Abstract
We study a scalar particle of mas ${m}_{1}$ decaying into two particles of mass ${m}_{2}$ during the radiation and matter dominated epochs of a standard cosmological model. An adiabatic approximation is introduced that is valid for degrees of freedom (d.o.f.) with typical wavelengths much smaller than the particle horizon ($\ensuremath{\propto}$ Hubble radius) at a given time. We implement a nonperturbative method that includes the cosmological expansion and obtain a cosmological Fermi's Golden Rule that enables one to compute the decay law of the parent particle with mass ${m}_{1}$, along with the build up of the population of daughter particles with mass ${m}_{2}$. The survival probability of the decaying particle is $P(t)={e}^{\ensuremath{-}{\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}}_{k}(t)t}$ with ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}}_{k}(t)$ being an effective momentum and time dependent decay rate. It features a transition timescale ${t}_{\mathrm{nr}}$ between the relativistic and nonrelativistic regimes and for $k\ensuremath{\ne}0$ is always smaller than the analogous rate in Minkowski spacetime, as a consequence of (local) time dilation and the cosmological redshift. For $t\ensuremath{\ll}{t}_{\mathrm{nr}}$ the decay law is a ``stretched exponential'' $P(t)={e}^{\ensuremath{-}(t/{t}^{*}{)}^{3/2}}$, whereas for the nonrelativistic stage with $t\ensuremath{\gg}{t}_{\mathrm{nr}}$, we find $P(t)={e}^{\ensuremath{-}{\mathrm{\ensuremath{\Gamma}}}_{0}t}(t/{t}_{\mathrm{nr}}{)}^{{\mathrm{\ensuremath{\Gamma}}}_{0}{t}_{\mathrm{nr}}/2}$, with ${\mathrm{\ensuremath{\Gamma}}}_{0}$ the Minkowski space time decay width at rest. The Hubble timescale $\ensuremath{\propto}1/H(t)$ introduces an energy uncertainty $\mathrm{\ensuremath{\Delta}}E\ensuremath{\sim}H(t)$ which relaxes the constraints of kinematic thresholds. This opens new decay channels into heavier particles for $2\ensuremath{\pi}{E}_{k}(t)H(t)\ensuremath{\gg}4{m}_{2}^{2}\ensuremath{-}{m}_{1}^{2}$, with ${E}_{k}(t)$ the (local) comoving energy of the decaying particle. As the expansion proceeds this channel closes and the usual two particle threshold restricts the decay kinematics.
Highlights
While in this study we focus on the radiation dominated (RD) era, our results can be extended to the subsequent matter dominated (MD) and dark energy dominated eras
In the standard cosmological picture and the majority of the most well-studied variants, most of the interesting particle physics processes occur during the RD era and so we focus most of our attention on this epoch; we contemplate the possibility of long-lived dark matter particles that would decay on timescales of the order of 1=H0
Most of the treatments of phenomenological consequences of particle decay in cosmology describe these processes in terms of a decay rate obtained via usual S-matrix theory in Minkowski space time
Summary
Particle decay is an ubiquitous process that has profound implications in cosmology, for baryogenesis [1,2], leptogenesis [3,4], CP violating decays [5], big bang nucleosynthesis (BBN) [6,7,8,9,10,11,12,13,14], and dark matter (DM) where large scale structure and supernova Ia luminosity distances constrain the lifetimes of potential, long-lived candidates [6,15,16,17,18,19]. Studies of quantum field theory in curved space-time revealed a wealth of unexpected novel phenomena, such as particle production from cosmological expansion [20,21,22,23,24,25,26,27,28,29] and the possibility of processes that are forbidden in Minkowski space time as a consequence of energy/momentum conservation. Brief summary of results: We combine a physically motivated adiabatic expansion with a nonperturbative method that is the quantum field theoretical version of the Wigner-Weisskopf theory of atomic line-widths [34] ubiquitous in quantum optics [35] This method is manifestly unitary, and has been implemented in both Minkowski spacetime and inflationary cosmology [36,37], and provides a systematic framework to obtain the decay law of the parent along with the production probability of the daughter particles. In Appendix A we implement the WignerWeisskopf method in Minkowski spacetime to provide a basis of comparison which will enable us to highlight the major differences with the cosmological setting
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