  In this letter, we argue and show numerically that the threshold to form
primordial black holes from an initial spherically symmetric perturbation is,
to an excellent approximation, universal, whenever given in terms of the
compaction function averaged over a sphere of radius $r_m$, where $r_m$ is the
scale on which the compaction function is maximum. This can be understood as
the requirement that, for a black hole to form, each shell of the averaged
compaction function should have an amplitude exceeding the so-called
Harada-Yoo-Kohri limit. For a radiation dominated universe we argued, supported
by the numerical simulations, that this limit is $\delta_c = 0.40$, which is
slightly below the one quoted in the literature. Additionally, we show that the
profile dependence of the threshold for the compaction function is only
sensitive to its curvature at the maximum. We use these results to provide an
analytic formula for the threshold amplitude of the compaction function at its
maximum in terms of the normalised compaction function curvature at $r_m$.
