:2) x (cid:0) s x + 12 s (3 s + 8) x + 26 x − (cid:1) − x (cid:0) s x + 21 sx + 1 (cid:1) − x (cid:0) s x (2 x + 1) + 3 s (cid:0) x + 4 x − (cid:1) + 40 x − (cid:1)(cid:3) / ∆ , (3.16) (cid:15) ϕ = − x (cid:0) sx + 6( s + 1) x + 1 (cid:1) / ∆ , (3.17)where the common denominator ∆ is given by∆ ≡ x (cid:0) x (2 sx + s ) + 3 s (cid:0) x + 4 x − (cid:1) + 40 x − (cid:1) + 2 x (2 x + 1) (cid:0) s x + 21 sx + 1 (cid:1) . (3.18)Hence, the equations of motion are now given in the form (3.11) and (3.12) with (cid:15) h and (cid:15) ϕ givenby the equations (3.16) and (3.17).The critical points are found by solving the equations (3.11) and (3.12) for x (cid:48) = 0 and x (cid:48) = 0.The number and stability properties of these fixed points are summarized in Table I. We seethat there are at most two stable fixed points corresponding to a de Sitter solution. To find theconditions on the parameters of the theory for these fixed points to exist, we have to study thesigns of x and x at the fixed points. It is immediate to show that • at the point C : x < x < s ; • at the point D : x > x < s > x < x > s < x and x (3.7), we see immediately that αx < X <
0) and βx >
0. As a consequence, we deduce that the fixed point C exists only if α > β < D exists only if µ < s = 0, i.e. µ = 0, corresponding to a DHOST theory that belongs tothe Horndeski subclass, the dynamical system admits a single fixed point given by the limit s → C , x = − , x = − , (3.19)whereas the limit of the fixed point D is ill-defined. points x x EigenvaluesA 0 x (0 ,
3) (unstable)B 3 0 ( − ,
3) (Saddle)C (3 − s − (cid:112) s + 2 s + 3)) / s ( − − s + (cid:112) s + 2 s + 3)) / s ( − , − − s + (cid:112) s + 2 s + 3) / s − (3 + 3 s + (cid:112) s + 2 s + 3)) / s ( − , − B. DHOST frame: Effective Friedmann equations
In the frame where matter is minimally coupled, it is always possible to write effectively theFriedmann equations in the usual form,3 H = ρ m + ρ DE , H + 3 H = P m + P DE , (3.20)where all new terms are “hidden” in the effective dark energy density and pressure, denoted ρ DE and P DE respectively. Hence, one can also define an equation of state parameter w DE for darkenergy as usual by the ratio w DE ≡ P DE ρ DE . (3.21)Moreover, one can define a global effective equation of state parameter as w eff ≡ P m + P DE ρ m + ρ DE = − −
23 ˙ HH . (3.22)For the models we are considering here, this parameter can be expressed in terms of the variablesintroduced earlier and reads w eff = − − H (cid:48) H = − − (cid:15) h + 4 s ( (cid:15) ϕ x (cid:48) + (cid:15) (cid:48) ϕ x )3(1 + 2 sx (cid:15) ϕ ) . (3.23)Using the fact that P m = 0 for non-relativistic matter, we can write, from (3.20) and (3.22), arelation between w DE and w eff given by w DE = w eff Ω DE , Ω DE ≡ ρ DE H . (3.24)The dynamical equations (3.8) and (3.9) can be solved numerically and the right amount ofnonrelativistic matter today, i.e. Ω m = ρ m / (3 H ) ≈ .
