Parameter-estimation bias induced by transient orbital resonances in extreme-mass-ratio inspirals

Author(s)

Levati, Edoardo, Cárdenas-Avendaño, Alejandro

Abstract

Given the multi-frequency nature of relativistic orbits, transient orbital resonances are expected to be ubiquitous during an extreme-mass-ratio inspiral (EMRI). At a resonance, the orbital dynamics is modified in a nontrivial way, imprinting an overall dephasing in the emitted gravitational waves and potentially impacting both the detection and parameter estimation of these sources. In this work, using a Fisher-matrix approach, we investigate the bias induced by transient orbital resonances in EMRI parameter estimation. We focus on the most dynamically significant low-order resonances, 3 : 2 and 2 : 1, as well as on the high-order, subdominant resonances 3 : 1 and 4 : 3. We find that, for most of the orbits considered, neglecting the effect of a resonance crossing leads to significant losses in signal-to-noise ratio and induces bias in parameter recovery. Furthermore, both the sign and the amplitude of the resonance-induced modifications to the integrals of motion play a crucial role and must be modeled accurately. Our results provide further evidence that failing to model transient orbital resonances accurately can hinder EMRI detection and parameter estimation, thereby limiting their scientific potential.

Figures

The last portion of the waveforms for the EMRI \textit{case (ii)} of Tab.~\ref{Table_1}, with ($a = 0.9$, $\eta = 10^{-5}$, $M = 10^6 M_{\odot}$, $p/M$ = $8.67$, $e$ = $0.30$, $\iota$ = $1.22$). We use the \texttt{fastlisaresponse}~\cite{Katz:2022yqe} code to project the time-domain signals onto the LISA second-generation time-delay interferometry (TDI) and construct the $\{A,E\}$ orthogonal channels (top and bottom panels, respectively), making use of the European Space Agency’s simulations of LISA orbits~\cite{lisatools}. The blue solid line corresponds to an adiabatic evolution without accounting for resonance effects. The yellow dashed line corresponds to an evolution where we implement the effective resonance model (ERM)~\cite{SperiGair}, activating the $3:2$ resonance with the coefficients provided in Ref.~\cite{FlanaganHughes}. We compute the loss in the recovered SNR and the mismatch between the two waveforms, finding $\rho_{\text{eff}}/\rho_{\text{opt}} = 0.24$ and $\mathcal{M} = 7.62 \times 10^{-1}$ at the end of the observation time, $T \approx 1$ year.
Caption The last portion of the waveforms for the EMRI \textit{case (ii)} of Tab.~\ref{Table_1}, with ($a = 0.9$, $\eta = 10^{-5}$, $M = 10^6 M_{\odot}$, $p/M$ = $8.67$, $e$ = $0.30$, $\iota$ = $1.22$). We use the \texttt{fastlisaresponse}~\cite{Katz:2022yqe} code to project the time-domain signals onto the LISA second-generation time-delay interferometry (TDI) and construct the $\{A,E\}$ orthogonal channels (top and bottom panels, respectively), making use of the European Space Agency’s simulations of LISA orbits~\cite{lisatools}. The blue solid line corresponds to an adiabatic evolution without accounting for resonance effects. The yellow dashed line corresponds to an evolution where we implement the effective resonance model (ERM)~\cite{SperiGair}, activating the $3:2$ resonance with the coefficients provided in Ref.~\cite{FlanaganHughes}. We compute the loss in the recovered SNR and the mismatch between the two waveforms, finding $\rho_{\text{eff}}/\rho_{\text{opt}} = 0.24$ and $\mathcal{M} = 7.62 \times 10^{-1}$ at the end of the observation time, $T \approx 1$ year.
