Massive gravity from bimetric gravity

TL;DR

This paper clarifies how ghost-free massive gravity emerges as a limit of ghost-free bimetric gravity and shows that the limit is subtle: the background metric must be externally specified by massive gravity, and the two dynamical metrics in the parent theory introduce extra constraints that prevent full continuity in parameter space. In the non-dynamical-background limit, the graviton-mass effect reduces to a cosmological-constant–like term, with no requirement that the background be flat, and ghost-free cosmologies with spherical symmetry fall into two main families of continuous solutions, plus two special cases, all yielding effective cosmological-constant–driven expansion under appropriate parameter relations. The two general solutions correspond to (i) proportional foreground/background metrics and (ii) more intricate relations between the metrics, both producing $T^{(\text{eff})\mu}{}_{ u}$ that acts as a cosmological constant with sign determined by theory parameters. A separate limit with background matter retained indicates the background sector remains indirectly observable in the foreground, but cannot be probed directly; overall, the results delineate when MG predictions align with BG in the limit and when they do not, informing how background and foreground geometries interrelate in modified gravity cosmologies.

Abstract

We discuss the subtle relationship between massive gravity and bimetric gravity, focusing particularly on the manner in which massive gravity may be viewed as a suitable limit of bimetric gravity. The limiting procedure is more delicate than currently appreciated. Specifically, this limiting procedure should not unnecessarily constrain the background metric, which must be externally specified by the theory of massive gravity itself. The fact that in bimetric theories one always has two sets of metric equations of motion continues to have an effect even in the massive gravity limit, leading to additional constraints besides the one set of equations of motion naively expected. Thus, since solutions of bimetric gravity in the limit of vanishing kinetic term are also solutions of massive gravity, but the contrary statement is not necessarily true, there is not complete continuity in the parameter space of the theory. In particular, we study the massive cosmological solutions which are continuous in the parameter space, showing that many interesting cosmologies belong to this class.