Nonlinear Effects in Black Hole Ringdown

We report evidence for nonlinear modes in the ringdown stage of the gravitational waveform produced by the merger of two comparable-mass black holes. We consider both the coalescence of black hole binaries in quasicircular orbits and high-energy, head-on black hole collisions. The presence of nonlinear modes in the numerical simulations confirms that general-relativistic nonlinearities are important and must be considered in gravitational-wave data analysis.

Figure 1

Evidence for nonlinear effects in the ringdown. Left: search for the $200\times 200$ mode in the $\ell m=40$ multipole of ultrarelativistic head-on mergers; center: $220\times 220$ mode in the $\ell m=44$ harmonic of quasicircular mergers with low mass ratio $q\le 1.5$; right: $220\times 330$ mode in the $\ell m=55$ harmonic of quasicircular mergers with $1.25\le q\le 2$. We highlight in brighter colors the results for $\gamma =1.5$ (left), $q=1.22$ (the “SXS:BBH:0305” simulation, center) and $q=1.88$ (“SXS:BBH:0403” simulation, right), while we plot the results for all other simulations in gray. Top row: search for the second-order mode frequency. We use a mode with a variable complex frequency in our fitting model to search for the expected second-order modes, and we use modes with fixed frequencies (black solid circles) to remove the contribution from linear modes when they are present. The color scale (top bar) represents different starting times of the fit. For quasicircular mergers, the labeled modes correspond to those of the remnant BH in the highlighted simulation. The location of the target mode for other simulations may be slightly different, because it depends on the remnant spin. Second row: fractional deviation $|\delta \omega |$ of the fitted complex frequency with respect to the expected second-order mode. Third row: amplitude of the second-order mode when $t_{\text{start}}$ is varied across a window of length $T_0$ centered around the value of minimum $|\delta \omega |$, $t_{\mathrm{opt}}$, and the second-order mode frequency is fixed to its expected value in the fitting model. Bottom row: same as the third row, but for the phase of the second-order mode.

Figure 2

Dependence of the second-order mode amplitude (left and middle columns) and phases (right column) on the amplitudes of the first-order modes sourcing them. The crosses are the amplitudes or phases extracted from simulations with different boost (left column) or mass ratio (center and right columns). The width and height of the crosses correspond to the errors. Blue crosses represent simulations where the two BHs are initially nonspinning, while golden crosses represent those with at least one spinning BH. The gray dotted line is the expected relationship between the first and second-order values with the slope fixed to either 1 or 2; the deep gray dashed line is a fit to the data with the slope unfixed. The phase dependence for head-on simulations is shown in the Supplemental Material [64].