Evolution of the black‐body spectrum in the Friedmann universe

A fully rigorous, but simple, derivation of the evolution of an initially black‐body spectrum in the closed Friedmann universe is given. Some authors have assumed that, strictly speaking, one cannot speak of thermal equilibrium of the radiation, since it is in a changing gravitational background. This is shown to be incorrect, and a simple analogy of vibrations on an expanding circle of string illustrates the reason. It is possible to define a very natural dimensionless equilibrium constant. Finally, the problem of finding the electromagnetic eigenmodes is solved efficiently with differential forms.