  We investigate the many-body behavior of polaritons formed from electron-hole
pairs strongly coupled to photons in a two-dimensional semiconductor
microcavity. We use a microscopic mean-field BCS theory that describes
polariton condensation in quasi-equilibrium across the full range of excitation
densities. In the limit of vanishing density, we show that our theory recovers
the exact single-particle properties of polaritons, while at low densities it
captures non-linear polariton-polariton interactions within the Born
approximation. For the case of highly screened contact interactions between
charge carriers, we obtain analytic expressions for the equation of state of
the many-body system. This allows us to show that there is a photon resonance
at a chemical potential higher than the photon cavity energy, where the
electron-hole pair correlations in the polariton condensate become universal
and independent of the details of the carrier interactions. Comparing the
effect of different ranged interactions between charge carriers, we find that
the Rytova-Keldysh potential (relevant to transition metal dichalcogenides)
offers the best prospect of reaching the BCS regime, where pairs strongly
overlap and the minimum pairing gap occurs at finite momentum. Finally, going
beyond thermal equilibrium, we argue that there are generically two polariton
branches in the driven-dissipative system and we discuss the possibility of a
density-driven exceptional point within our model.
