Stokes phenomenon and gravitational particle production -- How to evaluate it in practice

We revisit gravitational particle production from the Stokes phenomenon viewpoint, which helps us make a systematic way to understand asymptotic behavior of mode functions in time-dependent background. One of our purposes of this work is to make the method more practical for evaluation of non-perturbative particle production rate. In particular, with several examples of time-dependent backgrounds, we introduce some approximation methods that make the analysis more practical. Specifically, we consider particle production in simple expanding backgrounds, preheating after $R^2$ inflation, and a transition model with smoothly changing mass. As we find several technical issues in analyzing the Stokes phenomenon of each example, we discuss how to simplify the problems while showing the accuracy of analytic estimation under the approximations we make.