3, can be reached by tuning the initialconditions for x and x . We choose our initial conditions deep in the matter dominated era, i.e.when Ω m (cid:39)
1. According to the constraint (3.13), taking | x | (cid:28) | x | (cid:28) | x | (cid:28) | x | (cid:28) . (3.25)Indeed, in this regime, the dynamical system reduces to x (cid:48) ≈ x , x (cid:48) ≈ x / (9 s + 12) , (3.26) Ω DE Ω m w eff w DE - - - - - - N Ω DE Ω m w eff w DE - - - - - - N FIG. 1: Evolution of the parameters w DE , w eff , Ω DE , and Ω m as functions of N = ln( a ) for various choicesfor the parameter s and the initial conditions. Left: s = − x ( i )1 = − . × − and x ( i )2 = 10 − . Right: s = −
10 and initial conditions x ( i )1 = − . × − and x ( i )2 = 2 . × − . - - - - - - - N w D E - - - - - - - N w D E FIG. 2: Evolution of the dark energy ratio w DE = P DE /ρ DE as a function of N = ln( a ) for various choicesof s and initial conditions (set at N = − x ( i )2 = 10 − ; ( x ( i )1 , s ) = ( − . × − , −
2) for the greencurve, ( x ( i )1 , s ) = ( − . × − , −
4) for the red one and ( x ( i )1 , s ) = ( − . × − , −
10) for the black dashedone. Right: s = −
4; ( x ( i )1 , x ( i )2 ) = ( − . × − , − ) for the red curve, ( x ( i )1 , x ( i )2 ) = ( − . × − , − )for the blue one and ( x ( i )1 , x ( i )2 ) = ( − . × − , − ) for the green one. which shows that the system moves quickly away from the region where x and x are very small(this is not the case if we take | x | (cid:28) | x | (cid:28) DE isthen approximated, according to (3.13), byΩ DE ≈ − s + 5) x . (3.27)If we choose x > c T < s must satisfy s < − / m , Ω DE , w DE and w eff . We observea cosmological transition from the matter era to the dark energy era. We also observe that the darkenergy behaves like pressureless matter deep in the matter dominated era and like a cosmologicalconstant with w DE ≈ − w DE can even reach some significant positive values. IV. PERTURBATIVE LINEAR STABILITY
In this section, we study the linear stability of the models studied in the previous section.For that purpose, we work in the framework of the Effective Theory of Dark Energy developedin [25–27] and extended to DHOST theories in [19]. This effective approach relies on the ADMformulation where the metric is parametrized by the lapse function N , the shift vector N i and the0spatial metric h ij as follows, ds = − N dt + h ij ( dx i + N i dt )( dx j + N j dt ) . (4.1)In the ADM framework, the “velocity” of the spatial metric is encoded in the extrinsic curvaturetensor K ij defined by K ij ≡ N (cid:16) ˙ h ij − D i N j − D j N i (cid:17) , (4.2)where D i denotes the spatial covariant derivative associated to h ij . The DHOST action can bereformulated in terms of the ADM variables and the dynamics of the linear perturbations about anFLRW background is governed by the expansion of this action at quadratic order in the variables δN , δK ij and δh ij . After a long but straightforward calculation, one finds that the quadratic actionfor the perturbations is given by [19] S quad = (cid:90) d x dt a M (cid:26) δK ij δK ij − (cid:18) α L (cid:19) δK + (1 + α T ) (cid:18) R δ √ ha + δ R (cid:19) + H α K δN + 4 Hα B δKδN + (1 + α H ) RδN + 4 β δKδ ˙ N + β δ ˙ N + β a ( ∂ i δN ) (cid:27) , (4.3)where δ R stands for the second order term in the perturbative expansion of the Ricci scalar R and h is the determinant of the spatial metric. The coefficients M , α L , α T , α K , α B , α H , β , β and β , which fully characterize the quadratic action, are functions of time as they depend on thebackground. They can be expressed explicitly in terms of the functions entering the DHOST action(2.1), as recalled in the Appendix B.After integrating out the gauge degrees of freedom, and ignoring the coupling to matter for themoment, it has been shown in [19] that the quadratic action reduces to the sum of an action forthe curvature perturbation ζ , representing the scalar mode, S quad [ ζ ] = (cid:90) d x dt a M (cid:20) A ζ ˙ ζ − B ζ ( ∂ i ζ ) a (cid:21) , (4.4)and an action for the tensor modes γ ij , S quad [ γ ij ] = (cid:90) d x dt a M (cid:20) ˙ γ ij − c T a ( ∂ k γ ij ) (cid:21) . (4.5)The coefficients A ζ and B ζ that appear in the scalar action are given by A ζ = 1(1 + α B − ˙ β /H ) (cid:20) α K + 6 α B − a H M ddt (cid:0) a H M α B β (cid:1)(cid:21) , (4.6) B ζ = − α T ) + 2 aM ddt (cid:20) aM (1 + α H + β (1 + α T )) H (1 + α B ) − ˙ β (cid:21) , (4.7)while the speed of gravitational waves c T , which appears in the tensor action, is given by c T =1 + α T . Therefore the stability conditions for the linear perturbations are simply given by M > , A ζ > , B ζ > , c T > . (4.8)The expressions of these coefficients in terms of the dynamical variables (3.7) are given in AppendixB. As we will see, they will be useful for the numerical analysis of the linear stability of the model.1In the presence of matter, these stability conditions (for the scalar mode) are modified. Theyhave been derived explicitly in [19] for the simple case where matter is described by a scalar field σ whose dynamics is governed by a k-essence type action, S m = (cid:90) d x √− g K ( Y ) , Y ≡ g µν σ µ σ ν , (4.9)which is added to the DHOST action. The link with a perfect fluid description of matter, withenergy density ρ m , pressure P m and sound speed c m is given by the expressions ρ m = 2 Y K Y − K , P m = K , c m = K Y K Y + 2 Y K
