Halo abundance mass-budget bound ($\int M{n}_h(M)dM\le {\bar{\rho }}_{m,0}$)

A population of distinct, nonoverlapping dark-matter halos cannot contain more total mass per comoving volume than the cosmic mean matter density. A narrow mass population therefore obeys $n_c{M}_h\le {\bar{\rho }}_{m,0}$. Do not sum subhalo masses and their inclusive host masses in the same bound, because that double-counts material. This check can reveal incompatible inferred absorption incidence, object size and halo-mass assumptions.