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Tousif Islam

Publications and source records attributed to Tousif Islam.

At least 19 recordsLinked to original sources
arXiv2026

Unified remnant models for aligned-spin, precessing, and eccentric binary black hole mergers

Using approximately $5000$ numerical-relativity (NR) simulations spanning mass ratios up to $q=128$ and $1200$ black-hole-perturbation-theory (BHPT) simulations extending to $q=1000$, we present fully analytic models for the remnant properties of quasi-circular binary black hole mergers. The models gwModelRemS (for final mass, final spin, peak luminosity, and recoil kick) and gwModelRemP (for final mass, final spin, and peak luminosity) describe nonprecessing and precessing binaries, respectively. Our approach combines analytic insights from post-Newtonian theory and point-particle limits with a data-driven fitting framework which also includes validation-guided AI-agent-assisted optimization. The resulting fits outperform existing analytic models and achieve accuracies comparable to data-driven models within their domains of validity. We also construct gwModelRemP\_flow, a normalizing-flow model for recoil kicks from precessing binaries, marginalized over the in-plane spin orientations. This model additionally utilizes an enlarged low-fidelity training set, beyond the available NR and BHPT simulations, constructed through a waveform-based data-augmentation procedure. We also include simple eccentric extensions through leading-order dependence on eccentricity and orbital anomaly. The models span the equal-mass to extreme-mass-ratio regimes and are publicly available through the gwModels package for applications in gravitational-wave astronomy, astrophysical population studies, and cosmology.

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arXiv2026

Recoil kicks from binary black hole mergers in GWTC catalogs: implications for retention and hierarchical mergers

We infer recoil (kick) velocities for both individual binary black hole (BBH) mergers and candidate intermediate-mass black hole events, as well as for the BBH populations inferred from GWTC catalogs up to GWTC-5. We obtain informative recoil constraints for several events, including GW231028-153006 ($v_{\rm kick}=839^{+1018}_{-681}\,\mathrm{km\,s^{-1}}$) and GW231123-135430 ($v_{\rm kick}=974^{+944}_{-760}\,\mathrm{km\,s^{-1}}$), while finding that the majority of event-level recoil posteriors remain broad and only weakly informative. We further infer consistent population-level recoil distributions across GWTC-3, GWTC-4, and GWTC-5, with median kick velocities of approximately $300$--$330\,\mathrm{km\,s^{-1}}$. Using both event-level and population-level recoil estimates, we find typical retention probabilities of $\sim2$--$3\%$ for globular clusters, $\sim28$--$32\%$ for nuclear star clusters, $\sim25$--$29\%$ for dwarf galaxies, and $\sim92$--$94\%$ for elliptical galaxies. We also compute recoil-induced displacements and dynamical-friction return times, finding that retained remnants in globular clusters and nuclear star clusters can remain displaced from their host cores for extended periods. Our results show that retention alone is not sufficient to determine the prospects for hierarchical mergers: hierarchical-merger efficiency depends on both remnant retention and post-kick re-centering.

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arXiv2026

Accurate models for recoil velocity distribution in black hole mergers with comparable to extreme mass-ratios and their astrophysical implications

Modeling the remnant recoil velocity (kick) distribution from binary black hole mergers is crucial for understanding hierarchical mergers in active galactic nuclei or globular clusters. Existing analytic models often show large discrepancies with numerical relativity (NR) data, while data-driven models are limited to mass ratios of q<=8 (aligned spins) and q<=4 (precessing spins) and break down when extrapolated outside their training ranges. Using ~5000 of NR simulations from the SXS and RIT catalogs up to q=128 and ~100 black hole perturbation theory simulations up to q=200, we present two classes of models: (i) gwModel_kick_q200 (gwModel_kick_q200_GPR), an analytic (Gaussian process regression) model for aligned-spin binaries. (ii) gwModel_kick_prec_flow, a normalizing-flow model for kick distribution from precessing binaries with isotropic spins. Our approach combines analytic insights from post-Newtonian theory with data-driven techniques to ensure correct limiting behavior and high accuracy across parameter space. Both gwModel_kick_q200 and gwModel_kick_prec_flow are valid from comparable to extreme mass ratios. Extensive validation shows all three models outperform existing ones within their respective domains. Finally, using both back-of-the-envelope estimates and 1404 detailed star cluster simulations incorporating our kick models, we find that the black hole retention probability in low mass globular clusters can vary noticeably when the gwModel_kick_prec_flow model is employed. The models are publicly available through the gwModels package.

