The Early-Universe Dust Formation Crisis: A Stochastic Threshold Solution

The presence of substantial dust masses in galaxies at very high redshifts presents a major challenge to conventional theories of cosmic dust formation, which predict significantly longer timescales for dust enrichment. This discrepancy indicates that classical continuous and equilibrium-based models do not adequately capture the physical processes governing dust evolution in the early universe. The purpose of this study is to develop a mathematically rigorous and physically consistent framework capable of explaining rapid dust formation under realistic astrophysical conditions. A stochastic, non-equilibrium, multi-phase dynamical model is introduced to describe the coupled evolution of gas, metals, and dust in early galaxies. The interstellar medium is represented as a structured system consisting of diffuse, cold, and molecular phases, with transitions governed by metal enrichment and dust-mediated processes. Star formation is modeled as a burst-driven stochastic process, reflecting the intermittent nature of early galaxy evolution. Time-delay effects are incorporated to account for finite metal production and dust formation timescales. A central feature of the model is a threshold-driven mechanism in which dust growth shifts from inefficient stellar production to highly efficient grain accretion once the molecular gas fraction exceeds a critical level. The resulting system is formulated as a nonlinear stochastic delay differential equation with state-dependent switching. Analytical results establish the existence, positivity, and boundedness of solutions, and demonstrate the emergence of a threshold-induced transition from slow to rapid dust growth. The model predicts superlinear amplification of dust mass over short timescales once critical physical conditions are reached. Numerical simulations support the theoretical findings and show that the proposed mechanism reproduces rapid dust enrichment consistent with observed early-universe systems. The results provide a resolution to the dust formation problem without requiring extreme assumptions about stellar yields or hidden stellar populations. More broadly, the study highlights the importance of stochastic processes, phase transitions, and nonlinear feedback in shaping the evolution of astrophysical systems. The proposed framework offers a new mathematical paradigm for modeling non-equilibrium processes in galaxy formation and provides a basis for future observational and theoretical investigations.