We consider "brane-universes", where matter is confined to four-dimensional hypersurfaces (three-branes) whereas one extra compact dimension is felt by gravity only. We show that the cosmology of such branes is definitely different from standard cosmology and identify the reasons behind this difference. We give a new class of exact solutions with a constant five-dimensional radius and cosmologically evolving brane. We discuss various consequences.

Abstract.We consider the cosmology of a "3-brane universe" in a five dimensional (bulk) space-time with a cosmological constant. We show that Einstein's equations admit a first integral, analogous to the first Friedmann equation, which governs the evolution of the metric in the brane, whatever the time evolution of the metric along the fifth dimension. We thus obtain the cosmological evolution in the brane for any equation of state describing the matter in the brane, without needing the dependence of the metric on the fifth dimension. In the particular case p = wρ, (w = constant), we give explicit expressions for the time evolution of the brane scale factor, which show that standard cosmological evolution can be obtained (after an early non conventional phase) in a scenarioà la Randall and Sundrum, where a brane tension compensates the bulk cosmological constant. We also show that a tiny deviation from exact compensation leads to an effective cosmological constant at late time. Moreover, when the metric along the fifth dimension is static, we are able to extend the solution found on the brane to the whole spacetime.

Theories with higher order time derivatives generically suffer from ghost-like instabilities, known as Ostrogradski instabilities. This fate can be avoided by considering "degenerate" Lagrangians, whose kinetic matrix cannot be inverted, thus leading to constraints between canonical variables and a reduced number of physical degrees of freedom. In this work, we derive in a systematic way the degeneracy conditions for scalar-tensor theories that depend quadratically on second order derivatives of a scalar field. We thus obtain a classification of all degenerate theories within this class of scalar-tensor theories. The quartic Horndeski Lagrangian and its extension beyond Horndeski belong to these degenerate cases. We also identify new families of scalar-tensor theories with the property that they are degenerate despite the nondegeneracy of the purely scalar part of their Lagrangian. * Electronic address: langlois@apc.univ-paris7.fr † Electronic address: karim.noui@lmpt.univ-tours.fr 1 For a single variable q, a Lagrangian of the form L(q,q,q) is nondegenerate if ∂ 2 L/∂q 2 = 0. Multi-variable Lagrangians will be discussed in the main text.2 Another approach, familiar in the context of effective field theory, consists in simply discarding Ostrogradski instabilities when they arise from the perturbative part of the Lagrangian (see e.g.[25]).

We propose a minimal description of single field dark energy/modified gravity within the effective field theory formalism for cosmological perturbations, which encompasses most existing models. We start from a generic Lagrangian given as an arbitrary function of the lapse and of the extrinsic and intrinsic curvature tensors of the time hypersurfaces in unitary gauge, i.e. choosing as time slicing the uniform scalar field hypersurfaces. Focusing on linear perturbations, we identify seven Lagrangian operators that lead to equations of motion containing at most two (space or time) derivatives, the background evolution being determined by the time dependent coefficients of only three of these operators. We then establish a dictionary that translates any existing or future model whose Lagrangian can be written in the above form into our parametrized framework. As an illustration, we study Horndeski's-or generalized Galileon-theories and show that they can be described, up to linear order, by only six of the seven operators mentioned above. This implies, remarkably, that the dynamics of linear perturbations can be more general than that of Horndeski while remaining second order. Finally, in order to make the link with observations, we provide the entire set of linear perturbation equations in Newtonian gauge, the effective Newton constant in the quasi-static approximation and the ratio of the two gravitational potentials, in terms of the time-dependent coefficients of our Lagrangian.

We have recently proposed a new class of gravitational scalar-tensor theories free from Ostrogradski instabilities, in Ref. [1]. As they generalize Horndeski theories, or "generalized" galileons, we call them G 3 . These theories possess a simple formulation when the time hypersurfaces are chosen to coincide with the uniform scalar field hypersurfaces. We confirm that they contain only three propagating degrees of freedom by presenting the details of the Hamiltonian formulation. We examine the coupling between these theories and matter. Moreover, we investigate how they transform under a disformal redefinition of the metric. Remarkably, these theories are preserved by disformal transformations that depend on the scalar field gradient, which also allow to map subfamilies of G 3 into Horndeski theories.

We introduce a new class of scalar-tensor theories of gravity that extend Horndeski, or "generalized Galileon," models. Despite possessing equations of motion of higher order in derivatives, we show that the true propagating degrees of freedom obey well-behaved second-order equations and are thus free from Ostrogradski instabilities, in contrast to standard lore. Remarkably, the covariant versions of the original Galileon Lagrangians-obtained by direct replacement of derivatives with covariant derivatives-belong to this class of theories. These extensions of Horndeski theories exhibit an uncommon, interesting phenomenology: The scalar degree of freedom affects the speed of sound of matter, even when the latter is minimally coupled to gravity.

We present all scalar-tensor Lagrangians that are cubic in second derivatives of a scalar field, and that are degenerate, hence avoiding Ostrogradsky instabilities. Thanks to the existence of constraints, they propagate no more than three degrees of freedom, despite having higher order equations of motion. We also determine the viable combinations of previously identified quadratic degenerate Lagrangians and the newly established cubic ones. Finally, we study whether the new theories are connected to known scalartensor theories such as Horndeski and beyond Horndeski, through conformal or disformal transformations.

We review and extend a novel approach that we recently introduced , to describe general dark energy or scalar-tensor models. Our approach relies on an Arnowitt-Deser-Misner (ADM) formulation based on the hypersurfaces where the underlying scalar field is uniform. The advantage of this approach is that it can describe in the same language and in a minimal way a vast number of existing models, such as quintessence models, F (R) theories, scalar tensor theories, their Horndeski extensions and beyond. It also naturally includes Horava-Lifshitz theories. As summarized in this review, our approach provides a unified treatment of the linear cosmological perturbations about a Friedmann-Lemaître-Robertson-Walker (FLRW) universe, obtained by a systematic expansion of our general action up to quadratic order. This shows that the behaviour of these linear perturbations is generically characterized by five time-dependent functions. We derive the full equations of motion in the Newtonian gauge. In the Horndeski case, we obtainthe equation of state for dark energy perturbations in terms of these functions. Our unifying description thus provides the simplest and most systematic way to confront theoretical models with current and future cosmological observations.