  We study the double Compton (DC) process for a wide range of particle
energies, extending previous treatments well beyond the soft photon limit,
employing both numerical and analytical methods. This allows us to investigate
the physics of the DC process up to the highly relativistic regime relevant to
electromagnetic particle cascades in the early Universe and photon-dominated
astrophysical plasmas. Generalized exact analytic expressions for the DC
emissivity in the soft photon limit are obtained. These are compared to
existing approximations, for the first time studying the ultra-relativistic
regime. We also numerically integrate the full DC collision term calculating
the DC emissivity at general particle energies. A careful treatment of DC
infrared divergences inside astrophysical plasmas, including subtle effects
related to the presence of stimulated DC emission, is discussed. The obtained
results can be efficiently represented using the code DCpack, which also allows
one to compute average emissivities for general incoming electron and photon
distributions. This puts the modelling of the DC process inside astrophysical
plasmas on a solid footing and should find applications in particular for
computations of the cosmological thermalization problem in the early Universe.