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arXiv2026

Analysis of late-time tails in spin-aligned eccentric binary black hole mergers

We present a comprehensive analysis of late-time tails in gravitational radiation from merging spin-aligned eccentric binary black holes, using high-accuracy point-particle black hole perturbation theory simulations. We simulate the late-time evolution of 15 binary black hole mergers with mass ratio $q = 1000$, dimensionless spins $χ= [-0.9, -0.6, 0.0, 0.6, 0.9]$ and eccentricity at the last stable orbit $e_{\rm LSO} = [0.8, 0.9, 0.95]$. We track the tail amplitudes and exponents up to a retarded time coordinate $t = 9000M$ after merger for the six spin-weighted spherical harmonic modes $(2,1)$, $(2,2)$, $(3,2)$, $(3,3)$, $(4,3)$, and $(4,4)$ employing both frequentist and Bayesian approaches. We note that the tails are increasingly pronounced for binaries with high eccentricity $e_{\rm LSO}$ and large negative spin $χ$. We find that the overall late-time exponents closely approach their predicted asymptotic values ($p=-\ell-4$ for Weyl curvature scalar $ψ_{4,\ell m}$ where $\ell$ is the spin-weighted spherical harmonic index), while estimates restricted to the latest portion of the data exactly recover them. We further verify numerically that modes with the same spherical index $\ell$ share identical tail exponents, while variations in $m$ do not affect the tail behavior. Our analysis framework is publicly available through the gwtails Python package.

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arXiv2026

Adding higher-order spherical harmonics in non-spinning eccentric binary black hole merger waveform models

gwNRHME is a recently developed framework that seamlessly converts a multi-modal (i.e with several spherical harmonic modes) quasi-circular waveform into multi-modal eccentric waveform if the quadrupolar eccentric waveform is known. Here, we employ the gwNRHME framework to combine a multi-modal quasi-circular waveform model NRHybSur3dq8 and quadrupolar non-spinning eccentric waveform model EccentricIMR to construct multi-modal non-spinning eccentric model NRHybSur3dq8-gwNRHME. Using a total of 35 eccentric numerical relativity (NR) simulations obtained from the SXS and RIT catalogs, we demonstrate that NRHybSur3dq8-gwNRHME model predictions agree well with NR (with typical relative $L_2$ errors of ~0.01 for the dominant quadrupolar mode) for mass ratios $ 1 \leq q \leq 4$ and eccentricities up to ~0.2 measured about 10 cycles before the merger. To demonstrate the modularity of the gwNRHME framework, we further combine EccentricIMR with BHPTNRSur1dq1e4 model and develop a non-spinning eccentric models named BHPTNRSur1dq1e4-gwNRHME. Finally, we develop a different variant of these models by replacing EccentricIMR with EccentricTD. Both the gwNRHME framework and associated models are available through the gwModels package.

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arXiv2026

Consistency of spin effects between numerical relativity and perturbation theory for inspiraling comparable-mass black hole binaries

Numerical relativity (NR) provides the most accurate waveforms for comparable-mass binary black holes but becomes prohibitively expensive for increasingly asymmetric mass ratios. Point-particle black hole perturbation theory (ppBHPT), which expands the Einstein equations in the small-mass-ratio limit, offers a computationally efficient alternative but is expected to break down in the comparable-mass regime because it neglects nonlinear effects. Nonetheless, several recent studies have shown that ppBHPT can model non-spinning binaries with high accuracy when supplemented by simple calibrations or a first post-adiabatic (PA) correction. Here we assess the applicability of ppBHPT to quasi-circular binaries with a spinning primary by comparing waveform amplitudes, orbital frequencies, and orbital phases. We find that spin effects in ppBHPT waveforms (without additional spin information beyond adiabatic order) are in surprisingly close agreement with the corresponding NR calculation (outperforming some post-Newtonian models) over the last $\approx 20$ orbital cycles. This suggests that, after incorporating higher-order corrections into ppBHPT waveforms in the non-spinning limit -- via second-order self-force results or semi-analytical fits -- only modest spin-dependent adjustments may be required to achieve NR-faithful ppBHPT waveforms. We also show that combining non-spinning NR information with adiabatic ppBHPT can provide a reasonably accurate inspiral waveform for spins $χ\lesssim 0.5$ mass ratios $q \gtrsim 5$.

