General perturbations in D+1 standard and brane cosmology revisited

In the present work, we generalize the theory of perturbations in a dimensional space-time D + 1, where D is the number of spatial dimensions, with cosmological constant, studying scalar, vector and tensor perturbations, as well as its structure in Newtonian and synchronous gauge. We also show the theory of perturbations in the context of brane cosmology, where branes are embedded in a set of D-spatial dimensions, a temporal dimension, and an additional spatial dimension. In both standard and brane cosmology, unperturbed spacetime is provided with a Friedmann-Lemaitre-Robertson-Walker metric and arbitrary sectional curvature, the matter content has the shape of a perfect fluid. In addition, we consider the arbitrary sectional curvature, obtaining the respective equations in the Newtonian and synchronous norm. We highlight that the results presented in this article can be used to treat brane cosmology with a more efficient approach.