  In a non-standard cosmological scenario, heavy, long-lived particles, which
we call moduli, dominate the energy density prior to Big Bang Nucleosynthesis.
Weakly Interacting Massive Particles (WIMPs) may be produced non-thermally from
moduli decays. The final relic abundance then depends on additional parameters
such as the branching ratio of moduli to WIMPs and the modulus mass. This is of
interest for WIMP candidates, such as a bino-like neutralino, where thermal
production in standard cosmology leads to an overdensity. Previous works have
shown that the correct dark matter (DM) relic density can then still be
obtained if the moduli, with mass less than $10^{7}$ GeV, decay to WIMPs with a
branching ratio of less than $10^{-4}$. This upper bound could easily be
violated once higher order corrections, involving final states with more than
two particles, are included. We compute the branching ratios of three- and
four-body decays of a modulus into final states involving two DM particles for
general couplings. We then apply these expressions to sparticle production
within the Minimal Supersymmetric Standard Model (MSSM) with neutralino DM. We
find that this upper bound on the branching ratio can be satisfied in
simplified models through an appropriate choice of as yet undetermined
couplings. However, in the MSSM, it requires sparticle masses to be very close
to half the modulus mass, in contrast to the idea of weak-scale supersymmetry.
