Adiabatic regularization of functional determinants in cosmology and radiative corrections during inflation

Abstract

We express the in-in functional determinant giving the one-loop effective potential for a scalar field propagating in a cosmological spacetime in terms of the mode functions specifying the vacuum of the theory and then apply adiabatic regularization to make this bare potential finite. In this setup, the adiabatic regularization offers a particular renormalization prescription that isolates the effects of the cosmic expansion. We apply our findings to determine the radiative corrections to the classical inflaton potentials in scalar field inflationary models and also we derive an effective potential for the superhorizon curvature perturbation ζ encoding its scatterings with the subhorizon modes. Although the resulting modifications to the cosmological observables like nongaussianity turn out to be small, they distinctively appear after horizon crossing. I. INTRODUCTIONFunctional determinants arise in various instances in quantum field theory like in the calculations of the effective actions, gauge fixing Faddeev-Popov terms, semiclassical tunneling amplitudes and Jacobian factors (for a review see e.g. [1]). In general, they appear as results of Gaussian path integrals and there are different methods in evaluating them such as the heat-kernel expansion, zeta function regularization and Gel'fand-Yaglom theorem.The functional determinants also appear in quantum cosmology since Gaussian in-in/Schwinger-Keldysh path integrals are often encountered. For example, in certain models of preheating one assumes the existence of a reheating scalar that interacts with the inflaton, yet the action may still be quadratic in the reheating scalar yielding a Gaussian path integral in quantum theory (see [2,3]). Similarly, in single scalar field models one may consider integrating out the quadratic inflaton fluctuations about the inflating background to determine the quantum backreaction effects (see e.g. [4][5][6][7][8][9]). Obviously, the functional determinants in a cosmological setting are time dependent and they can naturally be interpreted as effective actions.

Renormalization of the inflationary perturbations revisited

“…The approach that implies a significant modification of P φ from renormalization was laid out already in [28][29][30] and is generally referred to as adiabatic regularization or adiabatic subtraction. This technique has since been established as one of the most popular approaches for renormalization of quantum fields in curved spaces, for examples see [31][32][33][34][35][36][37]. It relies on an adiabatic expansion of modes or more concretely a series in increasing number of derivatives which provides a counter term with the identical divergences to the ones generated by the full quantum correlations, which upon subtraction leads to quantities with the divergences removed.…”