A hierarchy of voids: more ado about nothing

We extend earlier work on the problem of estimating the void-volume function – the abundance and evolution of large voids which grow gravitationally in an expanding universe – in two ways. The first removes an ambiguity about how the void-in-cloud process, which erases small voids, should be incorporated into the excursion set approach. The main technical change here is to think of voids within a fully Eulerian, rather than purely Lagrangian, framework. The second accounts for correlations between different spatial scales in the initial conditions. We provide numerical and analytical arguments showing how and why both changes modify the predicted abundances substantially. In particular, we show that the predicted importance of the void-in-cloud process depends strongly on whether or not one accounts for correlations between scales. With our new formulation, the void-in-cloud process dramatically reduces the predicted abundances of voids if such correlations are ignored, but only matters for the smallest voids in the more realistic case in which the spatial correlations are included.