Particle production near the cosmological singularity

The method of definition of positive-frequency functions with the help of the causal propagator is applied to the calculation of particle production amplitude in the Robertson–Walker space-time with the flat space slice and in the Casner space-time. The propagator is defined as the Green's function with the minimal divergencies on the “Euclidean section” of the space-time. It is shown that the massless conformal invariant particles are produced in the Robertson–Walker space-time with the scale factor $a(t)=t^\lambda$, $\lambda<1$. It is demonstrated that the conformal transformation method gives the same prediction provided that the global properties of the space-time are taken into consideration. Due to catastrophical character of the particle production, Universe with the true singularity and the expansion law $a(t)=t^\lambda$, $\lambda<1$, turns out to be unstable.