Detection and characterization of spin-orbit resonances in the advanced gravitational wave detectors era

Spin-orbit resonances have important astrophysical implications as the evolution and subsequent coalescence of supermassive black hole binaries in one of these configurations may lead to low recoil velocity of merger remnants. It has also been shown that black hole spins in comparable mass stellar-mass black hole binaries could preferentially lie in a resonant plane when their gravitational waves (GWs) enter the advanced LIGO frequency band [1]. Therefore, it is highly desirable to investigate the possibility of detection and subsequent characterization of such GW sources in the advanced detector era, which can, in turn, improve our perception of their high mass counterparts. The current detection pipelines involve only nonprecessing templates for compact binary searches whereas parameter estimation pipelines can afford to use approximate precessing templates. In this paper, we test the performance of these templates in detection and characterization of spin-orbit resonant binaries. We use fully precessing time-domain SEOBNRv3 waveforms as well as four numerical relativity (NR) waveforms to model GWs from spin-orbit resonant binaries and filter them through IMRPhenomD, SEOBNRv4 and IMRPhenomPv2 approximants. We find that the nonprecessing approximants IMRPhenomD and SEOBNRv4 recover only $\sim 70\%$ of injections with fitting factor (FF) higher than 0.97 (or 90% of injections with $\mathrm{FF}>0.9$). This loss in signal-to-noise ratio is mainly due to the missing physics in these approximants in terms of precession and nonquadrupole modes. However, if we use a new statistic, i.e., maximizing the matched filter output over the sky-location parameters as well, the precessing approximant IMRPhenomPv2 performs magnificently better than their nonprecessing counterparts with recovering 99% of the injections with FFs higher than 0.97. Interestingly, injections with $\mathrm{\Delta }\varphi =180°$ have higher FFs ($\mathrm{\Delta }\varphi$ is the angle between the components of the black hole spins in the plane orthogonal to the orbital angular momentum) as compared to their $\mathrm{\Delta }\varphi =0°$ and generic counterparts. This is because $\mathrm{\Delta }\varphi =180°$ binaries are not as strongly precessing as $\mathrm{\Delta }\varphi =0°$ and generic binaries. This implies that we will have a slight observation bias towards $\mathrm{\Delta }\varphi =180°$ and away from $\mathrm{\Delta }\varphi =0°$ resonant binaries while using nonprecessing templates for searches. Moreover, all template approximants are able to recover most of the injected NR waveforms with FFs $>0.95$. For all the injections including NR, the systematic error in estimating chirp mass remains below $<10\%$ with minimum error for $\mathrm{\Delta }\varphi =180°$ resonant binaries. The symmetric mass-ratio can be estimated with errors below 15%. The effective spin parameter ${\chi }_{\mathrm{eff}}$ is measured with maximum absolute error of 0.13. The in-plane spin parameter ${\chi }_p$ is mostly underestimated indicating that a precessing signal will be recovered as a relatively less precessing signal. Based on our findings, we conclude that we not only need improvements in waveform models towards precession and nonquadrupole modes but also better search strategies for precessing GW signals.

Figure 1

Mismatch between different resolutions (N1,2,3) and the highest available resolution of NR waveforms for cases 1 (top), 2 (middle) and 3 (bottom), as a function of the total mass of the binary. We include higher GW modes up to $l=5$ and choose inclination angle $\iota$ and reference phase ${\mathrm{\Phi }}_{\mathrm{ref}}$ to be zero in these plots. The solid black line in the topmost panel represents the total mass when the lowermost frequency of the waveform is equal to 10 Hz. Such line for cases 2 and 3 lies beyond total mass of $100M_{\odot }$. The masses in the three panels are chosen such that the NR waveform completely covers the aLIGO frequency band [78].

Figure 2

The figure shows the 4 cases of the NR waveforms with total mass of $20M_{\odot }$, $\iota =0°$ and ${\mathrm{\Phi }}_{\mathrm{ref}}=0°$. We have included higher GW harmonics up to $l=5$ in these waveforms except in case 4 which has only (2,2) mode. The waveforms are shown in the NR (source) frame as well as the LAL wave frame. The quantity $\mathit{k}·({\mathit{s}}_1\times {\mathit{s}}_2)$ is plotted beneath each of the waveforms. The vertical dashed line represents the time when the orbital separation reaches 6M.

Figure 3

Comparison of performances of IMRPhenomD, SEOBNRv4 and IMRPhenomPv2 templates in recovering SEOBNRv3 injections with arbitrary orbital inclination. Cumulative histogram of FFs against the fraction of sources is plotted in log-scale. The FFs for $\mathrm{\Delta }\varphi =0°$ resonance family are depicted in red; $\mathrm{\Delta }\varphi =180°$ resonance family are depicted in blue and generic binaries are depicted in green. The dashed line represents IMRPhenomD templates, the translucent solid line SEOBNRv4 templates while dark solid line represents IMRPhenomPv2 templates.

