LISA Non-Linear Dynamics and Tilt-To-Length Coupling

Author(s)

Heisenberg, Lavinia, Inchauspé, Henri, Paczkowski, Sarah, Waibel, Ricardo

Abstract

For the LISA mission, Tilt-To-Length (TTL) coupling is expected to be one of the dominant instrumental noise contributions after laser frequency noise is suppressed based, on assumptions on the size of the coupling and angular jitter levels. This work uses for the first time a closed-loop, non-linear, and time-varying dynamics implementation to simulate detailed angular jitters for the spacecraft and optical benches. In turn, this gives an improved expectation of the TTL contribution to the interferometric output. It is shown that the TTL coupling impact is limited given current estimates on the size of coupling coefficients. A time-domain Least Squares estimator is used to infer the TTL parameters from the simulated measurements. The bias and correlations limit the estimator in the case of regular datasets with amplified TTL coefficients to a relative error of $10\%$, but the subtraction of the TTL signal still works well. For lower readout noises, the estimation error diverges, which can be mitigated using a regularization term. Alternatively, using sinusoidal maneuvers improves the inference to a high accuracy of $0.1\%$ for TTL coefficients around the expected level, removing all correlations in the inferred parameters. This validates the maneuver design by Wegener et al. (2025) in this closed-loop setting.

Figures

Figure cut of constellation geometry of \gls{lisa} in the triangle plane. The scheme for numbering \gls{sc}, \glspl{mosa}, light travel times $\tau$, and \gls{mosa} opening angles $\varphi_b$ are defined.
Caption Figure cut of constellation geometry of \gls{lisa} in the triangle plane. The scheme for numbering \gls{sc}, \glspl{mosa}, light travel times $\tau$, and \gls{mosa} opening angles $\varphi_b$ are defined.
Figure cut of \gls{sc} geometry. The definition of the $\mathcal{B}$, $\mathcal{H}_{1}$, and $\mathcal{H}_2$ frames is illustrated. The center-of-mass of the satellite forms the origin of the $\mathcal{B}$ frame, while the centers of the \gls{tm} housings form the origin of the $\mathcal{H}_1$ and $\mathcal{H}_2$ frames. Additionally to the basis vectors, the Cardan angles are defined, following the right-hand rule for their direction of rotation.
Caption Figure cut of \gls{sc} geometry. The definition of the $\mathcal{B}$, $\mathcal{H}_{1}$, and $\mathcal{H}_2$ frames is illustrated. The center-of-mass of the satellite forms the origin of the $\mathcal{B}$ frame, while the centers of the \gls{tm} housings form the origin of the $\mathcal{H}_1$ and $\mathcal{H}_2$ frames. Additionally to the basis vectors, the Cardan angles are defined, following the right-hand rule for their direction of rotation.
Representative \gls{sc} and \gls{mosa} jitter shapes are shown. The left subplots show the \glspl{asd} of the internal states of the simulator in terms of radians. The right subplots show the later used time derivatives. The \glspl{asd} are calculated from a \SI{e5}{\s} dataset with $N_\text{avg.}=5$ (c.f. App.~\myhyperref{app:lisanode}). Reference lines show jitters used in another publication \cite{hartig2025tilt}: solid lines in black or lighter colors are white jitter references with a low frequency roll off, while the dashed black lines are colored jitter. Note that for the colored jitter case, \gls{mosa} $\eta$ is set to zero. Additionally, for the \gls{mosa} jitter the dash-dotted reference lines have been added from case B in \cite{george2023fisher}. Their \glspl{asd} are defined in Eqs.~\myhyperref{eq:ref_white_sc_jitter}--\myhyperref{eq:ref_col_mosa_jitter2}.\\ The top left subplot has a plateau for low frequencies, which exactly corresponds to the white \gls{dws} readout noise (\SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}). The fall-off towards higher frequencies is a result of the moment-of-inertia. Without \gls{dws} noise, the fall-off would be a power law; but with the noise present, there is a change in the fall-off. The \gls{sc} thruster noise widens the peak.
Caption Representative \gls{sc} and \gls{mosa} jitter shapes are shown. The left subplots show the \glspl{asd} of the internal states of the simulator in terms of radians. The right subplots show the later used time derivatives. The \glspl{asd} are calculated from a \SI{e5}{\s} dataset with $N_\text{avg.}=5$ (c.f. App.~\myhyperref{app:lisanode}). Reference lines show jitters used in another publication \cite{hartig2025tilt}: solid lines in black or lighter colors are white jitter references with a low frequency roll off, while the dashed black lines are colored jitter. Note that for the colored jitter case, \gls{mosa} $\eta$ is set to zero. Additionally, for the \gls{mosa} jitter the dash-dotted reference lines have been added from case B in \cite{george2023fisher}. Their \glspl{asd} are defined in Eqs.~\myhyperref{eq:ref_white_sc_jitter}--\myhyperref{eq:ref_col_mosa_jitter2}.\\ The top left subplot has a plateau for low frequencies, which exactly corresponds to the white \gls{dws} readout noise (\SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}). The fall-off towards higher frequencies is a result of the moment-of-inertia. Without \gls{dws} noise, the fall-off would be a power law; but with the noise present, there is a change in the fall-off. The \gls{sc} thruster noise widens the peak.
Representative \gls{dws} outputs for \gls{mosa} 12. The left subplots show the \glspl{asd} of the output in terms of radians over time, the right subplots show the later used derivatives. The upper subplots show the result of a simulation with the standard \gls{dws} noise settings of \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}, and the lower subplots with half the noise. The \glspl{asd} are calculated from a \SI{e5}{\s} dataset with $N_\text{avg.}=5$ (c.f. App.~\myhyperref{app:lisanode}).\\ The left subplots show two peaks for the $\phi_{12}$ sensing output. The low-frequency peak comes from the contribution of the \gls{mosa} jitter, and the medium-frequency peak from the contribution of the \gls{sc} jitter. The $\eta_{12}$ sensing output only sees the \gls{sc} jitter contributions, as the \gls{mosa} jitter is subdominant, resulting in only one peak. The plateau towards higher frequencies corresponds directly to the level of \gls{dws} readout noise.
