Lorentzian quantum cosmology with $R^2$ correction

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.