Nowadays, the cosmological moduli problem (CMP) comes in three parts: 1. potential violation of Big-Bang nucleosynthesis (BBN) constraints from late decaying moduli fields, 2. the moduli-induced gravitino problem wherein gravitinos are overproduced and their decays violate BBN or dark matter overproduction bounds and 3. the moduli-induced lightest SUSY particle (LSP) overproduction problem. Also, the CMP may be regarded as either a problem or else a solution to scenarios with dark matter over- or under-production. 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 m3/2 vs. modulus mass mϕ 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 mϕ ≳ 2.5 × 103 TeV with mϕ< 2m3/2. For a potentially most propitious case B2 with ϕ decay to Higgs and matter turned off, then modulus branching fractions to SUSY and to gravitinos become highly suppressed at large mϕ. 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 mϕ< 2m3/2. In a pedagogical appendix, we present detailed calculations of modulus field two-body decay widths.

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Light moduli fields, gravitationally coupled scalar fields with no classical potential and which are expected to emerge as remnants from string theory compactification, are dangerous to cosmology in that 1. their late-time decays may disrupt successful Big Bang Nucleosynthesis (BBN), 2. they may decay into gravitino pairs which result in violation of BBN constraints or overproduction of lightest SUSY particles (LSPs, assumed to constitute at least a portion of the dark matter in the universe) and 3. they may decay directly into LSPs, resulting in gross DM overproduction. Together, these constitute the cosmological moduli problem (CMP). The combined effects require lightest modulus mass m φ 10 4 TeV, and if the lightest modulus mass m φ is correlated with the SUSY breaking scale m 3/2 , then the underlying SUSY model would be highly unnatural. We present a solution to the CMP wherein the lightest modulus initial field strength φ 0 is anthropically selected to be φ 0 ∼ 10 −7 m P by the requirement that the dark matter-to-baryonic matter ratio be not-too-far removed from its present value so that sufficient baryons are present in the universe to create observers. In this case, instead of dark matter overproduction via neutralino reannihilation at the modulus decay temperature, the neutralinos inherit the reduced moduli number density, thereby gaining accord with the measured dark matter relic density.

We study the production of dark matter and dark radiation after reheating in string inflation models where the Calabi-Yau has a fibred structure and the visible sector lives on D3 branes. We show how the interplay between different physical constraints from inflation, reheating, supersymmetry breaking and dark radiation, leads to distinct predictions for the nature of dark matter. In particular, in Fibre Inflation dark matter can only be primordial black holes or an open string QCD axion with an intermediate scale decay constant since WIMPs are always too heavy and ultralight closed string axions cannot behave as fuzzy dark matter due to strong isocurvature bounds. On the other hand, Kähler moduli inflation can allow for non-thermal WIMP dark matter at the TeV-scale.