Caption Representative \gls{dws} outputs for \gls{mosa} 12. The left subplots show the \glspl{asd} of the output in terms of radians over time, the right subplots show the later used derivatives. The upper subplots show the result of a simulation with the standard \gls{dws} noise settings of \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}, and the lower subplots with half the noise. The \glspl{asd} are calculated from a \SI{e5}{\s} dataset with $N_\text{avg.}=5$ (c.f. App.~\myhyperref{app:lisanode}).\\ The left subplots show two peaks for the $\phi_{12}$ sensing output. The low-frequency peak comes from the contribution of the \gls{mosa} jitter, and the medium-frequency peak from the contribution of the \gls{sc} jitter. The $\eta_{12}$ sensing output only sees the \gls{sc} jitter contributions, as the \gls{mosa} jitter is subdominant, resulting in only one peak. The plateau towards higher frequencies corresponds directly to the level of \gls{dws} readout noise.
\glspl{asd} of \gls{tdi}-X (second-generation Michelson variables) for different simulation scenarios. These include different settings for the \gls{ttl} coefficients and different \gls{dws} noise levels (low readout noise means half of the nominal settings). 'Random' refers to the \gls{ttl} coefficients being sampled from a uniform distribution. The black line gives the noise requirement of \cite{paczkowski_postprocessing_2022} (c.f. Eqs.\myhyperref{eq:req-tm-displacement},\myhyperref{eq:req-tm-acc}). The vertical dashed line gives the drag-free frequency bandwidth \cite{inchauspe23_dynamics}.\\ The variation in \gls{dws} readout noise only has an impact on frequencies around \SI{100}{\milli\Hz}. For equal \gls{ttl} coefficients of \SI[per-mode=symbol]{2.3}{\milli\metre\per\radian} the shape of the \gls{asd} is very close to the simulation with no \gls{ttl} contribution. The \glspl{asd} are calculated from a \SI{e6}{\s} dataset with $N_\text{avg.}=50$ (c.f. App.~\myhyperref{app:lisanode}).
Caption \glspl{asd} of \gls{tdi}-X (second-generation Michelson variables) for different simulation scenarios. These include different settings for the \gls{ttl} coefficients and different \gls{dws} noise levels (low readout noise means half of the nominal settings). 'Random' refers to the \gls{ttl} coefficients being sampled from a uniform distribution. The black line gives the noise requirement of \cite{paczkowski_postprocessing_2022} (c.f. Eqs.\myhyperref{eq:req-tm-displacement},\myhyperref{eq:req-tm-acc}). The vertical dashed line gives the drag-free frequency bandwidth \cite{inchauspe23_dynamics}.\\ The variation in \gls{dws} readout noise only has an impact on frequencies around \SI{100}{\milli\Hz}. For equal \gls{ttl} coefficients of \SI[per-mode=symbol]{2.3}{\milli\metre\per\radian} the shape of the \gls{asd} is very close to the simulation with no \gls{ttl} contribution. The \glspl{asd} are calculated from a \SI{e6}{\s} dataset with $N_\text{avg.}=50$ (c.f. App.~\myhyperref{app:lisanode}).
Inferred \num{24} \gls{ttl} parameters for two simulation, ordered in their categories as $\{12,13,23,21,31,32\}$. The top panels show results for the standard \gls{dws} noise settings of \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}} for the white noise in angles, the bottom panels with half the noise. The results are given in terms of a relative parameter deviation, i.e., the difference of the inferred parameter and the true parameter, normalized by the true parameter. The error bars are calculated from the standard deviations of \num{100} simulations (c.f. App.~\myhyperref{app:error}). The \gls{rms} errors reported are also split into contributions from the inferred $\eta$ and $\phi$ parameters. The results show that although the \gls{ttl} contribution is clearly visible for the larger \SI[per-mode=symbol]{23}{\milli\metre\per\radian} \gls{ttl} coefficients (c.f. Fig.~\myhyperref{fig:tdi-output}), the inference still has a substantial relative error of $11.3\% $ for the standard readout noise, and $5.2 \%$ for the low \gls{dws} noise.
Caption Inferred \num{24} \gls{ttl} parameters for two simulation, ordered in their categories as $\{12,13,23,21,31,32\}$. The top panels show results for the standard \gls{dws} noise settings of \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}} for the white noise in angles, the bottom panels with half the noise. The results are given in terms of a relative parameter deviation, i.e., the difference of the inferred parameter and the true parameter, normalized by the true parameter. The error bars are calculated from the standard deviations of \num{100} simulations (c.f. App.~\myhyperref{app:error}). The \gls{rms} errors reported are also split into contributions from the inferred $\eta$ and $\phi$ parameters. The results show that although the \gls{ttl} contribution is clearly visible for the larger \SI[per-mode=symbol]{23}{\milli\metre\per\radian} \gls{ttl} coefficients (c.f. Fig.~\myhyperref{fig:tdi-output}), the inference still has a substantial relative error of $11.3\% $ for the standard readout noise, and $5.2 \%$ for the low \gls{dws} noise.
The panels show the change in the relative error when varying equal \gls{ttl} coefficient levels in the simulations. Each data point corresponds to the mean of the \gls{ttl} coefficient estimates' \gls{rms} error over \num{100} simulations, with the error bars given by the standard deviation. The upper left panel shows simulation results with nominal \gls{dws} noise settings of \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}, the upper right panel with noise lowered to half. In the upper panels the results for the mean \gls{rms} values are also split into their contributions from the $\eta$ and $\phi$ parameters. Furthermore, the theoretical predictions for the bias of the \gls{ls} estimator are shown in lighter colors. The lower panel again shows the simulation with standard \gls{dws} noise levels, but now comparing the \gls{ls} results to the regularized \gls{ls} estimator, with the regularization parameter either chosen in the optimal way, or through the heuristic scheme described in Eq.~\myhyperref{eq:reg-heur-lam}.
