Reconciling thermal leptogenesis with the gravitino problem in SUSY models with mixed axion/axino dark matter

25 pages including 15 .eps figuresSuccessful implementation of thermal leptogenesis requires re-heat temperatures T_R\agt 2\times 10^9 GeV, in apparent conflict with SUSY models with TeV-scale gravitinos, which require much lower T_R in order to avoid Big Bang Nucleosynthesis (BBN) constraints. We show that mixed axion/axino dark matter can reconcile thermal leptogenesis with the gravitino problem in models with m_{\tG}\agt 30 TeV, a rather high Peccei-Quinn breaking scale and an initial mis-alignment angle \theta_i < 1. We calculate axion and axino dark matter production from four sources, and impose BBN constraints on long-lived gravitinos and neutralinos. Moreover, we discuss several SUSY models which naturally have gravitino masses of the order of tens of TeV. We find a reconciliation difficult in Yukawa-unified SUSY and in AMSB with a wino-like lightest neutralino. However, T_R\sim 10^{10}-10^{12} GeV can easily be achieved in effective SUSY and in models based on mixed moduli-anomaly mediation. Consequences of this scenario include: 1. an LHC SUSY discovery should be consistent with SUSY models with a large gravitino mass, 2. an apparent neutralino relic abundance \Omega_{\tz_1}h^2\alt 1, 3. no WIMP direct or indirect detection signals should be found, and 4. the axion mass should be less than \sim 10^{-6} eV, somewhat below the conventional range which is explored by microwave cavity axion detection experiments