Note on the chemical potential of decoupled matter in the Universe

TL;DR

This paper addresses a thermodynamic inconsistency in cosmology textbooks related to the chemical potential of decoupled matter, highlighting deviations from equilibrium values in free streaming conditions.

Contribution

It identifies and analyzes the deviation of the chemical potential from its equilibrium form in decoupled matter within the universe.

Findings

Chemical potential deviates from equilibrium value in free streaming matter.

Highlights thermodynamic inconsistency in standard cosmology textbooks.

Provides a theoretical analysis of chemical potential behavior in decoupled matter.

Abstract

Textbooks on cosmology exhibit a thermodynamic inconsistency for free streaming, decoupled matter. It is connected here to the chemical potential, which deviates from its equilibrium value $\mu=\alpha k_BT$, where $\alpha$ is the usual parameter of the Fermi-Dirac or Bose-Einstein distribution function.