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Thermodynamic stability in an Einstein universe

E. S. Moreira, Jr.1,* and J. P. A. Paula2,†

  • *Contact author: moreira@unifei.edu.br
  • †Contact author: almeida.paulo@ufabc.edu.br
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Phys. Rev. D 113, 105004 – Published 5 May, 2026

DOI: https://doi.org/10.1103/zcml-wwcz

Abstract

We calculate the Feynman propagator at finite temperature in an Einstein universe for a neutral massive scalar field arbitrarily coupled to the Ricci curvature. Then, the propagator is used to determine the mean square fluctuation, the internal energy, and pressure of a scalar blackbody radiation as functions of the curvature coupling parameter ξ. By studying thermodynamics of massless scalar fields, we show that the only value of ξ consistent with stable thermodynamic equilibrium at all temperatures and for all radii of the universe is 1/6, i.e., corresponding to the conformal coupling. Moreover, if electromagnetic and neutrino radiations are present at the regime of high temperatures and/or large radii, we show that at least one scalar field must also be present to ensure thermodynamic stability.

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Phys. Rev. D 113, 105004– Published 5 May, 2026
  • Received 2 February 2026
  • Accepted 10 April 2026
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