Fine structure of Choptuik’s mass-scaling relation

We conjecture (analytically) and demonstrate (numerically) the existence of a fine structure above the power-law behavior of the mass of black holes that form in the gravitational collapse of a spherical massless scalar field. The fine structure is a periodic function of the critical separation $\left(p-p^{*}\right)$. We predict that the period $\varpi$ is universal and that it depends on the previous universal parameters, the critical exponent $\beta$, and the echoing period $\Delta$ as $\varpi =\Delta /\left(2\mathrm{}\beta \right)$.