Freezing lakes as analogue models of $Λ$CDM cosmology and beyond

Abstract

We extend previous conduction-based analogies between ice growth in a lake and cosmological expansion by incorporating buoyancy-driven heat transport. Reformulating the Stefan problem with both conductive and convective fluxes yields an evolution equation for the ice thickness $s(t)$ that is structurally analogous to the Friedmann equations for the cosmological scale factor $a(t)$. Beyond reproducing radiation-, matter-, and curvature-like behaviors, we introduce a reduced description of convection in which the vertically integrated heat flux reaching the moving ice-water interface is modeled as a power-law function of the instantaneous liquid-layer thickness, generating two additional effective contributions. The first is a constant term, directly analogous to a cosmological constant, arising from the persistence of buoyancy-driven transport under geometric confinement. The second is a $s^{-1}$ contribution originating from the coupling between the moving ice boundary and the convective boundary layer. This term reflects the specific reduced flux-height Ansatz adopted, rather than a universal physical prediction. When expressed in Friedmann-like cosmological form, this term entails a fluid with negative energy density and equation-of-state parameter $w=-2/3$. In cosmology this term may be an effective one associated to a network of domain walls made of exotic energy/matter, but it might also arise from an energy exchange between cosmological components. Overall, the results should be interpreted as a structural analogy between evolution equations, showing how nonlinear transport mechanisms in a classical moving-boundary problem can reproduce the hierarchy of scaling terms familiar from cosmology within a reduced and analytically tractable framework.

Figures (3)

• Figure 1: Schematic representation of the temperatures. Here, $T_0$, $T_1$, $T_2$ and $T_3$ denote the bulk water, ice–water interface, ice-air interface and ambient air temperatures, respectively.
• Figure 2: Normalized ice thickness $s(t)/s_{\mathrm{end}}$ in the constant-convective (CC) ice-growth model (solid curve) compared with the normalized scale factor $a(t)/a_0$ in a matter–radiation cosmological expansion (CE) model with zero cosmological constant ($\Lambda = 0$), both plotted as functions of normalized time $\tau$. The analogy reproduces the successive radiation-, matter-, and curvature-like regimes but lacks late-time acceleration. The identified regime transitions for the CC model are: CC$\,1 \rightarrow 2$: $\tau \approx 0.0555$, $s \approx 2.0010$; and CC$\,2 \rightarrow 3$: $\tau \approx 0.7168$, $s \approx 8.0016$. For the CE model, the single transition is CE$\,1 \rightarrow 2$: $\tau \approx 0.0112$, $a \approx 0.0601$. The numbers (1, 2, 3) denote the dominant terms in each regime. Reference constants are provided in Appendix \ref{['app:sim']}.
• Figure 3: Normalized ice thickness $s(t)/s_{\mathrm{end}}$ in the Flux--Height (FH) ice growth convection model (solid curve) compared with the normalized scale factor $a(t)/a_0$ in the standard $\Lambda$CDM cosmology (dashed curve), both plotted as functions of normalized time $\tau$. The FH model reproduces radiation-, matter-, curvature-, and $\Lambda$-like regimes, while also exhibiting an additional $-s^{-1}$ contribution with no analogue in $\Lambda$CDM, leading to richer dynamical behavior. The observed transition values are as follows: for FH ice growth, $\tau \approx 0.026$, $s \approx 2.291$ (term $3 \rightarrow 4$) and $\tau \approx 0.899$, $s \approx 9.153$ (term $4 \rightarrow 5$); for $\Lambda$CDM expansion, $\tau \approx 0.009$, $a \approx 0.175$ (radiation $\rightarrow$ matter) and $\tau \approx 0.146$, $a \approx 0.931$ (matter $\rightarrow$ dark energy). Reference constants are provided in Appendix \ref{['app:sim']}.