A massless scalar field in Robertson-Walker spacetimes: Adiabatic regularization and Green's function*

We study adiabatic regularization of a coupling massless scalar field in general spatially flat Robertson-Walker (RW) spacetimes. For the conformal coupling, the 2nd-order regularized power spectrum and 4th-order regularized stress tensor are zero, and no trace anomaly exists in general RW spacetimes. This is a new result that exceeds those found in de Sitter space. For the minimal coupling, the regularized spectra are also zero in the radiation-dominant and matter-dominant stages, as well as in de Sitter space. The vanishing of these adiabatically regularized spectra is further confirmed by direct regularization of the Green's function. For a general coupling and general RW spacetimes, the regularized spectra can be negative under the conventional prescription. At a higher order of regularization, the spectra will generally become positive, but will also acquire IR divergence, which is inevitable for a massless field. To avoid the IR divergence, the inside-horizon regularization is applied. Through these procedures, nonnegative UV- and IR-convergent power spectrum and spectral energy density will eventually be achieved.