Nonlocal theory of massive gravity

Abstract

We construct a fully covariant theory of massive gravity which does not require the introduction of an external reference metric, and overcomes the usual problems of massive gravity theories (fatal ghosts instabilities, acausality and/or vDVZ discontinuity). The equations of motion of the theory are non-local, but respect causality. The starting point is the quadratic action proposed in the context of the degravitation idea. We show that it is possible to extended it to a fully non-linear covariant theory. This theory describes the five degrees of freedom of a massive graviton plus a scalar ghost. However, contrary to generic non-linear extensions of Fierz-Pauli massive gravity, the ghost has the same mass m as the massive graviton, independently of the background, and smoothly goes into a non-radiative degree of freedom for m → 0. As a consequence, for m ∼ H 0 the vacuum instability induced by the ghost is irrelevant even over cosmological time-scales. We finally show that an extension of the model degravitates a vacuum energy density of order M 4 Pl down to a value of order M 2 Pl m 2 , which for m = O(H 0 ) is of order of the observed value of the vacuum energy density.

Perturbations for massive gravity theories

“…Note that we cannot fix a particular gauge since the mass term in the action is not gauge invariant a priori (see, however, the discussion about the "hidden symmetry" for perturbations on Minkowski or de Sitter spacetime in Ref. [32]). Gauge invariance can be restored by means of the Stückelberg trick [37,38], but we are not doing this here.…”

“…In [32], it has been shown that on a fixed background the mass term (32) for α = −β indicates the presence of a ghost with mass…”

“…There is also the possibility to avoid the reference metric but at the cost of nonlocal terms like for example m 2 −1 G µν in the equations of motion [13]. Such theories are usually not ghost free, but recently a solution where massive gravity can mimic dark energy for such a theory has been found [14][15][16].…”

Stability analysis and future singularity of them2R□-2Rmodel of non-local gravity

“…Most probably, one could always find a set of initial conditions such that (1 −m 2 −2 R) < 0 in some space-time region, so we will rather discuss the realistic cases. The fact that the mass term is chosen of the order of the Hubble scale today, along with the fact that the theory exhibits no vDVZ discontinuity [1,18], implies thatm 2 −2 R 1 at solar system length and time scales 14 . The potential danger thus lies in cosmology.…”

“…(1.2) This model is currently receiving particular attention [2][3][4][5] because its phenomenology seems to privilege it among other non-local models that have been confronted with observations [6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Here the non-local term is controlled by a mass parameter m, in contrast with the non-local models of Deser and Woodard [6] and of Barvinsky [20], where no new scale is introduced in the theory.…”