Evolution of density perturbations in fractional Newtonian cosmology

Abstract: In this work, density perturbations are investigated within the framework of a fractional Newtonian cosmology. Focusing on the matter-dominated era and employing the fluid-flow approach, the growth equation for density perturbations is derived and solved analytically. No dynamical instability arises in the physically relevant parameter space. It is shown that both the growth equation and its solutions depend explicitly on the fractional parameter $α$, and reduce to their standard Newtonian and relativistic counterparts in the special limit $α= 1$. The existence of both growing and decaying perturbative modes is confirmed, and, in accordance with current cosmological observations, the analysis is restricted to the growing mode. Using observational relations, in particular the Sachs--Wolfe equation, an observational upper bound on the parameter $α$ is obtained, which is more restrictive than the bounds inferred from background dynamics and theoretical perturbative considerations. When combined with the independent constraints arising from the background analysis, these results confine the fractional parameter $α$ to a narrow and physically viable region of parameter space. Overall, the present study indicates that, although the background evolution of the fractional model may closely mimic that of $Λ$CDM, the associated density perturbations generically carry a distinct fractional signature that can, in principle, be tested observationally.