Parameter bias induced by the $4:3$, $3:2$, $2:1$ and $3:1$ resonances, from top to bottom, respectively, for the EMRIs of Tab.~\ref{Table_1} with $a = 0.9$, $\eta = 10^{-5}$ and $M = 10^6 M_{\odot}$. We use the resonance coefficients provided in Ref.~\cite{FlanaganHughes}, which are computed from Teukolsky-based calculations. We report the results for a specific choice of the signs of the coefficients, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $-$). For each orbit, we compute the loss in the recovered SNR, the mismatch at the end of the observation time between resonant and non-resonant crossing waveforms, and the ratio between the parameter bias and the corresponding statistical uncertainty, from Eq.~\ref{ratio}. These ranges are obtained from the Fisher-validation procedure described in Sec.~\ref{sec::FM}, reflecting the allowed variation in the finite-difference step $\epsilon$. In the case of the $3:1$ resonance, we do not show the results for the high-eccentricity, low-inclination orbit since we find that the kludge fluxes break down, and yield a trajectory that is not consistent with an adiabatic evolution. The horizontal dashed black line marks the threshold $\left|\delta\lambda_{\textrm{bias}}^{i}\right|/\Delta\lambda^{i}=1$.
Caption Parameter bias induced by the $4:3$, $3:2$, $2:1$ and $3:1$ resonances, from top to bottom, respectively, for the EMRIs of Tab.~\ref{Table_1} with $a = 0.9$, $\eta = 10^{-5}$ and $M = 10^6 M_{\odot}$. We use the resonance coefficients provided in Ref.~\cite{FlanaganHughes}, which are computed from Teukolsky-based calculations. We report the results for a specific choice of the signs of the coefficients, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $-$). For each orbit, we compute the loss in the recovered SNR, the mismatch at the end of the observation time between resonant and non-resonant crossing waveforms, and the ratio between the parameter bias and the corresponding statistical uncertainty, from Eq.~\ref{ratio}. These ranges are obtained from the Fisher-validation procedure described in Sec.~\ref{sec::FM}, reflecting the allowed variation in the finite-difference step $\epsilon$. In the case of the $3:1$ resonance, we do not show the results for the high-eccentricity, low-inclination orbit since we find that the kludge fluxes break down, and yield a trajectory that is not consistent with an adiabatic evolution. The horizontal dashed black line marks the threshold $\left|\delta\lambda_{\textrm{bias}}^{i}\right|/\Delta\lambda^{i}=1$.
Parameter bias induced by the $4:3$, $3:2$, $2:1$ and $3:1$ resonances, from top to bottom, respectively, for the EMRIs of Tab.~\ref{Table_1} with $a = 0.9$, $\eta = 10^{-5}$ and $M = 10^6 M_{\odot}$. We use the resonance coefficients provided in Ref.~\cite{FlanaganHughes}, which are computed from Teukolsky-based calculations. We report the results for a specific choice of the signs of the coefficients, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $-$). For each orbit, we compute the loss in the recovered SNR, the mismatch at the end of the observation time between resonant and non-resonant crossing waveforms, and the ratio between the parameter bias and the corresponding statistical uncertainty, from Eq.~\ref{ratio}. These ranges are obtained from the Fisher-validation procedure described in Sec.~\ref{sec::FM}, reflecting the allowed variation in the finite-difference step $\epsilon$. In the case of the $3:1$ resonance, we do not show the results for the high-eccentricity, low-inclination orbit since we find that the kludge fluxes break down, and yield a trajectory that is not consistent with an adiabatic evolution. The horizontal dashed black line marks the threshold $\left|\delta\lambda_{\textrm{bias}}^{i}\right|/\Delta\lambda^{i}=1$.
Caption Parameter bias induced by the $4:3$, $3:2$, $2:1$ and $3:1$ resonances, from top to bottom, respectively, for the EMRIs of Tab.~\ref{Table_1} with $a = 0.9$, $\eta = 10^{-5}$ and $M = 10^6 M_{\odot}$. We use the resonance coefficients provided in Ref.~\cite{FlanaganHughes}, which are computed from Teukolsky-based calculations. We report the results for a specific choice of the signs of the coefficients, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $-$). For each orbit, we compute the loss in the recovered SNR, the mismatch at the end of the observation time between resonant and non-resonant crossing waveforms, and the ratio between the parameter bias and the corresponding statistical uncertainty, from Eq.~\ref{ratio}. These ranges are obtained from the Fisher-validation procedure described in Sec.~\ref{sec::FM}, reflecting the allowed variation in the finite-difference step $\epsilon$. In the case of the $3:1$ resonance, we do not show the results for the high-eccentricity, low-inclination orbit since we find that the kludge fluxes break down, and yield a trajectory that is not consistent with an adiabatic evolution. The horizontal dashed black line marks the threshold $\left|\delta\lambda_{\textrm{bias}}^{i}\right|/\Delta\lambda^{i}=1$.
