The effective-action formalism used by Hartle, Hu, Fischetti, and Anderson for treating particle production and back reaction of free fields near the Planck time in the early Universe is extended here to interacting fields. We consider a massive (m) self-interacting λ${\phi }^4$ scalar field coupled (ξ) to a Friedmann-Robertson-Walker spacetime and study the effect of interaction on particle production and on the dynamics of the background geometry. A background-field splitting is introduced in the one-loop effective action which is calculated by perturbative expansions up to the second order in λ, $m^2$, and ξ. Ultraviolet divergences from the fluctuation field are removed by introducing counterterms via dimensional regularization. From the regularized effective action equations governing the effective geometry and the background field are derived and their solutions sought. We consider separately the conformal versus the nonconformal (ξ), and the massless versus the massive (m) cases. We also contrast the cases with or without the background field (χ) and those with or without interaction (λ). The dynamics of the scalar field and the scale factor and the probability of pair production in each case are calculated. In the discussion we present a qualitative explanation of these results, and draw the connection with related work. The classical limit of the present problem depicts the evolution of a Higgs field in curved space, its results being useful for the description of inflation and reheating processes in the grand-unified-theory epoch. The present problem is also the semiclassical limit of quantum cosmology. The results obtained here can be useful to tackling problems involving dynamical fields in curved space such as critical dynamics in the early Universe.

• Received 16 October 1987

DOI:https://doi.org/10.1103/PhysRevD.37.2151