  We study adiabatic regularization of a coupling massless scalar field in
general spatially flat Robertson-Walker (RW) spacetimes. For the
conformally-coupling, the 0th-order regularized power spectrum and 0th-order
regularized stress tensor are zero, and no trace anomaly exists in general RW
spacetimes. This is a new result which extents those found in de Sitter space.
For the minimally-coupling, the regularized spectra are also zero in the
radiation-dominant stage, the matter-dominant stage, and de Sitter space as
well. The vanishing of these adiabatically regularized spectra are also
confirmed by direct regularization of the Green's functions. For a general
coupling and general RW spacetimes, the regularized spectra can be negative
under the conventional prescription. By going to 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. By these procedures,
one will eventually achieve nonnegative, UV- and IR-convergent power spectrum
and spectral energy density.
