Spherical Freezing Analogy for Cosmology

Abstract: We develop a spherical freezing analogue of cosmological expansion based on an ice shell growing around an actively cooled isothermal core. The effective Stefan balance gives \(\dot R=\alpha_{\rm core}/R\) and therefore \(R(t)\propto t^{1/2}\), reproducing the radiation-dominated Friedmann scaling \(H_R^2\propto R^{-4}\). We then include buoyancy-driven water-side convection through a reduced Nusselt-factor model, which induces a weak axisymmetric deformation of the interface, \(R(\theta,t)=S(t)\left[1+\varepsilon(t)P_2(\cos\theta)\right]\). For \(\bar{Nu}(S)\propto S^{3\beta}\), the mean expansion obeys a modified Friedmann-like equation \(H^2=A_i^2S^{-4}+2A_iB\,S^{3\beta-4}+B^2S^{6\beta-4}\), while the anisotropic sector defines an LRS Bianchi-I-type shear. When the anisotropic forcing decays, \(\varepsilon\propto S^{-2}\), and in the convection-dominated case \(\beta=1/3\) the shear scales as \(\sigma^2\propto S^{-6}\), matching the standard Bianchi-I shear behavior.