Emergent inflation in fractional cosmology

Abstract

We investigate the evolution of the early universe within the proposed fractional cosmological framework. The underlying formulation is conceptually rooted in generalized measure constructions, closely related to fractal geometries and scale-dependent effective dimensions. Within this framework, the cosmological dynamics are governed by a single fractional deformation parameter $α$, without introducing additional propagating dynamical degrees of freedom. Guided by the physical requirements, we identify a minimal fractional potential that, together with the fractional time-dependent kernel, leads to a non-singular pre-inflationary phase, a stable quasi-de Sitter inflationary attractor, followed by a graceful exit and a smooth transition to the standard radiation-dominated era. We further establish an explicit relation between the number of e-folds and the fractional parameter, demonstrating that observationally viable inflationary solutions arise for a well-defined range of $α$, thereby providing a direct connection between fractional cosmological dynamics and early universe phenomenology while resolving the horizon problem. Unlike conventional inflationary scenarios requiring an external inflaton field, the inflationary phase emerges entirely from the fractional modification of the Newtonian framework. Together with the previously established fractional cosmological framework describing the subsequent matter-dominated and late-time accelerated epochs, the present work completes the background cosmological evolution within a unified framework extending from the pre-inflationary universe to the present cosmic acceleration.