Y Y , (4.10)where all terms are evaluated on a background solution.It has been shown in [19] that the conditions for the stability of scalar linear perturbationsare modified and more involved than the case without matter. Indeed, in addition to ζ , thereis an extra scalar degree of freedom that we denote δσ , and the dynamics of the two modes areentangled. The quadratic action for these two scalar perturbations takes the form [13] S quad = (cid:90) d x dt a M (cid:20) ˙ V T K ˙ V − a ∂ i V T G ∂ i V + . . . (cid:21) , (4.11)where the vector V T = ( ζ, H δσ ˙ σ ) contains the two scalar degrees of freedom and the dots stand forthe terms with fewer than two (space or time) derivatives, which are not relevant for the stabilitydiscussion. The kinetic and gradient matrices read (see [13] for details) K = A ζ + ρ m (1+ w m ) M c m ( H (1+ α B ) − ˙ β ) ρ m (1+ w m ) ( c m β − ) M c m H ( H (1+ α B ) − ˙ β ) ρ m (1+ w m ) ( c m β − ) M c m H ( H (1+ α B ) − ˙ β ) ρ m (1+ w m ) M c m H , (4.12) G = B ζ − ρ m (1+ w m )(1+ α H +(1+ α T ) β ) M H ( H (1+ α B ) − ˙ β ) − ρ m (1+ w m )(1+ α H +(1+ α T ) β ) M H ( H (1+ α B ) − ˙ β ) ρ m (1+ w m ) M H . (4.13)In order to avoid ghost and gradient instabilities, both matrices K and G must be positive definite.When matter satisfies c m (cid:28) w m (cid:28)
1, one can expand the expressions of the eigenvalues of K and G with respect to c m and w m and one obtains, at leading order, λ K = A ζ M H (1 + α B − β (cid:48) ) + 6 ρ m β M H (1 + (1 + α B − β (cid:48) ) ) , λ K = ρ m c m H M (cid:20) α B − β (cid:48) ) + 1 (cid:21) ,λ G ± = B ζ M (cid:34) ρ m H ± (cid:115) ρ m (1 + α H + (1 + α T ) β ) H (1 + α B − β (cid:48) ) + ( ρ m H − M B ζ ) (cid:35) , (4.14)where λ K , and λ G ± are the eigenvalues of K and G respectively. One thus finds that λ K isalways positive while the sign of the three other eigenvalues depends on the specific backgroundsolution.2All eigenvalues can be expressed in terms of x , x and Ω m . Moreover, the coefficients M and c which appear in the tensor action are given explicitly by M = 1 + 2 x , c = 11 + 2 x . (4.15)Deep in the matter dominated era when | x | (cid:28) | x | (cid:28)
1, the leading order behaviour of theeigenvalues λ K and λ G ± is given by λ K ≈ − s + 4) x , λ G + ≈ −
17 ( s + 1) x , λ G − ≈ − s + 1) x . (4.16)and they are all positive when we take 0 < x (cid:28) s < − /
3, as discussed below (3.27).In Fig. (3), we plot the time evolution of the eigenvalues, as well as c T . We have chosenparameters and initial conditions such that all eigenvalues remain positive and c T <
1. Withtheses choices, we see that the tensor and scalar perturbations remain stable from the matter erato the de Sitter era. - - - N λ G + - - - - - N λ G - - - N λ K - - - - - N c t FIG. 3: Evolution of the eigenvalues λ G ± (top), λ K (bottom left) and c T (at the bottom right) viewed asfunctions of N = ln( a ). Dotted curves: s = − x ( i )1 = − . × − and x ( i )2 = 10 − .Dot-Dashed curves: s = −
10 and initial conditions x ( i )1 = − . × − and x ( i )2 = 2 . × − . V. CONCLUSIONS
In this paper, we have studied the cosmology of DHOST theories. We have considered themost general action for quadratic DHOST theories and derived the general equations of motion inan isotropic and homogeneous background in the presence of a perfect fluid. We have presentedthese equations in full generality, without restricting ourselves to shift-symmetric Lagrangians inthe first part. Then, we have considered a particular family of shift-symmetric DHOST modelscharacterized by three parameters only. We have performed a dynamical system analysis andobtained the conditions for our models to admit self-accelerating solutions at late time. Then, wehave examined the linear stability of both tensor and scalar modes, in the presence of pressurelessmatter, and found that the models studied here are stable in some region of the parameters space.With the advent of stage IV cosmological probes (LSST, Euclid), the sharp increase in theamount of data will enable us to test gravitational laws on cosmological scales. In order to analyse3such a trove of data, it will be very useful to rely on a parametrized set of models that canquantify, in a flexible way, deviations from general relativity. DHOST theories, which describe themost general and simplest scalar-tensor theories (the simplest in the sense that they propagate asingle additional degree of freedom), are natural candidates to serve as benchmark models for theanalysis of future data.
