High frequency limit for gravitational perturbations of cosmological models in modified gravity theories

In general relativity, it has been shown that the effective gravitational stress-energy tensor for short-wavelength metric perturbations acts just like that for a radiation fluid, and thus, in particular, cannot provide any effects that mimic dark energy. However, it is far from obvious if this property of the effective gravitational stress-energy tensor is a specific nature held only in Einstein gravity, or holds also in other theories of gravity. In particular, when considering modified gravity theories that involve higher-order derivative terms, one may expect to have some non-negligible effects arising from higher-order derivatives of short-wavelength perturbations. In this paper, we argue that this is in general not the case in the cosmological context. We address this problem in a simple class of f (R) gravity theories on the assumptions that (i) the background, or coarse-grained metric averaged over several wavelengths, has Friedmann-Lemaitre-Robertson-Walker (FLRW) symmetry, and that (ii) when our f (R) theory reduces to Einstein gravity, the field equations of Einstein gravity should be reproduced. We show by explicit computation that the effective gravitational stress-energy tensor for a cosmological model in our f (R) theories, as well as that obtained in the corresponding scalar-tensor theory, takes a similar form to that in general relativity and is in fact traceless, hence acting again like a radiation fluid. If assumption (ii) above is dropped, then an undetermined integration constant appears and the resultant effective stress-energy tensor acquires a term that is in proportion to the background metric, hence being, in principle, able to describe a cosmological constant. Whether this effective cosmological constant term is positive and whether it has the right magnitude as dark energy depends upon the value of the integration constant.