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

Lorentzian quantum cosmology with $R^2$ correction