Quantum behavior of FRW radiation-filled universes

We study the quantum vacuum fluctuations around closed Friedmann-Robertson-Walker (FRW) radiation-filled universes with nonvanishing cosmological constant. These vacuum fluctuations are represented by a conformally coupled massive scalar field and are treated in the lowest order of perturbation theory. In the semiclassical approximation, the perturbations are governed by differential equations which, properly linearized, become generalized Lame equations. The wave function thus obtained must satisfy appropriate regularity conditions which ensure its finiteness for every field configuration. We apply these results to asymptotically anti de-Sitter Euclidean wormhole spacetimes and show that there is no catastrophic particle creation in the Euclidean region, which would lead to divergences of the wave function.