Figure 4

Depicting color coded SRF as a function of $\iota$ and ${\chi }_p$. The first, second and third columns are for IMRPhenomD, SEOBNRv4 and IMRPhenomPv2 templates, respectively. The first, second and third rows are for $\mathrm{\Delta }\varphi =0°$ injections, $\mathrm{\Delta }\varphi =180°$ injections and generic injections, respectively.

Figure 5

Depicting color coded SRF as a function of ${\theta }_1$ and ${\theta }_2$ for signals having $0°\le \iota \le 45°$ and $135°\le \iota \le 180°$. The first, second and third columns are for IMRPhenomD, SEOBNRv4 and IMRPhenomPv2 templates, respectively. The first, second and third rows are for $\mathrm{\Delta }\varphi =0°$ injections, $\mathrm{\Delta }\varphi =180°$ injections and generic injections, respectively.

Figure 6

Depicting color coded SRF as a function of ${\theta }_1$ and ${\theta }_2$ for signals having $45°\le \iota \le 135°$. The first, second and third columns are for IMRPhenomD, SEOBNRv4 and IMRPhenomPv2 templates, respectively. The first, second and third rows are for $\mathrm{\Delta }\varphi =0°$ injections, $\mathrm{\Delta }\varphi =180°$ injections and generic injections, respectively.

Figure 7

Depicting color coded SRF as a function of aligned-spin parameter ${\chi }_{\mathrm{eff}}$ and in-plane spin parameter ${\chi }_{\mathrm{p}}$ for signals having $0°\le \iota \le 45°$ and $135°\le \iota \le 180°$. The first, second and third columns are for IMRPhenomD, SEOBNRv4 and IMRPhenomPv2 templates, respectively. The first, second and third rows are for $\mathrm{\Delta }\varphi =0°$ injections, $\mathrm{\Delta }\varphi =180°$ injections and generic injections, respectively.

Figure 8

Depicting color coded SRF as a function of aligned-spin parameter ${\chi }_{\mathrm{eff}}$ and in-plane spin parameter ${\chi }_{\mathrm{p}}$ for signals having $45°\le \iota \le 135°$. The first, second and third columns are for IMRPhenomD, SEOBNRv4 and IMRPhenomPv2 templates, respectively. The first, second and third rows are for $\mathrm{\Delta }\varphi =0°$ injections, $\mathrm{\Delta }\varphi =180°$ injections and generic injections, respectively.

Figure 9

Comparison of the performances of three template approximants, IMRPhenomD, SEOBNRv4, and IMRPhenomPv2, in recovering NR injections. $x$ and $y$ axis represent the total mass (in $M_{\odot }$) and inclination angle (in degree) while FFs are given in color bar. Different markers are for different approximants. The solid vertical line on the top panel for case 1 represents the mass for which the lowest frequency of the NR waveform is equal to 10 Hz. All the masses on the right-hand side to this line have lowest frequency less than 10 Hz and hence we lose a few GW cycles while calculating FF for these NR waveforms as we keep $f_{\mathrm{low}}$ to be 10 Hz. Such a line for cases 2, 3 and 4 exists for total mass greater than $100M_{\odot }$ and hence cannot be seen in the plot. Similarly, the dashed vertical lines in all the four panels represent the total mass for which the lowest frequency of NR waveform is equal to 20 Hz.

Figure 10

Systematic error in $M_c$, $\eta$, ${\chi }_1$, ${\chi }_2$, $\widehat{\chi }$, ${\chi }_{\mathrm{eff}}$ and ${\chi }_{\mathrm{p}}$ for SEOBNRv3 injections. The quantities on the $x$ axis represent injected parameter values and are divided into equal bins. The mean (circle), standard deviations (error bars are symmetric with respect to mean) and median (cross) for systematic biases corresponding to each bin are plotted on the $y$ axis.

Figure 11

Systematic biases in the estimation of chirp mass $M_c$ for the all the four NR injections while using IMRPhenomD, SEOBNRv4 and IMRPhenomPv2 templates as a function of total mass and inclination angle $\iota$.

Figure 12

Systematic biases in the estimation of symmetric mass-ratio $\eta$ for all the four NR injections while using IMRPhenomD, SEOBNRv4 and IMRPhenomPv2 templates as a function of total mass and inclination angle $\iota$.

Figure 13

Systematic biases in the estimation of spin parameters for all the four NR injections while using IMRPhenomD and IMRPhenomPv2 templates as a function of total mass and inclination angle $\iota$. The spin components evolve with time and here in this figure the injected and recovered spin parameters are computed at $f_0$ (see Table 2).

Figure 14

Systematic biases in the estimation of spin parameters for all the four NR injections while using SEOBNRv4 templates as a function of total mass and inclination angle $\iota$. The spin components evolve with time and here in this figure the injected and recovered spin parameters are computed at $f_0$ (see Table 2).