Cosmological perturbation theory in generalized Einstein-Aether models

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 $\mathrm{\Lambda }\mathrm{CDM}$ 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.

Figure 1

Top left panel: Comparison of the evolution of $\mathcal{F}$ due to varying ${\mathcal{F}}_0$. In these models $M=H_0$ and $w_{\mathrm{de}}=-1$ are fixed. Top right panel: Comparison of the evolution of $\mathcal{F}$ due to the variation of $M$, as a multiple of $H_0$. In these models ${\mathcal{F}}_0=1$ and $w_{\mathrm{de}}=-1$ are fixed. Bottom left panel: Comparison of the evolution of $\mathcal{F}$ for varying $w_{\mathrm{de}}$ close to $-1$. In these models ${\mathcal{F}}_0=1$ and $M=H_0$ are fixed. Bottom right panel: Comparison of the evolution of $M^2\mathcal{F}$ for varying $M^2$ and ${\mathcal{F}}_0$, with $M^2{\mathcal{F}}_0/H_0^2=1$ and $w_{\mathrm{de}}=-1$ fixed.

Figure 2

Left panel: The spectrum of $\mathrm{\Phi }/\mathrm{\Psi }$, or $Z/Y$, at $a=1$ as a function of scale for varying ${\mathcal{F}}_0$ and $w_{\mathrm{de}}=-1$. Right panel: The spectrum of $\mathrm{\Phi }/\mathrm{\Psi }$ at $a=1$ as a function of scale for a general Einstein-Aether fluid with $w_{\mathrm{de}}$ varying around $-1$ and ${\mathcal{F}}_0=1$.

Figure 3

The evolution of the effective Newton’s constant, $G_{\mathrm{eff}}/G$, is shown for varying $w_{\mathrm{de}}$ around $-1$.

Figure 4

Top left panel: The evolution of the Newtonian potential, $\mathrm{\Phi }$, in $\mathrm{\Lambda }\mathrm{CDM}$ (black solid line) and for different scales in a designer $\mathcal{F}(\mathcal{K})$ model (dashed and dotted lines) for ${\mathcal{F}}_0=1$ and $w_{\mathrm{de}}=-1$. Note that the potential for the $\mathrm{\Lambda }\mathrm{CDM}$ model is scale independent. For comparison we also show the evolution of $\mathrm{\Phi }$ with the presence of a dark energy fluid with $w_{\mathrm{de}}=-1$ and a constant negative squared sound speed of $c_{\mathrm{s}}^2=-{10}^{-2}$ (red lines), calculated using (144). Top right panel: The evolution of $\mathrm{\Phi }$ in a generalized Einstein-Aether universe with varying ${\mathcal{F}}_0$ for $w_{\mathrm{de}}=-1$ and $K_0=1$ fixed. Bottom panel: The evolution of $\mathrm{\Phi }$ for a general Einstein-Aether fluid with $w_{\mathrm{de}}$ varying around $-1$, with ${\mathcal{F}}_0=1$ and $K_0=1$ fixed.