$\Delta N_{\text{eff}}$ and entropy production from early-decaying
  gravitinos
  Gravitinos are a fundamental prediction of supergravity, their mass ($m_{G}$)
is informative of the value of the SUSY breaking scale, and, if produced during
reheating, their number density is a function of the reheating temperature
($T_{\text{rh}}$). As a result, constraining their parameter space provides in
turn significant constraints on particles physics and cosmology. We have
previously shown that for gravitinos decaying into photons or charged particles
during the ($\mu$ and $y$) distortion eras, upcoming CMB spectral distortions
bounds are highly effective in constraining the $T_{\text{rh}}-m_{G}$ space.
For heavier gravitinos (with lifetimes shorter than a few $\times10^6$ sec),
distortions are quickly thermalized and energy injections cause a temperature
rise for the CMB bath. If the decay occurs after neutrino decoupling, its
overall effect is a suppression of the effective number of relativistic degrees
of freedom ($N_{\text{eff}}$). In this paper, we utilize the observational
bounds on $N_{\text{eff}}$ to constrain gravitino decays, and hence provide new
constaints on gravitinos and reheating. For gravitino masses less than $\approx
10^5$ GeV, current observations give an upper limit on the reheating scale in
the range of $\approx 5 \times 10^{10}- 5 \times 10^{11}$GeV. For masses
greater than $\approx 4 \times 10^3$ GeV they are more stringent than previous
bounds from BBN constraints, coming from photodissociation of deuterium, by
almost 2 orders of magnitude.
