Nonlinearities in Black Hole Ringdowns

The gravitational wave strain emitted by a perturbed black hole (BH) ringing down is typically modeled analytically using first-order BH perturbation theory. In this Letter, we show that second-order effects are necessary for modeling ringdowns from BH merger simulations. Focusing on the strain’s $(\ell ,m)=(4,4)$ angular harmonic, we show the presence of a quadratic effect across a range of binary BH mass ratios that agrees with theoretical expectations. We find that the quadratic $(4,4)$ mode’s amplitude exhibits quadratic scaling with the fundamental $(2,2)$ mode—its parent mode. The nonlinear mode’s amplitude is comparable to or even larger than that of the linear $(4,4)$ mode. Therefore, correctly modeling the ringdown of higher harmonics—improving mode mismatches by up to 2 orders of magnitude—requires the inclusion of nonlinear effects.

Figure 1

Relationship between the peak amplitudes of the linear $(2,2,0)$ and the quadratic $(2,2,0)\times (2,2,0)$ QNMs (top) as well as the linear $(4,4,0)$ QNM (bottom), at different model start times $u_0$. Colors show different mass ratios $q$, and circles and triangles denote systems with remnant dimensionless spin ${\chi }_f\approx 0.5$ and ${\chi }_f\approx 0.7$, respectively. Each blue curve is a pure quadratic fit with start time $u_0$, and the shaded region brackets every one of the individual fits.

Figure 2

Top: in black, the NR waveform for the SXS:BBH:0305 simulation and its comparison to the quadratic $(4,4)$ QNM model with start time $u_0=20M$ [total is dashed blue; yellow and green are contributions from individual QNMs, respectively the linear $(4,4,1)$ and the quadratic $(2,2,0)\times (2,2,0)$]. Bottom: residual in the $(4,4)$ mode when using the linear (solid red) or the quadratic (dashed blue) $(4,4)$ model. We also show a conservative estimate of the numerical error.

Figure 3

Top: mismatch in the $(4,4)$ mode for SXS:BBH:0305, as well as for every other simulation examined, and a comparison to the numerical error floor. Bottom: amplitudes of the three QNM terms in the quadratic $(4,4)$ QNM model as a function of the model start time $u_0$.

Figure 4

Contour plot of the mismatch between the SXS:BBH:0305 waveform and a $(4,4)$ model with three QNMs, in which two frequencies are fixed to the GR predictions of the linear $(4,4,0)$ and $(4,4,1)$ QNMs, but the third is varied. The contour lines are logarithmically spaced in $\mathcal{M}$ between ${10}^{-6}$ and ${10}^{-2}$. The start time of the model is taken to be $u_0=20M$.