Revisited apparent horizon entropy and GSL in modified gravity

TL;DR

This work introduces a universal revisited formalism for the apparent-horizon entropy in modified gravity by deriving it from the modified Friedmann equations in a non-flat FRW universe. The entropy comprises the standard Bekenstein–Hawking term plus an integral correction that encapsulates the effective energy–pressure from gravity modifications, ensuring consistency with the Clausius relation. A universal GSL expression is obtained and applied to $f(T)$ and $f(R)$ models, showing that the integral term can improve late-time thermodynamic viability in some cases (notably certain $f(T)$ and $f(R)$ models) while leaving others largely unchanged. The results reinforce a deep connection between horizon thermodynamics and gravity, providing a unified framework to test modified gravity scenarios against the generalized second law across cosmic evolution.

Abstract

This work presents a universal and revisited formalism for the entropy of the apparent horizon in modified gravity to investigate the validity of the Generalized Second Law (GSL) of thermodynamics. This revisited horizon entropy is constructed directly from the modified Friedmann equations in a non-flat Friedmann-Robertson-Walker (FRW) universe. The resulting entropy relation contains, beside the standard Bekenstein-Hawking term, an additional integral contribution that encodes the effective energy density and pressure generated by deviations from general relativity. Using this universal entropy formula, a compact expression for the GSL is derived. This formalism is then applied to some viable $f(T)$ and $f(R)$ gravity models, in order to re-evaluate the validity of the GSL as a function of redshift. The analysis demonstrates that including the integral term in the revisited entropy can relatively improve the late-time validity of the GSL for some of these models while living others unchanged, thereby reinforcing the profound connection between thermodynamics and gravity.

Figures (2)

• Figure 1: Evolutions of $T_{\rm A}\dot{S}_{\rm A}$ (\ref{['saft1']}), $T_{\rm A}\dot{S}_{\rm m}$ (\ref{['TdsmT']}), and $T_{\rm A}\dot{S}_{\rm tot}$ (\ref{['GSLt1']}) as functions of the redshift $z$ for $f(T)$ gravity (a) Model 1 and (b) Model 2. Black dashed curves show the entropy rates obtained with the standard formalism ($\alpha=0$), while magenta and cyan curves denote, respectively, the revisited total and horizon entropies including the integral term ($\alpha=18\pi/G$). The small panels highlight the late time behavior. The auxiliary parameters are $H_0=68.22~\rm km~s^{-1}~Mpc^{-1}$ and $\Omega_{\rm m_0}=0.3032$ACT. For Model 1 and Model 2, we set $n=0.04$Wu:2010blinder and $\beta=-0.02$Wu:2010b, respectively. These give $\mu_1=\left(\frac{1-\Omega_{\rm m_0}}{2n-1}\right)\left(6H_0^2\right)^{1-n}=-14043$ and $\mu_2=\frac{1-\Omega_{\rm m_0}}{1-(1-2\beta)e^{\beta}}=-35.9$.
• Figure 2: Evolutions of $T_{\rm A}\dot{S}_{\rm A}$ (\ref{['tsra-2']}), $T_{\rm A}\dot{S}_{\rm m}$ (\ref{['smr']}), and $T_{\rm A}\dot{S}_{\rm tot}$ (\ref{['gslr1']}) versus the redshift $z$ for (a) AB, (b) Starobinsky, (c) Exponential, (d) Hu--Sawicki, and (e) Tsujikawa $f(R)$ models. Black dashed curves show the entropy rates in the standard formalism without the integral term ($\beta=0$), whereas magenta and cyan curves denote, respectively, the revisited total and horizon entropies including the integral term ($\beta=\pi/G$). The insets highlight the late time domain. The auxiliary parameters are $H_0=68.22~\rm km~s^{-1}~Mpc^{-1}$ and $\Omega_{\rm m_0}=0.3032$ACT.