SaxiGUTs and their Predictions

We introduce a class of unified supersymmetric axion theories with unified and PQ symmetries broken by the same set of fields at a scale $\sim 2\times 10^{16}$ GeV. A typical domain wall number of order 30 leads to an axion decay constant $f_a$ of order $10^{15}$ GeV. Inflation generates a large saxion condensate giving a reheat temperature $T_R$ below the QCD scale for supersymmetry breaking of order $1-10$ TeV. Axion field oscillations commence in the saxion matter-dominated era near the QCD scale, and recent lattice computations of the temperature dependence of the axion mass in this era allow a controlled calculation of the axion dark matter abundance. A successful prediction of this abundance results for an initial axion misalignment angle of order unity, $\theta_i \sim 1$. A highly correlated set of predictions is discussed for $f_a$, $T_R$, the supersymmetric Higgs mass parameter $\mu$, the amount of dark radiation $\Delta N_{eff}$, the proton decay rate $\Gamma(p \rightarrow e^+ \pi^0)$, isocurvature density perturbations and the $B$-mode of the cosmic microwave background. The last two are particularly interesting when the energy scale of inflation is also of order $10^{16}$ GeV.