Perturbations in a spherically symmetric inhomogeneous cosmological model with the self-similar region

To clarify the evolution of inhomogeneous substructures in large-scale structures such as voids, we study the perturbations in a spherically symmetric inhomogeneous model with the self-similar region outside the inner low-density homogeneous region, first based on the Gerlach-Sengupta general-relativistic formulation for perturbations in spherically symmetric inhomogeneous models. Owing to self-similarity, the analysis for all kinds of perturbations at the early stage is simplified in a similar way to homogeneous models. Next we take the approximate treatment due to the local homogeneous model with the average density parameter which is considered at each point, in order to study the behavior of perturbations on small scales. It is found that the growth rate of density perturbations in the outside self-similar region can be larger by about $30-20\%$ (for $z=2\mathrm{}-3)$ than that in the corresponding ordinary low-density homogeneous model.