Cosmic alignment of the aether

In Einstein-aether theory and Horava gravity, a timelike unit vector is coupled to the spacetime metric. It has previously been shown that in an exponentially expanding homogeneous, isotropic background, small perturbations of the vector relax back to the isotropic frame. Here we investigate large deviations from isotropy, maintaining homogeneity. We find that, for generic values of the coupling constants, the aether and metric relax to the isotropic configuration if the initial aether hyperbolic boost angle and its time derivative in units of the cosmological constant are less than something of order unity. For larger angles or angle derivatives, the behavior is strongly dependent on the values of the coupling constants. Generally, there is runaway behavior, in which the anisotropy increases with time, and/or singularities occur.

Figure 1

Stream plot of the vector field $(\dot{\theta },\ddot{\theta })$ on the $(\theta ,\dot{\theta })$ plane, with $c_{+}=1/10$ and $c_{-}=1/40$, and $p_{\pm }=0$. In this case the determinant of $H_{ij}$ vanishes at ${\theta }_0\simeq 1.16$.

Figure 2

Stream plot with $c_{+}=1/2$ and $c_{-}=1/4$, and $p_{\pm }=0$. In this case ${\theta }_0\simeq 1.3$, but already for $\theta \gtrsim 1.1$ and $\dot{\theta }=0$ there is no solution to the constraint equation for $\dot{\alpha }$.