Filling the holes: Evolving excised binary black hole initial data with puncture techniques

We follow the inspiral and merger of equal-mass black holes (BHs) by the moving puncture technique and demonstrate that both the exterior solution and the asymptotic gravitational waveforms are unchanged when the initial interior solution is replaced by constraint-violating junk initial data. We apply this result to evolve conformal thin-sandwich (CTS) binary BH initial data by filling their excised interiors with arbitrary, but smooth, initial data and evolving with standard puncture gauge choices. The waveforms generated for both puncture and filled-CTS initial data are remarkably similar, and there are only minor differences between irrotational and corotational CTS BH binaries. Even the interior solutions appear to evolve to the same constraint-satisfying solution at late times, independent of the initial data.

Figure 1

Binary separation versus time and paths traced in coordinate space (inset) for runs P (thin solid), ${\mathrm{P}}_{\mathrm{FJ}}$ (thin dashed), ${\mathrm{P}}_{\mathrm{SJ}}$ (thin dotted), ${\mathrm{CP}}_{\mathrm{I}}$ (thick solid), and ${\mathrm{CP}}_{\mathrm{C}}$ (thick dashed). In the inset, we show the motion of one of the BHs in the equatorial plane, with points representing positions every $10M$ for runs P (open squares) and ${\mathrm{CP}}_{\mathrm{I}}$ (closed squares). $\pi$ symmetry is not enforced, but holds true numerically.

Figure 2

Real part of the $s=-2$ spin-weighted, $l=2$, $m=2$ component of the Weyl scalar ${\psi }_4$, evaluated on a sphere of radius $r=17.8M$. In the top panel we show results for puncture runs P (solid), ${\mathrm{P}}_{\mathrm{FJ}}$ (dashed), and ${\mathrm{P}}_{\mathrm{SJ}}$ (dotted), which differ in the form of the initial junk we impose within each BH horizon. In the bottom panel, we compare run P with our two runs started from CP initial data, both irrotational (dashed) and corotating (dotted). We see extremely close agreement between puncture and irrotational CP results, and only minor differences between these and the corotating CP results.

Figure 3

Waveforms for runs using the initial data of run ${\mathrm{CP}}_{\mathrm{I}}$ (top panel) evaluated at numerical resolutions of $M/24$ (solid), $M/20$ (dashed), and $M/16$ (dotted). Results are time and phase shifted so that the moment and phase of the maximum wave amplitudes coincide. In the bottom panel, we show the differences between the runs, properly rescaled to highlight second-order convergence.

Figure 4

Radial profiles of the quantity $r{\psi }^2$ as a function of coordinate radius, measured out from the BH center for runs P and ${\mathrm{CP}}_{\mathrm{I}}$. The evolution inevitably drives $\psi$ toward the power-law index $\psi \propto r^{-1/2}$ derived in 19. For the curves showing profiles at $t=20M$, we eliminate the two innermost points, whose values are inaccurate due to differencing across the puncture. Crosses denote the initial apparent horizons for each run. In the inset, we plot the value of $r{\psi }^2$ linearly extrapolated from the third and fourth innermost data points as a function of time for the same runs. The filled circles show the asymptotic value $1.3M$ determined analytically by 19.

Figure 5

Violation of the Hamiltonian ($\mathcal{H}=0$) and momentum (${\mathcal{M}}^i=0$) constraints. We show the normalized x-component of the momentum constraint violation, given by Eq. (60) of 30, and a similar quantity for the Hamiltonian constraint (bottom panel), summed over spherical shells about each puncture $0.2<r/M<0.3$, which extend into the BH interiors. The constraint-violating initial data are driven to constraint-satisfying solutions at late times (several $M$).