Spherical Freezing Analogy for Cosmology

Abstract: We develop a spherical freezing model as an analogue of a radiation-dominated Friedmann universe. Under quasi-static growth and the Stefan condition, the radius evolves as \(\dot{R} \propto 1/R\), giving \(R(t) \propto t^{1/2}\) and reproducing \(H^2 \propto a^{-4}\) with \(w=1/3\). Including buoyancy-driven convection breaks spherical symmetry and induces axisymmetric deformations, described via a Legendre expansion and a Nusselt-based transport model. The resulting coupled dynamics yield an effective equation of state \(w_{\mathrm{eff}} = 1/3 - 2\beta\), with anisotropic shear scaling as \(S^{-6}\) for \(\beta=1/3\), analogous to Bianchi-I cosmology. This provides an analogue for both isotropic and anisotropic cosmic expansion.