How Effective is $N_{eff}$ at Discovering Dark Radiation in a Cosmology with Heavy Particle Decay?

Any light relic which was in thermal equilibrium with the Standard Model before it freezes out results in a shift in the effective number of neutrino species, $N_{eff}$. This quantity is being measured with increasing precision, and planned experiments would seemingly rule out light particles beyond the Standard Model, even for rather high temperature light particle freeze out. Here we explore how these bounds are loosened if the energy density of the light particles is diluted with respect to that of Standard Model radiation, which can happen if a heavy particle decays into the Standard Model bath after the light particle freezes out. After calculating how heavy state decays alter $N_{eff}$ for light particles beyond the Standard Model, we focus in particular on the case that the heavy decaying particle is a gravitino, and use current bounds on $N_{eff}$ to place constraints on the gravitino mass and the branching ratio into light particles for different values of the reheating temperature of the Universe.