Quantum corrections to slow-roll inflation: scalar and tensor modes

Inflation is often described through the dynamics of a scalar field, slow-rolling in a suitable potential. Ultimately, this inflaton must be identified with the expectation value of a quantum field, evolving in a quantum effective potential. The shape of this potential is determined by the underlying tree-level potential, dressed by quantum corrections from the scalar field itself and the metric perturbations. Following [1], we compute the effective scalar field equations and the corrected Friedmann equations to quadratic order in both scalar field, scalar metric and tensor perturbations. We identify the quantum corrections from different sources at leading order in slow-roll, and estimate their magnitude in benchmark models of inflation. We comment on the implications of non-minimal coupling to gravity in this context.

Bach and Einstein’s equations in presence of a field

The aim of this paper is to introduce and justify a possible generalization of the classic Bach field equations on a four-dimensional smooth manifold [Formula: see text] in the presence of field [Formula: see text], given by a smooth map with source [Formula: see text] and target another Riemannian manifold. Those equations are characterized by the vanishing of a two times covariant, symmetric, traceless and conformally invariant tensor field, called [Formula: see text]-Bach tensor, that in absence of the field [Formula: see text] reduces to the classic Bach tensor, and by the vanishing another tensor related to the bi-energy of [Formula: see text]. Since solutions of the Einstein-massless scalar field equations, or more generally, of the Einstein field equations with source the wave map [Formula: see text] solves those generalized Bach’s equations, we include the latter in our analysis providing a systematic study for them, relying on the recent concept of [Formula: see text]-curvatures. We take the opportunity to discuss the related topic of warped product solutions.