The Generalized Second Law and the Spatial Curvature Index

TL;DR

This paper investigates whether the sign of the spatial curvature parameter $\Omega_k$ can be constrained by fundamental thermodynamics in an expanding FLRW universe within Einstein gravity. They derive a lower bound on the cosmic equation of state from the Dominant Energy Condition, $w \ge -1$, and a horizon-based bound from the Generalized Second Law, $w \ge \zeta = \frac{(1/3)\Omega_k - 1}{1-\Omega_k}$, together implying $1+q \ge \Omega_k$ and a stronger linear bound $1+q \ge g(\lambda)\Omega_k$. For $\Omega_k>0$ this cannot hold for all allowed $q$ and $\lambda$, effectively excluding hyperbolic spatial sections; flat ($\Omega_k=0$) and closed ($\Omega_k<0$) universes remain compatible. The result relies only on Einstein gravity, the DEC, and the GSL, and does not depend on a specific dark-energy model; the discussion places the finding in light of observational hints about curvature and argues that a hypothetical $\Omega_k>0$ would require a violation of one of the foundational assumptions.

Abstract

By applying the generalized second law to the apparent horizon of a homogeneous and isotropic universe and imposing that the equation of state is no less than $-1$, it is seen that universes with either flat or closed spatial sections are consistent with the joint consideration of the aforesaid law and the dominant energy condition, but not so universes with hyperbolic spatial sections