Computing First-Passage Times with the Functional Renormalisation Group

Abstract

We use Functional Renormalisation Group (FRG) techniques to analyse the behaviour of a spectator field, $\sigma$, during inflation that obeys an overdamped Langevin equation. We briefly review how a derivative expansion of the FRG can be used to obtain Effective Equations of Motion (EEOM) for the one- and two-point function and derive the EEOM for the three-point function. We show how to compute quantities like the amplitude of the power spectrum and the spectral tilt from the FRG. We do this explicitly for a potential with multiple barriers and show that in general many different potentials will give identical predictions for the spectral tilt suggesting that observations are agnostic to localised features in the potential. Finally we use the EEOM to compute first-passage time (FPT) quantities for the spectator field. The EEOM for the one- and two-point function are enough to accurately predict the average time taken $\left\langle \mathcal{N}\right\rangle$ to travel between two field values with a barrier in between and the variation in that time $\delta \mathcal{N}^2$. It can also accurately resolve the full PDF for time taken $\rho (\mathcal{N})$, predicting the correct exponential tail. This suggests that an extension of this analysis to the inflaton can correctly capture the exponential tail that is expected in models producing Primordial Black Holes.