Geometrical origin for the compaction function for primordial black hole formation

We propose a geometrical origin for the Shibata-Sasaki compaction function, which is known to be a reliable indicator of primordial black hole formation at least during radiation domination. In the long-wavelength limit, we identify it with a compactness function in the static spacetime obtained by removing the cosmological scale factor from the metric and this explains why it cannot be greater than $1/2$. If its maximum is below $1/2$, the perturbation is of type I. If its maximum equals $1/2$, it corresponds to an extremal surface, which is simultaneously a bifurcating trapping horizon and admits a circular photon orbit in the static spacetime. In the long-wavelength regime of the physical expanding Universe, the Shibata-Sasaki compaction reaches its maximum value of $1/2$ at maximal and minimal surfaces on the constant time spacelike hypersurface, which feature a type II perturbation and both correspond to photon spheres expanding along with the cosmological expansion. Thus, the Shibata-Sasaki compaction measures how close to the type II configuration the perturbed region is.