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We calculate the averaged number \nu of the lightest supersymmetric particles(LSPs) in a shower from the decay of superheavy particle X by generalized DGLAPequations. If the primary decayed particles have color charges and thevirtuality is around 10^13-10^14 GeV, the averaged number of the LSPs canbecome O(100). As the result, the upper limit of the mass of the superheavyparticle, whose decay can produce the observed abundance of the dark matter,can increase from 10^12 GeV to 10^14 GeV. Since the typical scale of theinflaton mass of the chaotic inflation is around 10^13 GeV, the decay of theinflaton can produce the observed dark matter abundance if the reheatingtemperature is of order 1 GeV. Even for the standard model particles withvirtuality Q\sim 10-100 TeV, the averaged number of the LSPs becomes O(0.1) forgluon, and O(0.01) for Higgs, which strongly constrains the scenario ofnon-thermal LSP production from the decay of moduli with 10-100 TeV mass.

Averaged Number of the Lightest Supersymmetric Particles in Decay of Superheavy Particle with Long Lifetime