  The study of temporal coherence in a Bose-Einstein condensate of photons can
be challenging, especially in the presence of correlations between the photonic
modes. In this work, we use a microscopic, multimode model of photonic
condensation inside a dye-filled microcavity and the quantum regression
theorem, to derive an analytical expression for the equation of motion of the
photon-photon correlation function. This allows us to derive the coherence time
of the photonic modes and identify a nonmonotonic dependence of the temporal
coherence of the condensed light with the cutoff frequency of the microcavity.
