The Void Hierarchy: Void-in-Void Merging and Void-in-Cloud Destruction Rates

Voids live inside larger environments, and their fate depends on the sign of that environment. A void inside a larger underdensity merges into it and survives as part of a bigger void; a void inside an overdensity is squeezed and eventually erased, the two modes that give the void population its hierarchy. This paper estimates both rates with a Gaussian environment model. For a void of 25 megaparsecs per h embedded in an environment of 80 megaparsecs per h, whose linear amplitude is 0.115, the probability of sitting in an underdense environment at all is 0.50, and the fraction meeting the adopted destruction criterion is 0.111. The destruction fraction falls from 0.169 at 15 megaparsecs per h to 0.042 at 40 megaparsecs per h: small voids are the vulnerable population, and the observed scarcity of small voids relative to naive counts is partly a destruction record rather than a formation record. Catalogues that count voids without modelling the destruction of the small ones will misread the small-radius end of the size function, whatever cosmology they assume. All values are computed from stated assumptions, not measured.