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## Finite-temperature Hartree-Fock-Bogoliubov theory for exciton-polaritons

Microcavity exciton-polaritons, known to exhibit nonequilibrium Bose condensation at high critical temperatures, can also be brought in thermal equilibrium with the surrounding medium and form a quantum degenerate Bose-Einstein distribution. It happens when their thermalization time in the regime of positive detunings – or, alternatively, for high-finesse microcavities – becomes shorter than their lifetime. Here we present the self-consistent finite-temperature Hartree–Fock–Bogoliubov description for such a system of polaritons, universally addressing the excitation spectrum, momentum-dependent interactions, condensate depletion, and the background population of dark excitons that contribute to the system’s chemical potential. Employing the derived expressions, we discuss the implications for the Bogoliubov sound velocity, confirmed by existing experiments, and define the critical temperatures of (quasi)condensation and the integral particle lifetime dependencies on the detuning. Large positive detunings are shown to provide conditions for the total lifetime reaching nanosecond timescales. This allows realization of *thermodynamically equilibrium *polariton systems with Bose-Einstein condensate forming at temperatures as high as tens of Kelvin.