Memory Effects in the Early Universe: Exact Perturbations in Fractional Scalar Field Cosmology

Standard inflationary models, while successful, often require fine-tuned potentials to satisfy current observational constraints on the spectral index (n_s) and the tensor-to-scalar ratio (r). In this work, we propose a generalized cosmological framework based on fractional calculus, where the effective action includes non-local memory terms arising from a modified gravitational coupling. We investigate the dynamics of a scalar field against an n-dimensional Friedmann–Lemaitre–Robertson–Walker (FLRW) background, demonstrating that the fractional order parameter, α, introduces a cumulative friction term, (1 - α)/H^-1, into the background equations.

Crucially, we extend this analysis to linear cosmological perturbations. By enforcing variational consistency, we derive the Fractional Mukhanov–Sasaki equation, which explicitly incorporates memory effects. We solve this equation analytically for power-law inflation, obtaining exact mode functions in terms of Hankel functions. The resulting power spectra reveal that the spectral index and tensor-to-scalar ratio are modified by the fractional parameter, taking the form n_s(α, m) and r(α). We show that these memory effects can naturally suppress the tensor-to-scalar ratio without requiring complex potentials, bringing power-law inflation back into agreement with recent Planck and BICEP/Keck data. This framework offers a novel, mathematically rigorous mechanism to address the "fine-tuning" problems of standard inflation through the lens of non-local gravity.