Some necessary conditions for allowing the PQ scale as high as M_{GUT} in SUSY models with an axino or neutralino LSP

We examine some conditions which are needed to allow the Peccei-Quinn scale (f_a) as high as the SUSY GUT scale (M_{GUT} \sim 10^{16} GeV) in the context of the PQMSSM with either an axino or a neutralino LSP. The main problem in non-SUSY models with f_a\sim M_{GUT} is the generation of an overabundance of axion dark matter (\sim f_a^{7/6}) due to vacuum misalignment. We show that once all the components of the axion supermultiplet are included, the upper limit on f_a can be evaded due to large entropy injection from saxion decays. This large entropy injection also dilutes all other quasi-stable relic densities, naturally evading the BBN constraints and solving the gravitino problem. We find that f_a\sim M_{GUT} can be allowed by relic density/BBN constraints provided that the saxion mass m_s> 50 TeV, the initial saxion field value is of order of the PQ scale and the initial axion mis-alignment angle \theta_i\alt 0.05 . These restrictions can be considerably loosened for f_a\sim 10^{15} GeV. The allowed range for the re-heat temperature (T_R) is strongly dependent on the nature of the LSP. For the axino LSP, T_R \gtrsim 10^8 GeV, while for neutralino LSP any value is allowed. In the latter case, f_a\sim M_{GUT} can be more easily accommodated than in the axino LSP scenario. For f_a\sim M_{GUT} in SUSY models, the dark matter abundance should be dominated by axions, albeit with mass m_a\sim 10^{-10} eV, far below the region currently probed by experiment.