Caption The panels show the change in the relative error when varying equal \gls{ttl} coefficient levels in the simulations. Each data point corresponds to the mean of the \gls{ttl} coefficient estimates' \gls{rms} error over \num{100} simulations, with the error bars given by the standard deviation. The upper left panel shows simulation results with nominal \gls{dws} noise settings of \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}, the upper right panel with noise lowered to half. In the upper panels the results for the mean \gls{rms} values are also split into their contributions from the $\eta$ and $\phi$ parameters. Furthermore, the theoretical predictions for the bias of the \gls{ls} estimator are shown in lighter colors. The lower panel again shows the simulation with standard \gls{dws} noise levels, but now comparing the \gls{ls} results to the regularized \gls{ls} estimator, with the regularization parameter either chosen in the optimal way, or through the heuristic scheme described in Eq.~\myhyperref{eq:reg-heur-lam}.
Demonstration of the suppression of the \gls{ttl} contribution to the noise. Both panels deal with the case of equal \gls{ttl} coefficients of \SI[per-mode=symbol]{23}{\milli\metre\per\radian}. The top panel has the standard \gls{dws} noise levels of \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}, the bottom one with noise lowered to half. Both panels demonstrate that the \gls{ttl} coupling noise can be removed to the same level as when using the true coefficients. The black line gives the noise requirement of \cite{paczkowski_postprocessing_2022} (c.f. Eqs.\myhyperref{eq:req-tm-displacement},\myhyperref{eq:req-tm-acc}). However, in the case of nominal \gls{dws} readout noise in the top panel, additional noise is added in high frequency regime above \SI{200}{\milli\Hz}, increasing the total noise beyond the level before subtraction. This can be improved upon with further filtering. The \glspl{asd} are calculated with $N_\text{avg.}=10$ (c.f. App.~\myhyperref{app:lisanode}).
Caption Demonstration of the suppression of the \gls{ttl} contribution to the noise. Both panels deal with the case of equal \gls{ttl} coefficients of \SI[per-mode=symbol]{23}{\milli\metre\per\radian}. The top panel has the standard \gls{dws} noise levels of \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}, the bottom one with noise lowered to half. Both panels demonstrate that the \gls{ttl} coupling noise can be removed to the same level as when using the true coefficients. The black line gives the noise requirement of \cite{paczkowski_postprocessing_2022} (c.f. Eqs.\myhyperref{eq:req-tm-displacement},\myhyperref{eq:req-tm-acc}). However, in the case of nominal \gls{dws} readout noise in the top panel, additional noise is added in high frequency regime above \SI{200}{\milli\Hz}, increasing the total noise beyond the level before subtraction. This can be improved upon with further filtering. The \glspl{asd} are calculated with $N_\text{avg.}=10$ (c.f. App.~\myhyperref{app:lisanode}).
Inferred \num{24} \gls{ttl} parameters for a single simulation with zero \gls{dws} readout noise, ordered in their categories as $\{12, 13, 23, 21, 31, 32\}$. The results are given in terms of a relative parameter deviation, i.e., the difference of the inferred parameter and the true parameter, normalized by the true parameter. The error bars are calculated from the standard deviations of \num{100} simulations (c.f. App.~\myhyperref{app:error}). The \gls{rms} errors reported are also split into contributions from the inferred $\eta$ and $\phi$ parameters. \\ Compared with non-zero \gls{dws} noise settings in Fig.~\myhyperref{fig:ttl-inference-result}, the inference for the $\eta$ parameters has improved due to a higher signal-to-noise ratio in the data. However, due to high correlations for the $\phi$ parameters, the estimator can constrain only certain combinations of parameters (c.f. App.~\myhyperref{app:corr}). With only subdominant \gls{mosa} jitter and no noise, the \gls{dws} $\phi$ outputs become virtually the same for the left and right \gls{mosa} of a single \gls{sc}. This leads to an ill-conditioned estimation.
Caption Inferred \num{24} \gls{ttl} parameters for a single simulation with zero \gls{dws} readout noise, ordered in their categories as $\{12, 13, 23, 21, 31, 32\}$. The results are given in terms of a relative parameter deviation, i.e., the difference of the inferred parameter and the true parameter, normalized by the true parameter. The error bars are calculated from the standard deviations of \num{100} simulations (c.f. App.~\myhyperref{app:error}). The \gls{rms} errors reported are also split into contributions from the inferred $\eta$ and $\phi$ parameters. \\ Compared with non-zero \gls{dws} noise settings in Fig.~\myhyperref{fig:ttl-inference-result}, the inference for the $\eta$ parameters has improved due to a higher signal-to-noise ratio in the data. However, due to high correlations for the $\phi$ parameters, the estimator can constrain only certain combinations of parameters (c.f. App.~\myhyperref{app:corr}). With only subdominant \gls{mosa} jitter and no noise, the \gls{dws} $\phi$ outputs become virtually the same for the left and right \gls{mosa} of a single \gls{sc}. This leads to an ill-conditioned estimation.
The panels show the change in the \gls{rms} error when varying \gls{dws} noise levels in the simulations. Each data point corresponds to the mean of the \gls{ttl} coefficient estimates' \gls{rms} error over \num{100} simulations, with the error bars given by the standard deviation. All simulations were run with equal \gls{ttl} coefficients of \SI[per-mode=symbol]{23}{\milli\metre\per\radian}. The left panel shows the results of the \gls{ls} estimator, also divided into the contributions from the $\eta$ and $\phi$ inference. The right panel compares the \gls{ls} results to the regularized \gls{ls} estimator, with the regularization parameter either chosen in the optimal way, or through the heuristic scheme described in Eq.~\myhyperref{eq:reg-heur-lam}. The left figure show the expected decrease of the inference in the $\eta$ coefficients with less \gls{dws} noise. The $\phi$ coefficient estimates increase dramatically due to increased correlations between the left and right \gls{mosa} in each \gls{sc}. The right panel shows that this can be prevented using a regularization term in the \gls{ls} estimator.
