Bose-Einstein condensation of photons in the matter-dominated universe

Abstract

In 1914, Planck introduced the concept of a white body. In nature, no true white bodies are known. We assume that the universe after last-scattering is an ideal white body that contains a tremendously large number of thermal photons and is at an extremely high temperature. Bose-Einstein condensation of photons in an ideal white body is investigated within the framework of quantum statistical mechanism. The computation shows that the transition temperature $T_c$ is a monotonically increasing function of the number density $n$ of photons. At finite temperature, we find that the condensate fraction $N_0(T)/N$ decreases continuously from unity to zero as the temperature increases from zero to the transition temperature $T_c$. Further, we study the radiation properties of an ideal white body. It is found that in the condensation region of $T<T_c$, the spectral intensity $I(\ddot{I}‰,T)$ of white body radiation is identical with Planck's law for blackbody radiation.