Phenomenological Model for Kinematic Description of the Universe

In the framework of the Robertson–Walker metric, we investigate the behavior of a subordinated cosmic scale factor a(t). The parent process a˜(τ) is modeled as a sharp phase transition between two Einstein universes, each with different total energies E1 and E2, at a specific moment τc of the operational time τ. To account for the effect of random clocks, which may arise from differences in the physical properties of dark matter and ordinary matter, this model incorporates a random operational time in the form of an inverse, strictly increasing Lévy-type subordinator. As an observable in physical time t, a(t) is defined as the mean of the parent process over an ensemble of realizations of the inverse subordinator. Specifically, we show that, under certain parameter regimes, the evolution of a(t) mimics primordial inflation, subsequent decelerated expansion, and cosmic acceleration. Employing energy conservation in a closed, expanding universe, we discuss the possibility of incorporating matter density evolution. Furthermore, we establish the emergence of dark energy. We believe that this work introduces a novel application of fractional calculus to cosmology.