Cosmology under the fractional calculus approach

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(-0.193)(+0.117) 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 Friedmann-Lemaitre-Robertson-Walker 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.