Scalar perturbations on the normal and self-accelerating branch of a DGP brane and ${\sigma }_8$

In this work we constrain the value of ${\sigma }_8$ for the normal and self-accelerating branch of a DGP brane embedded in a five-dimensional Minkowski space-time. For that purpose we first constrain the model parameters $H_0$, ${\mathrm{\Omega }}_{m0}$, ${\mathrm{\Omega }}_{r0}$ and $M$ by means of the Pantheon+ catalog and a mock catalog of gravitational waves. Then, we solve numerically the equation for dark matter scalar perturbations using the dynamical scaling solution for the master equation and assuming that $p=4$ for the matter dominated era. Finally, we found that the evolution of matter density perturbations in both branches is different from the $\mathrm{\Lambda }$CDM model and that the value of ${\sigma }_8=0.774\pm 0.027$ for the normal branch and ${\sigma }_8=0.913\pm 0.032$ for the self-accelerating branch.