Final velocity and radiated energy in numerical simulations of binary black holes

The evolution of global binary black holes variables such as energy or linear momentum are mainly obtained by applying numerical methods near coalescence, post-Newtonian (PN) expansions, or a combination of both. In this paper, we use a fully relativistic formalism presented several years ago that only uses global variables defined at null infinity together with the gravitational radiation emitted by the source to obtain the time evolution of such variables for binary black holes (BBH) systems. For that, we use the Rochester catalog composed of 776 BBHs simulations. We compute the final velocity, radiated energy, and intrinsic angular momentum predicted by the dynamical equations in this formalism for nonspinning, aligned and antialigned spins, and several different precessing configurations. We compare obtained values with reported values in numerical simulations. As BBHs parameter space is still not completely covered by numerical simulations, we fit phenomenological formulas for practical applications to the radiated energy and final velocities obtained. Also, we compare the fits with reported values. In conclusion, we see that our formulae and correlations for the variables described in this work are consistent with those found in the general literature.

Figure 1

Correlation between the final center-of-mass velocity, $V_f$ (in km/s) and the initial total angular momentum, $J_{\mathrm{in}}$.

Figure 2

Bar plots of the distribution from main variables of interest obtained after evolution in all the simulations.

Figure 3

Correlation between total radiated energy $E_{\mathrm{rad}}$ and the peak radiated energy ${\dot{M}}_{\max }$ for all simulations. The dashed line shows fit to data. The color bar indicates the value of mass ratio $q=m_1/m_2$.

Figure 4

Correlation between the radiated energy $E_{\mathrm{rad}}$ and the initial angular momentum $J_{\mathrm{in}}$ for all the simulations.

Figure 5

Correlation between the final velocity $V_f$ and mass ratio $q$. The color bar indicates the maximum energy lost ${\dot{M}}_{\max }$. The dashed line shows the Fitchett model fit.

Figure 6

Correlation between the total radiation emitted $E_{\mathrm{rad}}$ and the initial total angular momentum $J_{\mathrm{in}}$. The color bar indicates different levels of the initial mass ratio $q$. Letter $k$ indicates polynomial degree of the fit.

Figure 7

Correlation between the peak energy radiated ${\dot{M}}_{\max }$ and mass quotient $q$. The color bar indicates values of energy radiated that are proportional to the peak energy loss. The dashed line shows the model fit to data.

Figure 8

Correlation between final velocity $V_f$ and mass ratio $q$ for aligned binaries. (a) Red and blue colors denote alignment of spin $S_2$ with respect to orbital angular momentum L_in. Green colors represent binaries whose $S_2$ alignment with respect to L_in can not be established or dierentiated. Dierent markers are used to denote alignment type between spins $S_1$ and $S_2$, (b) Red and blue colors denote alignment of spin $S_2$ with respect to orbital angular momentum L_in. Yellow color has $S_2=0$ and thus no alignment can be establish with respect to L_in. Green dots are indierent as $S_1=S_2$. Markers clear out the bigger value of the spin in simulations.

Figure 9

Correlation between $V_f$ and $q$ for aligned binaries. The color bar shows levels of peak energy loss ${\dot{M}}_{\max }$. Dashed line indicates border limit to final velocities (in $\mathrm{km}/\mathrm{s}$).

Figure 10

Correlation between initial total angular momentum $J_{\mathrm{in}}$ and total radiation emitted $E_{\mathrm{rad}}$ for aligned binaries. Mass ratio intervals have been divided and selected to be studied and are illustrated with different colors.

Figure 11

Correlation between initial total spin $\overline{\chi }$ and total radiation emitted $E_{\mathrm{rad}}$. The dashed gray line shows the fit to our data and below its residuals. The dashed light blue line shows Reisswig polynomial for comparison purposes.

Figure 12

Detailed description of the many variable dependence for the precessing binaries. (a) Correlation between initial total angular momentum $J_{\mathrm{in}}$ and total radiation emitted $E_{\mathrm{rad}}$, (b) Correlation between nal velocity $V_f$ (in km/s) with the initial mass ratio $q$. Colorbar shows dierent levels of peak energy loss.