Quantum critical superconductivity in a dense fermi dilute bose mixture close to mechanical instability

We couple a metal to an uncondensed bose gas with repulsive bose-fermi interactions in two dimensions. Within the random phase approximation, we show that the density fluctuations of the bosons, which mediate the effective fermion-fermion attraction, become long ranged on appropriate tuning of interactions. This leads to parametrically enhanced superconducting temperatures $T_c\propto g$ as opposed to $T_c\propto \exp (-1\mathrm{/}g)$ for the BCS case. The dilute bose gas density acts as the primary bottleneck to the enhancement. We discuss the conceptual difficulties of implementing this idea in electron-exciton mixtures.