Quantum effects in the early universe. V. Finite particle production without trace anomalies

The production of scalar particles in a homogeneous isotropic spatially flat universe containing classical baryons and radiation is studied for regulated massless free-field theories without trace anomalies. The pair-production probability is evaluated to lowest nonvanishing order in the parameter $\xi =\frac{l{\rho }_b}{{{\rho }_r}^{\frac{3}{4}}}$, where $l$ is the Planck length, ${\rho }_b$ is the baryon energy density, and ${\rho }_r$ is the radiation energy density, all in units where $\hslash =c=1\mathrm{}$. For our universe $\xi \sim {10}^{-27\mathrm{}}$. The back reaction regulates the production of scalar particles near the singularity so that the probability to produce a pair in the volume occupied by one baryon over the history of the universe is finite, of order of magnitude $\xi m_{b}l$, where $m_b$ is the mass of a typical baryon, and therefore very small.