Caption The panels show the change in the \gls{rms} error when varying \gls{dws} noise levels in the simulations. Each data point corresponds to the mean of the \gls{ttl} coefficient estimates' \gls{rms} error over \num{100} simulations, with the error bars given by the standard deviation. All simulations were run with equal \gls{ttl} coefficients of \SI[per-mode=symbol]{23}{\milli\metre\per\radian}. The left panel shows the results of the \gls{ls} estimator, also divided into the contributions from the $\eta$ and $\phi$ inference. The right panel compares the \gls{ls} results to the regularized \gls{ls} estimator, with the regularization parameter either chosen in the optimal way, or through the heuristic scheme described in Eq.~\myhyperref{eq:reg-heur-lam}. The left figure show the expected decrease of the inference in the $\eta$ coefficients with less \gls{dws} noise. The $\phi$ coefficient estimates increase dramatically due to increased correlations between the left and right \gls{mosa} in each \gls{sc}. The right panel shows that this can be prevented using a regularization term in the \gls{ls} estimator.
The panels show how $\eta$ (left) and $\phi$ (right) maneuvers are induced. The top panels display the actual rotations induced in the \gls{sc} and \gls{mosa} \gls{dof}. The bottom panels show the resulting \gls{dws} readout, with excitations only present on the left \gls{mosa}. Details on the required excitations can be found in Sec.~\myhyperref{sec:exp-2-man}. Maneuver duration is \SI{600}{\s}. The shift between top and bottom panels in the start of the excitations is due to the delay caused by the \gls{adc}. A bandpass filter has been applied to the displayed data with frequencies \SI{15}{\milli\Hz} and \SI{70}{\milli\Hz}.
Caption The panels show how $\eta$ (left) and $\phi$ (right) maneuvers are induced. The top panels display the actual rotations induced in the \gls{sc} and \gls{mosa} \gls{dof}. The bottom panels show the resulting \gls{dws} readout, with excitations only present on the left \gls{mosa}. Details on the required excitations can be found in Sec.~\myhyperref{sec:exp-2-man}. Maneuver duration is \SI{600}{\s}. The shift between top and bottom panels in the start of the excitations is due to the delay caused by the \gls{adc}. A bandpass filter has been applied to the displayed data with frequencies \SI{15}{\milli\Hz} and \SI{70}{\milli\Hz}.
The \gls{dws} outputs for \gls{mosa} 12 during a full maneuver as defined in Sec.~\myhyperref{sec:exp-2-man}. The maneuver consists of two single \SI{600}{\s} phases after each other, with a waiting time of \SI{100}{\s}. For the first part, the left \gls{dws} channel is excited in $\eta$ and the right in $\phi$, and in the second phase vice-versa. The frequencies of the excitations have been carefully chosen according to \cite{wegener2025design}. A bandpass filter has been applied to the displayed data with frequencies \SI{15}{\milli\Hz} and \SI{70}{\milli\Hz}. The vertical dashed lines correspond to the time interval chosen for parameter inference.
Caption The \gls{dws} outputs for \gls{mosa} 12 during a full maneuver as defined in Sec.~\myhyperref{sec:exp-2-man}. The maneuver consists of two single \SI{600}{\s} phases after each other, with a waiting time of \SI{100}{\s}. For the first part, the left \gls{dws} channel is excited in $\eta$ and the right in $\phi$, and in the second phase vice-versa. The frequencies of the excitations have been carefully chosen according to \cite{wegener2025design}. A bandpass filter has been applied to the displayed data with frequencies \SI{15}{\milli\Hz} and \SI{70}{\milli\Hz}. The vertical dashed lines correspond to the time interval chosen for parameter inference.
Inferred \num{24} \gls{ttl} parameters for a single simulation with a full maneuver, ordered in their categories as $\{12, 13, 23, 21, 31, 32\}$. The results are given in terms of a relative parameter deviation, i.e., the difference of the inferred parameter and the true parameter, normalized by the true parameter. The error bars are calculated from the standard deviations of \num{100} simulations (c.f. App.~\myhyperref{app:error}). The \gls{rms} errors reported are also split into contributions from the inferred $\eta$ and $\phi$ parameters. Compared with simulations without maneuvers in Fig.~\myhyperref{fig:ttl-inference-result}, the inference works far better with the \gls{ls} estimator due to the high signal-to-noise ratios in both the interferometer and \gls{dws} outputs during the maneuver time. This works even though the equal \gls{ttl} coefficients have been reduced to the nominal value of \SI[per-mode=symbol]{2.3}{\milli\metre\per\radian}.
Caption Inferred \num{24} \gls{ttl} parameters for a single simulation with a full maneuver, ordered in their categories as $\{12, 13, 23, 21, 31, 32\}$. The results are given in terms of a relative parameter deviation, i.e., the difference of the inferred parameter and the true parameter, normalized by the true parameter. The error bars are calculated from the standard deviations of \num{100} simulations (c.f. App.~\myhyperref{app:error}). The \gls{rms} errors reported are also split into contributions from the inferred $\eta$ and $\phi$ parameters. Compared with simulations without maneuvers in Fig.~\myhyperref{fig:ttl-inference-result}, the inference works far better with the \gls{ls} estimator due to the high signal-to-noise ratios in both the interferometer and \gls{dws} outputs during the maneuver time. This works even though the equal \gls{ttl} coefficients have been reduced to the nominal value of \SI[per-mode=symbol]{2.3}{\milli\metre\per\radian}.
Demonstration of the removal of the \gls{ttl} contribution in the case of a full maneuver. The data is shown in the time domain (with a bandpass filter applied, using frequencies \SI{15}{\milli\Hz} and \SI{70}{\milli\Hz}), with the left panels showing the time during the maneuver, and the right panels some later time without any injected signal. The top panels show the interferometric output after \gls{tdi} but before subtraction, the bottom panels show the result after subtraction of the \gls{ttl} contribution. Outside the injection time, the subtraction with the \gls{ls}-estimated coefficients is nearly identical to the case when using the true coefficients. During the maneuver, even these small deviations get amplified by the large signal and thus the subtractions differ (during injection: \gls{rms} deviation of \SI{0.28}{\micro\Hz}; maximum deviation of \SI{1.1}{\micro\Hz}). For comparison, the lower right panel also shows the amplitude of the \gls{ttl} signal that is subtracted. For this simulation with equal \gls{ttl} coefficients of \SI[per-mode=symbol]{2.3}{\milli\metre\per\radian}, the \gls{ttl} signal is relevant, but not dominant.
