[Paper Review] Horizon entropy consistent with FLRW equations for general modified theories of gravity and for all EoS of the matter field

Shin’ichi Nojiri, Sergei D. Odintsov+2 more|arXiv (Cornell University)|Jul 11, 2023

Cosmology and Gravitation Theories4 citations

TL;DR

This paper proposes a modified thermodynamic law for the apparent horizon in cosmology, $TdS = -dE + \rho dV$, which resolves inconsistencies in the standard $TdS = -dE + WdV$ formulation for non-relativistic matter ($\omega \neq -1$). The new law enables a generalized horizon entropy that consistently links Friedmann equations to any modified gravity theory across all equations of state, unifying thermodynamics with cosmology beyond Einstein and Gauss-Bonnet gravity.

ABSTRACT

The question that continues to hinge the interrelation between cosmology and thermodynamics is broadly described as -- what is the form of horizon entropy that links the Friedmann equations for a "$general$" gravity theory with the underlying thermodynamics of the apparent horizon? The answer to this question was known only for Einstein's gravity and for $(n+1)$ dimensional Gauss-Bonnet gravity theory, but not for a general modified theory of gravity (for instance, the $F(R)$ gravity). In the present work, we take this issue and determine a general form of entropy that connects the Friedmann equations for any gravity theory with the apparent horizon thermodynamics given by $TdS = -dE + WdV$ (the symbols have their usual meaning in the context of entropic cosmology and $W = \left(ρ- p ight)/2$ is the work density of the matter fields represented by $ρ$ and $p$ as the energy density and the pressure, respectively). Using such generalized entropy, we find the respective entropies for several modified theories of gravity (including the $F(R)$ gravity). Further, it turns out that besides the above-mentioned question, the thermodynamic law $TdS = -dE + WdV$ itself has some serious difficulties for certain values of $ω$ (the EoS of matter field). Thus we propose a modified thermodynamic law of apparent horizon, given by $TdS = -dE + ρdV$, that is interestingly free from such difficulties. The modified law proves to be valid for all EoS of the matter field and thus is considered to be more general compared to the previous one which, however, is a limiting case of the modified law for $p = -ρ$. Based on such modified thermodynamics, we further determine a generalized entropy that can provide the Friedmann equations of any general gravity theory for all values of EoS of the matter field. The further implications are discussed.

Motivation & Objective

• To resolve the inconsistency in the standard thermodynamic law $TdS = -dE + WdV$ for non-relativistic matter fields ($\omega \neq -1$) in apparent horizon thermodynamics.
• To derive a generalized horizon entropy that consistently connects Friedmann equations of any gravity theory to the thermodynamics of the apparent horizon.
• To establish a unified framework valid for all equations of state ($\omega$), including $\omega > 1/3$ during reheating, where previous formulations fail.
• To demonstrate that the new thermodynamic law $TdS = -dE + \rho dV$ reduces to the standard form only for $\omega = -1$ (cosmological constant-like matter).

Proposed method

• Derive the generalized horizon entropy by assuming the modified thermodynamic law $TdS = -dE + \rho dV$ as the fundamental postulate.
• Use the conservation law $\dot{\rho} + 3H(\rho + p) = 0$ and the FLRW equations to relate entropy evolution to the Hubble parameter and scale factor.
• Apply the condition that the entropy must reproduce the Friedmann equations for arbitrary gravity theories, including $F(R)$, Gauss-Bonnet, and higher-curvature models.
• Show that the standard $WdV$ term leads to inconsistencies for $\omega \neq -1$, especially for $\omega > 1/3$, by deriving non-physical results in entropy evolution.
• Demonstrate that the new law $\rho dV$ term avoids these inconsistencies and allows consistent entropy derivation across all $\omega$.
• Verify that for $\omega = -1$, the new entropy reduces to the Bekenstein-Hawking form with an $\omega$-dependent prefactor, confirming consistency in the limiting case.

Experimental results

Research questions

• RQ1 Why does the standard thermodynamic law $TdS = -dE + WdV$ fail for matter fields with $\omega \neq -1$ in apparent horizon thermodynamics?
• RQ2 Can a unified thermodynamic law be formulated that consistently links Friedmann equations to horizon thermodynamics across all equations of state in general gravity theories?
• RQ3 What is the correct generalized horizon entropy that reproduces the Friedmann equations for any modified gravity theory and all $\omega$?
• RQ4 How does the new thermodynamic law $TdS = -dE + \rho dV$ differ from the standard $WdV$ formulation, and why is it more universally valid?
• RQ5 Does the new entropy formulation reduce to the Bekenstein-Hawking entropy in the Einstein gravity limit, and if so, under what conditions?

Key findings

• The standard thermodynamic law $TdS = -dE + WdV$ fails for $\omega > 1/3$, leading to inconsistencies in entropy derivation, particularly during reheating in scalar field cosmology.
• The proposed modified law $TdS = -dE + \rho dV$ is free from such inconsistencies and is valid for all values of the equation of state parameter $\omega$, including $\omega = 0$ (dust) and $\omega = 1/3$ (radiation).
• For Einstein gravity, the new entropy formula yields a Bekenstein-Hawking-like form but with an $\omega$-dependent prefactor, indicating a departure from the standard entropy in non-relativistic regimes.
• The generalized entropy derived from the new thermodynamic law successfully reproduces the Friedmann equations for any modified gravity theory, including $F(R)$ and Gauss-Bonnet gravity, for all $\omega$.
• The new formulation unifies thermodynamics and cosmology beyond the $\omega = -1$ limit, providing a consistent framework for dark energy and early-universe cosmology.
• The entropy for $F(R)$ gravity and other modified theories derived from the new law does not resemble black hole entropy forms, indicating a fundamental shift in the thermodynamic interpretation of horizons in modified gravity.