Renormalization in theories with modified dispersion relations: weak gravitational fields

We consider a free quantum scalar field satisfying modified dispersion relations in curved spacetimes, within the framework of Einstein-Aether theory. Using a power counting analysis, we study the divergences in the adiabatic expansion of <\phi^2> and <T_{\mu\nu}>, working in the weak field approximation. We show that for dispersion relations containing up to $2s$ powers of the spatial momentum, the subtraction necessary to renormalize these two quantities on general backgrounds depends on $s$ in a qualitatively different way: while <\phi^2> becomes convergent for a sufficiently large value of $s$, the number of divergent terms in the adiabatic expansion of <T_{\mu\nu}> increases with $s$. This property was not apparent in previous results for spatially homogeneous backgrounds.