Fusing numerical relativity and deep learning to detect higher-order multipole waveforms from eccentric binary black hole mergers

We determine the mass ratio, eccentricity and binary inclination angles that maximize the contribution of the higher-order waveform multipoles $(\ell ,|m|)=\{(2,2),(2,1),(3,3),(3,2),(3,1),(4,4),(4,3),(4,2),(4,1)\}$ for the gravitational wave detection of eccentric binary black hole mergers. We carry out this study using numerical relativity waveforms that describe nonspinning black hole binaries with mass ratios $1\le q\le 10$, and orbital eccentricities as high as $e_0=0.18$ fifteen cycles before merger. For stellar-mass, asymmetric mass-ratio, binary black hole mergers, and assuming LIGO’s zero detuned high power configuration, we find that in regions of parameter space where black hole mergers modeled with $\ell =|m|=2$ waveforms have vanishing signal-to-noise ratios, the inclusion of $(\ell ,|m|)$ modes enables the observation of these sources with signal-to-noise ratios that range between 30% and 45% of the signal-to-noise ratio of optimally oriented binary black hole mergers modeled with $\ell =|m|=2$ numerical relativity waveforms. Having determined the parameter space where $(\ell ,|m|)$ modes are important for gravitational wave detection, we construct waveform signals that describe these astrophysically motivated scenarios and demonstrate that these topologically complex signals can be detected and characterized in real LIGO noise with deep learning algorithms.

Figure 1

Amplitude $A_{\ell m}={|h_{\ell m}{h}_{\ell m}^{*}|}^{1/2}$ of the most significant $(\ell ,|m|)$ modes for the NR simulation P0009.

Figure 2

$\mathrm{\Delta }\mathcal{A}(\theta ,\varphi )$, as defined in Eq. (7), constrains the regions of $(\theta ,\varphi )$ parameter space that maximize the contribution of $(\ell ,|m|)$ modes to the waveform amplitude near the merger. These $(\theta ,\varphi )$ maps are constructed using the Mollweide projection: $(\vartheta ,\phi )\to (\pi /2-\theta ,\varphi -\pi )$.

Figure 3

$\mathrm{\Delta }\mathcal{B}$, as defined in Eq. (11), identifies the $(\theta ,\varphi )$ regions where the integrated amplitude of NR waveforms that include $(\ell ,|m|)$ modes is greater than their $\ell =|m|=2$ counterparts. These $(\theta ,\varphi )$ maps are constructed using the Mollweide projection: $(\vartheta ,\phi )\to (\pi /2-\theta ,\varphi -\pi )$.

Figure 4

Top panels: $\mathrm{\Delta }\mathrm{SNR}$ distributions [see Eq. (12)], using the maxima of metric $\mathrm{\Delta }{\mathcal{A}}^{(\ell ,|m|)}$. Bottom panels: As before, but now using the maxima of metric $\mathrm{\Delta }{\mathcal{B}}^{(\ell ,|m|)}$. The $\mathrm{\Delta }\mathrm{SNR}$ distributions are presented as a function of the source’s sky location $(\alpha ,\beta$) mapped into a Mollweide projection: $(\vartheta ,\phi )\to (\pi /2-\alpha ,\beta -\pi )$. We have set $\psi =\pi /4$ in these calculations.

Figure 5

Top panels: $\mathrm{\Delta }\mathrm{SNR}$ distributions [see Eq. (12)], using the maxima of metric ${\mathcal{B}}^{(\ell ,|m|)}$. The SNR distributions are presented as a function of the source’s sky location ($\alpha ,\beta$) mapped into a Mollweide projection: $(\vartheta ,\phi )\to (\pi /2-\alpha ,\beta -\pi )$. We have set $\psi =\pi /4$ in these calculations. Bottom panels: Comparison between NR waveforms that include either all $(\ell ,|m|)$ modes or the $\ell =|m|=2$ mode only, using $(\theta ,\varphi )$ values that correspond to the maxima of metric ${\mathcal{B}}^{(\ell ,|m|)}$. Visualizations for these $\mathrm{\Delta }\mathrm{SNR}$ distributions for a continuous range of $\psi$ values can be found at [73].

Figure 6

Pairwise comparisons of BBH systems (P0017, P0020) and (P0009, P0024). (P0017, P0020) have mass ratio $q=8$, whereas (P0009, P0024) have mass ratio $q=10$. From each pair, P0017 and P0009 are the least eccentric. Top panels: $(\theta ,\varphi )$ regions that maximize the contribution of higher-order modes for GW detection using metric $\mathrm{\Delta }{\mathcal{B}}^{(\ell ,|m|)}$. Compare these results to the angular distributions for (P0020, P0024) in Fig. 3. Bottom panels: $\mathrm{\Delta }\mathrm{SNR}$ distributions assuming $\psi =\pi /4$. Compare these results to those presented in Fig. 5.

Figure 7

Increase in detection range due to the inclusion of higher-order waveform multipoles for the eccentric binary black hole mergers described by the NR waveforms P0020 and P0024. These panels were produced setting $\psi =\pi /4$. However, we have also produced visualizations that show the increase in $D_{\mathrm{obs}}$ using a continuous range of $\psi$ values. These are available at [73].

Figure 8

Left panel: Sensitivity of detecting higher-order waveform multipole signals injected in simulated LIGO noise. Right panel: As in left panel, but now injecting higher-order waveform multipole signals in real Advanced LIGO noise. The binary black hole systems used in this study describe a broad range of eccentricities and mass ratios to quantify the accuracy of our neural network models to extract waveforms in which higher-order modes have a marginal impact (comparable mass ratios and moderate values of eccentricity) to configurations (high mass ratio and large eccentricities) in which the injected signals are very distinct to our training dataset. In both scenarios, our neural network models reach 100% sensitivity to identify higher-order waveform multipole signals with $\mathrm{SNR}\ge 10$.

Figure 9

The panels present mean percentage error of estimated total mass for each eccentric BBH population. These results were obtained using 600 different noise realizations. Left panels: Total mass recovery assuming simulated Gaussian noise for five different BBH populations, with three different total mass combinations. Right panels: As left panels but now using real Advanced LIGO noise. We notice that in both cases, our neural network models can infer the total mass of the system better than 20% for BBH systems with $\mathrm{SNR}\gtrsim 15$.

Figure 10

Higher-order mode gravitational waves whitened with real Advanced LIGO noise taken from its first observing run.

Figure 11

Higher-order mode gravitational waves whitened with the target, simulated zero detuned high power spectral density of the Advanced LIGO detectors [27].