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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 <ϕ^2> and <T_{μν}>, 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 <ϕ^2> becomes convergent for a sufficiently large value of $s$, the number of divergent terms in the adiabatic expansion of <T_{μν}> increases with $s$. This property was not apparent in previous results for spatially homogeneous backgrounds.