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Late time cosmological evolution in DHOST models

We study the late cosmological evolution, from the nonrelativistic matter dominated era to the dark energy era, in modified gravity models described by Degenerate Higher-Order Scalar-Tensor (DHOST) theories. They represent the most general scalar-tensor theories propagating a single scalar degree of freedom and include Horndeski and Beyond Horndeski theories. We provide the homogeneous evolution equations for any quadratic DHOST theory, without restricting ourselves to theories where the speed of gravitational waves coincides with that of light since the present constraints apply to wavelengths much smaller than cosmological scales. To illustrate the potential richness of the cosmological background evolution in these theories, we consider a simple family of shift-symmetric models, characterized by three parameters and compute the evolution of dark energy and of its equation of state. We also identify the regions in parameter space where the models are perturbatively stable.

We study the late cosmological evolution, from the nonrelativistic matter dominated era to the dark energy era, in modified gravity models described by Degenerate Higher-Order Scalar-Tensor (DHOST) theories. They represent the most general scalar-tensor theories propagating a single scalar degree of freedom and include Horndeski and Beyond Horndeski theories. We provide the homogeneous evolution equations for any quadratic DHOST theory, without restricting ourselves to theories where the speed of gravitational waves coincides with that of light since the present constraints apply to wavelengths much smaller than cosmological scales. To illustrate the potential richness of the cosmological background evolution in these theories, we consider a simple family of shift-symmetric models, characterized by three parameters and compute the evolution of dark energy and of its equation of state. We also 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 that gravity is modified on cosmological scales. Concrete realisations of this idea often rely on scalartensor theories, which represent the simplest extension of general relativity since a scalar degree of freedom is added to the usual tensor modes of general relativity. The most general family of scalartensor 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 the Einstein-Hilbert action [4]. In the present work, we consider the whole family of quadratic DHOST theories, 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) which have been fully classified in [8].
Most of the literature has recently concentrated on DHOST theories where the speed of gravitational waves coincides with that of light, following the observation of a neutron star binary merger that 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 of gravitational waves, yielding a further tight constraint on DHOST theories [10,11]. The cosmology of DHOST theories satisfying either the first or both of the above constraints has been studied in [12][13][14][15][16][17].
However, it should be stressed that the LIGO-Virgo measurements probe wavelengths of order 10 3 km, which are many orders of magnitude smaller than cosmological scales, and an effective theory describing cosmological scales might not be adequate to describe physics on much smaller length scales, as those probed by LIGO-Virgo (see [18] for a discussion on this point). In the present work, we adopt the point of view that DHOST theories apply only to cosmological scales and cannot be extrapolated down to astrophysical scales within the same framework 1 . In this perspective, all the constraints derived from GW170817 mentioned above are not directly relevant and it thus makes sense to study models that can lead to distinct propagation velocities for light and gravitational waves on cosmological scales.
The outline of the paper is the following. In section II, starting from the most general action for 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 auxiliary scale factor that makes the equations manifestly second-order. The second part of the paper is devoted to the study of a simple subfamiliy of quadratic DHOST theories characterized by a few parameters. In section III, we write the equations of motion in the form of a dynamical system and identify the fixed points and their nature. In section IV, we turn to the linear perturbations in 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 the notations used throughout this paper. Then, we focus on the case of a homogeneous and isotropic universe and provide the cosmological evolution equations for the whole family of quadratic DHOST theories.

A. Quadratic DHOST theories
The most general theory of quadratic DHOST theory is described by the action 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 are defined by 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 extra scalar 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 classes which are stable under general disformal transformations, i.e. transformations of the metric of the form where C and D are arbitrary functions (provided that the metricg µν remains regular).
The theories belonging to the first class, named class Ia in [6], can be mapped into a Horndeski form 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 free functions F, A 1 and A 3 (in addition to P and Q) and the three remaining functions are given by the relations [1]

