Central density cusps in the Lema\^{i}tre-Tolman solutions

The character of the central density profile in the Lema^{i}tre-Tolman (LT) solutions plays a fundamental role in their application as cosmological models. This same character is studied here for these solutions used to model complete gravitational collapse. A necessary condition for the development of a black hole (not even locally naked singularities) is developed. This condition allows central density cusps, the central feature of the LT solutions when used to match cosmological observations without invoking the cosmological constant.