Consistent Extension of Quasidilaton Massive Gravity

This paper is devoted to the Hamiltonian analysis of extension of the quasidilaton massive gravity as was proposed recently in [arXiv:1306.5502]. We show that, for given formulation of the theory, the additional primary constraint that is responsible for the elimination of the Boulware-Deser ghost is missing. We compare this situation with the quasidilaton massive gravity. Finally, we propose the ghost-free extension of quasidilaton massive gravity. 1. Introduction and Summary Recently, new version of the full nonlinear massive gravity was found by de Rham, Gabadadze and Toley (dRGT) [1, 2] that provides the positive answer to the question of whether graviton can have a nonzero mass. In fact, among many remarkable properties, there is the crucial one which is the absence of the Boulware-Deser ghost [3, 4]. The consistent massive gravity could also provide a possible explanation of the observed acceleration of the cosmic expansion which is one of the greatest mysteries in modern cosmology. It is tempting to speculate that the finite graviton mass could be a source of the accelerated expansion of the universe. For that reason, it is of great interest to formulate theoretically consistent cosmological scenario in massive gravity that is also in agreement with the observations. Unfortunately, it was recently shown that all homogeneous and isotropic cosmological solutions in dRGT theory are unstable [5]; see also [6–8]. In order to resolve this problem, we have two possible options: either to break homogeneity [9] or isotropy [10, 11] or to extend the theory as in [12, 13]. Recently in [14], de Felice and Mukohyama proposed new extension of the quasidilaton massive gravity that could provide stable and self-accelerating homogeneous and isotropic cosmological solution. They further argued that given extension belongs to the class of models studied in [15] that are free from the Boulware-Deser ghosts. However, the explicit Hamiltonian analysis of given theory was not performed in [14]. The goal of this paper is to reconsider the problem of the Boulware-Deser ghost in the model [14]. We present an evidence that, for the action that was introduced in [14], the Boulware-Deser ghost cannot be eliminated. More precisely, performing the Hamiltonian analysis of this model with the time dependent quasidilaton, we find that the primary constraint which is responsible for the elimination of the ghost in Stückelberg formulation of nonlinear massive gravity [16–19] is missing (for related work, see [20]). This result implies that generally Boulware-Deser ghost is present.