Alternatives to standard puncture initial data for binary black hole evolution

Standard puncture initial data have been widely used for numerical binary black hole evolutions despite their shortcomings, most notably the inherent lack of gravitational radiation at the initial time that is later followed by a burst of spurious radiation. We study the evolution of three alternative initial data schemes. Two of the three alternatives are based on post-Newtonian expansions that contain realistic gravitational waves. The first scheme is based on a second-order post-Newtonian expansion in Arnowitt, Deser, and Misner transverse-traceless (ADMTT) gauge that has been resummed to approach standard puncture data at the black holes. The second scheme is based on asymptotic matching of the 4-metrics of two tidally perturbed Schwarzschild solutions to a first-order post-Newtonian expansion in ADMTT gauge away from the black holes. The final alternative is obtained through asymptotic matching of the 4-metrics of two tidally perturbed Schwarzschild solutions to a second-order post-Newtonian expansion in harmonic gauge away from the black holes. When evolved, the second scheme fails to produce quasicircular orbits (and instead leads to a nearly head-on collision). This failure can be traced back to inaccuracies in the extrinsic curvature due to low-order matching. More encouraging is that the latter two alternatives lead to quasicircular orbits and show gravitational radiation from the onset of the evolution, as well as a reduction of spurious radiation. Current deficiencies compared to standard punctures data include more eccentric trajectories during the inspiral and larger constraint violations, since the alternative data sets are only approximate solutions of Einstein’s equations. The eccentricity problem can be ameliorated by adjusting the initial momentum parameters.

Figure 1

The real part of the dominant $\ell =2$, $m=2$ waveform mode of ${\psi }_4$ over time extracted at a distance of $r=90M$. The lack of a gravitational wave signature for the initial phase of the evolution is a direct consequence of prescribing conformal flatness. The burst behavior (visible as the waveform transitions from zero to a regular chirp signal) is indicative of a so-called spurious radiation signal.

Figure 2

This figure illustrates the separation of the spacetime into specific zones: the inner zones, near zone, and far zone. In each zone one can use a specific approximation. The corresponding buffer zones are the regions where two zones overlap.

Figure 3

The (2,2) mode of ${\psi }_4$ extracted at a radius of $r=90M$ for standard punctures and ADMTT-match initial data sets. Both waveforms show no initial wave signature at $t=0$, due to their reliance on conformal flatness for the 3-metric expressions. It can also be seen that ADMTT-match merges much faster (within $250M$). This is evidence that the orbits were not quasicircular at all.

Figure 4

Trajectories of both black holes in the ADMTT-match approach. A nearly head-on collision between the two holes indicates there is insufficient tangential momenta to satisfy a PN quasicircular orbit starting from $t=0$.

Figure 5

The (2,2) mode of ${\psi }_4$ extracted at a radius of $r=90M$ for standard punctures, harmonic-match, and ADMTT-PN data sets.

Figure 6

Coordinate separation for standard punctures, harmonic-match, unmodified ADMTT-PN, and modified ADMTT-PN.

Figure 7

Eccentricity estimation for standard punctures, harmonic-match, unmodified ADMTT-PN, and modified ADMTT-PN. The range between $100M$ and $600M$ is the range to analyze all sets simultaneously while guaranteeing no errors due to incomplete information. (Before $100M$ our eccentricity estimator which averages over an orbital time is unreliable, and after $600M$ most sets have begun merger so that one cannot define an eccentricity.)

Figure 8

Apparent horizon mass of one of the black holes in the inspiral phase under the standard punctures, harmonic-match, and ADMTT-PN approach.

Figure 9

Apparent horizon masses of the final black hole after merger under the standard punctures, harmonic-match, and ADMTT-PN approach.

Figure 10

The $L^2$ norm of the Hamiltonian constraint violation for the standard punctures, harmonic-match, and ADMTT-PN approach. The $y$ axis is plotted in ${log}_{10}$. The inset shows the violations at the beginning of the evolution.

Figure 11

The $L^2$ norm of the $x$ component of the momentum constraint violation, for the standard punctures, harmonic-match, and ADMTT-PN approach. The $y$ axis is plotted in ${log}_{10}$. The inset shows the violations at the beginning of the evolution.