Caption Demonstration of the removal of the \gls{ttl} contribution in the case of a full maneuver. The data is shown in the time domain (with a bandpass filter applied, using frequencies \SI{15}{\milli\Hz} and \SI{70}{\milli\Hz}), with the left panels showing the time during the maneuver, and the right panels some later time without any injected signal. The top panels show the interferometric output after \gls{tdi} but before subtraction, the bottom panels show the result after subtraction of the \gls{ttl} contribution. Outside the injection time, the subtraction with the \gls{ls}-estimated coefficients is nearly identical to the case when using the true coefficients. During the maneuver, even these small deviations get amplified by the large signal and thus the subtractions differ (during injection: \gls{rms} deviation of \SI{0.28}{\micro\Hz}; maximum deviation of \SI{1.1}{\micro\Hz}). For comparison, the lower right panel also shows the amplitude of the \gls{ttl} signal that is subtracted. For this simulation with equal \gls{ttl} coefficients of \SI[per-mode=symbol]{2.3}{\milli\metre\per\radian}, the \gls{ttl} signal is relevant, but not dominant.
The panels show the change in the \gls{rms} error when varying duration and amplitude of a full maneuver. Each data point corresponds to the mean of the \gls{ttl} coefficient estimates' \gls{rms} error over \num{100} simulations, with the error bars given by the standard deviation. All simulations were run with equal \gls{ttl} coefficients of \SI[per-mode=symbol]{2.3}{\milli\metre\per\radian}. The left panel shows the results when changing the maneuver duration, the right panel the reaction to a changing amplitude. In both cases the results are displayed also in terms of the separate $\eta$ and $\phi$ contribution to the total \gls{rms} value. Also, a theoretical scaling curve is displayed in the lighter color. The amplitude of this prediction is fixed at the nominal cases of a duration of \SI{600}{\s} and the standard amplitude. The horizontal dashed line gives the usual target accuracy of the coupling coefficients at \SI[per-mode=symbol]{0.1}{\milli\metre\per\radian}.\\ The scaling law describes the reaction to a changing amplitude maneuver well, but underestimates the error for smaller maneuver duration, where the additional effect of a smaller effective amplitude increases the error. In the right panel, the normalized maneuver amplitude is with respect to the nominal injection amplitude, chosen such that a $H$ excitation amplitude of \SI{100}{\nano\radian} is achieved. The vertical dashed line gives the maximum possible maneuver amplitude given the expected thruster power. The rise towards higher amplitudes can be attributed to a cross-coupling with the \gls{tm} \gls{dof}.
Caption The panels show the change in the \gls{rms} error when varying duration and amplitude of a full maneuver. Each data point corresponds to the mean of the \gls{ttl} coefficient estimates' \gls{rms} error over \num{100} simulations, with the error bars given by the standard deviation. All simulations were run with equal \gls{ttl} coefficients of \SI[per-mode=symbol]{2.3}{\milli\metre\per\radian}. The left panel shows the results when changing the maneuver duration, the right panel the reaction to a changing amplitude. In both cases the results are displayed also in terms of the separate $\eta$ and $\phi$ contribution to the total \gls{rms} value. Also, a theoretical scaling curve is displayed in the lighter color. The amplitude of this prediction is fixed at the nominal cases of a duration of \SI{600}{\s} and the standard amplitude. The horizontal dashed line gives the usual target accuracy of the coupling coefficients at \SI[per-mode=symbol]{0.1}{\milli\metre\per\radian}.\\ The scaling law describes the reaction to a changing amplitude maneuver well, but underestimates the error for smaller maneuver duration, where the additional effect of a smaller effective amplitude increases the error. In the right panel, the normalized maneuver amplitude is with respect to the nominal injection amplitude, chosen such that a $H$ excitation amplitude of \SI{100}{\nano\radian} is achieved. The vertical dashed line gives the maximum possible maneuver amplitude given the expected thruster power. The rise towards higher amplitudes can be attributed to a cross-coupling with the \gls{tm} \gls{dof}.
Results of inference when running simulations with randomly sampled \gls{ttl} coefficients from a uniform distribution on the interval $[-23,23]\,\unit[per-mode=symbol]{\milli\metre\per\radian}$. Each bar corresponds to the mean of the \gls{ttl} coefficient estimates' \gls{rms} error over \num{100} simulations, with the error bars given by the standard deviation. The result is split up in terms of the contributions of $\eta$ and $\phi$ to the total \gls{rms} error, and the different estimators: regular \gls{ls}, and regularized \gls{ls} with the optimally and heuristically chosen regularization parameter $\lambda$. The three panels show the results with respect to different \gls{dws} readout noise levels in the simulations: nominal (\SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}), lowered to half, and no readout noise. The panels show that for random \gls{ttl} coefficients, the inference task becomes somewhat harder. The coefficient estimate is worse than for the corresponding simulation with equal coefficients (c.f. Fig.~\myhyperref{fig:ttl-level-dependence}), and the regularization does not improve the result anymore. However, regularized \gls{ls} still is better in the highly-correlated, no \gls{dws} noise case, improving the inference result drastically.
Caption Results of inference when running simulations with randomly sampled \gls{ttl} coefficients from a uniform distribution on the interval $[-23,23]\,\unit[per-mode=symbol]{\milli\metre\per\radian}$. Each bar corresponds to the mean of the \gls{ttl} coefficient estimates' \gls{rms} error over \num{100} simulations, with the error bars given by the standard deviation. The result is split up in terms of the contributions of $\eta$ and $\phi$ to the total \gls{rms} error, and the different estimators: regular \gls{ls}, and regularized \gls{ls} with the optimally and heuristically chosen regularization parameter $\lambda$. The three panels show the results with respect to different \gls{dws} readout noise levels in the simulations: nominal (\SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}), lowered to half, and no readout noise. The panels show that for random \gls{ttl} coefficients, the inference task becomes somewhat harder. The coefficient estimate is worse than for the corresponding simulation with equal coefficients (c.f. Fig.~\myhyperref{fig:ttl-level-dependence}), and the regularization does not improve the result anymore. However, regularized \gls{ls} still is better in the highly-correlated, no \gls{dws} noise case, improving the inference result drastically.
