How empty are the voids?

Abstract

We find an analytical solution for the minimal matter density of a void, its central density. It turns out that the voids are not so empty: most of the voids have the central underdensity $\Delta_c \sim -50\%$ (which means that the matter density in their centers is only two times lower than in the Universe on average). For small voids (of radius $R_0\simeq 5-10$~{Mpc}), the underdensity can be significantly greater, but the number of voids decreases rapidly with increasing of $|\Delta_c|$ over $50\%$, and voids with $\Delta_c < -80\%$ are practically absent. The large voids ($R_0\ge 40$~{Mpc}) always have $|\Delta_c| < 50\%$.