Impact of gravitational radiation higher order modes on single aligned-spin gravitational wave searches for binary black holes

Current template-based gravitational wave searches for compact binary coalescences use waveform models that omit the higher order modes content of the gravitational radiation emitted, considering only the quadrupolar $(\ell ,|m|)=(2,2)$ modes. We study the effect of such omission for the case of aligned-spin compact binary coalescence searches for equal-spin (and nonspinning) binary black holes in the context of two versions of Advanced LIGO: the upcoming 2015 version, known as early Advanced LIGO (eaLIGO) and its zero-detuned high-energy power version, which we will refer to as Advanced LIGO (AdvLIGO). In addition, we study the case of a nonspinning search for initial LIGO (iLIGO). We do this via computing the effectualness of the aligned-spin SEOBNRv1 reduced order model waveform family, which only considers quadrupolar modes, toward hybrid post-Newtonian/numerical relativity waveforms which contain higher order modes. We find that for all LIGO versions losses of more than 10% of events occur in the case of AdvLIGO for mass ratio $q\ge 6$ and total mass $M\ge 100M_{\odot }$ due to the omission of higher modes, this region of the parameter space being larger for eaLIGO and iLIGO. Moreover, while the maximum event loss observed over the explored parameter space for AdvLIGO is of 15% of events, for iLIGO and eaLIGO, this increases up to (39,23)%. We find that omission of higher modes leads to observation-averaged systematic parameter biases toward lower spin, total mass, and chirp mass. For completeness, we perform a preliminar, nonexhaustive comparison of systematic biases to statistical errors. We find that, for a given signal-to-noise ratio, systematic biases dominate over statistical errors at much lower total mass for eaLIGO than for AdvLIGO.

Figure 1

Amplitude of the $(\ell ,m)$ modes of a $(q,\chi )=(8,0)$ system during the last orbits of the coalescence in the logarithmic scale. The modes are the result of hybridizing post-Newtonian Taylor T1 and numerical relativity data [see Sec. 4].

Figure 2

Relative T1 (top) and NR (bottom) amplitude of the higher modes (3,3) and (4,4) relative to the dominant (2,2) mode as a function of the frequency for several nonspinning systems. Note how the larger the mass ratio $q$, which increases from bottom to top in the plots, the larger the contribution from higher order modes. As in Ref. [23], we notice the differences between the predicted amplitudes by both PN and NR mainly due to PN truncation error. Note that the merger, taken as coincident with the amplitude maximum of the (2,2) mode, occurs between $M{\omega }_0\simeq 0.30$ for $q=8$ and $M{\omega }_0\simeq 0.36$ for $q=2$ in the bottom plots.

Figure 3

Relative T1 (top) and NR (bottom) amplitude of the (3,3) and (4,4) modes relative to the dominant (2,2) mode as a function of the frequency for several $q=3$ spinning systems.

Figure 4

The upper panels show the absolute value of the Fourier transform ${\tilde{h}}_{\ell ,m}(f)\times \mathrm{\Delta }{f}^{1/2}$ of the (2,2), (3,3), and (4,4) modes of a nonspinning $q=8$ binary and the three noise curves considered in this paper. The modes have been rescaled by an arbitrary factor $(\mathrm{\Delta }{f}^{1/2}/d_L)$ to clearly stand out from the noise curves, since we are only interested in their relative values. The vertical line marks the 30 Hz cutoff of eaLIGO and iLIGO. Note how for the case of $M=100M_{\odot }$ the (2,2) mode clearly dominates at the sweet spot of the different noise curves, while this is not the case when $M=200M_{\odot }$, resulting in a higher contribution of the HM in the latter case. The bottom panels show the corresponding whitened templates for eaLIGO (red) and AdvLIGO (black). The lower low-frequency cutoff of AdvLIGO, together with its flatter sensitivity curve, makes the detector sensitive to a much longer inspiral, clearly dominated by the (2,2) mode and whose amplitude (unlike for eaLIGO) dominates that of the higher mode peaks corresponding to the merger stage. This makes contribution from the HM weaker for AdvLIGO. Also, it can be noticed how for the $200M_{\odot }$ eaLIGO case the peaks of all whitened templates have similar amplitudes due to the (2,2) mode being clearly out of the sweet spot while the HM are in, which does not happen for AdvLIGO.

Figure 5

Value of the ratio $I_{3,3}/I_{2,2}$ and $I_{4,4}/I_{2,2}$ as a function of the total mass of a nonspinning $q=8$ binary for the cases of eaLIGO and AdvLIGO. Note how the (2,2) mode is more dominant for AdvLIGO while the contribution of the HM is larger for the case of eaLIGO for all the mass range.

Figure 6

Left: Fractional volume loss in % for nonspinning $q=(4,6,8)$ systems (in dotted, dashed, and solid lines). Right: same for the $q=3$, $\chi =(-0.5,+0.5,0)$ systems (in dotted, dashed, and solid lines), using the same color-detector code as in the left panel. Recall that for the case of iLIGO we have not considered spinning targets.

Figure 7

Systematic biases obtained for the total mass (left) and effective spin (right) for a $q=8$ nonspinning system for eaLIGO (top) and AdvLIGO (bottom) as a function of the location of the detector on the (upper hemisphere) sky of the source. Note that the different interaction of the modes as a function of the angle $\phi$ generates biases to either larger or lower values, which in general grow (in absolute value) as $\theta$ does. Biases toward low masses are more common due to the higher frequency content of the signal for most values $\phi$, has to be imitated by low-mass templates. Last, note that $\theta =0$ corresponds to the center of the plot, while its perimeter corresponds to $\theta =\pi /2$.

Figure 8

Top: $M$, ${\mathcal{M}}_c$, and $\chi$ systematic bias for the $q=(4,6,8)$ nonspinning cases. Bottom: Same for the $(q;\chi )=(3;0,\pm 0.5)$ and the $(q,\chi )=(1,\pm 0.2)$ cases for eaLIGO.

Figure 9

Comparison between systematic errors and statistical uncertainties. We show the minimum SNR ${\rho }_0$ at which systematic biases due to the omission of the HM dominate those due to statistical uncertainties for the studied sources.

Figure 10

Fractional volume loss in % a for $q=8$ nonspinning system and an unequal-spin source with dimensionless spins $({\chi }_1,{\chi }_2)=(0,-0.5)$.