New counterterms induced by trans-Planckian physics in semiclassical gravity

We consider free and self-interacting quantum scalar fields satisfying modified dispersion relations in the framework of Einstein-Aether theory. Using adiabatic regularization, we study the renormalization of the equation for the mean value of the field in the self-interacting case, and the renormalization of the semiclassical Einstein-Aether equations for free fields. In both cases we consider Bianchi type I background spacetimes. Contrary to what happens for free fields in flat Robertson-Walker spacetimes, the self-interaction and/or the anisotropy produce nonpurely geometric terms in the adiabatic expansion, i.e. terms that involve both the metric $g_{\mu \nu }$ and the aether field $u_{\mu }$. We argue that, in a general spacetime, the renormalization of the theory would involve new counterterms constructed with $g_{\mu \nu }$ and $u_{\mu }$, generating a fine-tuning problem for the Einstein-Aether theory.