We construct four-dimensional covariant non-linear theories of massive gravity which are ghostfree in the decoupling limit to all orders. These theories resum explicitly all the nonlinear terms of an effective field theory of massive gravity. We show that away from the decoupling limit the Hamiltonian constraint is maintained at least up to and including quartic order in non-linearities, hence, excluding the possibility of the Boulware-Deser ghost up to this order. We also show that the same remains true to all orders in a similar toy-model. Introduction:Whether there exist a consistent extension of General Relativity by a mass term is a basic question of a classical field theory. A small graviton mass could also be of a significant physical interest, notably for the cosmological constant problem.A ghost-free linear theory of massive spin-2 -the FierzPauli (FP) model [1] -had been notoriously hard to generalize to the nonlinear level [2]: the Hamiltonian constraint gets lost in general and, as a result, the sixth degree of freedom -the Boulware-Deser (BD) ghostemerges as a mode propagating on otherwise physically meaningful local backgrounds (e.g., on a background of a lump of matter). Part of this problem can be seen in the effective field theory (EFT) approach to massive gravity [3] in the decoupling limit [3,4]. There, the problem manifests itself in the Lagrangian for the helicity-0 component of the massive graviton. This Lagrangian generically contains nonlinear terms with more than two time derivatives. The latter give rise to the sixth degree of freedom on local backgrounds, while in general, these terms lead to the loss of well-posedness of the Cauchy problem for the helicity-0 field theory [3,4].A step forward has been made recently in [5] where it was shown that: (a) the coefficients of the EFT can be chosen so that the decoupling limit Lagrangian is ghostfree; this involves choosing the "appropriate coefficients" order-by-order, and an algorithm was set for this procedure to an arbitrary order; (b) once the "appropriate coefficients" are chosen in the effective Lagrangian, in the decoupling limit only a few terms up to the quartic order survive, all the higher order terms vanish identically. Moreover, the surviving terms are unique as their structure is fixed by symmetries [5,6].In the present work we build on the above two points, and go far beyond them. In particular: (1) We construct Lagrangians that automatically produce the "appropriate coefficients" once expanded in powers of the fields; these give rise to theories that are ghost-free automatically to all orders in the decoupling limit. (2) Using the obtained Lagrangians we study the issue of the BD ghost away from the decoupling limit; we show that the Hamiltonian constraint is maintained at least up to and including

We consider the Lagrangian of gravity covariantly amended by the mass and polynomial interaction terms with arbitrary coefficients, and reinvestigate the consistency of such a theory in the decoupling limit, up to the fifth order in the nonlinearities. We calculate explicitly the self-interactions of the helicity-0 mode, as well as the nonlinear mixing between the helicity-0 and -2 modes. We show that ghost-like pathologies in these interactions disappear for special choices of the polynomial interactions, and argue that this result remains true to all orders in the decoupling limit. Moreover, we show that the linear, and some of the nonlinear mixing terms between the helicity-0 and -2 modes can be absorbed by a local change of variables, which then naturally generates the cubic, quartic, and quintic Galileon interactions, introduced in a different context. We also point out that the mixing between the helicity-0 and 2 modes can be at most quartic in the decoupling limit. Finally, we discuss the implications of our findings for the consistency of the effective field theory away from the decoupling limit, and for the Boulware-Deser problem.

We review recent progress in massive gravity. We start by showing how different theories of massive gravity emerge from a higher-dimensional theory of general relativity, leading to the Dvali-Gabadadze-Porrati model (DGP), cascading gravity, and ghost-free massive gravity. We then explore their theoretical and phenomenological consistency, proving the absence of Boulware-Deser ghosts and reviewing the Vainshtein mechanism and the cosmological solutions in these models. Finally, we present alternative and related models of massive gravity such as new massive gravity, Lorentz-violating massive gravity and non-local massive gravity.

Euclid is a European Space Agency medium-class mission selected for launch in 2020 within the cosmic vision 2015–2025 program. The main goal of Euclid is to understand the origin of the accelerated expansion of the universe. Euclid will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and red-shifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky. Although the main driver for Euclid is the nature of dark energy, Euclid science covers a vast range of topics, from cosmology to galaxy evolution to planetary research. In this review we focus on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. We discuss five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions and questions of methodology in the data analysis. This review has been planned and carried out within Euclid’s Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the Euclid mission.

We derive the relativistic generalization of the Galileon, by studying the brane position modulus of a relativistic probe brane embedded in a fivedimensional bulk. In the appropriate Galilean contraction limit, we recover the complete Galileon generalization of the DGP decoupling theory and its conformal extension. All higher order interactions for the Galileon and its relativistic generalization naturally follow from the brane tension, induced curvature, and the Gibbons-Hawking-York boundary terms associated with all bulk Lovelock invariants. Our approach makes the coupling to gravity straightforward, in particular allowing a simple rederivation of the nonminimal couplings required by the Covariant Galileon. The connection with the Lovelock invariants makes the well-defined Cauchy problem manifest, and gives a natural unification of four dimensional effective field theories of the DBI type and the Galileon type.