The panels show the impact of the bandpass filters and how the frequency band was chosen. The top panels show the $\phi$ and $\eta$ total \gls{mosa} jitters, i.e., the \gls{dws} measurement outputs for \gls{mosa} 12, split up into the internal state and the readout noise. The bottom panels show the jitters after applying the bandpass filter with frequencies \SI{15}{\milli\Hz} and \SI{70}{\milli\Hz}. These cut-offs were chosen such that a broad frequency range is selected with a good signal-to-noise ratio. The \glspl{asd} are calculated from a \SI{e5}{\s} dataset with $N_\text{avg.}=5$.
Caption The panels show the impact of the bandpass filters and how the frequency band was chosen. The top panels show the $\phi$ and $\eta$ total \gls{mosa} jitters, i.e., the \gls{dws} measurement outputs for \gls{mosa} 12, split up into the internal state and the readout noise. The bottom panels show the jitters after applying the bandpass filter with frequencies \SI{15}{\milli\Hz} and \SI{70}{\milli\Hz}. These cut-offs were chosen such that a broad frequency range is selected with a good signal-to-noise ratio. The \glspl{asd} are calculated from a \SI{e5}{\s} dataset with $N_\text{avg.}=5$.
The panels show the normalized covariances, i.e., correlations, between the \gls{dws} channels in a single simulation. The channels are order as $\{12,13,23,21,31,32\}$. The first panel is the standard simulation with \gls{dws} readout noise at \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}. The second is a simulation without any \gls{dws} noise. The last panel looks at the situation when full maneuvers are present. The first two panels show how removing the \gls{dws} noise increases the correlation between the $\phi$ \gls{dws} channels of the same \gls{sc}. The corresponding $\eta$ correlations are weaker and remain largely unaffected by the \gls{dws} noise level. When introducing maneuvers, new strong (anti-)correlations are present, due to the design of the maneuver. Figure~\myhyperref{fig:ls-correlations} shows that these correlations are not carried over into the \gls{ttl} coefficient estimates.
Caption The panels show the normalized covariances, i.e., correlations, between the \gls{dws} channels in a single simulation. The channels are order as $\{12,13,23,21,31,32\}$. The first panel is the standard simulation with \gls{dws} readout noise at \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}. The second is a simulation without any \gls{dws} noise. The last panel looks at the situation when full maneuvers are present. The first two panels show how removing the \gls{dws} noise increases the correlation between the $\phi$ \gls{dws} channels of the same \gls{sc}. The corresponding $\eta$ correlations are weaker and remain largely unaffected by the \gls{dws} noise level. When introducing maneuvers, new strong (anti-)correlations are present, due to the design of the maneuver. Figure~\myhyperref{fig:ls-correlations} shows that these correlations are not carried over into the \gls{ttl} coefficient estimates.
The panels show the normalized covariances, i.e., correlations, between the estimated \gls{ttl} coefficient deviations of \num{100} simulations, i.e., correlations of $\hat{C}_\text{TTL}-C_\text{TTL}$. For each parameter subset, the order is $\{12,13,23,21,31,32\}$. The top row looks at the case of equal \gls{ttl} coefficients in the simulation, the middle for coefficients drawn from a uniform distribution. The bottom row investigates the output of the regularized \gls{ls} estimator for random \gls{ttl} coefficients. The first column is the standard simulation with \gls{dws} readout noise at \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}. The second column is a simulation without any \gls{dws} noise. The last column looks at the situation when full maneuvers are present.\\ The panels show that the maneuvers manage to get rid of cross-correlations between coefficients, while the regular simulations have quite strongly coupled $\phi$ coefficients. The regularization scheme does not alleviate these, it can actually introduce additional correlations for $\eta$.
Caption The panels show the normalized covariances, i.e., correlations, between the estimated \gls{ttl} coefficient deviations of \num{100} simulations, i.e., correlations of $\hat{C}_\text{TTL}-C_\text{TTL}$. For each parameter subset, the order is $\{12,13,23,21,31,32\}$. The top row looks at the case of equal \gls{ttl} coefficients in the simulation, the middle for coefficients drawn from a uniform distribution. The bottom row investigates the output of the regularized \gls{ls} estimator for random \gls{ttl} coefficients. The first column is the standard simulation with \gls{dws} readout noise at \SI[power-half-as-sqrt,per-mode=symbol]{0.2}{\nano\radian\per\Hz\tothe{0.5}}. The second column is a simulation without any \gls{dws} noise. The last column looks at the situation when full maneuvers are present.\\ The panels show that the maneuvers manage to get rid of cross-correlations between coefficients, while the regular simulations have quite strongly coupled $\phi$ coefficients. The regularization scheme does not alleviate these, it can actually introduce additional correlations for $\eta$.
The panels show how $\phi$ (left) and $\eta$ (right) maneuvers are induced in both \glspl{mosa} on a single \gls{sc} at different frequencies \SI[parse-numbers=false]{43.\bar{3}}{\milli\Hz} and \SI[parse-numbers=false]{41.\bar{6}}{\milli\Hz}. This can be used to construct a full type 2 maneuver, with exciting all $\phi$ channels in phase \num{1}, and then all the $\eta$ channels in phase \num{2}. The top panels display the actual rotations induced in the \gls{sc}/\gls{mosa} degrees-of-freedom. The bottom panels show the resulting \gls{dws} readout, with excitations present in both \glspl{mosa}. Details on the required excitations can be found in Sec.~\myhyperref{sec:exp-2-man}. Maneuver duration is \SI{600}{\s}. The shift between top and bottom panels in the start of the excitations is due to the delay caused by the \gls{adc}. A bandpass filter has been applied to the data with frequencies \SI{15}{\milli\Hz} and \SI{70}{\milli\Hz}.
