Quasi-Normal Mode Ringing of Binary Black Hole Mergers in Scalar-Gauss-Bonnet Gravity

Abstract

Observations of gravitational waves (GWs) generated by binary black hole (BBH) mergers provide us with a powerful way to explore the strong and highly dynamical regime of gravity theories. The ringdown of BBH merger, consisting of a series of quasi-normal modes (QNMs), is of particular interest for both the black hole (BH) spectroscopy and the inspiral-merger-ringdown consistency check. Unlike the QNM frequencies that only depend on the properties of the remnant BH, the excitation amplitudes and phases of QNMs depend on the progenitor system, and calculating them is beyond the perturbative approach. In this paper, by performing self-consistent fully non-linear simulations of BBH merger in shift-symmetric scalar-Gauss-Bonnet (sGB) gravity as well as in sGB gravity allowing for scalarization, and extracting the QNM excitation, we explore the possible deviations from GR at the ringdown stage. We numerically verify that the mode frequencies are consistent with the theory prediction, and provide the fitting results of mode amplitudes and phases. We find relatively small changes in the mode excitation, considering that the largest coupling we used in the simulations is close to the limit of loss of hyperbolicity. To demonstrate that our results are robust against the eccentricity caused by the imperfect initial data, we also perform eccentricity reduction and estimate the effect caused by the initial eccentricity. These studies are useful for understanding the ringdown in sGB gravity.

Figures (17)

• Figure 1: The waveform of $\Psi^{33}_4$ extracted at $r=90\, M$ and the average Hamiltonian constraint violation at the AH for a single perturbed BH simulation in shift-symmetric sGB gravity with an initial axial $l=3$, $m=3$ perturbation. $u=t-r$ is the Bondi time, such that one can roughly compare the quantities in the two panels. It can be seen that, when the ringdown starts, the constraint violation is largely damped.
• Figure 2: Convergence of the waveform ${\rm Re}\,\Psi_4^{33}$ for simulations of a single perturbed BH. The three simulations have finest grid resolutions $\Delta x=M/32$, $M/40$, and $M/48$, and they are denoted by the subscripts $128$, $160$, and $192$, respectively. The multiplying coefficient corresponding to a second-order convergence is $Q_2=(1/128^2-1/160^2)/(1/160^2-1/192^2)$. The results show that a second-order convergence is observed. The figure only shows the time window that has a significant ringdown signal.
• Figure 3: Fitting results of the QNM frequencies for $\Psi_4^{33}$ of the single perturbed BH simulation that has an initial axial $l=3$, $m=3$ perturbation. The dots with the same color show the evolution of the fitting result of a single mode. The black circles denote the tensorial mode frequencies of a Kerr BH, except for $s330$ which denotes the scalar-led $330$ mode in the test-field limit. The blue and orange circles show the axial and polar modes in the shift-symmetric sGB theory. For this figure, we use $N=8$.
• Figure 4: Similar figure as Fig. \ref{['fig:A_33']} but with an initial polar $l=3$, $m=3$ perturbation. It is clear that at this time, the excited $330$ mode has a frequency that is consistent with the theoretical predictions for the polar mode.
• Figure 5: Similar figure as Fig. \ref{['fig:A_33']} but with an initial $l=3$, $m=3$ perturbation that contains both polar and axial parts. In the right panel, we show the fitting result of the real and imaginary parts of the mode frequencies as a function of the fitting starting time $t_0$. The dashed lines show the estimation of the mode frequencies. Compared to Fig. \ref{['fig:A_33']} and Fig. \ref{['fig:P_33']} that contain only one mode centering at the estimated frequency, this figure shows a significantly more complex fitting result. Considering the additional mode, we use $N=9$ for the fitting.