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 $zsimeq 2times 10^{6}$ (or roughly $6times 10^6{rm s}$), 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.
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