Searching for precessing binary systems with mode-by-mode filtering and marginalization

Abstract

Nearly all previous binary black hole searches in LIGO--Virgo--KAGRA (LVK) gravitational wave data have assumed that the component spins are aligned with the orbital angular momentum, thereby neglecting spin-precession effects in the waveform, which can lead to potentially missing interesting signals. Precessing searches are challenging, because the extra degrees of freedom due to misaligned spins lead to: $(i)$ a much larger number of templates compared to the aligned-spin configurations, $(ii)$ an increased rate of background triggers. To address this, we develop novel precessing signal template banks using mode-by-mode filtering and marginalization methods. We use the precession harmonic decomposition from Fairhurst et al. (2019) and filter each precessing harmonic separately with the data. We then marginalize over the SNRs from different harmonics in our detection statistic. We also use machine learning methods to improve our search efficiency: $(i)$ we use singular value decomposition together with random forest regressor to reduce redundancy in the dominant precessing-harmonic templates; $(ii)$ we use normalizing flows to generate optimal prior samples for harmonic SNRs for the marginalized statistic. We show that marginalizing (instead of maximizing) over the harmonic mode SNRs increases the search sensitive volume by $\sim 10\%$. Results from searching in LVK data using this framework will be reported in a companion paper.

Figures (9)

• Figure 1: Neglecting precession harmonics in the template waveforms of a matched-filtering search can reduce the overlap with a fully precessing signal and thereby decrease the sensitive (detection) volume. In this figure we quantify this effect across the parameter space shown by plotting the fractional volume loss (calculated by the cube of the waveform overlap in this simplistic scenario). The four panels correspond to increasingly complete the precession harmonics in the templates: nonprecessing (NP), and precession-harmonics ${\rm P}_0$, $\rm P_1$, and $\rm P_2$. As more harmonics are included, the mismatch-driven loss is progressively reduced. For mass-ratio $q>0.2$, the first two harmonics play the important role while the third one only gives marginal improvement.
• Figure 2: The configuration of a binary black hole system with orbital angular momentum $\boldsymbol{L}$ and total angular momentum $\boldsymbol{J}$ misaligned. In the $L$-frame (blue), $\hat{\boldsymbol{z}} \parallel \hat{\boldsymbol{L}}$, and $\hat{\boldsymbol{x}}$ is defined as the unit vector pointing from $m_2$ to $m_1$. The $J$-frame (red) is defined with $\hat{\boldsymbol{z}}' \parallel \hat{\boldsymbol{J}}$, and the line-of-sight vector $\hat{\boldsymbol{N}}$ lies in the $x'-z'$ plane. The three Euler angles $(\alpha,\beta,\gamma=-\epsilon)$ describe the active rotation from the $J$-frame to the $L$-frame in the $(z,y,z)$ convention Pratten:2020ceb. Here $\tilde{x}$ denotes the line of nodes (intersection of the two equatorial planes), $\alpha$ is measured in the $J$-frame equatorial plane, $\beta$ is the opening angle between $\hat{\boldsymbol{J}}$ and $\hat{\boldsymbol{L}}$, and $\gamma$ is measured in the $L$-frame equatorial plane.
• Figure 3: The template banks in our analysis are split a according to the normalized waveform amplitudes of the dominant precessing mode (the different banks are roughly distinguished by the cutoff frequencies of the waveforms). We show here the physical parameters $M_{\rm tot},q,\chi_{\rm eff},\chi_p$ correspond to different banks and the bank indices are shown with different colors.
• Figure 4: We model the phases of dominant-mode templates using the top three basis vectors from an SVD-based decomposition of the full GW waveforms (see Eq. \ref{['eq:SVD_decomposition']}). We show here the SVD coefficients for phases in one of our banks BBH-2. The coefficients lie on an intrinsically lower-dimensional manifold with a non-linear structure. We therefore use a non-linear machine learning tool called random forest regressor (RF) to model the higher-order coefficients from the lowest-order and thus compress the dimensionality of our banks (see Eq. \ref{['eq:RF']}).
• Figure 5: Each of the dominant-mode template in our banks is accompanied by $(i)$ normalized sub-dominant mode templates and $(ii)$ mode-SNR-ratio samples corresponding to the chunk of parameter phase space associated with the dominant template [$R^\perp_{k} = h^\perp_k(f_{\rm ref})/h^\perp_0(f_{\rm ref})$ for precession harmonic $k$, see Eq. \ref{['eq:mode_ratio_perp']}]. We model the distribution of the mode-ratio samples $\boldsymbol{R}$ using a conditional normalizing flow (NF). We show the mode ratio amplitude distribution $p_\theta(|\boldsymbol{R}|\, \vert\, \boldsymbol{c})$ (in blue) learned from NF. The blue line denotes the point true value of $\boldsymbol{R}$ for a particular choice of intrinsic parameters in one of our banks BBH-7. Compared with the $|\boldsymbol{R}|$ distribution corresponding to the entire bank, NF significantly shrinks the prior domain for the marginalization integral over $|\boldsymbol{R}|$, which improves the efficiency of marginalization.