Detection and parameter estimation of gravitational waves from compact binary inspirals with analytical double-precessing templates

We study the performance of various analytical frequency-domain templates for detection and parameter estimation of gravitational waves from spin-precessing, quasicircular, compact binary inspirals. We begin by assessing the extent to which nonspinning, spin-aligned, and the new (analytical, frequency-domain, small-spin) double-precessing frequency-domain templates can be used to detect signals from such systems. For effective, dimensionless spin values above 0.2, the use of nonspinning or spin-aligned templates for detection purposes will result in a loss of up to 30% of all events, while in the case of the double-precessing model, this never exceeds 6%. Moreover, even for signals from systems with small spins, nonspinning and spin-aligned templates introduce large biases in the extracted masses and spins. The use of a model that encodes spin-induced precession effects, such as the double-precessing model, improves the mass and spin extraction by up to an order of magnitude. The additional information encoded in the spin-orbit interaction is invaluable if one wishes to extract the maximum amount of information from gravitational wave signals.

Figure 1

Median faithfulness (left panel) and median drop in detection rates (right panel) between a numerical PN waveform and a full double-precessing waveform (magenta dot-dashed line), a restricted double-precessing waveform (blue dotted line), a restricted spin-aligned waveform (green dashed line), and a restricted nonspinning waveform (red sold line) for NSNS binaries as a function of the symmetric spin. The shaded areas give the 1-$\sigma$ confidence regions and the black solid line represents the 98% threshold. In the case of detection rates this threshold corresponds to the loss of 6% of all events.

Figure 2

Median parameter bias for the chirp mass (top left), the total mass (top right), and the absolute value of the dimensionless effective spin parameter (bottom) for full double-precessing waveforms (magenta dot-dashed line), restricted double-precessing waveforms (blue dotted line), restricted spin-aligned waveforms (green dashed line), and restricted nonspinning waveforms (red sold line) for NSNS binaries as a function of the symmetric spin parameter. The shaded areas give the 1-$\sigma$ confidence regions. The bias from using the nonspinning, or spin-aligned templates is about an order of magnitude larger than the bias from the double-precessing templates. Also, the similar performance of the full and the restricted double-precessing templates makes the computationally less expensive restricted templates ideal for the parameter estimation studies of Sec. 4.

Figure 3

Median faithfulness (left panel) and median drop in detection rates (right panel) between a numerical PN waveform and a full double-precessing waveform (magenta dot-dashed line), a restricted double-precessing waveform (blue dotted line), a restricted spin-aligned waveform (green dashed line), and a restricted nonspinning waveform (red sold line) for BHBH binaries as a function of the symmetric spin. The shaded areas give the 1-$\sigma$ confidence regions and the black solid line represents a value of 98% (corresponding to a 6% drop in detection rates). The use of nonspinning or spin-aligned templates for the detection of generically spinning BHBH binaries will result in a the loss of most highly spinning systems. On the other hand, both double-precessing models are capable of capturing most of the systems.

Figure 4

Median parameter bias for the chirp mass (top left), the total mass (top right), and $|{\chi }_{\text{eff}}|$ (bottom) for full double-precessing waveforms (magenta dot-dashed line), restricted double-precessing waveforms (blue dotted line), restricted spin-aligned waveforms (green dashed line), and restricted nonspinning waveforms (red sold line) for BHBH binaries. The shaded areas give the 1-$\sigma$ confidence regions. All four models induce a significant amour of bias, even though the double-precessing ones perform much better.

Figure 5

Log likelihood as a function of the chirp mass (left) and of the total mass (right) for a BHBH system. The vertical dashed lines correspond to the injected values $\mathcal{M}=7.23M_{\odot }$ and $M=16.7M_{\odot }$.

Figure 6

Real bias (blue dotted line) and theoretical bias (magenta dot-dashed line) for the chirp mass (left panel) and the total mass (right panel) for a BHBH binary as a function of the injected spin. The shaded regions give the 1-$\sigma$ confidence intervals.

Figure 7

Example 2D scatter plots for ${\chi }_1-{\chi }_2$ for the pilot run (main plot) and the focused run (inset) to illustrate the two-stage analysis. The red box in the pilot run indicates the size of the region $(0,{\chi }_1^{\max },0,{\chi }_2^{\max })$ of the focused run. This region contains $\sim 10\%$ of the total points of the pilot run.

Figure 8

BF as a function of the injected ${\chi }_{\text{eff}}$ between nonspinning and spinning models for System 1 of Table 1 and for spin-aligned (black) and double-precessing (red) templates, assuming an injection with SNR 30 (solid) and 60 (dashed) as seen by LIGO3.

Figure 9

BF as a function of the injected ${\chi }_{\text{eff}}$ between nonspinning and spinning models with double-precessing templates, where ${\chi }_{\text{eff}}$ has been updated through a change in the spin magnitudes (red lines, System 1 of Table 1) and a change in the spin angles (blue lines, System 2 of Table 1). The injected signal has a SNR of 10 (solid lines) and 20 (dashed lines) and is measured by aLIGO. At the same SNR value, both techniques of increasing ${\chi }_{\text{eff}}$ give similar results, confirming the hypothesis that it is this combination that affects the gravitational waveform.

Figure 10

Left panel: Scatter plot of the 90% probability quantile in $(m_1,m_2)$ for System 3 of Table 1 and SNR 10 with different masses extracted with spin-aligned (black) and double-precessing (red) templates. The short lines that cut across the scatter plots mark the boundaries of the quantiles in the direction orthogonal to the chirp mass. Right panel: Scatter plot showing the 90% probability quantile in the $({\chi }_m,m_1)$ plane for the $(1.36,1.34)M_{\odot }$ system from the left panel. The mass-spin correlation is far more pronounced for the spin-aligned model (black) than for the double-precessing (red) waveform model.

Figure 11

Prior distribution for ${\chi }_{\text{eff}}$ for the priors we have used in the two different models: uniform spin magnitudes and uniform priors on the unit sphere for all direction angles for the precessing model (solid line), and uniform spin magnitudes for the aligned model (dotted line). In the range of interest $[-0.5,0.5]$ the priors differ by less than a factor of $\sim 3-4$, demonstrating that it is not a difference in priors that results in the increased measurement accuracy of the double-precessing model.

Figure 12

Maximum posterior value for ${\chi }_{\text{eff}}$ for System 1 of Table 1 as a function of the injected value of ${\chi }_{\text{eff}}$. The top panels correspond to a signal measured by aLIGO, while the bottom ones are for LIGO3. The left panels show signals recovered with the double-precessing templates, while the right ones with the spin-aligned ones. The shaded regions indicate the minimum interval that contains 68% of the posterior distribution. The black dashed line indicates ${\chi }_{\text{eff}}=0$, while the black solid line gives the ${\chi }_{\text{eff}}={\chi }_{\text{eff}}^{\mathrm{inj}}$ curve. The use of double-precessing templates reduces the spin extraction error by about an order of magnitude compared to what one would get from spin-aligned templates. Also, for the same model the accuracy is essentially independent on the injected spin value, and depends mainly on the SNR value.