5)
where F X denotes the derivative of F (X, ϕ) with respect to X. Similarly F ϕ will denote the partial derivative 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 that guarantee only one scalar degree of freedom is present [1,20]. In conclusion, this means that all the DHOST theories we study here are characterized by five free functions of X and ϕ, which are P , Q, F , A 1 and A 3 . Notice that we have implicitly supposed the condition F + XA 2 = 0. Theories where F + XA 2 = 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 where the lapse function N (t) and the scale factor a(t) depend on time only. As a consequence of the spacetime symmetries, the scalar field must also be homogeneous and therefore depends only on time.
Substituting the above metric (2.7) into the action (2.1), and taking into account the degeneracy conditions (2.4-2.6), one finds that the corresponding homogeneous action can be written as a functional of N (t), a(t) and of the homogeneous scalar field ϕ(t). It reads where we have introduced the new functions, and, everywhere, the expression of X is explicitly given by The Euler-Lagrange equations derived from the above action (2.8) lead to equations of motion that appear higher than second order. However, due to the degeneracy of the theory, these equations can be recast into a second order system. As done in [13], this can be demonstrated explicitly by introducing an auxiliary scale factor b, defined by the relation where λ satisfies the condition so that the terms quadratic inφ in the action (2.8) are reabsorbed in the derivatives of the new scale factor. It is also convenient to use a Hubble parameter associated with this auxiliary scale factor, defined by 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 using this "Horndeski frame" is that matter is no longer minimally coupled, as it was assumed in the inital 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 (2.14) The coupling to matter is described by adding to L hom a matter Lagrangian L m , and the total Lagrangian is denoted L ≡ L hom + L m . We get the equations of motion by writing the Euler-Lagrange equations for N , b and ϕ. The first 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 equation of the form Once we have derived the equations of motion, we fix the time coordinate such that N = 1 (and thusṄ = 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-momentum tensor of the fluid, defined as usual by As a consequence, in the DHOST frame, where matter is minimally coupled, the variation of the matter Lagrangian is immediately given by where ρ m and P m are the fluid energy density and pressure, respectively. Using which follows from the definition (2.11) of b, one finds that the variation of the matter Lagrangian in the Horndeski frame is given by Inserting the above variation of the matter Lagrangian into the Euler-Lagrange equations, we obtain the set of equations of motion for the cosmological dynamics. The analogs of the two Friedmann equations (with N = 1) take the form where all the coefficients g i can be written explicitly in terms of the functions that appear in the Lagrangian and of λ. They are given in Appendix A.
Finally the scalar field equation can be written as where we have defined The two functions U and J are of the form where the corresponding coefficients g i are also given explicitly in Appendix A. As usual, the equation for the scalar field is not independent and can be obtained from the two Friedmann equations.
One recovers the equations of motion given in [13] when the theory is shift symmetric, i.e. when the functions in the Lagrangian are invariant under the transformation ϕ → ϕ + c and thus depend only 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 three constant parameters only. These Lagrangians are shift symmetric and characterized by the simple polynomial functions where α, β and µ are arbitrary constants. Hence, from (2.12) and (2.9), we deduce The expressions of A 4 and A 5 are easily obtained from the degeneracy conditions (2.5) and (2.6). One thus gets 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 to Furthermore, the equation of motion for the scalar field becomes 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 following similar treatments for dark energy models (see e.g. [21][22][23] in the context of Galileons and [24] for a recent review). These variables are not independent and one can easily see that where ln b plays the role of time. The two equations above can also be formulated in terms of the time ln a instead of ln b by using the relation between the two Hubble constants, which follows from (2.13). The two previous relations (3.8) and (3.9) then become 11) 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 2(1 + 2x 2 ) h + 4x 2 (4 + 3s + 6sx 2 ) ϕ + (3 + x 1 + 6x 2 ) = 0 , (3.14) These equations can be seen as constraints for the dynamical system (3.11-3.12). The first constraint, Eq. (3.13), involves the matter density, whereas the last two equations (3.14) and (3.15) can be used to determine ϕ and h in terms of x 1 and x 2 . After a straightforward calculation, one gets where the common denominator ∆ is given by  Table I. We see that there are at most two stable fixed points corresponding to a de Sitter solution. To find the conditions on the parameters of the theory for these fixed points to exist, we have to study the signs of x 1 and x 2 at the fixed points. It is immediate to show that • at the point C: x 1 < 0 and x 2 < 0 for all values of s; • at the point D: x 1 > 0 and x 2 < 0 for s > 0 whereas x 1 < 0 and x 2 > 0 for s < 0.
From the definition of x 1 and x 2 (3.7), we see immediately that αx 1 < 0 (because X < 0) and βx 2 > 0. As a consequence, we deduce that the fixed point C exists only if α > 0 and β < 0 whereas the fixed point D exists only if µ < 0. Notice that, in the limit s = 0, i.e. µ = 0, corresponding to a DHOST theory that belongs to the Horndeski subclass, the dynamical system admits a single fixed point given by the limit s → 0 of C, whereas the limit of the fixed point D is ill-defined.

