On (dis)agreement between different methods of calculation of the imaginary part of the effective action in expanding space-times

TL;DR

The paper analyzes two nonperturbative routes to the imaginary part of the effective action in expanding spacetimes: the in-out amplitude from Bogolyubov coefficients and the functional-integral (Feynman propagator) approach. It shows that for backgrounds that expand for a finite time these methods agree, but for eternal expansion they diverge, with the mismatch traced to vacuum-functionals in the out-state and to how the functional integral is defined. By examining the expanding Poincaré patch and global de Sitter space, it demonstrates concrete differences between $W_B$ and $W_P$, and argues that correct treatment requires accounting for vacuum contributions or reframing the calculation in a Keldysh-Schwinger framework to compute observables like the energy-momentum tensor. The work highlights subtle but crucial distinctions in defining particle production and effective actions in curved spacetime, guiding practitioners to careful, state-dependent, and observer-friendly formulations.

Abstract

We consider two approaches to calculate imaginary parts of effective actions in expanding space-times. While the first approach uses Bogolyubov coefficients, the second one uses the functional integral or the Feynman propagator. In eternally expanding space-times these two approaches give different answers for the imaginary parts. The origin of the difference can be traced to the presence if the wave-functionals for the initial and final states in the functional integral. We show this explicitly on the example of the expanding Poincare patch of the de Sitter space-time.