Here are a few hasty random observations on the issues put forward by Grifols, Mohapatra, Riotto. They discuss three points.

1. SN energy loss via γe → S/P + e. Here they do not take into account the thermal masses, but I suspect their effect would only be minor. The phase spaces would get changed a little, but since p ∼ 3T ≫ m(T) always, this would not affect the order of magnitude of the energy loss.

Regarding the decay S/P → γγ, here the situation is different. I would have expected that S/P in fact could not decay inside SN because of the plasmon masses of the final state photons. The thermal mass of S/P itself should be very small. However, it might be in equilibrium with photons (this should be the case for small gravitino masses). Therefore I wonder about their limit m_{S/P} ≤ 10 MeV.

2. Stellar cooling. Here again energy loss argument applies. GMR claim that for small m_{3/2}, photon flux is not turned into S/P because the latter decay back into photons. This happens for m_{3/2} < 3 × 10^{−6}. In the EEN model of light gravitinos this would correspond to S/P with masses close to MeV. Hence they would not equilibrate and their masses would be greater than the plasmon masses. Hence this argument seems sound.

3. Nucleosynthesis. Consider again S/P → γγ. Although S/P might be out of equilibrium at t = 1 s, they are not at rest with respect to thermal bath and can decay. The rate is

Γ(S/P → γγ) = \frac{κ^{2}}{96π} (\frac{Mγ̃}{m_{3/2}} )^{2} m_{S/P} .  (1)

Writing

m_{S/P} = \frac{Λ_{QCD}^{2}}{m_{3/2}} \frac{M_{P}}{Mγ̃}  (2)

one notices that the S/P life time can be quite short: for p = 1.9, corresponding to m_{3/2} = 4 × 10^{−5} eV one obtains τ_{3/2} ∼ 10 s (with m_{S/P} ∼ 0.3 MeV).

Here a possibly interesting point is the following: could one perhaps allow for S/P to be in equilibrium at t = 1 s if they decay shortly after? This way you would overproduce neutrons, but if S/P were to decay so that they would reheat the photon temperature above the deuteron binding temperature, the excess neutrons might have time to decay away. This might be a way round the GMR nucleosynthesis bound m_{3/2} > 1 eV. (Also, if m_{S/P} is close to MeV one should be careful). I admit that it does not sound terribly attractive.

The calculation by Gherghetta on gravitino equilibration seems fine and well done and we probably can believe his limit m_{3/2} > 10^{−6} eV, provided 1 extra neutrino dof is not allowed.