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 (p-p^*). We predict that the period is universal and that it depends on the previous universal parameters, the critical exponent, and the echoing period as =/ (2).