Emergent spontaneous symmetry breaking and emergent symmetry restoration in rippling gravitational background

We study effects of a rippling gravitational background on a scalar field with a double well potential, focusing on the analogy with the well known dynamics of the Kapitza’s pendulum. The ripples are rendered as infinitesimal but rapidly oscillating perturbations of the scale factor. We find that the resulting dynamics crucially depends on a value of the parameter $\xi$ in the $\xi R{\varphi }^2$ vertex. For the time-dependent perturbations of a proper form the resulting effective action is generally covariant, and at a high enough frequency at $\xi <0$ and at $\xi >1\mathrm{/}6$ the effective potential has a single minimum at zero, thereby restoring spontaneously broken symmetry of the ground state. On the other side, at $0<\xi <1\mathrm{/}6$ spontaneous symmetry breaking emerges even when it is absent in the unperturbed case.