Using Cosmological Perturbation Theory to Distinguish between GR and Unimodular Gravity

Unimodular gravity is one of the oldest geometric gravity theories and alternatives to general relativity. Essentially, it is based on the Einstein–Hilbert Lagrangian with an additional constraint on the determinant of the metric. It can be explicitly shown that unimodular gravity can be recast as general relativity in the presence of a cosmological constant. This fact has led to many discussions on the equivalence of both theories at the classical and quantum levels. Here, we present an analysis focused on the classical scalar perturbations around a cosmological background. We focus on the unusual situation in which the typical conservation laws are not adopted. The discussion is extended to the case where a non-minimal coupled scalar field is introduced. We also present a gauge-invariant analysis showing that perturbations in unimodular gravity display instabilities. Our results reinforce that the equivalence is not verified completely at a cosmological perturbative level.