Acknowledgments
We thank Marco Crisostomi for instructive discussions and for providing the full expressions forthe parameters α K and α B in terms of the functions of the general Lagrangian. H. B. would liketo thank APC for their hospitality during his two stays when this project was initiated and thencontinued. Appendix A: Coefficients in the cosmological equations
The coefficients entering in the Friedmann equations (2.20) and (2.21) are given by g = P + X (cid:2) − (cid:0) F λ ϕ + F ϕ λ ϕ + P λ X (cid:1) − P X + Q ϕ (cid:3) − X (cid:2) λ ϕ (2 F X + 6 F λ X − A ) + λ ϕ (2 F ϕX + 4 F λ Xϕ + Q X ) − F ϕϕ λ X − Q ϕ λ X (cid:3) +12 λ ϕ X [ λ ϕ (3 A λ X + A X ) + 2 A λ Xϕ ] ,g = 6 { F λ ϕ + F ϕ + X [ Q X + 2 F ϕX + 4 F λ Xϕ + λ ϕ (4 F X + 12 F λ X − A )] − X [ λ ϕ (3 A λ X + A X ) + A λ Xϕ ] (cid:9) ,g = 6 F + 6 X (6 F λ X + 2 F X − A ) − X (3 A λ X + A X ) ,g = P − X (cid:0) F λ ϕϕ + 6 F λ ϕ + 4 F ϕ λ ϕ + 2 F ϕϕ + Q ϕ (cid:1) + X (cid:2) (4 λ ϕϕ + 6 λ ϕ ) A + 4 λ ϕ A ϕ (cid:3) ,g = 2( F − A X ) ,g = 4 [ F ϕ − XA ϕ + 3 λ ϕ ( F − A X )] ,g = 2 F ϕ + 4 λ ϕ F + 2 X [ Q X + 2 F Xϕ + 4 λ Xϕ F + λ ϕ (4 F X + 12 λ X F − A )] − X ( λ ϕ ( A X + 3 λ X A ) + λ Xϕ A ) ,g = − F X + 3 λ X F − A − X ( A X + 3 λ X A )] , The coefficients entering in the scalar equation (2.22) through the functions (2.24) are given by g = P ϕ + 3 λ ϕ P + 3 X (cid:0) F ϕϕ λ ϕ + 4 F λ ϕ λ ϕϕ + 2 F ϕ λ ϕϕ + 6 F λ ϕ + 8 F ϕ λ ϕ + 3 Qλ ϕ + λ ϕ Q ϕ + Qλ ϕϕ (cid:1) − X λ ϕ (cid:0) A λ ϕϕ + 3 A λ ϕ + A ϕ λ ϕ (cid:1) ,g = − (cid:0) F λ ϕ + 10 F ϕ λ ϕ + 4 F λ ϕϕ + 2 F ϕϕ + Q ϕ + 3 λ ϕ Q (cid:1) + 12 X (3 A λ ϕ + A ϕ λ ϕ + A λ ϕϕ ) ,g = − F ϕ + 3 λ ϕ F ) + 6 X ( A ϕ + 3 λ ϕ A ) ,g = − ( Q ϕ + 3 λ ϕ Q ) − X (6 F ϕ λ ϕ λ X + 2 F ϕϕ λ X + 2 F ϕ λ Xϕ + Q ϕ λ X + 3 Qλ ϕ λ X + Qλ Xϕ ) ,g = 12 λ ϕ ( F ϕ + λ ϕ F ) + 2( P X + 3 λ X P ) − Q ϕ + 3 λ ϕ Q +6 X ( − A λ ϕ + 2 F Xϕ λ ϕ + 2 F X λ ϕ + 6 F λ ϕ λ X − F ϕϕ λ X + 4 F λ ϕ λ Xϕ + λ ϕ Q X − Q ϕ λ X ) − X ( A X λ ϕ + 3 A λ ϕ λ X + 24 A λ ϕ λ Xϕ ) ,g = 12 F λ ϕ + 6 F ϕ + 6 X ( − A λ ϕ + 2 F Xϕ + 4 F X λ ϕ + 12 F λ ϕ λ X + 4 F λ Xϕ + Q X ) − X ( A X λ ϕ + 3 A λ ϕ λ X + A λ Xϕ ) ,g = −
12 [ F X + 3 λ X F − A − X ( A X + 3 λ X A )] . Appendix B: Effective parameters in the quadratic action of perturbations
