Constructing a gravitational wave analysis pipeline for extremely large mass ratio inspirals

Author(s)

Wang, Tian-Xiao, Wang, Yan, Torres-Orjuela, Alejandro, Lin, Yi-Ren, Fan, Hui-Min, Vázquez-Aceves, Verónica, Hu, Yi-Ming

Abstract

Extremely large mass-ratio inspirals (XMRIs), consisting of a brown dwarf orbiting a supermassive black hole, emit long-lived and nearly monochromatic gravitational waves in the millihertz band and constitute a promising probe of strong-field gravity and black-hole properties. However, dedicated data-analysis pipelines for XMRI signals have not yet been established. In this work, we develop, for the first time, a hierarchical semi-coherent search pipeline for XMRIs tailored to space-based gravitational-wave detectors, with a particular focus on the TianQin mission. The pipeline combines a semi-coherent multi-harmonic $\mathcal{F}$-statistic with particle swarm optimization, and incorporates a novel eccentricity estimation method based on the relative power distribution among harmonics. We validate the performance of the pipeline using simulated TianQin data for a Galactic center XMRI composed of a brown dwarf and Sgr A*. For a three-month observation, the pipeline successfully recovers the signal and achieves high-precision parameter estimation, including fractional uncertainties of $<10^{-6}$ in the orbital frequency, $\lesssim10^{-3}$ in the eccentricity, $\lesssim2\times10^{-3}$ in the black-hole mass, and $\lesssim10^{-3}$ in the black-hole spin. Our framework establishes a practical foundation for future XMRI searches with space-based detectors and highlights the potential of XMRIs as precision probes of stellar dynamics and strong-field gravity in the vicinity of supermassive black holes.

