Approximate initial data for binary black holes

We construct approximate analytical solutions to the constraint equations of general relativity for binary black holes of arbitrary mass ratio in quasicircular orbit. We adopt the puncture method to solve the constraint equations in the transverse-traceless decomposition and consider perturbations of Schwarzschild black holes caused by boosts and the presence of a binary companion. A superposition of these two perturbations then yields approximate, but fully analytic binary black hole initial data that are accurate to first order in the inverse of the binary separation and the square of the black holes’ momenta.

Figure 1

The top panel shows the ADM energy (solid line) and irreducible mass (dashed) for a boosted black hole. The lines are drawn according to Eqs. (22, 25); the points are numerical results, with error bars smaller than the points. The bottom panel shows the difference between our analytical and numerical results. The dotted lines have slopes of 4, indicating that the errors in our analytical results are of order ${ϵ}_P^4$. The deviation from this scaling for very small values of $P/\mathcal{M}$ is caused by the truncation error in the numerical data.

Figure 2

The top panel shows perturbative [solid line, Eq. (27)] and numerical (dots) results for the ADM energy of a boosted black hole. The dashed line shows the ADM energy of a properly boosted Schwarzschild black hole, Eq. (28). The bottom panel shows the deviation $(E_{\mathrm{num}}-E_{\mathrm{Sch}})/M$ introduced by representing the boosted black hole with conformally flat, maximally sliced puncture data (triangles) as well as the error $|E_{\mathrm{ana}}-E_{\mathrm{num}}|/M$ introduced by our perturbative solution (squares). The dotted lines have slopes of 4, indicating that all errors scale as ${ϵ}_P^4$.

Figure 3

The top panel shows the irreducible mass for a black hole with a companion a coordinate distance $s$ away. Two cases with the same total bare mass ${\mathcal{M}}_{\mathrm{tot}}={\mathcal{M}}_1+{\mathcal{M}}_2$ but different reduced bare masses ${\mu }_{\mathrm{bare}}={\mathcal{M}}_1{\mathcal{M}}_2/{\mathcal{M}}_{\mathrm{tot}}$ are shown: ${\mathcal{M}}_1={\mathcal{M}}_2=1$ and ${\mathcal{M}}_1=\frac{3}{2}$, ${\mathcal{M}}_2=\frac{1}{2}$. The line drawn is the prediction of Eq. (33), and is the same for each case. The points are numerical results: squares for the equal mass case, and circles and triangles for the larger and small masses, respectively. The bottom panel shows the difference between our analytical and numerical results. The dotted lines have slopes of $-5$, indicating that the errors in our analytical results are of order ${ϵ}_s^5$.

Figure 4

The deviation in binding energy $E_b/\mu$ of an equal mass binary between our calculation (which yields the Newtonian result) and more sophisticated approaches as a function of orbital frequency. We compare with the post-Newtonian expansion of [41] as well as numerical relativity results [6, 10]. The line for “1PN” has precisely the slope of our error term, $(M_{\mathrm{tot}}\Omega {)}^{4/3}$, confirming that our analysis finds the correct scaling.