Type II critical collapse of a self-gravitating nonlinear $\sigma$ model

We report on the existence and phenomenology of type II critical collapse within the one-parameter family of SU(2) σ models coupled to gravity. Numerical investigations in spherical symmetry show discretely self-similar (DSS) behavior at the threshold of black hole formation for values of the dimensionless coupling constant η ranging from 0.2 to 100; at 0.18 we see small deviations from DSS. While the echoing period Δ of the critical solution rises sharply towards the lower limit of this range, the characteristic mass scaling has a critical exponent γ which is almost independent of η, asymptoting to 0.1185±0.0005 at large η. We also find critical scaling of the scalar curvature for near-critical initial data. Our numerical results are based on an outgoing–null-cone formulation of the Einstein-matter equations, specialized to spherical symmetry. Our numerically computed initial-data critical parameters $p^{*}$ show second order convergence with the grid resolution, and after compensating for this variation in $p^{*},$ our individual evolutions are uniformly second order convergent even very close to criticality.