Getting in shape with minimal energy. A variational principle for protohaloes

In analytical models of structure formation, protohalos are routinely assumed to be peaks of the smoothed initial density field, with the smoothing filter being spherically symmetric. This works reasonably well for identifying a protohalo’s center of mass, but not its shape. To provide a more realistic description of protohalo boundaries, one must go beyond the spherical picture. We suggest that this can be done by looking for regions of fixed volume, but arbitrary shape, that minimize the enclosed energy. Such regions are surrounded by surfaces over which (a slightly modified version of) the gravitational potential is constant. We show that these equipotential surfaces provide an excellent description of protohalo shapes, orientations and associated torques.