Twist and higher modes of a complex scalar field at the threshold of collapse

We investigate the threshold of collapse of a massless complex scalar field in axisymmetric spacetimes under the ansatz of Choptuik et al. 2004, in which a symmetry depending on the azimuthal parameter $m$ is imposed on the scalar field. This allows for both non-vanishing twist and angular momentum. We extend earlier work to include higher angular modes. Using the pseudospectral code bamps with a new adapted symmetry reduction method, which we call $m$-cartoon, and a generalized twist-compatible apparent horizon finder, we evolve near-critical initial data to the verge of black hole formation for the lowest nontrivial modes, $m=1$ and $m=2$. For $m=1$ we recover discrete self-similarity with echoing period $Δ\simeq0.42$ and power-law scaling with exponent $γ\simeq0.11$, consistent with earlier work. For $m=2$ we find that universality is maintained within this nonzero fixed-$m$ symmetry class but with smaller period and critical exponents, $Δ\simeq0.09$ and $γ\simeq0.035$, establishing an explicit dependence of the critical solution on the angular mode. Analysis of the relation between the angular momentum and the mass of apparent horizons at the instant of formation, $J_{\mathrm{AH}}{-}M_{\mathrm{AH}}$, shows that the effect of angular momentum is minimal at the threshold, with $χ_{\mathrm{AH}}=J_{\mathrm{AH}}/M_{\mathrm{AH}}^2\to0$, and, therefore, excludes extremal black holes for the families under consideration. Our results demonstrate that while universality and DSS hold within each $m$-sector, the critical universal values vary with $m$, and neither extremality nor bifurcation occur in the complex scalar field model within the families considered here.