Determine whether FLRW trajectories intersect the thermodynamic critical or coexistence loci

Determine whether a physical FLRW cosmological trajectory intersects the critical locus or the first-order coexistence locus of the fixed-curvature apparent-horizon thermodynamic ensemble.

Background

The paper constructs an equilibrium thermodynamic description of a nonflat FLRW apparent horizon by introducing the dimensionless curvature variable $\chi=kA^2/a^2$ and restricting thermodynamic variations to fixed-$\chi$ slices. Within this ensemble, the holographic-type dark-energy model possesses a critical point and, for the quadratic model, a first-order coexistence curve.

A physical FLRW solution generally evolves through the enlarged state space with $T=T(t)$, $v=v(t)$, and $\chi=\chi(t)$, rather than remaining on a fixed-$\chi$ thermodynamic slice. Consequently, the existence of equilibrium criticality does not establish that cosmological evolution reaches it. Resolving the problem requires solving the Friedmann and continuity equations for specified matter content and initial conditions, and determining whether the resulting trajectory intersects the critical or coexistence locus.