Cosmology under the fractional calculus approach

TL;DR

This paper explores fractional calculus modifications to cosmology, deriving new Friedmann equations, constraining the fractional order with observational data, and showing potential for late-time cosmic acceleration without dark energy.

Contribution

It introduces a fractional calculus framework into cosmology, deriving modified equations, constraining the fractional order, and analyzing stability and acceleration without dark energy.

Findings

Estimated fractional order μ ≈ 2.84 with observational data.

Identified late-time attractor solutions corresponding to cosmic acceleration.

Demonstrated acceleration without dark energy in fractional cosmology.

Abstract

Fractional cosmology modifies the standard derivative to Caputo's fractional derivative of order $\mu$, generating changes in General Relativity. Friedmann equations are modified, and the evolution of the species densities depends on $\mu$ and the age of the Universe $t_U$. We estimate stringent constraints on $\mu$ using cosmic chronometers, Type Ia supernovae, and joint analysis. We obtain $\mu=2.839^{+0.117}_{-0.193}$ within the $1\sigma$ confidence level providing a non-standard cosmic acceleration at late times; consequently, the Universe would be older than the standard estimations. Additionally, we present a stability analysis for different $\mu$ values. This analysis identifies a late-time attractor corresponding to a power-law decelerated solution for $\mu < 2$. Moreover, a non-relativistic critical point exists for $\mu > 1$ and a sink for $\mu > 2$. This solution is a decelerated power-law if $1 < \mu < 2$ and an accelerated power-law solution if $\mu > 2$, consistent with the mean values obtained from the observational analysis. Therefore, for both flat FLRW and Bianchi I metrics, the modified Friedmann equations provide a late cosmic acceleration under this paradigm without introducing a dark energy component. This approach could be a new path to tackling unsolved cosmological problems.