Cosmological perturbations of brane-induced gravity and the vDVZ discontinuity on FLRW space-times

We investigate the cosmological perturbations of the brane-induced (Dvali-Gabadadze-Porrati) model which exhibits a van Dam-Veltman-Zakharov (vDVZ) discontinuity when linearized over a Minkowski background. We show that the linear brane scalar cosmological perturbations over an arbitrary Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time have a well defined limit when the radius of transition between 4D and 5D gravity is sent to infinity with respect to the background Hubble radius. This radius of transition plays for the brane-induced gravity model a role equivalent to the Compton wavelength of the graviton in a Pauli-Fierz theory, as far as the vDVZ discontinuity is concerned. This well defined limit is shown to obey the linearized 4D Einstein's equations whenever the Hubble factor is non vanishing. This shows the disappearance of the vDVZ discontinuity for general FLRW background, and extends the previously know result for maximally-symmetric space-times of non vanishing curvature. Our reasoning is valid for matter with simple equation of state such as a scalar field, or a perfect fluid with adiabatic perturbations, and involves to distinguish between space-times with a vanishing scalar curvature and space-times with a non vanishing one. We also discuss the validity of the linear perturbation theory, in particular for those FLRW space-times where the Ricci scalar is vanishing only on a set of zero measure. In those cases, we argue that the linear perturbation theory breaks down when the Ricci scalar vanishes (and the radius of transition is sent to infinity), in a way similar to what has been found to occur around sources on a Minkowski background.Comment: 36 pages, v.2, typos correcte

A formal introduction to Horndeski and Galileon theories and their generalizations

We review different constructions of Galileon theories in both flat and curved space, and for both single scalar field models as well as multi-field models. Our main emphasis is on the formal mathematical properties of these theories and their construction.Comment: 19 page

The Boulware-Deser mode in Zwei-Dreibein gravity

Massive gravity in three dimensions accepts several different formulations. Recently, the 3-dimensional bigravity dRGT model in first order form, Zwei-Dreibein gravity, was considered by Bergshoeff {\it et al.} and it was argued that the Boulware-Deser mode is killed by extra constraints. We revisit this assertion and conclude that there are sectors on the space of initial conditions, or subsets of the most general such model, where this mode is absent. But, generically, the theory does carry 3 degrees of freedom and thus the Boulware-Deser mode is still active. Our results also sheds light on the equivalence between metric and vierbein formulations of dRGT model.Comment: 4 page

Can Hamiltonians be boundary observables in Parametrized Field Theories?

It has been argued that holography in gravitational theories is related to the existence of a particularly useful Gauss Law that allows energy to be measured at the boundary. The present work investigates the extent to which such Gauss Laws follow from diffeomorphism invariance. We study parametrized field theories, which are a class of diffeomorphism-invariant theories without gravity. We find that the Hamiltonian for parametrized field theories vanishes on shell even in the presence of a boundary and under a variety of boundary conditions. We conclude that parametrized theories have no useful Gauss Law, consistent with the absence of holography in these theories.Comment: 28 pages, LaTeX, references added, citations clarified, typos correcte

Improving relativistic MOND with Galileon k-mouflage

We propose a simple field theory reproducing the MOND phenomenology at galaxy scale, while predicting negligible deviations from general relativity at small scales thanks to an extended Vainshtein ("k-mouflage") mechanism induced by a covariant Galileon-type Lagrangian. The model passes solar-system tests at the post-Newtonian order, including those of local Lorentz invariance, and its anomalous forces in binary-pulsar systems are orders of magnitude smaller than the tightest experimental constraints. The large-distance behavior is obtained as in Bekenstein's tensor-vector-scalar (TeVeS) model, but with several simplifications. In particular, no fine-tuned function is needed to interpolate between the MOND and Newtonian regimes, and no dynamics needs to be defined for the vector field because preferred-frame effects are negligible at small distances. The field equations depend on second (and lower) derivatives, and avoid thus the generic instabilities related to higher derivatives. Their perturbative solution around a Schwarzschild background is remarkably simple to derive. We also underline why the proposed model is particularly efficient within the class of covariant Galileons.Comment: 6 pages, 1 figure, RevTeX4 forma

Field equations and cosmology for a class of nonlocal metric models of MOND

We consider a class of nonlocal, pure-metric modified gravity models which were developed to reproduce the Tully-Fisher relation without dark matter and without changing the amount of weak lensing predicted by general relativity. Previous work gave only the weak field limiting form of the field equations specialized to a static and spherically symmetric geometry. Here we derive the full field equations and specialize them to a homogeneous, isotropic and spatially flat geometry. We also discuss the problem of fitting the free function to reproduce the expansion history. Results are derived for models in which the MOND acceleration a_0 ~ 1.2 x 10^{-10} m/s^{2} is a fundamental constant and for the more phenomenologically interesting case in which the MOND acceleration changes with the cosmological expansion rate.Comment: 15 pages, no figures, uses revtex4, dedicated to Stanley Deser on the occasion of his 83rd birthda

On Partially Massless Theory in 3 Dimensions

We analyze the first-order formulation of the ghost-free bigravity model in three-dimensions known as zwei-dreibein gravity. For a special choice of parameters, it was argued to have an additional gauge symmetry and give rise to a partially massless theory. We provide a thorough canonical analysis and identify that whether the theory becomes partially massless depends on the form of the stability condition of the secondary constraint responsible for the absence of the ghost. Generically, it is found to be an equation for a Lagrange multiplier implying that partially massless zwei-dreibein gravity does not exist. However, for special backgrounds this condition is identically satisfied leading to the presence of additional symmetries, which however disappear at quadratic order in perturbations.Comment: 22 pages; added references and few comments in the conclusions; added few clarification

Counting the degrees of freedom of generalized Galileons

We consider Galileon models on curved spacetime, as well as the counterterms introduced to maintain the second-order nature of the field equations of these models when both the metric and the scalar are made dynamical. Working in a gauge invariant framework, we first show how all the third-order time derivatives appearing in the field equations -- both metric and scalar -- of a Galileon model or one defined by a given counterterm can be eliminated to leave field equations which contain at most second-order time derivatives of the metric and of the scalar. The same is shown to hold for arbitrary linear combinations of such models, as well as their k-essence-like/Horndeski generalizations. This supports the claim that the number of degrees of freedom in these models is only 3, counting 2 for the graviton and 1 for the scalar. We comment on the arguments given previously in support of this claim. We then prove that this number of degrees of freedom is strictly less that 4 in one particular such model by carrying out a full-fledged Hamiltonian analysis. In contrast to previous results, our analyses do not assume any particular gauge choice of restricted applicability.Comment: 27 pages, no figure; v2: short explanation added below Eq. (42), improved Sec. II.B.

An introduction to the Vainshtein mechanism

We introduce the Vainshtein mechanism which plays a crucial role in massive gravities, as well as in related theories such as Galileons and their extensions. This mechanism, also known as k-mouflage, allows to hide via non linear effects - typically for source distances smaller than a so-called Vainshtein radius which depends on the source and on the theory considered - some degrees of freedom whose effects are then only left important at large distances, e.g. for cosmology. It is introduced here in non linear Fierz-Pauli theories (massive gravities), including the dRGT theories, in their decoupling limits, as well as in other models such as DGP model or generalized Galileons. This presentation is self-contained and before discussing the Vainshtein mechanism we introduce some useful results and concepts concerning massive gravity, such as the vDVZ discontinuity, the decoupling limits or the Boulware-Deser ghost.Comment: 28 pages, invited review for CQG focus issue; v2: matches the published versio