  If local supersymmetry is the correct extension of the standard model of
particle physics, then following Inflation the early universe would have been
populated by gravitinos produced from scatterings in the hot plasma during
reheating. Their abundance is directly related to the magnitude of the
reheating temperature. The gravitino lifetime is fixed as a function of its
mass, and for gravitinos with lifetimes longer than the age of the universe at
redshift $z\simeq 2\times 10^{6}$ (or roughly $6\times 10^6{\rm s}$), decay
products can produce spectral distortion of the cosmic microwave background.
Currently available COBE/FIRAS limits on spectral distortion can, in certain
cases, already be competitive with respect to cosmological constraints from
primordial nucleosynthesis for some gravitino decay scenarios. We show how the
sensitivity limits on $\mu$ and $y$ distortions that can be reached with
current technology would improve constraints and possibly rule out a
significant portion of the parameter space for gravitino masses and Inflation
reheating temperatures.