Parameter bias induced by the $4:3$, $3:2$, $2:1$ and $3:1$ resonances, from top to bottom, respectively, for the EMRIs of Tab.~\ref{Table_1} with $a = 0.9$, $\eta = 10^{-5}$ and $M = 10^6 M_{\odot}$. We use the resonance coefficients provided in Ref.~\cite{FlanaganHughes}, which are computed from Teukolsky-based calculations. We report the results for a specific choice of the signs of the coefficients, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $-$). For each orbit, we compute the loss in the recovered SNR, the mismatch at the end of the observation time between resonant and non-resonant crossing waveforms, and the ratio between the parameter bias and the corresponding statistical uncertainty, from Eq.~\ref{ratio}. These ranges are obtained from the Fisher-validation procedure described in Sec.~\ref{sec::FM}, reflecting the allowed variation in the finite-difference step $\epsilon$. In the case of the $3:1$ resonance, we do not show the results for the high-eccentricity, low-inclination orbit since we find that the kludge fluxes break down, and yield a trajectory that is not consistent with an adiabatic evolution. The horizontal dashed black line marks the threshold $\left|\delta\lambda_{\textrm{bias}}^{i}\right|/\Delta\lambda^{i}=1$.
Caption Parameter bias induced by the $4:3$, $3:2$, $2:1$ and $3:1$ resonances, from top to bottom, respectively, for the EMRIs of Tab.~\ref{Table_1} with $a = 0.9$, $\eta = 10^{-5}$ and $M = 10^6 M_{\odot}$. We use the resonance coefficients provided in Ref.~\cite{FlanaganHughes}, which are computed from Teukolsky-based calculations. We report the results for a specific choice of the signs of the coefficients, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $-$). For each orbit, we compute the loss in the recovered SNR, the mismatch at the end of the observation time between resonant and non-resonant crossing waveforms, and the ratio between the parameter bias and the corresponding statistical uncertainty, from Eq.~\ref{ratio}. These ranges are obtained from the Fisher-validation procedure described in Sec.~\ref{sec::FM}, reflecting the allowed variation in the finite-difference step $\epsilon$. In the case of the $3:1$ resonance, we do not show the results for the high-eccentricity, low-inclination orbit since we find that the kludge fluxes break down, and yield a trajectory that is not consistent with an adiabatic evolution. The horizontal dashed black line marks the threshold $\left|\delta\lambda_{\textrm{bias}}^{i}\right|/\Delta\lambda^{i}=1$.
Parameter bias induced by the $4:3$, $3:2$, $2:1$ and $3:1$ resonances, from top to bottom, respectively, for the EMRIs of Tab.~\ref{Table_1} with $a = 0.9$, $\eta = 10^{-5}$ and $M = 10^6 M_{\odot}$. We use the resonance coefficients provided in Ref.~\cite{FlanaganHughes}, which are computed from Teukolsky-based calculations. We report the results for a specific choice of the signs of the coefficients, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $-$). For each orbit, we compute the loss in the recovered SNR, the mismatch at the end of the observation time between resonant and non-resonant crossing waveforms, and the ratio between the parameter bias and the corresponding statistical uncertainty, from Eq.~\ref{ratio}. These ranges are obtained from the Fisher-validation procedure described in Sec.~\ref{sec::FM}, reflecting the allowed variation in the finite-difference step $\epsilon$. In the case of the $3:1$ resonance, we do not show the results for the high-eccentricity, low-inclination orbit since we find that the kludge fluxes break down, and yield a trajectory that is not consistent with an adiabatic evolution. The horizontal dashed black line marks the threshold $\left|\delta\lambda_{\textrm{bias}}^{i}\right|/\Delta\lambda^{i}=1$.
