Thermodynamic stability in an Einstein universe

TL;DR

This work derives the renormalized finite-temperature Feynman propagator for a neutral scalar field with curvature coupling $\xi$ in an Einstein universe and uses it to compute $\langle\phi^2\rangle$ and $\langle T^{\mu}_{\nu}\rangle$ decomposed into vacuum, mixed, and thermal parts. By analyzing the resulting energy density $\rho$ and its dependence on temperature $T$, radius $a$, and $\xi$, the authors derive stringent stability criteria: for massless scalar radiation, thermodynamic equilibrium at all $T$ and $a$ is achieved only at the conformal coupling $\xi=1/6$, with other $\xi$ leading to instability in different asymptotic regimes ($Ta\to0$ or $Ta\to\infty$). The paper also discusses mixed radiations and shows that at least one scalar field with suitable $\xi$ is required to ensure stability when neutrinos and photons are present, highlighting nontrivial constraints on the composition of early-universe radiation. Overall, the results connect finite-temperature QFT in curved backgrounds with fundamental thermodynamic stability, offering insights for early-uncosmology and possible generalizations to more general cosmological spacetimes.

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.

Figures (5)

• Figure 2: $\left<\phi^{2}\right>_{{\tt thermal}}$ vs. $T$, for $M=0$ and $a=1$. On the right: the lower (blue) plot, the middle (green) plot, and the upper (yellow) plot correspond, respectively, to $\xi=0.1666$, $\xi=0.1667$, and $\xi=1/6$. Note that these asymptotic behaviors as $T\rightarrow 0$ differ radically from each other though their associated values of $\xi$ are not that different.
• Figure 3: Massless $\left<\phi^{2}\right>_{{\tt thermal}}$ around $\xi =1/6$, for unitary $a$ and $T$. On the left (blue) and on the right (yellow) are the plots corresponding to eqs. (\ref{['tphi2-5']}) and (\ref{['tphi2-6']}), respectively.
• Figure 4: $\rho_{{\tt vacuum}}$ vs. $\xi$, for $\xi>0$, $M=0$ and $a=1$. The vacuum energy density vanishes at $\xi\simeq 0.054$, changes sign there, and as $\xi\rightarrow\infty$ it approaches zero from above.
• Figure 5: $\rho_{{\tt vacuum}}$ vs. $\xi$, for $\xi>-1/2$, $M=0$ and $a=1$. At $\xi=0$, $\rho_{{\tt vacuum}}\simeq -0.009$. The cusp-like pattern showing in the plot repeats with increasing amplitude as $\xi\rightarrow -\infty$.
• Figure 6: $\rho_{{\tt thermal}}$ vs. $T$, for $M=0$ and $a=1$. On the right: the lower (green) plot, the middle (yellow) plot, and the upper (blue) plot correspond, respectively, to $\xi=0.1667$, $\xi=1/6$, and $\xi=0.1666$. Again, note that these asymptotic behaviors as $T\rightarrow 0$ differ radically from each other in spite of their associated values of $\xi$ are not that different.