In this section, we recall the expressions of the effective parameters entering in the quadraticaction of the perturbations about a FLRW background, S quad = (cid:90) d x dt a M (cid:26) δK ij δK ij − (cid:18) α L (cid:19) δK + (1 + α T ) (cid:18) R δ √ ha + δ R (cid:19) + H α K δN + 4 Hα B δKδN + (1 + α H ) RδN + 4 β δKδ ˙ N + β δ ˙ N + β a ( ∂ i δN ) (cid:27) , (B1)in terms of the functions (evaluated in the background solution) entering in the DHOST action, S = (cid:90) d x √− g (cid:32) P ( X, ϕ ) + Q ( X, ϕ ) (cid:50) ϕ + F ( X, ϕ ) R + (cid:88) i =1 A i ( X, ϕ ) L i (cid:33) . (B2)We restrict ourselves to shift-symmetric theories where all the functions in the action above dependon X only.All parameters but α K and α B depend on F and A i only, and they were given in [19], M F − A X , M α T ) = F , M α H ) = F − XF X ,M (cid:18) α L (cid:19) = F + A X , M β = − X (cid:0) A + A + ( A + A ) X + A X (cid:1) , M β = X (4 F X + 2 A + A X ) , M β = − X (4 F X − A − A X ) , (B3)where the right-hand side quantities are evaluated on the homogeneous and isotropic background.The expressions of α K and α B are much more complicated and they involve, in addition to F and A i , the functions P and Q . A long calculation gives,2 HM α B = (4 HX + ˙ X ) A + 2(3 HX + ˙ X ) A + 32 X ( − HX + ˙ X ) A − X ˙ XA − X ˙ XA + 4 HX A X + 2 X (6 HX + ˙ X ) A X + X ˙ XA X + ( − HX + 6 ˙ X ) F X + 2 √− XXQ X + 4 X ˙ XP XX , (B4) M H α K = ( − H X + 3 H ˙ X − X X + 2 ¨ X ) A + (9 ˙ HX + 3 H ˙ X − X X + 2 ¨ X ) A + 34 (18 H X + 10 ˙ HX + 8 HX ˙ X − ˙ X + 4 X ¨ X ) A + (6 HX ˙ X − X X ¨ X )+ X (9 HX ˙ X + ˙ X + 4 X ¨ X ) A + ( − H X + 3 HX ˙ X + 3 ˙ X X ¨ X ) A X + ( − H X + 3 HX ˙ X + 3 ˙ X X ˙ H + X ¨ X ) A X + X HX ˙ X + 7 ˙ X + 4 X ¨ X ) A X + (9 H X + 3 ˙ HX + 3 HX ˙ X + 7 X ˙ X X ¨ X ) A X + X HX ˙ X + 11 ˙ X + 4 X ¨ X ) A X + ( − H X + X ˙ X A XX + ( − H X + X ˙ X A XX + X ˙ X A XX + X ˙ X A XX + X ˙ X A XX + 6 X (2 H + 3 ˙ H ) F X + 12 X (2 H + ˙ H ) F XX + 2 X Q XX − HX √− XQ XX . (B5)5When applied to the model we are considering in the paper, P = αX, Q = 0 , F = 12 , A = βX, A = − β + 2 µ ) − βµX ,A = 2( β + 2 µ − µ X ) , A = 8 µX ( β + 2 µ + 3 βµX ) , (B6)the expressions of (B3) yield M = 1 + 2 βX , α T = α H = − βX βX , α L = 0 ,β = − µX , β = − µ X , β = − µX ( − µX )1 + 2 βX , (B7)while the expressions for α B and α K simplify into α B = 2 ˙ ϕ (cid:0) β ˙ ϕH + µ (cid:0) βX ( ˙ ϕH − ϕ ) + 3 ˙ ϕH − ϕ (cid:1) − µ X (cid:0) βX + 1 (cid:1) ¨ ϕ (cid:1) βX H + H , (B8) α K = − XH (2 βX + 1) (6 X (2 µ X (24 ˙ ϕH (3 βX + 1) ¨ ϕ + 2 ˙ ϕ ... ϕ (14 βX + 5)+ (126 βX + 25)( ¨ ϕ ) ) + µ (6 βX (7 H + 3 ˙ H ) + 9 H + 5 ˙ H ) + 6 βH ) − α ) , (B9)where we have used X = − ˙ ϕ , ˙ X = − ϕ ¨ ϕ and ¨ X = −
