Sensitivity of gravitational wave searches to the full signal of intermediate-mass black hole binaries during the first observing run of Advanced LIGO

The sensitivity of gravitational wave searches for binary black holes is estimated via the injection and posterior recovery of simulated gravitational wave signals in the detector data streams. When a search reports no detections, the estimated sensitivity is then used to place upper limits on the coalescence rate of the target source. In order to obtain correct sensitivity and rate estimates, the injected waveforms must be faithful representations of the real signals. Up to date, however, injected waveforms have neglected radiation modes of order higher than the quadrupole, potentially biasing sensitivity and coalescence rate estimates. In particular, higher-order modes are known to have a large impact in the gravitational waves emitted by intermediate-mass black holes binaries. In this work, we evaluate the impact of this approximation in the context of two search algorithms run by the LIGO Scientific Collaboration in their search for intermediate-mass black hole binaries in the O1 LIGO Science Run data: a matched filter–based pipeline and a coherent unmodeled one. To this end, we estimate the sensitivity of both searches to simulated signals for nonspinning binaries including and omitting higher-order modes. We find that omission of higher-order modes leads to biases in the sensitivity estimates which depend on the masses of the binary, the search algorithm, and the required level of significance for detection. In addition, we compare the sensitivity of the two search algorithms across the studied parameter space. We conclude that the most recent LIGO-Virgo upper limits on the rate of coalescence of intermediate-mass black hole binaries are conservative for the case of highly asymmetric binaries. However, the tightest upper limits, placed for nearly equal-mass sources, remain unchanged due to the small contribution of higher modes to the corresponding sources.

Figure 1

Impact of higher modes in a time domain waveform: We show, in geometrical units, the last few cycles of the full GW signal (black) and only the (2,2) mode alone (red) of a mass ratio $q=6$ nonspinning source, when this is face on (top panel) and edge on (bottom panel). Note that both signals are indistinguishable when the source is face on, while higher modes cause huge discrepancies in the last cycles of the edge-on case. The simulation used is GT0604 from the Georgia Tech waveform catalog [50].

Figure 2

Impact of higher modes in the PyCBC analysis: The plots show SNR and reduced chi-squared ${\chi }_r^2$ for background noise triggers (black dots) and triggers corresponding to simulated signals with $(q=7,M=300M_{\odot })$ (colored dots). The dashed lines denote contours of equal single-detector ranking statistic, roughly representing contours of equal detection significance. The color scale indicates the inclination $\iota$ of the simulated source: blue denotes edge-on sources, and red denotes face-on/-off ones. In the left panel, simulated signals only consider the quadrupolar radiation mode, while in the right panel, several higher-order modes are also included. In the latter case, the mismatch between the signal and template raises the value of ${\chi }_r^2$, moving the triggers to regions of lower significance, especially in the case of edge-on sources.

Figure 3

PyCBC: (2,2) mode vs higher-order modes: $\mathrm{\Delta }V$ values for PyCBC. The left plot shows results for $\mathrm{IFAR}=1\mathrm{yr}$, while the right one corresponds to $\mathrm{IFAR}={10}^3\mathrm{yr}$. Values close to zero denote a negligible impact. At low IFAR, the additional signal power in the higher modes makes the search more sensitive, leading to negative $\mathrm{\Delta }V$ values. At large IFAR, the dominant effect is the worsening of the ${\chi }_r^2$ due to the mismatch between the signals and the quadrupolar templates, producing the opposite effect. In particular, values of $\mathrm{\Delta }V=200\%$ are reached in the large-mass-ratio, large-mass limit. Note that $\mathrm{\Delta }V=-50\%$ corresponds to $⟨VT{⟩}_{HM}/⟨VT{⟩}_{NoHM}=2$, meaning that the search is twice as sensitive to the corresponding source when higher modes are included in the injected signals.

Figure 4

cWB: (2,2) mode vs higher-order modes: $\mathrm{\Delta }V$ values for cWB. The left plot shows results for $\mathrm{IFAR}=1\mathrm{yr}$, while the right one corresponds to $\mathrm{IFAR}={10}^3\mathrm{yr}$. Values close to zero denote a negligible impact. Note that for almost all cases, and unlike for PyCBC, the increment of the GW signal power due to the inclusion of higher modes enhances the sensitivity of cWB, leading to negative $\mathrm{\Delta }V$ values for all IFARs. Again, this effect is larger in the large-mass-ratio, large-mass limit. Note that $\mathrm{\Delta }V=-61\%$ corresponds to $⟨VT{⟩}_{HM}/⟨VT{⟩}_{NoHM}=2.56$, meaning that the search is 2.56 times as sensitive to the corresponding source when higher modes are included in the injected signals.

Figure 5

PyCBC vs CWB including higher modes: Comparison of sensitive volumes of cWB and PyCBC when higher modes are included. Here, we take $\mathrm{\Delta }V=100(V_{PyCBC}/V_{cWB}-1)$. At low IFAR, the PyCBC search always outperforms cWB. The effect of higher modes on PyCBC and cWB causes cWB to outperform PyCBC by a larger margin than in the case in which higher modes are omitted (see Fig. 6). Errors on the numbers are typically $\pm 2\%$ and at most $\pm 6\%$. Note that $\mathrm{\Delta }V=-97\%$ corresponds to $V_{cWB}/V_{PyCBC}=33$, meaning that cWB is 33 times as sensitive as PyCBC to the corresponding source at the given IFAR.

Figure 6

PyCBC vs CWB omitting higher modes: Comparison of sensitive volumes of cWB and PyCBC when higher modes are omitted. Here, we take $\mathrm{\Delta }V=100(V_{PyCBC}/V_{cWB}-1)$. At low IFAR, the PyCBC search always outperforms cWB. At large IFAR, cWB is always more sensitive than PyCBC to sources with $M>200M_{\odot }$. Errors on the numbers are typically $\pm 2\%$ and at most $\pm 6\%$. Note that $\mathrm{\Delta }V=-71\%$ corresponds to $V_{cWB}/V_{PyCBC}=3.44$, meaning that cWB is 3.44 times as sensitive as PyCBC to the corresponding source at the given IFAR.

Figure 7

Mismatch, or unfaithfulness, of the modes computed by EOBNRv2HM toward the corresponding SXS numerical relativity ones. The noise curve used is early Advanced LIGO (EA) with $f_{\mathrm{low}}=20\mathrm{Hz}$ for the dashed lines and zero-detuned high-energy-power (ZD) with $f_{\mathrm{low}}=10\mathrm{Hz}$ for the solid ones.

Figure 8

Last cycles of EOBNRv2HM (black) and SXS modes (red) for the case of a $q=8$ binary. The SXS simulation is SXS:BBH:0063, the axes are in geometrical units, and $t_c$ denotes the coalescence time, which we make coincide with the peak of the (2,2) mode.