In this paper we investigate the case of nonminimal coupling in the (extended) nonlinear massive gravity theories. We first consider massive gravity in the Brans-Dicke background such that the graviton mass is replaced by $A^2(\sigma )m$ where $\sigma$ is the Brans-Dicke field and $A(\sigma )$ is the conformal coupling and show that there is no viable thermal history of the Universe in this case. We then invoke a cubic Galileon term as the nonlinear completion of the $\sigma$ Lagrangian and show that there is a stable de Sitter solution in this case. However, the de Sitter is blocked by the matter phase, which is also a simultaneous attractor of the dynamics. The de Sitter phase can, however, be realized by invoking unnatural fine-tunings. We next investigate the cosmology of quasidilaton nonlinear massive gravity with nonminimal coupling. As a generic feature of the nonminimal coupling, we show that the model exhibits a transient phantom phase that is otherwise impossible. While performing the observational data analysis on the models, we find that a small value of the coupling constant is allowed for quasidilaton nonlinear massive gravity. For both of the cases under consideration, it is observed that we have an effective pressure of matter that comes from the constraint equation. For mass-varying nonlinear massive gravity in the Brans-Dicke background, the effective pressure of matter is nonzero, which affects the evolution of the Hubble parameter, thereby spoiling the consistency of the model with data. As for the quasidilaton nonlinear massive gravity, the effective pressure of matter can be kept around zero by controlling the coupling constant, and the model is shown to be fit well with observations.

• Received 2 October 2013

DOI:https://doi.org/10.1103/PhysRevD.89.083518