Search

Recently, a new field of study called fractional cosmology has emerged. It uses fractional calculus to modify the standard derivative equations and change the Friedmann equations. The evolution of cosmic species densities is also affected by the $\mu$ fractional parameter and the age of the Universe $t_0$. This new approach to cosmology modifies the Friedmann equations and allows for a late cosmic acceleration without the need for a dark energy component. This could be a breakthrough in solving longstanding problems in cosmology. By analyzing observational Hubble data and Type Ia supernovae, we have been able to place strict constraints on the fractional and cosmological parameters. Our results suggest that the Universe may be older than previously estimated. We also explore whether fractional cosmology can help resolve the $H_0$ tension., Comment: We have gathered information from arXiv:2207.00878 and arXiv:2303.16409 to create a report on the topic of [gr-qc]. The report focuses on the presentation given by Genly Leon at the Tensions in Cosmology Corfu2022 conference, titled "Cosmology under the fractional calculus approach: a possible $H_0$ tension resolution?" (limit of 15 pages)

This paper investigates exact solutions of cosmological interest in fractional cosmology. Given $\mu$, the order of Caputo's fractional derivative, and $w$, the matter equation of state, we present specific exact power-law solutions. We discuss the exact general solution of the Riccati Equation, where the solution for the scale factor is a combination of power laws. Using cosmological data, we estimate the free parameters. An analysis of type Ia supernovae (SNe Ia) data and the observational Hubble parameter data (OHD), also known as cosmic chronometers, and a joint analysis with data from SNe Ia + OHD leads to best-fit values for the free parameters calculated at $1\sigma$, $2\sigma$ and $3\sigma$ confidence levels (CLs). On the other hand, these best-fit values are used to calculate the age of the Universe, the current deceleration parameter (both at $3\sigma$ CL) and the current matter density parameter at $1\sigma$ CL. Finding a Universe roughly twice as old as the one of $\Lambda$CDM is a distinction of fractional cosmology. Focusing our analysis on these results, we can conclude that the region in which $\mu>2$ is not ruled out by observations. This parameter region is relevant because fractional cosmology gives a power-law solution without matter, which is accelerated for $\mu>2$. We present a fractional origin model that leads to an accelerated state without appealing to $\Lambda$ or dark energy., Comment: 56 pages, 10 figures. This article belongs to the Special Issue Fractional Gravity/Cosmology in Classical and Quantum Regimes in Fractal and Fractional

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