Caption The panels show how $\phi$ (left) and $\eta$ (right) maneuvers are induced in both \glspl{mosa} on a single \gls{sc} at different frequencies \SI[parse-numbers=false]{43.\bar{3}}{\milli\Hz} and \SI[parse-numbers=false]{41.\bar{6}}{\milli\Hz}. This can be used to construct a full type 2 maneuver, with exciting all $\phi$ channels in phase \num{1}, and then all the $\eta$ channels in phase \num{2}. The top panels display the actual rotations induced in the \gls{sc}/\gls{mosa} degrees-of-freedom. The bottom panels show the resulting \gls{dws} readout, with excitations present in both \glspl{mosa}. Details on the required excitations can be found in Sec.~\myhyperref{sec:exp-2-man}. Maneuver duration is \SI{600}{\s}. The shift between top and bottom panels in the start of the excitations is due to the delay caused by the \gls{adc}. A bandpass filter has been applied to the data with frequencies \SI{15}{\milli\Hz} and \SI{70}{\milli\Hz}.
Results of inference when running simulations with maneuvers with different scenarios. Each bar corresponds to the mean of the \gls{ttl} coefficient estimates' \gls{rms} error over \num{100} simulations, with the error bars given by the standard deviation. The result is split up in terms of the contributions of $\eta$ and $\phi$ to the total \gls{rms} error. The four different scenarios are: maneuvers of type \num{1} (mixed $\eta$ and $\phi$ excitations), type \num{2} (separated $\eta$ and $\phi$), maneuver type \num{1} with setting the breathing angle to $\pi/3$ in the \gls{ttl} contribution calculation inside the simulation, and lastly maneuver type \num{1} with randomly sampled \gls{ttl} coefficients from the interval $[-2.3,2.3]\,\unit[per-mode=symbol]{\milli\metre\per\radian}$.
Caption Results of inference when running simulations with maneuvers with different scenarios. Each bar corresponds to the mean of the \gls{ttl} coefficient estimates' \gls{rms} error over \num{100} simulations, with the error bars given by the standard deviation. The result is split up in terms of the contributions of $\eta$ and $\phi$ to the total \gls{rms} error. The four different scenarios are: maneuvers of type \num{1} (mixed $\eta$ and $\phi$ excitations), type \num{2} (separated $\eta$ and $\phi$), maneuver type \num{1} with setting the breathing angle to $\pi/3$ in the \gls{ttl} contribution calculation inside the simulation, and lastly maneuver type \num{1} with randomly sampled \gls{ttl} coefficients from the interval $[-2.3,2.3]\,\unit[per-mode=symbol]{\milli\metre\per\radian}$.
The panels show the breathing angle of the constellation, and the angular velocity induced by the orbit, varying over the course of a year. On the left the positive breathing half-angle $\varphi_b^+$ is shown for two pre-computed orbits: Kepler-type and \gls{esa}-optimized with the \gls{lisa} constellation trailing Earth. The right panel shows the variation of the components of the angular velocity vector induced by the orbital motion.
Caption The panels show the breathing angle of the constellation, and the angular velocity induced by the orbit, varying over the course of a year. On the left the positive breathing half-angle $\varphi_b^+$ is shown for two pre-computed orbits: Kepler-type and \gls{esa}-optimized with the \gls{lisa} constellation trailing Earth. The right panel shows the variation of the components of the angular velocity vector induced by the orbital motion.
References
  • [1] H. Wegener, S. Paczkowski, M.-S. Hartig, M. Hewitson, G. Heinzel, and G. Wanner, Design of dedicated tilt-tolength calibration maneuvers for LISA, Physical Review D 112, 023006 (2025).
  • [2] M. Colpi, K. Danzmann, M. Hewitson, K. HolleyBockelmann, P. Jetzer, G. Nelemans, A. Petiteau, D. Shoemaker, C. Sopuerta, R. Stebbins, N. Tanvir, H. Ward, W. J. Weber, I. Thorpe, A. Daurskikh, et al., LISA definition study report (2024), arXiv:2402.07571 [astro-ph.CO].
  • [3] B. P. Abbott, R. Abbott, T. Abbott, S. Abraham, F. Acernese, K. Ackley, C. Adams, V. Adya, C. Affeldt, M. Agathos, et al., Prospects for observing and localizing gravitational-wave transients with Advanced LIGO, Advanced Virgo and KAGRA, Living reviews in relativity 23, 3 (2020).
  • [4] J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, R. Adhikari, et al., Advanced LIGO, Classical and quantum gravity 32, 074001 (2015).
  • [5] F. Acernese, M. Agathos, K. Agatsuma, D. Aisa, N. Allemandou, A. Allocca, J. Amarni, P. Astone, G. Balestri, G. Ballardin, et al., Advanced Virgo: a second-generation interferometric gravitational wave detector, Classical and Quantum Gravity 32, 024001 (2014).
  • [6] T. Akutsu, M. Ando, K. Arai, Y. Arai, S. Araki, A. Araya, N. Aritomi, Y. Aso, S. Bae, Y. Bae, et al., Overview of KAGRA: Detector design and construction history, Progress of Theoretical and Experimental Physics 2021, 05A101 (2020).
  • [7] K. L. Dooley, J. Leong, T. Adams, C. Affeldt, A. Bisht, C. Bogan, J. Degallaix, C. Gräf, S. Hild, J. Hough, et al., GEO 600 and the GEO-HF upgrade program: successes and challenges, Classical and Quantum Gravity 33, 075009 (2016).
  • [8] G. Agazie, A. Anumarlapudi, A. M. Archibald, Z. Arzoumanian, P. T. Baker, B. Bécsy, L. Blecha, A. Brazier, P. R. Brook, S. Burke-Spolaor, et al., The NANOGrav 15 yr data set: Evidence for a gravitational-wave background, The Astrophysical Journal Letters 951, L8 (2023).
  • [9] J. Antoniadis, P. Arumugam, S. Arumugam, S. Babak, M. Bagchi, A.-S. B. Nielsen, C. Bassa, A. Bathula, A. Berthereau, M. Bonetti, et al., The second data release from the European Pulsar Timing Array-III. search for gravitational wave signals, Astronomy & Astrophysics 678, A50 (2023).
