The cosmological moduli problem and naturalness

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.