Massive gravity, the elasticity of space-time and perturbations in the dark sector

TL;DR

This paper explores modified gravity models where space-time's elastic properties are characterized by a tensor similar to elasticity, linking massive gravity to space-time rigidity and analyzing implications for dark energy perturbations.

Contribution

It introduces a tensor-based framework for massive gravity models, connecting them to elasticity theory and analyzing symmetry constraints and degrees of freedom in cosmological contexts.

Findings

Elastic tensor analogy for massive gravity models.

Reparameterization invariance restricts models to a cosmological constant.

Time translation invariance yields elastic dark energy with specific degrees of freedom.

Abstract

We consider a class of modified gravity models where the terms added to the standard Einstein-Hilbert Lagrangian are just a function of the metric only. For linearized perturbations around an isotropic space-time, this class of models is entirely specified by a rank 4 tensor that encodes possibly time-dependent masses for the gravitons. This tensor has the same symmetries as an elasticity tensor, suggesting an interpretation of massive gravity as an effective rigidity of space-time. If we choose a form for this tensor which is compatible with the symmetries of FRW and enforce full reparameterization invariance, then the only theory possible is a cosmological constant. However, in the case where the theory is only time translation invariant, the ghost-free massive gravity theory is equivalent to the elastic dark energy scenario with the extra Lorentz violating vector giving rise to 2 transverse and 1 longitudinal degrees of freedom, whereas when one demands spatial translation invariance one is left with scalar field theory with a non-standard kinetic term.