We study the evolution of linear density perturbations in a large spherical void universe which accounts for the acceleration of the cosmic volume expansion without introducing dark energy. The density contrast of this void is not large within the light cone of an observer at the center of the void. Therefore, we describe the void structure as a perturbation with a dimensionless small parameter $kappa$ in a homogeneous and isotropic universe within the region observable for the observer. We introduce additional anisotropic perturbations with a dimensionless small parameter $epsilon$, whose evolution is of interest. Then, we solve perturbation equations up to order $kappa epsilon$ by applying second-order perturbation theory in the homogeneous and isotropic universe model. By this method, we can know the evolution of anisotropic perturbations affected by the void structure. We show that the growth rate of the anisotropic density perturbations in the large void universe is significantly different from that in the homogeneous and isotropic universe. This result suggests that the observation of the distribution of galaxies may give a strong constraint on the large void universe model.
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