Cosmology under the fractional calculus approach

Abstract

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

Revisiting Fractional Cosmology

“…Instead of following [116], we present an alternative approach using the Riccati Equation (17) while assuming that the matter components have equations of state p i = w i ρ i , where…”

“…On the other hand, the best-fit µ-value is obtained from the reconstruction of H(z) using Equations ( 19), (20), and (22). The best-fit values (µ * , t * U ) for different priors of µ were derived in [116], and are summarized in Table IX. These best-fit values are used in the numerics.…”

“…The natural generalization of the model studied in [116] is, investigating the influence of the fractional order of the derivative in a fractional theory of gravity, including a scalar field minimally coupled to gravity. Below, we review known results and discuss new results in the context of cosmologies with a scalar field used in the fractional formulation of gravity.…”

Cosmological impact of microwave background temperature measurements

“…Finally, we consider the recently proposed fractional cosmology model [32]. This is based on mathematical formalism of fractional calculus [33]; for our purposes, it suffices to note that it is a parametric extension of the usual concepts of differentiation and integration.…”

“…these recover the standard ΛCDM behaviour for α = 1. A recent analysis [32], using the Pantheon supernova and cosmic chronometers data and assuming Ω Λ = 0, finds α = 2.8 ± 0.2 and Ω m = 0.23 ± 0.04 (implying an age of the universe of t 0 ∼ 33.6 Gyr). That work does not use baryon acoustic oscillation data, and treats the Hubble constant as an additional parameter, In our analysis, we do include baryon acoustic oscillation data (which is part of the Hubble parameter compilation), as well as CMB temperature data, and analytically marginalize H 0 and T 0 .…”