Energy-momentum tensor of particles created in an expanding universe

Abstract

We present a general formulation of the time-dependent initial value problem for a quantum scalar field of arbitrary mass and curvature coupling in a Friedmann-Robertson-Walker (FRW) cosmological model. We introduce an adiabatic number basis which has the virtue that the divergent parts of the quantum expectation value of the energy-momentum tensor $〈{T}_{\mathrm{ab}}〉$ are isolated in the vacuum piece of $〈{T}_{\mathrm{ab}}〉,$ and may be removed using adiabatic subtraction. The resulting renormalized $〈{T}_{\mathrm{ab}}〉$ is conserved, independent of the cutoff, and has a physically transparent, quasiclassical form in terms of the average number of created adiabatic ``particles.'' By analyzing the evolution of the adiabatic particle number in de Sitter spacetime we exhibit the time structure of the particle creation process, which can be understood in terms of the time at which different momentum scales enter the horizon. A numerical scheme to compute $〈{T}_{\mathrm{ab}}〉$ as a function of time with arbitrary adiabatic initial states (not necessarily de Sitter invariant) is described. For minimally coupled, massless fields, at late times the renormalized $〈{T}_{\mathrm{ab}}〉$ goes asymptotically to the de Sitter invariant state previously found by Allen and Folacci, and not to the zero mass limit of the Bunch-Davies vacuum. If the mass m and the curvature coupling $\ensuremath{\xi}$ differ from zero, but satisfy ${m}^{2}+\ensuremath{\xi}R=0,$ the energy density and pressure of the scalar field grow linearly in cosmic time demonstrating that, at least in this case, back reaction effects become significant and cannot be neglected in de Sitter spacetime.