Charge scaling and universality in critical collapse

Consider any one-parameter family of initial data such that data with a parameter value $p>\mathrm{}{p}_{*}$ form black holes, and data with $p<\mathrm{}{p}_{*}$ do not. As $p\to p_{*}$ from above ("critical collapse"), the black hole mass scales as $M\sim {(p-p_{*})}^{\gamma }$, where the critical exponent $\gamma$ is the same for all such families of initial data. So far critical collapse has been investigated only for initial data with zero charge and zero angular momentum. Here, we allow for U(1) charge. In scalar electrodynamics coupled to gravity, with action $R+{\left|\right(\partial +iqA\left)\phi \right|}^2+F^2$, we consider initial data with spherical symmetry and a nonvanishing charge. From dimensional analysis and a previous calculation of Lyapunov exponents, we predict that in critical collapse the black hole mass scales as $M\sim {(p-p_{*})}^{\gamma }$, and the black hole charge as $Q\sim {(p-p_{*})}^{\delta }$, with $\gamma =0.374\mathrm{}\pm 0.001$ (as for the real scalar field) and $\delta =0.883\mathrm{}\pm 0.007$. We conjecture that, where there is no mass gap, this behavior generalizes to other charged matter models, with $\delta >~2\gamma$. We suggest the existence of universality classes with respect to parameters such as $q$.