Unveiling the spectrum of inspiralling binary black holes

The higher multipoles of gravitational wave signals from coalescing compact binaries play a vital role in the accurate reconstruction of source properties, bringing about a deeper and nuanced understanding of fundamental physics and astrophysics. Their effect is most pronounced in systems with asymmetric masses having an orbital geometry that is not face-on. The detection of higher multipoles of gravitational wave signals from any single, isolated merger event is challenging, as there is much less power in comparison to the dominant quadrupole mode. In this paper, we present a new method for their detection by combining multiple events observed in interferometric gravitational wave detectors. Sub-dominant modes present in (the inspiral part of) the signal from separate events are stacked using time-frequency spectrogram of the data. We demonstrate that this procedure enhances the signal-to-noise ratio of the higher-multipole components and thereby leads to increased chances of their detection. From Monte Carlo simulations we estimate that a combination of $\sim 100$ events observed in two-detector coincidence can lead to the detection of the higher-multipole components with a $\ge 95\%$ detection probability. The Advanced LIGO detectors are expected to record these many binary black hole merger events within a month of operation at design sensitivity. We also present results from the analysis of data from first and second observing runs containing previously detected events using our new method.

Figure 1

Template vectors $S(\alpha )$ for three different nonprecessing asymmetric BBH systems generated using the $\text{SEOBNRv}4\text{HM}$ waveform model [19] consisting of (2, 1), (2, 2) (3, 3), (4, 4), and (5, 5) multipoles. Peaks at $\alpha =1.0$ and $\alpha =1.5$ indicate the relative energy of the $m=2$ and $m=3$ modes.

Figure 2

The plot of ${[⟨\beta ⟩/⟨{\beta }_j⟩]}^2$ versus a number of events follows a straight line with unit slope, where all the events are identical. Ensemble averages are taken over 300 realizations of ideal aLIGO noise. The plot implies that the stacking algorithm is coherent where with the average detection static (after stacking $n_0$ identical events) scales as $\sqrt{n_0}$.

Figure 3

Distributions of the single-event detection statistic ${\beta }_j$ (dashed-blue trace) from the entire set of simulated events observed in a aLIGO detector and of the combined statistic $\beta$ (in red), after optimally stacking a subset of $n_0=100$ randomly chosen events. This should be contrasted with the filled green histogram showing the detection probability obtained from choosing the maximum ${\beta }_j$ out of the same subset of 100 events without stacking. There is a 95% probability of detecting higher multipoles after stacking 100 events.

Figure 4

Plot showing the increase in detection probability ($P_D$) of higher multipoles as more events are stacked using the method outlined in the text. As can be seen, stacking $n_0$ events is more efficient in comparison to choosing the loudest single event statistic. From the plot, we find that a nominal value of $P_D=95\%$ is reached after stacking 100 events. In comparison, the same target is reached in the latter case by considering 130 events.

Figure 5

Analysis of GWTC-1 events: $p$ values determined from the background distribution are shown for two events (GW150914, GW170814), and after stacking three events (GW170814, GW170818, GW170104). The vertical dashed lines indicate the detection statistic in each case, which can be used to directly infer the significance from corresponding $p$-value traces. Stacking events in the GWTC-1 catalog leads to a marginal increase of the detection statistic (and improved significance).

Figure 6

Numerically evaluated noise covariance matrix $\mathrm{\Sigma }(\alpha ,{\alpha }^{\prime })$ for the ensemble of noise vectors $N(\alpha )$. Each noise vector $N(\alpha )$ in the ensemble is calculated from the scaleogram of a distinct realization of synthetic aLIGO noise. The scaleogram pixels are summed along time-frequency tracks scaled with respect to a fiducial quadrupole-mode trajectory of a BBH system with component masses $[35,6]M_{\odot }$ and effective spin $-0.3$.

Figure 7

Distribution of the detection statistic $\beta$ for two cases: ${\mathcal{H}}_2$ true, i.e., when data contains only the dominant quadrupole component of the signal (red-histogram, to the left) and ${\mathcal{H}}_3$ true, when the next higher ($m=3$) multipoles are also present in the signal (blue-histogram, to the right). The background distribution $p(\beta |{\mathcal{H}}_2)$ is shown to agree with a Gaussian distribution $\mathcal{N}(0,{1.7}^2)$. In presence of higher-multipoles of the signal, the distribution $p(\beta |{\mathcal{H}}_3)$ is shown to agree with a Gaussian distribution with mean equal to the optimum signal norm ${\gamma }_3^{\mathrm{inj}}$. Results shown are obtained from BBH signal injections with component masses $[32,6]M_{\odot }$, effective spin $-0.3$, inclination angle 97° and fixed SNR of 36.7.