The energy shear of protohaloes

As it collapses to form a halo, the shape of a protohalo patch is deformed by the initial shear field. This deformation is often modeled using the ‘deformation’ tensor, constructed from second derivatives of the gravitational potential, whose trace gives the initial overdensity. However, especially for lower mass protohaloes, this matrix is not always positive definite: one of its eigenvalues has a different sign from the others. We argue that the evolution of a patch is better described by the ‘energy shear’ tensor, which is actually positive definite and plays a direct role in the evolution, and test our analytical result against N-body simulations. We discuss the implications of this positive-definiteness for analytical models of halo abundances, assembly and of the cosmic web.