Friedmann equations from nonequilibrium thermodynamics of the Universe: A unified formulation for modified gravity

Inspired by the Wald-Kodama entropy $S=A/(4G_{\text{eff}})$ where $A$ is the horizon area and $G_{\text{eff}}$ is the effective gravitational coupling strength in modified gravity with field equation $R_{\mu\nu}-Rg_{\mu\nu}/2=$ $8\pi G_{\text{eff}} T_{\mu\nu}^{\text{(eff)}}$, we develop a unified and compact formulation in which the Friedmann equations can be derived from thermodynamics of the Universe. The Hawking and Misner-Sharp masses are generalized by replacing Newton's constant $G$ with $G_{\text{eff}}$, and the unified first law of equilibrium thermodynamics is supplemented by a nonequilibrium energy dissipation term $\mathcal{E}$ which arises from the revised continuity equation of the perfect-fluid effective matter content and is related to the evolution of $G_{\text{eff}}$. By identifying the mass as the total internal energy, the unified first law for the interior and its smooth transit to the apparent horizon yield both Friedmann equations, while the nonequilibrium Clausius relation with entropy production for an isochoric process provides an alternative derivation on the horizon. We also analyze the equilibrium situation $G_{\text{eff}}=G=\text{constant}$, provide a viability test of the generalized geometric masses, and discuss the continuity/conservation equation. Finally, the general formulation is applied to the FRW cosmology of minimally coupled $f(R)$, generalized Brans-Dicke, scalar-tensor-chameleon, quadratic, $f(R,\mathcal{G})$ generalized Gauss-Bonnet and dynamical Chern-Simons gravity. In these theories we also analyze the $f(R)$-Brans-Dicke equivalence, find that the chameleon effect causes extra energy dissipation and entropy production, geometrically reconstruct the mass $\rho_m V$ for the physical matter content, and show the self-inconsistency of $f(R,\mathcal{G})$ gravity in problems involving $G_{\text{eff}}$.