Stress-Energy Tensor of Adiabatic Vacuum in Friedmann-Robertson-Walker Spacetimes

TL;DR

This paper calculates the leading finite contribution to the stress-energy tensor of a quantum scalar field in Friedmann-Robertson-Walker spacetimes using adiabatic regularization, with implications for cosmological models.

Contribution

It explicitly determines the sixth-order adiabatic terms for the stress-energy tensor in arbitrary FRW universes, extending previous lower-order analyses.

Findings

Leading order finite stress-energy tensor terms are sixth-order in adiabatic expansion.

Massive modes contribute H^6/m^2, massless modes contribute H^6a^2/k*^2.

The equation of state parameter w depends on the power-law index n of the scale factor.

Abstract

We compute the leading order contribution to the stress-energy tensor corresponding to the modes of a quantum scalar field propagating in a Friedmann-Robertson-Walker universe with arbitrary coupling to the scalar curvature, whose exact mode functions can be expanded as an infinite adiabatic series. While for a massive field this is a good approximation for all modes when the mass of the field m is larger than the Hubble parameter H, for a massless field only the subhorizon modes with comoving wave-numbers larger than some fixed k* obeying k*/a>H can be analyzed in this way. As infinities coming from adiabatic zero, second and fourth order expressions are removed by adiabatic regularization, the leading order finite contribution to the stress-energy tensor is given by the adiabatic order six terms, which we determine explicitly. For massive and massless modes these have the magnitudes H^6/m^2 and H^6a^2/k*^2, respectively, and higher order corrections are suppressed by additional powers of (H/m)^2 and (Ha/k*)^2. When the scale factor in the conformal time \eta is a simple power a(\eta)=(1/\eta)^n, the stress-energy tensor obeys P=w\rho with w=(n-2)/n for massive and w=(n-6)/(3n) for massless modes. In that case, the adiabaticity is eventually lost when 0<n<1 for massive and when 0<n<3/2 for massless fields since in time H/m and Ha/k* become order one. We discuss the implications of these results for de Sitter and other cosmologically relevant spaces.