Evolution of density perturbations in fractional cosmology

We investigate the evolution of matter density perturbations within a fractional cosmological framework inspired by fractal space-time constructions in field theory, where a deformation of the integration measure induces non-locality and memory effects in the dynamics. Working in the matter-dominated era and adopting a covariant fluid-flow approach, we derive the modified growth equation for the density contrast and obtain exact analytical solutions. The resulting dynamics depends explicitly on the fractional parameter $\alpha$ and smoothly reduces to the corresponding standard case in the limit $\alpha=1$. We show that the model admits both growing and decaying modes, and we identify the parameter range in which structure formation is physically viable. Focusing on the growing mode, we compute the evolution of density fluctuations from recombination to the present epoch. By confronting theoretical predictions with observational constraints from large-scale structure, in particular the $\sigma_8$ normalization and the Sachs-Wolfe effect, we derive a stringent upper bound on the fractional parameter, $\alpha\lesssim1.07$, which significantly improves upon previous constraints obtained at the background level. Our results show that the growth of density perturbations exhibits distinct fractional signatures, providing a sensitive observational probe of the underlying framework.