Lorentzian quantum cosmology with $R^2$ correction

TL;DR

This paper applies Picard-Lefschetz theory to Lorentzian quantum cosmology with $R^2$ corrections, computing transition amplitudes perturbatively and analyzing the effects of higher-derivative modifications in the early universe.

Contribution

It introduces a perturbative approach using Picard-Lefschetz theory to compute quantum cosmological transition amplitudes with $R^2$ modifications.

Findings

Computed transition amplitudes to first order in $R^2$ correction.

Analyzed saddle points and integration contours in Lorentzian quantum cosmology.

Identified regimes where perturbative approximation is valid.

Abstract

Quantum mechanical transition amplitudes directly tells the probability of each transition and which one is more favourable. Path-integrals offers a systematic methodology to compute this quantum mechanical process in a consistent manner. Although it is not complicated in simple quantum mechanical system but defining path-integral legitimately becomes highly nontrivial in the context of quantum-gravity, where apart from usual issues of renormalizability, regularisation, measure, gauge-fixing, boundary conditions, one still has to define the sensible integration contour for convergence. Picard-Lefschetz (PL) theory offers a unique way to find a contour of integration based on the analysis of saddle points and the steepest descent/ascent flow lines in the complex plane. In this paper we make use of PL-theory to investigate Lorentzian quantum cosmology where the gravity gets modified in the ultraviolet with the $R^2$ corrections. We approach the problem perturbatively and compute the transition amplitude in the saddle point approximation to first order in higher-derivative coupling. This perturbative approximation is valid in certain regimes but the approximation cannot be used to address issues of very early Universe or no-boundary proposal.