Cosmic string loop collapse in full general relativity

We present the first fully general relativistic dynamical simulations of Abelian Higgs cosmic strings using $3+1\mathrm{D}$ numerical relativity. Focusing on cosmic string loops, we show that they collapse due to their tension and can either (i) unwind and disperse or (ii) form a black hole, depending on their tension $G\mu$ and initial radius. We show that these results can be predicted using an approximate formula derived using the hoop conjecture, and argue that it is independent of field interactions. We extract the gravitational waveform produced in the black hole formation case and show that it is dominated by the $l=2$ and $m=0$ mode. We also compute the total gravitational wave energy emitted during such a collapse, being $0.5\pm 0.2\%$ of the initial total cosmic string loop mass, for a string tension of $G\mu =1.6\times {10}^{-2}$ and radius $R=100M_{\mathrm{Pl}}^{-1}$. We use our results to put a bound on the production rate of planar cosmic strings loops as $N\lesssim {10}^{-2}{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$.

Figure 1

GW for a BH formed from circular cosmic string loop collapse: We plot the real part of the dominant $l=2$ $m=0$ mode of $r{\mathrm{\Psi }}_4$ over time. The loop has tension $G\mu =1.6\times {10}^{-2}$ and an initial radius $R=100M_{\mathrm{Pl}}^{-1}$. The grey shaded area of the plot are mixed with stray GWs that arise as artifacts of the initial data. The x-axis $t_{\mathrm{ret}}=t-r_{\mathrm{ext}}$ is the retarded time where $r_{\mathrm{ext}}$ is the extraction radius.

Figure 2

Overview of simulations: The loop can either form a BH or unwind and radiate all its mass. The analytical expression derived from the hoop conjecture accurately predicts the outcome. Movie links for the evolution over time of the collapse are available for the dispersion [18, 19] and black hole [19, 20] cases.

Figure 3

Critical collapse: We plot the logarithm of the mass of the black hole vs the logarithm of the difference between the initial and the theoretical(star)/observed(cross) critical radius for $G\mu =1.6\times {10}^{-2}$. As we argued in the text, our simulation showed that the actual $R_{*}^{\mathrm{ob}}>R_{*}^{\mathrm{th}}$, resulting in a critical index within $0.17<\gamma <0.39$, where the error is due to the uncertainty in determining numerically $R_{*}^{\mathrm{th}}<R_{*}<R_{*}^{\mathrm{ob}}$. Note that we only use the first 7 points to compute the critical index for $R\le 0.05R_{*}$ as the critical relation is only expected to hold perturbatively.

Figure 4

Gauss constraint for static string: We run the same simulation for am infinite static string with $G\mu =1.6\times {10}^{-2}$ ($\eta =0.05M_{\mathrm{Pl}}$) with and without damping. We find that the damping stabilizes the linear growth in violation.

Figure 5

$L^2$ norm of constraints: Loop with $G\mu =1.6\times {10}^{-2}$ and $R=100M_{\mathrm{Pl}}^{-1}$ remains stable throughout evolution, even after black hole formation. The initial Hamiltonian constraint is smaller than it can be maintained by the evolution scheme. The momentum constraints violation are negligible throughout.

Figure 6

Toroidal coordinates encode the symmetry of our cosmic string loops. They are used to generate the initial field configuration, where R defines the radius of the loop.

Figure 7

Initial relative violation: Slice through initial data for loop from center through string with $G\mu =1.6\times {10}^{-2}$ and initial radius $R=100M_{\mathrm{Pl}}^{-1}$. The green region indicates where the string is located. We find that there is an error of at most 0.3%.

Figure 8

Convergence in $r{\mathrm{\Psi }}_4$ between low, mid and high resolutions giving an overall 2nd-3rd order convergence. The x-axis $t_{\mathrm{ret}}=t-r_{\mathrm{ext}}$ is the retarded time where $r_{\mathrm{ext}}$ is the extraction radius.

Figure 9

Radial profile of $\alpha$ and $f$ for an infinite static string with gravity in the critical coupling limit ($e=1$, $\lambda =2$) and $\eta =0.05M_{\mathrm{Pl}}$ ($G\mu =1.6\times {10}^{-2}$).

Figure 10

Comparison with Nambu-Goto for loop with $G\mu =1.6\times {10}^{-2}$ and initial radius $R=100M_{\mathrm{Pl}}^{-1}$ shows agreement.