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We derive a phase diagram for the pseudogap onset temperature $T^*$(associated with the breakdown of the Fermi liquid state, due to strong pairingcorrelations) and the superconducting instability, $T_c$, as a function ofvariable pairing strength. Our diagrammatic approach to the BCS - Bose-Einsteincross-over problem self consistently treats the coupling between the singleparticle and pair propagators, and leads to a continuous evolution of thesepropagators into the standard $T<T_c$ counterparts. A rich structure is foundin $T_c$ which reflects the way in which the superconducting instability at$T_c$ is affected by the pseudogap $\Delta_{pg}$. An important consequence ofCooper-pair- induced pseudogaps is that the magnitude of $T_c$ is sustained,even when $\Delta_{pg}>T_c$.

Superconductivity from a pseudogapped normal state: A mode-coupling approach to precursor superconductivity