  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.
