Higgs effective potential in a perturbed Robertson-Walker background

We calculate the one-loop effective potential of a scalar field in a Robertson-Walker background with scalar metric perturbations. A complete set of orthonormal solutions of the perturbed equations is obtained by using the adiabatic approximation for comoving observers. After analyzing the problem of renormalization in inhomogeneous backgrounds, we get the explicit contribution of metric perturbations to the effective potential. We apply these results to the Standard Model Higgs field and evaluate the effects of metric perturbations on the Higgs mass and on its vacuum expectation value. Space-time variations are found, which are proportional to the gravitational slip parameter, with a typical amplitude of the order of $\Delta\phi/\phi\simeq 10^{-11}$ on cosmological scales. We also discuss possible astrophysical signatures in the Solar System and in the Milky Way that could open new possibilities to explore the symmetry breaking sector of the electroweak interactions.