Figures

Geometry of an eccentric \ac{XMRI} system in the black-hole spin–aligned frame $(X,Y,Z)$ and the orbital frame $(X',Y',Z')$. $Z$-axis is parallel to the spin of Sgr~A* ($\vec{S}$), while $Z'$-axis aligned with the orbital angular momentum $\vec{L}$, where the $X'$-axis points towards the pericenter $P$. The misalignment between $\vec{L}$ and $\vec{S}$ is characterized by the inclination angle $\iota$. The unit vector $\hat{k}$ denotes the propagation direction of the \acp{GW} toward Earth, with $(\theta, \phi)$ being its corresponding polar and azimuthal angles in the orbital frame, and $\delta$ being its angle relative to $\vec{S}$. The geometry is further specified by the longitude of the ascending node (point $A$) $\alpha$ and the argument of pericenter $\gamma$.
Caption Geometry of an eccentric \ac{XMRI} system in the black-hole spin–aligned frame $(X,Y,Z)$ and the orbital frame $(X',Y',Z')$. $Z$-axis is parallel to the spin of Sgr~A* ($\vec{S}$), while $Z'$-axis aligned with the orbital angular momentum $\vec{L}$, where the $X'$-axis points towards the pericenter $P$. The misalignment between $\vec{L}$ and $\vec{S}$ is characterized by the inclination angle $\iota$. The unit vector $\hat{k}$ denotes the propagation direction of the \acp{GW} toward Earth, with $(\theta, \phi)$ being its corresponding polar and azimuthal angles in the orbital frame, and $\delta$ being its angle relative to $\vec{S}$. The geometry is further specified by the longitude of the ascending node (point $A$) $\alpha$ and the argument of pericenter $\gamma$.
The detection statistic $2\mathcal{F}_n$ versus harmonic index $n$. Noiseless results (red dots) coincide with the optimal \ac{SNR} squared $\rho_n^2$ (blue open circles), while noisy realizations are shown as green crosses.
Caption The detection statistic $2\mathcal{F}_n$ versus harmonic index $n$. Noiseless results (red dots) coincide with the optimal \ac{SNR} squared $\rho_n^2$ (blue open circles), while noisy realizations are shown as green crosses.
Comparison between the \texttt{GWSpace} template and the analytic template, obtained via direct Fourier transform of the time-domain signals. Upper panel: The strain of the $n=3$ harmonic for the \texttt{GWSpace} template (blue), the analytic template (grey), and their resulting strain residual (red). The vertical green dashed line marks the frequency $3f_0$, where $f_0$ denotes the orbital frequency of the \ac{XMRI} system, while the frequency sidebands surrounding the peak result from the combined effects of the \ac{XMRI} orbital precession and the periodic motion of TianQin. Lower panel: The phase evolution of the two templates and their corresponding phase residual (red). The close agreement between the templates, especially near the peak frequency, demonstrates the reliability of the analytic approximation used in our search pipeline.
Caption Comparison between the \texttt{GWSpace} template and the analytic template, obtained via direct Fourier transform of the time-domain signals. Upper panel: The strain of the $n=3$ harmonic for the \texttt{GWSpace} template (blue), the analytic template (grey), and their resulting strain residual (red). The vertical green dashed line marks the frequency $3f_0$, where $f_0$ denotes the orbital frequency of the \ac{XMRI} system, while the frequency sidebands surrounding the peak result from the combined effects of the \ac{XMRI} orbital precession and the periodic motion of TianQin. Lower panel: The phase evolution of the two templates and their corresponding phase residual (red). The close agreement between the templates, especially near the peak frequency, demonstrates the reliability of the analytic approximation used in our search pipeline.
Schematic of the hierarchical semi-coherent search pipeline for \ac{XMRI} signals. The top panels illustrate the foundational components: (left) the generation of the $\mathcal{F}$-statistic using downsampled data and analytic \ac{TDI} templates; (center) the evolution of the $2\mathcal{F}$ peak as the coherence time $T_c$ increases from 15 to 90 days. The broader peaks at shorter $T_c$ provide a larger capture range that facilitates the initial global search, while the narrower peaks at longer $T_c$ yield significantly higher parameter precision, motivating our hierarchical semi-coherent approach; and (right) the use of the multi-harmonic power distribution $2\mathcal{F}_n$ to provide an initial probe of the initial eccentricity $e_0$. The bottom panel details the three-stage hierarchical \ac{PSO} workflow. Stage I ($T_c = 15$\,d) focuses on the initial localization of orbital frequency $f_0$ and initial eccentricity $e_0$. Stage II ($T_c = 30$\,d) performs an intermediate refinement to break parameter degeneracies among the intrinsic parameters. Stage III ($T_c = 90$\,d) executes the final fully coherent search to obtain high-precision estimates for the complete parameter set, including the analytically inferred extrinsic parameters. At each transition, the parameter space is refined based on a $t$-distribution analysis of the best-performing \ac{PSO} runs.
Caption Schematic of the hierarchical semi-coherent search pipeline for \ac{XMRI} signals. The top panels illustrate the foundational components: (left) the generation of the $\mathcal{F}$-statistic using downsampled data and analytic \ac{TDI} templates; (center) the evolution of the $2\mathcal{F}$ peak as the coherence time $T_c$ increases from 15 to 90 days. The broader peaks at shorter $T_c$ provide a larger capture range that facilitates the initial global search, while the narrower peaks at longer $T_c$ yield significantly higher parameter precision, motivating our hierarchical semi-coherent approach; and (right) the use of the multi-harmonic power distribution $2\mathcal{F}_n$ to provide an initial probe of the initial eccentricity $e_0$. The bottom panel details the three-stage hierarchical \ac{PSO} workflow. Stage I ($T_c = 15$\,d) focuses on the initial localization of orbital frequency $f_0$ and initial eccentricity $e_0$. Stage II ($T_c = 30$\,d) performs an intermediate refinement to break parameter degeneracies among the intrinsic parameters. Stage III ($T_c = 90$\,d) executes the final fully coherent search to obtain high-precision estimates for the complete parameter set, including the analytically inferred extrinsic parameters. At each transition, the parameter space is refined based on a $t$-distribution analysis of the best-performing \ac{PSO} runs.
Recovered values of the orbital inclination angle $\iota$ (blue circles) and Sgr~A* spin $s$ (red squares) from ten independent PSO runs for Stage I (open) and Stage II (filled). The results obtained in Stage II exhibit a substantially reduced scatter and cluster more closely around the injected parameters (vertical dashed lines), although a residual spread remains.
Caption Recovered values of the orbital inclination angle $\iota$ (blue circles) and Sgr~A* spin $s$ (red squares) from ten independent PSO runs for Stage I (open) and Stage II (filled). The results obtained in Stage II exhibit a substantially reduced scatter and cluster more closely around the injected parameters (vertical dashed lines), although a residual spread remains.
