Cosmology under the fractional calculus approach

Abstract

Fractional cosmology modifies the standard derivative to Caputo's fractional derivative of order $\hat{I}¼$, generating changes in General Relativity. Friedmann equations are modified, and the evolution of the species densities depends on $\hat{I}¼$ and the age of the Universe $t_U$. We estimate stringent constraints on $\hat{I}¼$ using cosmic chronometers, Type Ia supernovae, and joint analysis. We obtain $\hat{I}¼=2.839_{-0.193}^{+0.117}$ within the $1\ddot{I}ƒ$ 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 $\hat{I}¼$ values. This analysis identifies a late-time attractor corresponding to a power-law decelerated solution for $\hat{I}¼<2$. Moreover, a non-relativistic critical point exists for $\hat{I}¼>1$ and a sink for $\hat{I}¼>2$. This solution is a decelerated power-law if $1<\hat{I}¼<2$ and an accelerated power-law solution if $\hat{I}¼>2$, 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.

15 pages, 8 figures. Moderate revision. Matches the version accepted for publication in MNRAS