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 〈ϕ2〉 and 〈Tμν〉, 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 〈ϕ2〉 becomes convergent for a sufficiently large value of s, the number of divergent terms in the adiabatic expansion of 〈Tμν〉 increases with s. This property was not apparent in previous results for spatially homogeneous backgrounds.