Figure 1

Values of ${log}_{10}({\Delta }_l)$ versus $l$. In this figure we can see that ${\Delta }_l$ decreases almost exponentially with $l$. The different lines represent different families of initial data. Assuming universality, the differences between the values calculated for the different families provides one measure of error in our determination of ${\Delta }_l$.Reuse & Permissions

Figure 2

Values of ${log}_{10}({\gamma }_l)$ versus $l$, where ${\gamma }_l$ is the scaling exponent defined by (35). As for the case of the echoing exponent, ${\Delta }_l$, ${\gamma }_l$ also decreases approximately exponentially with $l$. We note that due to lack of numerical precision we can only reliably compute ${\gamma }_l$ for $l\le 6.5$.Reuse & Permissions

Figure 3

Evolution in time of the maximum in $r$ of the function $2M(t,r)/r$ for four different critical solutions with increasing value of $l$ ($l=0$, $l=1$, $l=2$, and $l=4$). The plot shows the evolution during the period of time when each solution shows discrete self-similarity. The time coordinate is rescaled by ${\Delta }_l$ for visualization purposes and is shifted so that the function values coincide at $t=0$. We note how the solutions tend to periodicity with increasing values of $l$.Reuse & Permissions

Figure 4

$\underset{t}{max}\{\underset{r}{max}\{2M(t,r)/r\}\}$ in the critical regime as a function of $l$ (solid line) and the same for $\underset{t}{min}\{\underset{r}{max}\{2M(t,r)/r\}\}$ (dashed line). We see how both the maximum and minimum values of $2M/r$ increase with $l$. On the other hand the amplitude of oscillation, given by their difference, apparently tends to zero with increasing $l$.Reuse & Permissions

Figure 5

In the top pane we show the spatial profiles (in the region of self-similarity) of the scalar field $\psi$ for different families of initial data, but for fixed angular momentum parameter $l=9$. In particular we show the solutions ${\psi }_F$ calculated from initial data types $F=1,\dots ,6$ (see Table ) at times when ${\psi }_F$ reaches maximum amplitude. Each solution is shifted by an amount proportional to its family number for better visualization, with $F=1$ the bottom curve, and $F=6$ the top. The $r$ coordinate is rescaled for each family by a constant factor $K_F$, which is family dependent, in such a way that the difference with respect to the profile obtained for the initial data labeled with $F=1$ (for which we consider $K_1=1$) is minimized. In the bottom pane we show the differences between the rescaled profiles for $F=2,\dots ,6$ and the profile for $F=1$, divided by the ${\ell }_2$ norm of the solution. The maximum relative difference is of the order of a few percent, providing strong evidence that the critical solution is universal.Reuse & Permissions

Figure 6

$T_l^{\star }{\Delta }_l$ as a function of $l$. The fact that these products remain finite as $T_l^{\star }\to \infty$ and ${\Delta }_l\to 0$ is evidence that the critical solutions tend to a periodic solution in the limit $l\to \infty$.Reuse & Permissions

Figure 7

Fit of the times $T_n$ at which $\underset{r}{max}\{2M(t,r)/r\}$ reaches its maximum in time (triangles, left scale) assuming a periodic ansatz. Initial data type $F=1$ was used with angular momentum parameter $l=10$. We also plot the residuals of each data point with respect to the best fit (pentagons, right scale).Reuse & Permissions

Figure 8

Fit of the times $T_n$ at which $\underset{r}{max}\{2M(t,r)/r\}$ reaches its maximum in time (triangles, left scale) assuming a self-similar ansatz. As in the previous plot, initial data type $F=1$ was used with angular momentum parameter $l=10$. Again, we also plot the residuals of each data point with respect to the best fit (pentagons, right scale). Notice that the errors in the fit are of the same order as the errors in the fit that assumes periodicity (Fig. 7), indicating that from our numerical results we are unable to distinguish between the two types of solutions for $l\ge 10$.Reuse & Permissions