B. DHOST frame: Effective Friedmann equations
In the frame where matter is minimally coupled, it is always possible to write effectively the Friedmann equations in the usual form, 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 dark energy as usual by the ratio Moreover, one can define a global effective equation of state parameter as For the models we are considering here, this parameter can be expressed in terms of the variables introduced earlier and reads Using the fact that P m = 0 for non-relativistic matter, we can write, from (3.20) and (3.22), a relation between w DE and w eff given by The dynamical equations (3.8) and (3.9) can be solved numerically and the right amount of nonrelativistic matter today, i.e. Ω m = ρ m /(3H 2 ) ≈ 0.3, can be reached by tuning the initial conditions for x 1 and x 2 . We choose our initial conditions deep in the matter dominated era, i.e. when Ω m 1. According to the constraint (3.13), taking |x 1 | 1 and |x 2 | 1 initially guarantees that we are deep in the matter dominated era. Moreover, in order to observe a relatively rapid transition from the matter era to de Sitter era, we take initial conditions such that Indeed, in this regime, the dynamical system reduces to  which shows that the system moves quickly away from the region where x 1 and x 2 are very small (this is not the case if we take |x 2 | |x 1 | 1 instead). The initial dark energy parameter Ω DE is then approximated, according to (3.13), by (3.27) If we choose x 2 > 0 initially, in order to get c T < 1 (as we will see in Eq. (4.15) of the next section), then the parameter s must satisfy s < −5/3. We have plotted some illustrative examples of numerical results in (Fig. 1) and (Fig. 2). The first figure shows the evolution of the cosmological parameters Ω m , Ω DE , w DE and w eff . We observe a cosmological transition from the matter era to the dark energy era. We also observe that the dark energy behaves like pressureless matter deep in the matter dominated era and like a cosmological constant with w DE ≈ −1 at very late times, with a transition going through an intermediate regime where 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 developed in [25][26][27] and extended to DHOST theories in [19]. This effective approach relies on the ADM formulation where the metric is parametrized by the lapse function N , the shift vector N i and the spatial metric h ij as follows, (4.1) In the ADM framework, the "velocity" of the spatial metric is encoded in the extrinsic curvature tensor K ij defined by where D i denotes the spatial covariant derivative associated to h ij . The DHOST action can be reformulated in terms of the ADM variables and the dynamics of the linear perturbations about an FLRW 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 action for the perturbations is given by [19] where δ 2 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 1 , β 2 and β 3 , which fully characterize the quadratic action, are functions of time as they depend on the background. 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 the moment, it has been shown in [19] that the quadratic action reduces to the sum of an action for the curvature perturbation ζ, representing the scalar mode, and an action for the tensor modes γ ij , The coefficients A ζ and B ζ that appear in the scalar action are given by while the speed of gravitational waves c T , which appears in the tensor action, is given by c 2 T = 1 + α T . Therefore the stability conditions for the linear perturbations are simply given by The expressions of these coefficients in terms of the dynamical variables (3.7) are given in Appendix B. As we will see, they will be useful for the numerical analysis of the linear stability of the model.
In the presence of matter, these stability conditions (for the scalar mode) are modified. They have 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, which is added to the DHOST action. The link with a perfect fluid description of matter, with energy density ρ m , pressure P m and sound speed c m is given by the expressions where all terms are evaluated on a background solution.
It has been shown in [19] that the conditions for the stability of scalar linear perturbations are modified and more involved than the case without matter. Indeed, in addition to ζ, there is an extra scalar degree of freedom that we denote δσ, and the dynamics of the two modes are entangled. The quadratic action for these two scalar perturbations takes the form [13] where the vector V T = (ζ, H δσ σ ) contains the two scalar degrees of freedom and the dots stand for the terms with fewer than two (space or time) derivatives, which are not relevant for the stability discussion. The kinetic and gradient matrices read (see [13] for details) (4.13) In order to avoid ghost and gradient instabilities, both matrices K and G must be positive definite. When matter satisfies c m 1 and w m 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, (4.14) where λ K 1,2 and λ G ± are the eigenvalues of K and G respectively. One thus finds that λ K 2 is always positive while the sign of the three other eigenvalues depends on the specific background solution.
All eigenvalues can be expressed in terms of x 1 , x 2 and Ω m . Moreover, the coefficients M 2 and c 2 T which appear in the tensor action are given explicitly by Deep in the matter dominated era when |x 1 | |x 2 | 1, the leading order behaviour of the eigenvalues λ K 1 and λ G ± is given by (4. 16) and they are all positive when we take 0 < x 2 1 and s < −5/3, as discussed below (3.27). In Fig. (3), we plot the time evolution of the eigenvalues, as well as c 2 T . We have chosen parameters and initial conditions such that all eigenvalues remain positive and c 2 T < 1. With theses choices, we see that the tensor and scalar perturbations remain stable from the matter era to the de Sitter era.

V. CONCLUSIONS
In this paper, we have studied the cosmology of DHOST theories. We have considered the most general action for quadratic DHOST theories and derived the general equations of motion in an isotropic and homogeneous background in the presence of a perfect fluid. We have presented these equations in full generality, without restricting ourselves to shift-symmetric Lagrangians in the first part. Then, we have considered a particular family of shift-symmetric DHOST models characterized by three parameters only. We have performed a dynamical system analysis and obtained the conditions for our models to admit self-accelerating solutions at late time. Then, we have examined the linear stability of both tensor and scalar modes, in the presence of pressureless matter, 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 the amount of data will enable us to test gravitational laws on cosmological scales. In order to analyse such a trove of data, it will be very useful to rely on a parametrized set of models that can quantify, in a flexible way, deviations from general relativity. DHOST theories, which describe the most general and simplest scalar-tensor theories (the simplest in the sense that they propagate a single additional degree of freedom), are natural candidates to serve as benchmark models for the analysis of future data. which follows from (3.10).
These are the expressions we use to plot Fig. (3) and to express the eigenvalues (4.16) in the matter era where |x 1 | |x 2 |