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arXiv2026

Post-Newtonian theory-inspired framework for characterizing eccentricity in gravitational waveforms

Characterizing eccentricity in gravitational waveforms in a consistent manner is crucial to facilitate parameter estimation, astrophysical population studies, as well as searches for these rare systems. We present a framework to characterize eccentricity directly from gravitational waveforms for non-precessing eccentric binary black hole (BBH) mergers using common modulations that eccentricity induces in all spherical harmonic modes of the signals. Our framework is in the spirit of existing methods that use frequency modulations in the waveforms, but we refine the approach by connecting it to state-of-the-art post-Newtonian calculations of the time evolution of the eccentricity. Using 39 numerical relativity (NR) simulations from the SXS and RIT catalogs, as well as waveforms obtained from the post-Newtonian approximation and effective-one-body (EOB) formalism, we show that our framework provides eccentricity estimates that connect smoothly into the relativistic regime (even up to $\sim 2M$ before merger). We also find that it is necessary to carry existing post-Newtonian calculations to an extra $0.5$PN order to adequately characterize existing NR simulations, and provide fits to the extra coefficient for existing simulations. We make the framework publicly available through the Python-based \texttt{gwModels} package.

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arXiv2026

Progenitor of the recoiling super-massive black hole RBH-1 identified using HST/JWST imaging

Using a combination of \textit{Hubble Space Telescope} and \textit{James Webb Space Telescope} imaging, a runaway supermassive black hole (RBH-1) was recently identified with an inferred velocity of $954^{+110}_{-126}\,\mathrm{km\,s^{-1}}$, likely ejected from a compact star-forming galaxy (denoted as GX) at $z \approx 0.96$. Assuming the runaway black hole was the outcome of the gravitational-wave-driven merger of two black holes, we use its measured runaway velocity together with gravitational-wave recoil predictions from numerical relativity and black hole perturbation theory to constrain the mass ratio and spin configuration of the progenitor SMBHs that overcame the final-parsec problem and merged $\sim 70$~Myr ago. We find that the progenitor binary must have been precessing, with a mass ratio $m_1/m_2\lesssim 6$, and that the more massive SMBH must have possessed a high spin (dimensionless spin magnitude $\sim 0.75$) in order to generate a recoil of this magnitude. This has important astrophysical implications as similar SMBH mergers can be an interesting source population for the upcoming LISA mission with signal-to-noise ratios $\gtrsim$ 1000. Furthermore, the progenitor SMBH properties imply that GX was likely formed through a major, gas-rich (``wet'') merger between two galaxies of comparable mass, with a mass ratio $\lesssim 4$.

astro-ph.HE↗
arXiv2026

Universal phenomenological relations between spherical harmonic modes in non-precessing eccentric binary black hole merger waveforms

Using publicly available numerical relativity (NR) simulations for non-spinning eccentric binary black hole (BBH) mergers, Ref \cite{Islam:2024rhm} demonstrated that the eccentricity-induced modulations in the amplitudes and frequencies of different spherical harmonic modes are mutually consistent and can be modeled using a single time series modulation. We extend the validity of the results to all non-precessing binaries by using 83 NR simulations from the SXS, RIT, and MAYA catalogs for aligned-spin eccentric BBH mergers with mass ratios ranging from $1:1$ to $1:4$. Based on these phenomenological relations, we provide a framework named gwNRXHME to compute multi-modal eccentric non-precessing waveforms using two inputs: quadrupolar eccentric waveforms, and the corresponding multi-modal quasi-circular non-precessing waveforms. Furthermore, we compute an overall degree of departure in SXS, RIT, and MAYA NR data from these relations and find that SXS NR simulations generally adhere to these relations more strictly than RIT and MAYA data. We also show that these relations can offer a cost-effective way to filter out noisy higher-order spherical harmonic modes extracted from NR data. Our framework is publicly available through the gwModels package.

