[Paper Review] The thermodynamics of a gravitating vacuum

M. Heyl, H. J. Fahr+1 more|arXiv (Cornell University)|Dec 9, 2014

Cosmology and Gravitation Theories4 citations

TL;DR

This paper challenges the standard assumption of constant vacuum energy density in cosmology, arguing instead that thermodynamic consistency requires vacuum energy density to scale as $\epsilon_{\text{vac}} \sim R^{-2}$ with cosmic scale factor $R$. The authors derive this scaling from energy conservation in a gravitating vacuum, showing it resolves the cosmological constant problem by naturally yielding the observed vacuum energy density when scaled from Planck-era conditions.

ABSTRACT

In the present days of modern cosmology it is assumed that the main ingredient to cosmic energy presently is vacuum energy with an energy density $ε_\mathrm{vac}$ that is constant over the cosmic evolution. In this paper here we show, however, that this assumption of constant vacuum energy density is unphysical, since it conflicts with the requirements of cosmic thermodynamics. We start from the total vacuum energy including the negatively valued gravitational binding energy and show that cosmic thermodynamics then requires that the cosmic vacuum energy density can only vary with cosmic scale $R=R(t)$ according to $ε_\mathrm{vac}\sim R^{-ν}$ with only two values of $ν$ being allowed, namely $ν_\mathrm{1}=2$ and $ν_\mathrm{2}=5/2$. We then discuss these two remaining solutions and find, when requiring a universe with a constant total energy, that the only allowed power index is $ν_\mathrm{1}=2$. We discuss the consequences of this scaling of $ε_\mathrm{vac}$ and show the results for a cosmic scale evolution of a quasi-empty universe like the one that we are presently faced by.

Motivation & Objective

• To challenge the standard cosmological assumption of constant vacuum energy density.
• To investigate thermodynamic consistency of vacuum energy in an expanding universe.
• To derive a physically valid scaling law for vacuum energy density that respects energy conservation.
• To resolve the cosmological constant problem by showing $\epsilon_{\text{vac}} \sim R^{-2}$ naturally yields the observed vacuum energy density.

Proposed method

• Derives the total energy of a comoving volume, including gravitational binding energy.
• Applies cosmic thermodynamics to require energy conservation in the expanding universe.
• Uses the Robertson-Walker metric to express proper volume and energy density scaling.
• Derives the condition $\epsilon_{\text{vac}} \sim R^{-\nu}$ and identifies allowed $\nu$ values as $\nu = 2$ and $\nu = 5/2$.
• Imposes the constraint of constant total energy to select $\nu = 2$ as the only viable solution.
• Scales the vacuum energy density from Planck time to present using $R^{-2}$, matching observed values.

Experimental results

Research questions

• RQ1 Is the assumption of constant vacuum energy density thermodynamically consistent in an expanding universe?
• RQ2 What scaling of vacuum energy density $\epsilon_{\text{vac}}$ with cosmic scale $R(t)$ satisfies energy conservation and thermodynamic constraints?
• RQ3 Can the observed vacuum energy density be naturally derived from a fundamental scaling law?
• RQ4 Why does the $R^{-2}$ scaling resolve the cosmological constant problem?
• RQ5 What is the physical significance of the Planck-scale vacuum density in this model?

Key findings

• The vacuum energy density must scale as $\epsilon_{\text{vac}} \sim R^{-2}$ to satisfy cosmic thermodynamics and energy conservation.
• The $R^{-2}$ scaling yields a present-day vacuum energy density of $\rho_{\text{vac},0} \approx 10^{-26}~\text{kg/m}^3$, matching observational estimates.
• The model predicts a total vacuum mass of $M_{\text{vac}} \approx 10^{53}~\text{kg}$, consistent with the visible universe's mass content.
• At the Planck time, the vacuum energy density reaches $\rho_{\text{vac}}(t_p) = \rho_p = \frac{3}{8\pi} \frac{c^5}{\hbar G^2}$, the Planck density.
• The ratio $\rho_{\text{vac},0}/\rho_p \approx 10^{-122}$ is naturally explained by the $R^{-2}$ scaling, resolving the cosmological constant discrepancy.
• The $R^{-2}$ scaling provides a physically consistent alternative to the standard $\Lambda$CDM assumption of constant $\epsilon_{\text{vac}}$.