论文标题
分数演算方法下的宇宙学
Cosmology under the fractional calculus approach
论文作者
论文摘要
分数宇宙学将标准衍生物修改为Caputo的分数$μ$的分数导数，从而产生了一般相对论的变化。修改了弗里德曼方程，物种密度的演变取决于$μ$和宇宙的年龄$ t_u $。我们使用宇宙天元素，IA型超新星和关节分析对$μ$估计严格的约束。我们在$1σ$置信度内获得$μ= 2.839^{+0.117} _ { - 0.193} $，在后期提供非标准的宇宙加速度；因此，宇宙将比标准估计年龄更古老。此外，我们为不同的$ $ $值提供了稳定性分析。该分析确定了与$μ<2 $的幂律减速解决方案相对应的延迟吸引子。此外，以$ u> 1美元的价格存在非相关性的关键点，$μ> 2 $的水槽存在。如果$ 1 <μ<2 $，则该解决方案是减速的幂律，如果$μ> 2 $，则是加速的幂律解决方案，这与从观测分析获得的平均值一致。因此，对于平坦的FLRW和Bianchi I指标，修改的Friedmann方程在此范式下提供了晚期宇宙加速度，而无需引入暗能量。这种方法可能是解决未解决的宇宙学问题的新途径。
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 $t_U$. We estimate stringent constraints on $μ$ using cosmic chronometers, Type Ia supernovae, and joint analysis. We obtain $μ=2.839^{+0.117}_{-0.193}$ 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.