Caption Parameter bias induced by the $4:3$, $3:2$, $2:1$ and $3:1$ resonances, from top to bottom, respectively, for the EMRIs of Tab.~\ref{Table_1} with $a = 0.9$, $\eta = 10^{-5}$ and $M = 10^6 M_{\odot}$. We use the resonance coefficients provided in Ref.~\cite{FlanaganHughes}, which are computed from Teukolsky-based calculations. We report the results for a specific choice of the signs of the coefficients, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $-$). For each orbit, we compute the loss in the recovered SNR, the mismatch at the end of the observation time between resonant and non-resonant crossing waveforms, and the ratio between the parameter bias and the corresponding statistical uncertainty, from Eq.~\ref{ratio}. These ranges are obtained from the Fisher-validation procedure described in Sec.~\ref{sec::FM}, reflecting the allowed variation in the finite-difference step $\epsilon$. In the case of the $3:1$ resonance, we do not show the results for the high-eccentricity, low-inclination orbit since we find that the kludge fluxes break down, and yield a trajectory that is not consistent with an adiabatic evolution. The horizontal dashed black line marks the threshold $\left|\delta\lambda_{\textrm{bias}}^{i}\right|/\Delta\lambda^{i}=1$.
Parameter bias for the EMRI \textit{case (i)} of Tab.~\ref{Table_1}, for which two independent sets of resonance coefficients are available. We compare the bias induced by the $4:3$, $3:2$, $2:1$, and $3:1$ resonances when using the coefficients reported in Ref.~\cite{FlanaganHughes}, obtained from Teukolsky-based calculations, and those computed using a self-force-based model, as provided in Ref.~\cite{Lynch:2024ohd}. The choice of the signs of the coefficients is the same as in Fig.~\ref{fig2}. While the two prescriptions are in overall agreement for most of the resonances considered, they differ significantly in one case, namely the $4:3$ resonance, where the induced bias differs by several orders of magnitude.
Caption Parameter bias for the EMRI \textit{case (i)} of Tab.~\ref{Table_1}, for which two independent sets of resonance coefficients are available. We compare the bias induced by the $4:3$, $3:2$, $2:1$, and $3:1$ resonances when using the coefficients reported in Ref.~\cite{FlanaganHughes}, obtained from Teukolsky-based calculations, and those computed using a self-force-based model, as provided in Ref.~\cite{Lynch:2024ohd}. The choice of the signs of the coefficients is the same as in Fig.~\ref{fig2}. While the two prescriptions are in overall agreement for most of the resonances considered, they differ significantly in one case, namely the $4:3$ resonance, where the induced bias differs by several orders of magnitude.
Gaussian posterior approximation from the Fisher matrix for the EMRI \textit{case (iii)} of Tab.~\ref{Table_1}, where we activate the $4:3$ resonance. The purple line represents the injected parameter values. We compare two different choices for the signs of the resonance coefficients. The blue contours illustrate the induced parameter bias when the modifications to the constants of motion are all in phase, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $-$). The orange contours refer to the case where the modifications in $\mathcal{E}$ and $\mathcal{L}_z$ are in phase, while that in $\mathcal{Q}$ is out of phase, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $+$). For this second choice, the bias induced in $a$ and $p/M$ is lower and falls within the $1\sigma$ threshold.
Caption Gaussian posterior approximation from the Fisher matrix for the EMRI \textit{case (iii)} of Tab.~\ref{Table_1}, where we activate the $4:3$ resonance. The purple line represents the injected parameter values. We compare two different choices for the signs of the resonance coefficients. The blue contours illustrate the induced parameter bias when the modifications to the constants of motion are all in phase, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $-$). The orange contours refer to the case where the modifications in $\mathcal{E}$ and $\mathcal{L}_z$ are in phase, while that in $\mathcal{Q}$ is out of phase, i.e., $\mathrm{sgn}(\mathcal{C}_{\mathcal{E}}$, $\mathcal{C}_{\mathcal{L}_z}$, $\mathcal{C}_{\mathcal{Q}})$ $=$ ($-$, $-$, $+$). For this second choice, the bias induced in $a$ and $p/M$ is lower and falls within the $1\sigma$ threshold.
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