Quasi-Homogeneous Thermodynamics and Microscopic Structure of the Quantum-Corrected FLRW Universe

TL;DR

The paper develops a quasi-homogeneous thermodynamic framework for the quantum-corrected FLRW horizon by promoting the GUP deformation parameter $β$ to a thermodynamic variable, resolving inconsistencies with Hayward’s unified first law. It derives a fundamental entropy relation $S(U,β,σ)$ and corresponding temperature, energy, and scaling relations, showing that phase transitions akin to van der Waals behavior arise for $β>0$ while maintaining thermodynamic consistency. Using Geometrothermodynamics, the authors map the thermodynamic phase structure to curvature singularities in the equilibrium space, finding a genuine phase transition at $U≃1.94\, ext{√}β$ and a critical exponent $\\zeta\approx1$, largely independent of dimension. The GTD analysis further reveals attractive and repulsive microstructure interactions and demonstrates a universal thermodynamic geometric signature linking cosmological horizons and black-hole-like systems. Overall, the work strengthens the case for thermodynamic universality in gravitational systems and provides a geometrically robust toolkit to probe the microstructure of spacetime.

Abstract

The analysis of phase transitions in cosmological spacetimes shows that their existence requires a time-dependent apparent horizon radius, which in turn implies an equation of state different from that of a dark energy fluid. This condition is not compatible with the simultaneous fulfillment of Hayward's unified gravitational first law and the fundamental thermodynamic equation of the apparent horizon. To solve this problem, we introduce an alternative formulation in which the cosmological horizon is modeled as a quasi-homogeneous thermodynamic system. We apply this approach to the Friedmann-Lemaître-Robertson-Walker (FLRW) universe under quantum gravity corrections encoded by the Generalized Uncertainty Principle (GUP), promote the deformation parameter to a thermodynamic variable, and obtain a consistent thermodynamic description without relying on the usual pressure-volume interpretation. Using Geometrothermodynamics (GTD), we show that fluctuations of the GUP parameter can induce phase transitions closely resembling those of black hole configurations. Finally, we perform a numerical analysis of the behavior of the GTD scalar curvature near the phase transition point, where we find a scaling behavior characterized by the critical exponent close to 1, independently of the dimension of the equilibrium space. This reveals that quantum gravity corrections not only modify the thermodynamic consistency of cosmological models but also strengthen the notion of thermodynamic universality across gravitational systems. Our findings confirm GTD as a powerful geometric tool to unveil the emergent thermodynamic microstructure of spacetime.

Figures (16)

• Figure 1: (a) Entropy profile for various values of $\beta$ with $\sigma = 1$. (b) $S$--$U$ diagram for $\sigma = 1$, showing curves corresponding to $\beta = 1$ (red), $\beta = 0$ (black) and $\beta = -10$ (blue). The dashed red curve represents the $R_A^{-}$ branch given by Eq. \ref{['eqr']}, associated with negative temperature $(T < 0)$.
• Figure 2: (a) Energy profile for different $\beta\geq0$ and $\sigma=1$. (b) $U$--$S$ plot for $\sigma=1$. The dashed red curve highlights regions where $T < 0$. A nonzero $U_{\min}$ emerges for $\beta > 0$.
• Figure 3: (a) Thermodynamic temperature profile for different values of $\beta$. (b) Geometric temperature profile for $\beta>0$ and different values of $\omega$.
• Figure 4: Heat capacity for different values of GUP parameter: $\beta = 2$ (red curve), $\beta=0$ (black curve), and $\beta = -1.5$ (blue curve). (b) Phase structure for $\beta > 0$. The horizon exhibits distinct thermodynamic regimes: SFLRW, LFLRW, and GFLRW.
• Figure 5: $S$--$T$ plot for $\beta = 1$ and $\sigma = 1$. The dashed red line represents the GFLRW phase.