The cosmological moduli problem and naturalness

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