On energy and particle production in cosmology: the particular case of the gravitino

It is well-known that the number of particles produced in cosmology, commonly defined in the literature from the Fock space of the instantaneous hamiltonian of the canonically normalized fields, is ambiguous. On the other hand, the energy computed from the energy-momentum tensor should be physical. We compare the corresponding Fock spaces and relate them through a Bogolyubov transformation. We find that for particles of spin 0, 1 and 3/2 the two Fock spaces are different, whereas they are the same for spin 1/2 fermions. For spin 0 and 1, for particles of high momenta the two Fock spaces align, as intuitively expected. For the spin 3/2, one finds two puzzles. The first one is that the two corresponding Fock spaces do not match even in the limit of high momenta. The second is that whereas we provide evidence for the equivalence theorem between longitudinal gravitinos and the goldstino in terms of an exact matching between the lagrangians and the instantaneous hamiltonians for the canonically normalized fields, the energy operator computed from the Rarita-Schwinger action does not seem to be captured in a simple way by the goldstino action. Our results suggest a re-analysis of non-thermal gravitino production in cosmology.