Evolution of the black-body spectrum in the Friedmann universe

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Abstract
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.

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