New approach to template banks of gravitational waves with higher harmonics: Reducing matched-filtering cost by over an order of magnitude

TL;DR

A new strategy to include HM in template banks that exploits the natural connection between the modes is developed, which leads to a HM search pipeline whose matched-filtering cost is just 3 times that of a quadrupole-only search.

Abstract

Searches for gravitational wave events use models, or templates, for the signals of interest. The templates used in current searches in the LIGO-Virgo-Kagra (LVK) data model the dominant quadrupole mode $(\ell,|m|)=(2,2)$ of the signals, and omit sub-dominant higher-order modes (HM) such as $(\ell,|m|)=(3,3)$, $(4,4)$, which are predicted by general relativity. This omission reduces search sensitivity to black hole mergers in interesting parts of parameter space, such as systems with high masses and asymmetric mass-ratios. We develop a new strategy to include HM in template banks: instead of making templates containing a combination of different modes, we separately store normalized templates corresponding to $(2,2)$, $(3,3)$ and $(4,4)$ modes. To model aligned-spin $(3,3)$, $(4,4)$ waveforms corresponding to a given $(2,2)$ waveform, we use a combination of post-Newtonian formulae and machine learning tools. In the matched filtering stage, one can filter each mode separately with the data and collect the timeseries of signal-to-noise ratios (SNR). This leads to a HM template bank whose matched-filtering cost is just $\approx 3\times$ that of a quadrupole-only search (as opposed to $\approx\! 100 \times$ in previously proposed HM search methods). Our method is effectual and generally applicable for template banks constructed with either stochastic or geometric placement techniques. New GW candidate events that we detect using our HM banks and details for combining the different SNR mode timeseries are presented in accompanying papers: Wadekar et al. [1] and [2] respectively. Additionally, we discuss non-linear compression of $(2,2)$-only geometric-placement template banks using machine learning algorithms.

Figures (9)

• Figure 1: In our template banks, for each set of intrinsic parameters (i.e., masses and spins [$m_1,m_2, s_{1z}, s_{2z}$]), we generate and store the normalized (2,2), (3,3) and (4,4) mode waveforms separately. We then filter the data with the individual modes, which results in a cost increase of just $3\times$ compared to the (2,2)-only case. Note that the $3\times$ factor is significantly smaller than a factor of $\sim 100\times$ occurring when the templates are made with combinations of different harmonics (in the combined case, templates need to span the larger 6D space: [$m_1,m_2, s_{1z}, s_{2z}, \iota, \phi_0$], where $\iota$ is the inclination and $\phi_0$ is the initial reference phase of the binary) Cha22Sch23_NF_TemplateBankHar18). We store the output SNR timeseries of each mode and later combine them by marginalizing over $\iota$ and $\phi_\mathrm{ref}$ (details of the marginalization algorithm are given in our companion paper: Ref. Wad23_Pipeline, and we also provide a brief overview in Section \ref{['sec:matched_filtering']}).
• Figure 2: Ignoring HM in the waveform templates can cause a loss of waveform overlap with the true signal, which in turn leads to a reduction in the detection volume of systems. In this figure, we show the fractional detection volume of systems (calculated by the cube of the waveform overlap in a simplistic scenario) in particular regions of parameter space. In the left panel, we include only the dominant $(2,2)$ mode in the templates and see that the detection volume loss is larger for high-mass and asymmetric mass ratio systems (see Fig. \ref{['fig:HM_PSD']} for the reasoning behind the high mass behavior). The center and right panels show the behavior when additional $\ell = |m|$ modes are included in addition to $(2,2)$ (omitting the $\ell\neq |m|$ modes: $(2,1)$, $(3,2)$). We find that the fractional loss in volume is $\lesssim 4$% in the given parameter space, showing that adding only the (3,3) and (4,4) modes to our template banks is roughly sufficient. In all the panels, we have averaged over inclination accounting for the brightness of the 22 waveform. Note that the range of the color bar is significantly different in the three panels.
• Figure 3: As seen in Fig. \ref{['fig:SNR_Mtot_q']}, the fractional contribution of HM to SNR increases significantly for high-mass binaries. This is because the noise ASD (amplitude spectral density) increases at low-frequencies, which leads to preferential downweighting of the (2,2) mode SNR, especially for high masses where the (2,2) signal is dominated by lower frequencies (while the HM, being at higher frequencies, are less affected). The reference noise ASD shown here corresponds to the O3 run and has been used throughout this paper. The parameters of the system used to generate the mode curves are shown in the title of the plot.
• Figure 4: The template banks in our analysis are split a according to the normalized waveform amplitudes of the (2,2) mode. Top: Normalized amplitude profiles corresponding to our banks, where we see that the different banks are roughly distinguished by the cutoff frequencies of the waveforms. Bottom: Physical parameters corresponding to the different banks.
• Figure 5: We model the phases of our 22 templates using a low-dimensional basis constructed using SVD (see Eq. \ref{['eq:phases_22']}). The dimensionality of our banks is set by the number of SVD coefficients $c_i$ needed to get effectualness above a particular threshold. Top: The top three SVD coefficients corresponding to physical waveforms follow a curved hypersurface (showing that $c_0, c_1, c_2$ are not independent, but there is a redundancy which leads to a spurious increase in number of dimensions of our subbanks). Bottom: Histograms of waveform matches for different case of $c_i$ dimensions. Upon modeling $c_{i>2}$ as a function of $c_0, c_1$ using a machine learning tool called random forest regressor (RF), we find that all our quadrupole subbanks can be compressed to two dimensions. Overall, using the RF enables us to search with $\sim 40$% times fewer templates while keeping a similar level of effectualness.