The spatial Functional Renormalization Group and Hadamard states on cosmological spacetimes

TL;DR

This paper introduces a spatial variant of the Functional Renormalization Group on cosmological spacetimes, emphasizing its state dependence and connection to Hadamard states, with applications to scalar fields in cosmology.

Contribution

It develops a new spatial FRG framework on Lorentzian spacetimes, linking it to Hadamard states and analyzing its behavior in cosmological models.

Findings

The FRG flow must be state dependent and based on Hadamard states.

Matching the flow to one-loop renormalization involves a tower of potentials.

Infrared fixed point equations are derived for cosmological scale factors.

Abstract

A spatial variant of the Functional Renormalization Group (FRG) is introduced on (Lorentzian signature) globally hyperbolic spacetimes. Through its perturbative expansion it is argued that such a FRG must inevitably be state dependent and that it should be based on a Hadamard state. A concrete implementation is presented for scalar quantum fields on flat Friedmann-Lema\^{i}tre spacetimes. The universal ultraviolet behavior of Hadamard states allows the flow to be matched to the one-loop renormalized flow (where strict removal of the ultraviolet cutoff requires a tower of potentials, one for each power of the Ricci scalar). The state-dependent infrared behavior of the flow is investigated for States of Low Energy, which are Hadamard states deemed to be viable vacua for a pre-inflationary period. A simple time-dependent infrared fixed point equation (resembling that in Minkowski space) arises for any scale factor, with analytically computable corrections coding the non-perturbative ramifications of the Hadamard property in the infrared.