Two-Channel Lévy Model of Cosmological Kinematics

Within the framework of the Robertson–Walker metric, we propose a fractional-calculus-based phenomenological model for the dynamics of the cosmological scale factor. The parent process a˜(τ) is represented as a sharp transition between two Einstein Universes with different total energies, E1 and E2, occurring at a critical operational time τc. To incorporate the effect of random internal clocks, possibly associated with distinct matter components, the physical-time evolution is obtained through an inverse, strictly increasing Lévy-type subordinator. The observable scale factor a(t) is defined as the ensemble mean of the parent process over realizations of the inverse subordinator. We show that, for suitable parameter regimes, the resulting dynamics can qualitatively reproduce key stages of cosmic expansion, including an inflation-like early phase, subsequent decelerated expansion, late-time acceleration, and possible future deceleration. Using energy conservation in a closed expanding universe, we discuss the associated evolution of matter density and the emergence of an effective dark-energy component. The results suggest that fractional-calculus methods may provide a useful mathematical framework for exploring nonstandard cosmological scenarios.