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arXiv2026

Data-driven extraction, phenomenology and modeling of eccentric harmonics in binary black hole merger waveforms

Newtonian and post-Newtonian (PN) calculations suggest that each spherical harmonic mode of the gravitational waveforms (radiation) emitted by eccentric binaries can be further decomposed into several eccentricity-induced modes (indexed by $j=1$ to $j=\infty$), referred to as eccentric harmonics. These harmonics exhibit monotonically time-varying amplitudes and instantaneous frequencies, unlike the full eccentric spherical harmonic modes. However, computing or extracting these harmonics are not straightforward in current numerical relativity (NR) simulations and eccentric waveform models. To address this, Patterson \textit{et al} have developed a framework to extract the eccentric harmonics directly from effective-one-body formalism waveforms. In this paper, we build on the ideas presented in Patterson \textit{et al} and propose a data-driven framework, utilizing singular-value decomposition (SVD), that incorporates additional features based on PN intuition to ensure monotonicity in the extracted harmonics. We further demonstrate that the phase (frequency) of these harmonics is simply $jϕ_λ+ϕ_{\rm ecc}$ ($jf_λ+f_{\rm ecc}$) where $ϕ_λ$ ($f_λ$) is related to the secular orbital phase (frequency) and $ϕ_{\rm ecc}$ ($f_{\rm ecc}$) is an additional phase (frequency) that only depends on the eccentricity. We also provide simple analytical fits to obtain the harmonics as a function of the mean anomaly. These relations may prove useful in constructing faithful models that can be employed in cheap and efficient searches and parameter estimation of eccentric mergers. Our framework is modular and can be extended for any other eccentric waveform models or simulation frameworks. The framework is available through the \texttt{gwMiner} package.

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arXiv2026

Merger remnant and eccentricity dynamics surrogates for eccentric nonspinning black hole binaries

Accurate models of merger remnants are increasingly important for gravitational-wave science, including precision tests of gravity with ringdown, inference of black-hole populations, and modeling hierarchical mergers. For eccentric binaries, remnant mass, spin, and recoil carry nontrivial imprints of eccentricity that are both physically informative and more challenging to model, yet remain less developed than in the quasi-circular case. We present two new models trained on numerical-relativity (NR) simulations of unequal-mass, non-spinning eccentric binary black holes: NRSurE_q4NoSpin_Remnant, which predicts remnant properties, and NRSurE_q4NoSpin_Dynamics, a time-domain surrogate for the evolution of eccentricity and mean anomaly. Both models are trained on NR simulations over a three-dimensional parameter space with mass ratios $q \leq 4$, eccentricity $e < 0.23$, and mean anomaly $\ell \in [0,2π)$ radians, where both $e$ and $\ell$ defined at $t=-1000M$ relative to peak amplitude and $M$ is the total mass. We highlight some applications, including the phenomenological impact of eccentricity on remnant properties and the enhancement or suppression of recoil. We also provide error estimates for all modeled quantities, supporting reliable use in current and future gravitational-wave parameter-estimation analyses. Both models will be made available through open-source codes.

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arXiv2026

Gravitational wave surrogate model for spinning, intermediate mass ratio binaries based on perturbation theory and numerical relativity

We present BHPTNRSur2dq1e3, a reduced order surrogate model of gravitational waves emitted from binary black hole (BBH) systems in the comparable to large mass ratio regime with aligned spin ($χ_1$) on the heavier mass ($m_1$). We trained this model on waveform data generated from point particle black hole perturbation theory (ppBHPT) with mass ratios varying from $3 \leq q \leq 1000$ and spins from $-0.8 \leq χ_1 \leq 0.8$. The waveforms are $13,500 \ m_1$ long and include all spin-weighted spherical harmonic modes up to $\ell = 4$ except the $(4,1)$ and $m = 0$ modes. We find that for binaries with $χ_1 \lesssim -0.5$, retrograde quasi-normal modes are significantly excited, thereby complicating the modeling process. To overcome this issue, we introduce a domain decomposition approach to model the inspiral and merger-ringdown portion of the signal separately. The resulting model can faithfully reproduce ppBHPT waveforms with a median time-domain mismatch error of $8 \times 10^{-5}$. We then calibrate our model with numerical relativity (NR) data in the comparable mass regime $(3 \leq q \leq 10)$. By comparing with spin-aligned BBH NR simulations at $q = 15$, we find that the dominant quadrupolar (subdominant) modes agree to better than $\approx 10^{-3} \ (\approx 10^{-2})$ when using a time-domain mismatch error, where the largest source of calibration error comes from the transition-to-plunge and ringdown approximations of perturbation theory. Mismatch errors are below $\approx 10^{-2}$ for systems with mass ratios between $6 \leq q \leq 15$ and typically get smaller at larger mass ratio. Our two models - both the ppBHPT waveform model and the NR-calibrated ppBHPT model - will be publicly available through gwsurrogate and the Black Hole Perturbation Toolkit packages.

