  The cosmological moduli problem (CMP) comes in three parts: 1. potential
violation of BBN constraints from late decaying moduli fields, 2. the
moduli-induced gravitino problem wherein gravitinos are overproduced and 3. the
moduli-induced lightest SUSY particle (LSP) overproduction problem. We examine
the cosmological moduli problem and its connection to electroweak naturalness.
We calculate the various two-body decay widths of a light modulus field into
MSSM particles and gravitinos within general supersymmetric models. We include
both phase space and mixing effects. We examine cases without and with helicity
suppression of modulus decays to gravitinos (cases 1 & 2) and/or gauginos
(cases A & B). For case B1, we evaluate regions of gravitino mass m_{3/2} vs.
modulus mass m_\phi parameter space constrained by BBN, by overproduction of
gravitinos and by overproduction of neutralino dark matter, along with
connections to naturalness. For this case, essentially all of parameter space
is excluded unless m_\phi >~ 2.5\times 10^3 TeV with m_\phi<2m_{3/2}. For a
potentially most propitious case B2 with \phi decay to Higgs and matter turned
off, then modulus branching fractions to SUSY and to gravitinos become highly
suppressed at large m_\phi. But since the modulus number density increases
faster than the branching fractions decrease, there is still gross
overproduction of neutralino dark matter. We also show that in this scenario
the thermally produced gravitino problem is fixed by huge entropy dilution, but
non-thermal gravitino production from moduli decay remains a huge problem
unless it is kinematically suppressed with m_\phi < 2m_{3/2}. In a pedagogical
appendix, we present detailed calculations of modulus field two-body decay
widths.
