Cosmological solutions with massive gravitons in the bigravity theory

TL;DR

This work studies cosmological solutions in a ghost-free bimetric gravity framework (the HR3 generalization of RGT) with two dynamical metrics $g_{\mu\nu}$ and $f_{\mu\nu}$ coupled through ${\cal L}_{\rm int}$ and matter coupled to $g_{\mu\nu}$. Using a tetrad formulation, the authors derive the field equations and classify solutions into decoupled-metrics and generic coupled cases; decoupled solutions yield FRW evolution for $g_{\mu\nu}$ with an effective late-time term $m^2(C-1)$ and an AdS-like $f_{\mu\nu}$, while generic solutions produce two branches for the total energy density $\rho_\ast(\rho)=m^2T^0{}_0+\rho$, including physical and exotic branches that can drive self-acceleration or even avoid singularities. They show that in the limit $\eta\to0$, only the decoupled-metrics solutions have a nontrivial RGT limit with flat $f_{\mu\nu}$, whereas the generic solutions do not, indicating rich beyond-GR phenomenology in the full bimetric theory. The paper also discusses non-accelerating branches and notes stability concerns due to possible negative-energy contributions from the graviton mass. Overall, the results illuminate diverse cosmological behaviors in massive bimetric gravity and the conditions under which RGT-like limits are recovered.

Abstract

We present solutions describing homogeneous and isotropic cosmologies in the massive gravity theory with two dynamical metrics recently proposed in arXiv:1109.3515 and claimed to be ghost free. These solutions can be spatially open, closed, or flat, and at early times they are sourced by the perfect fluid, while the graviton mass typically manifests itself at late times by giving rise to a cosmological term. In addition, there are also exotic solutions, for which already at early times, when the matter density is high, the contribution of the graviton mass to the energy density is negative and large enough to screen that of the matter contribution. The total energy can then be negative, which may result in removing the initial singularity. For special parameter values there are also solutions for which the two metrics effectively decouple and evolve independently of each other. In the limit where one of the gravitational coupling constant vanishes, such special solutions reduce to those found in arXiv:1107.5504 within the theory where one of the metrics is flat.

Figures (4)

• Figure 1: The total energy density $\rho_\ast=m^2T^0_0+\rho$ versus the matter energy density $\rho$ for the physical and exotic branches for $c_3=0.9$, $c_4=-1$, $\eta=1$ (left) and $c_3=0.9$, $c_4=1$, $\eta=1$ (right).
• Figure 2: The total energy density $\rho_\ast$ versus $\rho$ for the physical and exotic branches for $c_3=c_4=1$, $\eta=-1$ (left) and $c_4=-c_3=1$, $\eta=1$ (right).
• Figure 3: The effective potential ${\rm U}({\bf a})$\ref{['U']} for the physical (left) and exotic (right) branches for solutions shown in Fig.1 and Fig.2.
• Figure 4: The effective potential ${\rm U}({\bf a})$\ref{['U']} for the physical solutions with $c_3=0.9$, $c_4=1$ (left) and exotic solutions with $c_3=0.9$, $c_4=-1$ (right) for several values of $\eta$.