Propagators and pair production in a homogeneous isotropic universe

A covariant method of defining positive-frequency functions in an external gravitational field based on the postulate that the causal propagator propagates particles into the future and antiparticles into the past is applied to the homogeneous isotropic model of the Universe with scale factor $a(t)=t$. Postive- and negative-frequency wave functions are found that define the vacuums of scalar particles in the future asymptotic region and near the singularity. The probability of pair production in such a model is calculated. The cases when the spatial section of space-time is a fiat space or a three-dimensional sphere are considered.