Dark matter and dark radiation from the early universe with a modulus coupled to the PQMSSM

The supersymmetrized DFSZ axion model is especially compelling in that it contains 1. the SUSY solution to the gauge hierarchy problem, 2. the Peccei-Quinn (PQ) solution to the strong CP problem and 3. the Kim-Nilles solution to the SUSY mu problem. In a string setting, where a discrete R-symmetry ({\bf Z}{24}^R for example) may emerge from the compactification process, a high-quality accidental axion (accion) can emerge from the accidental, approximate remnant global U(1){PQ} symmetry where the decay constant f_a is linked to the SUSY breaking scale, and is within the cosmological sweet zone. In this setup, one also expects the presence of stringy remnant moduli fields φ_i. Here, we consider the situation of a single light modulus φcoupled to the PQMSSM in the early universe, with mixed axion plus higgsino-like WIMP dark matter. We evaluate dark matter and dark radiation production via nine coupled Boltzmann equations and assess the severity of the cosmological moduli problem (CMP) along with dark matter and dark radiation production rates. We find that typically the light modulus mass should be m_φ>~ 10^4 TeV to avoid the moduli-induced dark matter overproduction problem. If one is able to (anthropically) tune the modulus field amplitude, we find a value of φ_0 <~ 10^{-7}m_P would be required to solve the overall CMP.