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arXiv2026

gwBenchmarks: Stress-Testing LLM Agents on High-Precision Gravitational Wave Astronomy

Modern gravitational wave astronomy relies on modeling tasks that often require months of graduate-level effort, including building fast waveform surrogates from expensive numerical relativity simulations, modeling orbital dynamics of black holes, fitting merger remnant properties and constructing template banks. These problems demand extreme precision to support detection and parameter inference, with state-of-the-art models achieving $\lesssim 10^{-4}$ relative error. We study whether state-of-the-art LLM coding agents can perform such end-to-end scientific modeling, where success requires constructing models with stringent accuracy criteria and reasoning about physical systems. We introduce gwBenchmarks, a suite of eight tasks grounded in gravitational wave analytic calculations and numerical simulations collectively representing over $10^8$ core-hours of compute. The tasks span interpolation, regression, and high-dimensional time-series modeling, requiring a combination of numerical methods, machine learning, and physics-informed approaches. In preliminary experiments, agents frequently relied on proxy metrics, partial evaluation, or fabricated results to spuriously complete tasks. We therefore implement an external pre-defined framework to gauge agent progress. Evaluating twelve coding agents, we find no consistent winner. On the easiest task, multiple agents converge to the same cubic spline solution, with one rediscovering a coordinate transformation widely used in the literature. On harder tasks like analytic waveform modeling, all agents fall 1-2 orders of magnitude short of domain requirements and exhibit systematic failures, including metric misuse, constraint violations, and result fabrication. Our code, data, and website are publicly available.

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arXiv2026

Discovery of Interpretable Surrogates via Agentic AI: Application to Gravitational Waves

Fast surrogate models for expensive simulations are now essential across the sciences, yet they typically operate as black boxes. We present \texttt{GWAgent}, a large language model (LLM)-based workflow that constructs interpretable analytic surrogates directly from simulation data. Surrogate modeling is well suited to agentic workflows because candidate models can be quantitatively validated against ground-truth simulations at each iteration. As a demonstration, we build a surrogate for gravitational waveforms from eccentric binary black hole mergers. We show that providing the agent with a physics-informed domain ansatz substantially improves output model accuracy. The resulting analytic surrogate attains a median Advanced LIGO mismatch of $6.9\times10^{-4}$ together with an $\sim 8.4\times$ speedup in waveform evaluation, surpassing both symbolic regression and conventional machine learning baselines. Beyond producing an accurate model, the workflow identifies compact physical structure from the learned representation. As an astrophysical application, we use \texttt{GWAgent} to analyze the eccentricity of GW200129 and infer $e_{20\mathrm{Hz}}=0.099^{+0.063}_{-0.044}$. These results show that validation-constrained agentic workflows can produce accurate, fast, and interpretable surrogates for scientific simulations and inference.

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arXiv2026

Including higher-order modes in a quadrupolar eccentric numerical relativity surrogate using universal eccentric modulation functions

