One of the motivations of non-local theories of massive gravity is to formulate the theory without any reference metric.34 This is the main idea behind the non-local theory of massive gravity introduced in [328*].35
Starting with the linearized equation about flat space-time of the Fierz–Pauli theory
The theory propagates what looks like a ghost-like instability irrespectively of the exact formulation chosen in (15.2*). However, it was recently argued that the would-be ghost is not a radiative degree of freedom and therefore does not lead to any vacuum decay. It remains an open question of whether the would be ghost can be avoided in the full nonlinear theory.
The cosmology of this model was studied in [395, 228]. The new contribution (15.2*) in the Einstein equation can play the role of dark energy. Taking the second formulation of (15.2*) and setting the graviton mass to , where is the Hubble parameter today, reproduces the observed amount of dark energy. The mass term acts as a dark fluid with effective time-dependent equation of state , where is the scale factor, and is thus phantom-like.
Since this theory is formulated at the level of the equations of motion and not at the level of the action and since it includes non-local operators it ought to be thought as an effective classical theory. These equations of motion should not be used to get some insight on the quantum nature of the theory nor on its quantum stability. New physics would kick in when quantum corrections ought to be taken into account. It remains an open question at the moment of how to embed nonlocal massive gravity into a consistent quantum effective field theory., 15.2*) causality requires to represent the retarded one.
We stress, however, that this theory should be considered as a classical theory uniquely and not be quantized. It is an interesting question of whether or not the ghost reappears when considering quantum fluctuations like the ones that seed any cosmological perturbations. We emphasize for instance that when dealing with any cosmological perturbations, these perturbations have a quantum origin and it is important to rely of a theory that can be quantized to describe them.