Particle decay in post inflationary cosmology

We study scalar particle decay during the radiation and matter dominated epochs of a standard cosmological model. An adiabatic approximation is introduced that is valid for degrees of freedom with typical wavelengths much smaller than the particle horizon ($\propto$~Hubble radius) at a given time. We implement a non-perturbative method that includes the cosmological expansion and obtain a cosmological Fermi's Golden Rule that enables one to compute the decay law of a parent particle of mass $m_1$, along with the build up of the population of daughter particles of mass $m_2$. The survival probability of the decaying particle is $P(t)=e^{-\widetilde{\Gamma}_k(t)\,t}$ with $\widetilde{\Gamma}_k(t)$ being an \emph{effective momentum and time dependent decay rate}. It features a transition time scale $t_{nr}$ between the relativistic and non-relativistic regimes and for $k \neq 0$ is always smaller than the analogous rate in Minkowski spacetime, as a consequence of (local) time dilation and the cosmological redshift. For $t \ll t_{nr}$ the decay law is a "stretched exponential" $P(t) = e^{-(t/t^*)^{3/2}}$, whereas for the non-relativistic stage with $t \gg t_{nr}$, we find $P(t) = e^{-\Gamma_0 t}\,(t/t_{nr})^{\Gamma_0\,t_{nr}/2}$. The Hubble time scale $\propto 1/H(t)$ introduces an energy uncertainty $\Delta E \sim H(t)$ which relaxes the constraints of kinematic thresholds. This opens new decay channels into heavier particles for $2\pi E_k(t) H(t) \gg 4m^2_2-m^2_1$, with $E_k(t)$ the (local) comoving energy of the decaying particle. As the expansion proceeds this channel closes and the usual two particle thresholds restrict the decay kinematics.