\texttt{gwNRHME} is a framework that converts multi-modal (i.e., containing several spherical harmonic modes) quasi-circular waveforms into their eccentric counterparts, provided the quadrupolar eccentric mode is known, by exploiting universal eccentric modulation functions. Leveraging this framework, we combine the quasi-circular NR surrogate model \texttt{NRHybSur3dq8} with the quadrupolar, non-spinning, eccentric surrogate \texttt{NRSurE\_q4NoSpin\_22} to construct a multi-modal, non-spinning, eccentric model, denoted as \model{}, which includes nine modes: $(2,\{1,2\})$, $(3,\{1,2,3\})$, $(4,\{2,3,4\})$, and $(5,5)$. When compared against 156 eccentric SXS NR waveforms, \model{} achieves median frequency-domain mismatches (computed using the Advanced LIGO design sensitivity) of $\sim 9\times 10^{-5}$, with a standard deviation of $\sim 2 \times 10^{-4}$. To demonstrate the modularity of the framework, we further combine \texttt{NRSurE\_q4NoSpin\_22} with effective-one-body (EOB) models \texttt{SEOBNRv5HM} and \texttt{TEOBResumS-Dali} in their non-spinning limits, yielding eccentric waveforms with median mismatches of $\sim 2\times10^{-4}$ and $\sim 10^{-3}$, respectively, with standard deviation of $\sim 2 \times 10^{-3}$ and $\sim 2 \times 10^{-2}$ respectively. Finally, we provide both a surrogate model, \texttt{gwEccEvolve\_q4NoSpin\_Sur}, and an analytical model, \texttt{gwEccEvNSv2}, for the eccentricity evolution up to $2M$ before merger, based on eccentricity definitions derived from the universal modulation functions. The \texttt{gwNRHME} framework is publicly available through the \texttt{gwModels} package, and the resulting waveform models will be released via the \texttt{gwsurrogate} package.

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arXiv2026

GW190711_030756 and GW200114_020818: astrophysical interpretation of two asymmetric binary black hole mergers in the IAS catalog

We provide a comprehensive analysis of GW190711_030756 and GW200114_020818, two of the most significant binary black hole merger candidates in the IAS catalog, with probabilities of astrophysical origin $p_{\rm astro}=0.99$ and $0.71$, respectively, and signal-to-noise ratios of approximately $10.0$ and $13.4$. We employ numerical relativity surrogate models to infer both the source properties and the remnant properties of these two candidates. We find that both GW190711_030756 and GW200114_020818 are asymmetric-mass binaries, with inferred mass ratios of $0.35^{+0.32}_{-0.15}$ and $\leq 0.20$. In addition, GW200114_020818 is inferred to have a source-frame total mass of approximately $220M_{\odot}$ and highly spinning black holes, with primary (secondary) dimensionless spin magnitudes of $0.96^{+0.03}_{-0.07}$ ($0.84^{+0.13}_{-0.34}$), closely resembling GW231123_135430. We further find that GW200114\_020818 has a confidently negative effective inspiral spin of $χ_{\rm eff}=-0.60^{+0.22}_{-0.13}$ and exhibits strong spin precession, characterized by an effective precession parameter of $χ_{\rm p}=0.60^{+0.21}_{-0.19}$. GW200114_020818 (when considered alongside GW231123_135430) points towards an emerging population of massive, rapidly spinning BBH mergers. While GW231123_135430 is consistent with mergers in globular clusters, producing systems like GW200114_020818 in such environments remains difficult even under hierarchical merger scenarios. The probability that the remnant black hole of GW190711_030756 (GW200114_020818) is retained in its host environment is $0.079$ ($0.0002$), $0.62$ ($0.965$), and $0.997$ ($1$) if the merger occurred in a globular cluster, a nuclear star cluster, or an elliptical galaxy, respectively.

astro-ph.HE↗
arXiv2026

Kick matters: The impact of a new recoil model on the retention of hierarchical black-hole remnants in globular clusters

In globular clusters, hierarchical mergers are among the most promising pathways to forming massive black holes such as GW231123. A key factor determining whether a merger-remnant black hole will be retained in these environments and thus participate in subsequent hierarchical mergers is the recoil kick velocity. Analytic models for the recoil velocity are currently employed in nearly all population-synthesis frameworks. We instead use a state-of-the-art recoil-kick model gwModel_flow_prec developed from a combination of numerical-relativity and black-hole perturbation-theory data, together with data-driven techniques such as normalizing flows and the post-Newtonian structure of the kick. Employing both back-of-the-envelope estimates and detailed N-body as well as semi-analytical cluster simulations, we show that gwModel_flow_prec leads to a noticeable increase in the retention probability of hierarchical-merger remnants compared to the previously used analytic model and changes the mass and spin distribution of the black holes formed through hierarchical mergers. Additionally, we discuss the implications of our results in the context of massive binaries such as GW231123.

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arXiv2026

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

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.

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