Cosmological perturbation theory in generalized Einstein-Aether models

Abstract

We investigate the evolution of cosmological perturbations in models of dark energy described by a timelike unit normalized vector field specified by a general function $\mathcal{F}(\mathcal{K})$, so-called generalized Einstein-Aether models. First we study the background dynamics of such models via a designer approach in an attempt to model this theory as dark energy. We find that only one specific form of this designer approach matches $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ at background order, and we also obtain a differential equation which $\mathcal{F}(\mathcal{K})$ must satisfy for general $w\mathrm{CDM}$ cosmologies, where CDM refers to cold dark matter. We also present the equations of state for perturbations in generalized Einstein-Aether models, which completely parametrize these models at the level of linear perturbations. A generic feature of modified gravity models is that they introduce new degrees of freedom. By fully eliminating these we are able to express the gauge invariant entropy perturbation and the scalar, vector, and tensor anisotropic stresses in terms of the perturbed fluid variables and metric perturbations only. These can then be used to study the evolution of perturbations in the scalar, vector, and tensor sectors, and we use these to evolve the Newtonian gravitational potentials.