Evolution of the fully coherent $2\mathcal{F}$ statistic as a function of PSO iteration for the optimal chain in Stage III ($T_c = 90$~d). The solid blue curve represents the best-fit likelihood encountered by the swarm, while the red dashed line denotes the theoretical $2\mathcal{F}_{\mathrm{theory}}$ for the injected signal. The monotonic increase and subsequent saturation toward the expected value confirm that the PSO has successfully navigated the high-dimensional likelihood surface to the global maximum.
Caption Evolution of the fully coherent $2\mathcal{F}$ statistic as a function of PSO iteration for the optimal chain in Stage III ($T_c = 90$~d). The solid blue curve represents the best-fit likelihood encountered by the swarm, while the red dashed line denotes the theoretical $2\mathcal{F}_{\mathrm{theory}}$ for the injected signal. The monotonic increase and subsequent saturation toward the expected value confirm that the PSO has successfully navigated the high-dimensional likelihood surface to the global maximum.
Multistage parameter localization for the Sgr A* spin $s$ (top) and mass $M$ (bottom) across the hierarchical pipeline. From left to right, the columns illustrate the progressive contraction of the search range and the corresponding enhancement in parameter precision as $T_c$ increases from 15 to 90 days. The blue solid curves show the PSO trajectories as a function of iteration number. The red dotted lines indicate the injected parameters, $s = 0.9$ and $M = 4\times10^{6}\,\mathrm{M_\odot}$. In Stages~I and~II, the green dashed lines represent the search ranges adopted for the subsequent stage: the bounds are inherited from the previous stage when no further restriction is applied, and are tightened once sufficient localization is achieved. Gray dashed connectors illustrate how these search ranges are propagated between successive stages. Gray dashed lines visualize the propagation of search ranges between successive stages. The convergence toward the injected parameters (red dotted lines) underscores the robustness of the strategy in mitigating parameter degeneracies through incremental coherence.
Caption Multistage parameter localization for the Sgr A* spin $s$ (top) and mass $M$ (bottom) across the hierarchical pipeline. From left to right, the columns illustrate the progressive contraction of the search range and the corresponding enhancement in parameter precision as $T_c$ increases from 15 to 90 days. The blue solid curves show the PSO trajectories as a function of iteration number. The red dotted lines indicate the injected parameters, $s = 0.9$ and $M = 4\times10^{6}\,\mathrm{M_\odot}$. In Stages~I and~II, the green dashed lines represent the search ranges adopted for the subsequent stage: the bounds are inherited from the previous stage when no further restriction is applied, and are tightened once sufficient localization is achieved. Gray dashed connectors illustrate how these search ranges are propagated between successive stages. Gray dashed lines visualize the propagation of search ranges between successive stages. The convergence toward the injected parameters (red dotted lines) underscores the robustness of the strategy in mitigating parameter degeneracies through incremental coherence.
One-dimensional profiles of $2\mathcal{F}$ obtained by varying a single parameter while keeping all other parameters fixed at their injected parameters. The vertical red lines indicate the injected parameter values. (a) $2\mathcal{F}$ as a function of the \ac{BD} mass $m$, showing a flat distribution. (b) $2\mathcal{F}$ as a function of the initial argument of pericenter $\gamma_0$, exhibiting a secondary peak at $\gamma_{0,\mathrm{true}} + \pi$ resulting from the inherent phase-shift degeneracy.
Caption One-dimensional profiles of $2\mathcal{F}$ obtained by varying a single parameter while keeping all other parameters fixed at their injected parameters. The vertical red lines indicate the injected parameter values. (a) $2\mathcal{F}$ as a function of the \ac{BD} mass $m$, showing a flat distribution. (b) $2\mathcal{F}$ as a function of the initial argument of pericenter $\gamma_0$, exhibiting a secondary peak at $\gamma_{0,\mathrm{true}} + \pi$ resulting from the inherent phase-shift degeneracy.
One-dimensional profiles of $2\mathcal{F}$ obtained by varying a single parameter while keeping all other parameters fixed at their injected parameters. The vertical red lines indicate the injected parameter values. (a) $2\mathcal{F}$ as a function of the \ac{BD} mass $m$, showing a flat distribution. (b) $2\mathcal{F}$ as a function of the initial argument of pericenter $\gamma_0$, exhibiting a secondary peak at $\gamma_{0,\mathrm{true}} + \pi$ resulting from the inherent phase-shift degeneracy.
Caption One-dimensional profiles of $2\mathcal{F}$ obtained by varying a single parameter while keeping all other parameters fixed at their injected parameters. The vertical red lines indicate the injected parameter values. (a) $2\mathcal{F}$ as a function of the \ac{BD} mass $m$, showing a flat distribution. (b) $2\mathcal{F}$ as a function of the initial argument of pericenter $\gamma_0$, exhibiting a secondary peak at $\gamma_{0,\mathrm{true}} + \pi$ resulting from the inherent phase-shift degeneracy.
Two-dimensional $2\mathcal{F}$ surfaces evaluated over selected intrinsic-parameter subspaces. The surfaces are obtained by scanning the parameters shown on the axes, while keeping all remaining parameters fixed at their injected values. The cyan plus signs indicate the injected parameter locations. (a) $2\mathcal{F}$ surface in the $e_0$--$M$ plane. (b) $2\mathcal{F}$ surface in the $e_0$--$s$ plane.
Caption Two-dimensional $2\mathcal{F}$ surfaces evaluated over selected intrinsic-parameter subspaces. The surfaces are obtained by scanning the parameters shown on the axes, while keeping all remaining parameters fixed at their injected values. The cyan plus signs indicate the injected parameter locations. (a) $2\mathcal{F}$ surface in the $e_0$--$M$ plane. (b) $2\mathcal{F}$ surface in the $e_0$--$s$ plane.
Two-dimensional $2\mathcal{F}$ surfaces evaluated over selected intrinsic-parameter subspaces. The surfaces are obtained by scanning the parameters shown on the axes, while keeping all remaining parameters fixed at their injected values. The cyan plus signs indicate the injected parameter locations. (a) $2\mathcal{F}$ surface in the $e_0$--$M$ plane. (b) $2\mathcal{F}$ surface in the $e_0$--$s$ plane.
Caption Two-dimensional $2\mathcal{F}$ surfaces evaluated over selected intrinsic-parameter subspaces. The surfaces are obtained by scanning the parameters shown on the axes, while keeping all remaining parameters fixed at their injected values. The cyan plus signs indicate the injected parameter locations. (a) $2\mathcal{F}$ surface in the $e_0$--$M$ plane. (b) $2\mathcal{F}$ surface in the $e_0$--$s$ plane.
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