Effects of quantum fields on singularities and particle horizons in the early universe. II

Abstract

The back-reaction problem for conformally invariant free quantum fields in homogeneous and isotropic spacetimes containing classical radiation is solved for spacetimes with nonzero spatial curvatures and/or nonzero cosmological constants. The solutions depend upon two regularization parameters which we call $\ensuremath{\alpha}$ and $\ensuremath{\beta}$. Only solutions which at late times approach the appropriate solution to the field equations of general relativity are considered. The results are much the same as those found previously for spatially flat spacetimes with zero cosmological constants. Thus, if $\ensuremath{\beta}>3\ensuremath{\alpha}>0$, there is always one solution which undergoes a "time-symmetric bounce" and which contains no singularities, if $\ensuremath{\alpha}$,$\ensuremath{\beta}>0$ there is a family of solutions with particle horizons and no singularities, and if $\ensuremath{\alpha}>0$ there is always at least one solution with an initial singularity but no particle horizons. The differences caused by the spatial curvature and cosmological constant include the initial behavior of the time-symmetric bounce solution and, if the spatial curvature is nonzero, the initial behavior of many solutions for the cases $\ensuremath{\beta}=3\ensuremath{\alpha}>0$ and $\ensuremath{\beta}\ensuremath{\approx}3\ensuremath{\alpha}<0$.