Inflationary potentials from the exact renormalisation group

Abstract

We show that an inflationary slow-roll potential can be derived as an IR limit of the non-perturbative exact renormalisation group equation for a scalar field within the mean-field approximation. The result follows without having to specify a Lagrangian in the UV, which we take to be somewhere below the Planck scale to avoid discussing quantum gravity effects. We assume that the theory contains a scalar mode with suppressed coupling to other fields, and that higher derivative couplings are suppressed. In this framework the exact RG equation becomes a one-dimensional Schrödinger equation, which we solve. The effective IR potential is then dominated by the eigen-states of the RG Hamiltonian with the highest eigenvalues. We find that these potentials can generically give rise to slow-roll inflation, which is fully consistent with recent observations. As an example of how the proposed renormalisation group procedure works, we perform an explicit calculation in the ϕ4 theory in an appendix.