2( ¨ ϕ + ˙ ϕ ... ϕ ).In terms of the variables introduced in (3.7), these coefficients become M = 1 + 2 x , α T = α H = − x x + 1 , α L = 0 ,β = − sx , β = − s x , β = − sx ( sx − x + 1 ,α B = − x (5 (2 sx + s ) (cid:15) ϕ − sx − s − x + 1) (2 sx (cid:15) ϕ + 1) ,α K = 2(2 x + 1)(2 sx (cid:15) ϕ + 1) (cid:0) x + (cid:15) ϕ (cid:0) x (cid:0) s (cid:15) h − s (cid:1) + 24 sx (5 s(cid:15) h + 3 s − (cid:1) − ( ˙ H/H ) (cid:0)(cid:0) s x + 120 s x (cid:1) (cid:15) ϕ + (cid:0) s x + 120 s x (cid:1) (cid:15) ϕ + 108 sx + 30 sx (cid:1) + (cid:15) (cid:48) ϕ (cid:0)(cid:0) s x + 240 s x (cid:1) (cid:15) ϕ + 336 s x + 120 s x (cid:1) + (cid:0) s x + 24 s (15 s + 71) x + 420 s x (cid:1) (cid:15) ϕ − sx − (54 s + 36) x (cid:1) . (B10)Notice that in the last equation, we have used the relation ... ϕ = (cid:15) (cid:48) ϕ (1 + 2 sx (cid:15) ϕ ) + (cid:15) ϕ + (cid:15) ϕ (cid:15) h , whichcan be deduced from the relations ˙ X = − ϕ ¨ ϕ = 2 (cid:15) ϕ H b X , (B11)which comes from the definition of (cid:15) ϕ , and¨ X = −
2( ¨ ϕ + ˙ ϕ ... ϕ ) = 2 XH b (cid:18) HH b (cid:15) (cid:48) ϕ + (cid:15) ϕ (cid:15) h + 2 (cid:15) ϕ (cid:19) . (B12)We could also replace ˙ H/H in the expression for α K by using the relation˙ HH = (cid:15) h + 2 s ( x (cid:48) (cid:15) ϕ + x (cid:15) (cid:48) ϕ )1 + 2 sx (cid:15) ϕ , (B13)6which follows from (3.10).Finally, these results allow us to express the coefficients A ζ in (4.6) and B ζ in (4.7), entering inthe quadratic action for the scalar perturbation, in terms of the dynamical variables in the form, A ζ = A ( x ) + x A ( x ) A ( x ) , (B14) B ζ = B ( x ) + x B ( x ) + x B ( x ) B ( x ) + x B ( x ) , (B15)where the functions A i and B i are polynomials of the variable x only (which depends on theparameter s ) given by A ( x ) = 6 x (cid:0) s x + (cid:0) s + 84 s + 40 (cid:1) x + 36 s (6 s + 5) x − s + 4) (cid:1) , A ( x ) = 2 (2 x + 1) (cid:0) − s x + 15 sx + 1 (cid:1) , A ( x ) = (cid:0) sx + 2(3 s + 5) x + 1 (cid:1) , B ( x ) = 3 x (cid:0) s x + 576 s (132 s + 125) x + 288 s (cid:0) s + 344 s + 215 (cid:1) x + 16 s (cid:0) s + 2538 s + 2784 s + 1775 (cid:1) x − (cid:0) s + 1176 s + 1154 s + 500 (cid:1) x + (cid:0)
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