Final spin and radiated energy in numerical simulations of binary black holes with equal masses and equal, aligned or antialigned spins

The behavior of merging black holes (including the emitted gravitational waves and the properties of the remnant) can currently be computed only by numerical simulations. This paper introduces ten numerical relativity simulations of binary black holes with equal masses and equal spins aligned or antialigned with the orbital angular momentum. The initial spin magnitudes have $|{\chi }_i|\lesssim 0.95$ and are more concentrated in the aligned direction because of the greater astrophysical interest of this case. We combine these data with five previously reported simulations of the same configuration, but with different spin magnitudes, including the highest spin simulated to date, ${\chi }_i\approx 0.97$. This data set is sufficiently accurate to enable us to offer improved analytic fitting formulas for the final spin and for the energy radiated by gravitational waves as a function of initial spin. The improved fitting formulas can help to improve our understanding of the properties of binary black hole merger remnants and can be used to enhance future approximate waveforms for gravitational wave searches, such as effective-one-body waveforms.

Figure 1

Effect of spin on horizon geometry. This image shows the intrinsic Ricci scalar on the apparent horizons in simulation $S_{0.85}^{++}$. The proper separation of the horizons along the line connecting their centers is about $1.7M$. Both spin effects (gradients as a function of polar angle) and tidal bulges (dark regions near the intersection with the line connecting the horizon centers) can be seen.

Figure 2

Normalized constraint violations for $S_{0.9}^{--}$. For each resolution level $k$, we plot $\Vert \mathcal{C}\Vert$, the volume-averaged $L^2$-norm of the generalized harmonic constraint energy divided by the volume-averaged $L^2$-norm of the dynamical field gradients. This measure is defined in Eq. (71) of Ref. 50. As the resolution level increases, the constraints decrease. Jumps in the constraints are attributed to changes in the domain structure, and the spike around $t\sim 3500M$ corresponds to the merger.

Figure 3

Plots of the apparent horizon quantities as a function of time for a representative case, $S_{0.9}^{++}$. The top panels display the dimensionless spin and the bottom panels display the Christodoulou mass. From left to right, the panels display the inspiral, merger, and ringdown. We normalize the $y$ scales separately so that the differences between each resolution can be clearly seen. The discontinuity in the middle panel indicates where we begin to measure the mass and spin on the common horizon. The dots in the early inspiral identify our choice of $t_i$ for each resolution level.

Figure 4

Differences in the final masses and spins between resolution levels. For each case, we compare ${\chi }_f$ and $M_f$ of the highest resolution to the two lower resolutions. Note that, except for the older $S_{0.44}^{\pm \pm }$ simulations, all differences are $\lesssim {10}^{-4}$. Differences in the initial masses and spins behave similarly.

Figure 5

Convergence of the final spin for $S_{0.9}^{--}$ as a function of the $\ell$ of the horizon finder. We plot the difference between ${\chi }_f(\ell )$ and ${\chi }_f(\ell =20)$ for each resolution. The adaptive horizon finder for this case chose $\ell =8$ at the final time of the simulation.

Figure 6

In the top panel, we plot our preferred fitting formula (solid line), the fourth-order polynomial in Eq. (6), and a comparison with a second-order polynomial (dashed line) for ${\chi }_f$ as a function of ${\chi }_i$. Our data points are plotted as polygons, where more sides indicates higher resolution level. In the bottom panel, we plot the residuals of the fourth-order polynomial. We indicate our fit parameter (dotted line) and total prediction (dashed line) uncertainties (defined in the Appendix), which in this case are nearly identical. Note that the residuals for the two lower resolution runs for $S_{0.44}^{++}$ are too large to fit in this panel.

Figure 7

Final spin as a function of initial spin. In the left panels, we plot our data (circles) along with the fitting formula (red line) with error estimates (dashed line) from several other studies. The top panel is from Ref. 29, the middle is from Ref. 27, and the bottom is from Ref. 28. In the right panels, we plot the difference between our data and the corresponding fitting formula on the left. The value $r$ quantifies the size of the systematic error compared to the fourth-order polynomial.

Figure 8

In the top panel, we plot our preferred fitting formula (solid line), the hyperbolic function in Eq. (9), and a comparison with a second-order polynomial (dashed line) for $E_{\mathrm{rad}}$ as a function of ${\chi }_i$. Our data points are plotted as polygons, where more sides indicates higher resolution level. In the bottom panel, we plot the residuals of the hyperbolic function. We give our fit parameter (dotted line) and total prediction (dashed line) uncertainties (defined in the Appendix).

Figure 9

$E_{\mathrm{rad}}$ as a function of initial spin. In the left panels, we plot our data (circles) along with the fitting formula (red line) with error estimates (dashed line) from several other studies. The top panel is from Ref. 29, the middle is from Ref. 30, and the bottom is from Ref. 31. In the right panels, we plot the difference between our data and the corresponding fitting formula on the left. The value $r$ quantifies the size of the systematic error compared to the three-parameter hyperbola.

Figure 10

Directed acyclic graph displaying the conditional dependence structure of the Bayesian nonlinear measurement error model adopted for the fitting function analysis. See text for a detailed description.