  • [10] D. J. Reardon, A. Zic, R. M. Shannon, G. B. Hobbs, M. Bailes, V. Di Marco, A. Kapur, A. F. Rogers, E. Thrane, J. Askew, et al., Search for an isotropic gravitational-wave background with the Parkes Pulsar Timing Array, The Astrophysical Journal Letters 951, L6 (2023).
  • [11] H. Xu, S. Chen, Y. Guo, J. Jiang, B. Wang, J. Xu, Z. Xue, R. N. Caballero, J. Yuan, Y. Xu, et al., Searching for the nano-hertz stochastic gravitational wave background with the Chinese pulsar timing array data release I, Research in Astronomy and Astrophysics 23, 075024 (2023).
  • [12] J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, et al., TianQin: a space-borne gravitational wave detector, Classical and Quantum Gravity 33, 035010 (2016).
  • [13] Z. Luo, Y. Wang, Y. Wu, W. Hu, and G. Jin, The Taiji program: A concise overview, Progress of Theoretical and Experimental Physics 2021, 05A108 (2021).
  • [14] M. Armano, H. Audley, G. Auger, J. T. Baird, M. Bassan, P. Binetruy, M. Born, D. Bortoluzzi, N. Brandt, M. Caleno, et al., Sub-femto-g free fall for space-based gravitational wave observatories: LISA pathfinder results, Physical review letters 116, 231101 (2016).
  • [15] M. Armano, H. Audley, J. Baird, P. Binetruy, M. Born, D. Bortoluzzi, E. Castelli, A. Cavalleri, A. Cesarini, A. Cruise, et al., Beyond the required LISA free-fall performance: New LISA pathfinder results down to 20 µ hz, Physical review letters 120, 061101 (2018).
  • [16] M. Armano, H. Audley, J. Baird, P. Binetruy, M. Born, D. Bortoluzzi, N. Brandt, E. Castelli, A. Cavalleri, A. Cesarini, A. M. Cruise, K. Danzmann, M. de Deus Silva, I. Diepholz, G. Dixon, et al., Sensor noise in LISA Pathfinder: In-flight performance of the optical test mass readout, Phys. Rev. Lett. 126, 131103 (2021).
  • [17] K. Abich, A. Abramovici, B. Amparan, A. Baatzsch, B. B. Okihiro, D. C. Barr, M. P. Bize, C. Bogan, C. Braxmaier, M. J. Burke, K. C. Clark, C. Dahl, K. Dahl, K. Danzmann, M. A. Davis, et al., In-orbit performance of the GRACE Follow-on laser ranging interferometer, Phys. Rev. Lett. 123, 031101 (2019).
  • [18] M. Tinto and S. V. Dhurandhar, Time-delay interferometry, Living Reviews in Relativity 24, 1 (2020).
  • [19] M. Muratore, D. Vetrugno, and S. Vitale, Revisitation of time delay interferometry combinations that suppress laser noise in LISA, Classical and Quantum Gravity 37, 185019 (2020).
  • [20] J.-B. Bayle, O. Hartwig, and M. Staab, Adapting timedelay interferometry for LISA data in frequency, Physical Review D 104, 023006 (2021).
  • [21] G. Wanner, S. Shah, M. Staab, H. Wegener, and S. Paczkowski, In-depth modeling of tilt-to-length coupling in LISA’s interferometers and TDI michelson observables, Physical Review D 110, 022003 (2024).
  • [22] M.-S. Hartig, J. Marmor, D. George, S. Paczkowski, and J. Sanjuan, Tilt-to-length coupling in LISA—uncertainty and biases, Classical and Quantum Gravity 42, 045004 (2025).
  • [23] M.-S. Hartig, S. Schuster, and G. Wanner, Geometric tilt-to-length coupling in precision interferometry: mechanisms and analytical descriptions, Journal of Optics 24, 065601 (2022).
  • [24] M.-S. Hartig, S. Schuster, G. Heinzel, and G. Wanner, Non-geometric tilt-to-length coupling in precision interferometry: mechanisms and analytical descriptions, Journal of Optics 25, 055601 (2023).
  • [25] M. Armano, H. Audley, J. Baird, P. Binetruy, M. Born, D. Bortoluzzi, N. Brandt, E. Castelli, A. Cavalleri, A. Cesarini, A. Cruise, K. Danzmann, M. de Deus Silva, I. Diepholz, G. Dixon, et al., Sensor noise in LISA Pathfinder: An extensive in-flight review of the angular and longitudinal interferometric measurement system, Physical Review D 106, 082001 (2022).
  • [26] D. George, J. Sanjuan, P. Fulda, and G. Mueller, Calculating the precision of tilt-to-length coupling estimation and noise subtraction in LISA using Fisher information, Phys. Rev. D 107, 022005 (2023).
  • [27] S. Paczkowski, R. Giusteri, M. Hewitson, N. Karnesis, E. Fitzsimons, G. Wanner, and G. Heinzel, Postprocessing subtraction of tilt-to-length noise in LISA, Physical Review D 106, 042005 (2022).
  • [28] N. Houba, S. Delchambre, G. Hechenblaikner, T. Ziegler, and W. Fichter, Time-delay interferometry infinity for tilt-to-length noise estimation in LISA, Classical and Quantum Gravity 40, 107001 (2023).
  • [29] Z. Wang, S. Yang, K. Wu, X. Wang, H. Duan, X. Zhang, H.-C. Yeh, and Y. Liang, Postprocessing of tilt-to-length noise with coefficient drifts in TianQin using a null timedelay interferometry channel, Physical Review D 111, 042004 (2025).
  • [30] H.-K. Chen, P.-P. Wang, and C.-G. Shao, Characteristics of tilt-to-length coupling coefficients for different time-delay interferometry combinations for LISA, Classical and Quantum Gravity 42, 185008 (2025).
  • [31] M.-S. Hartig, S. Paczkowski, M. Hewitson, G. Heinzel, and G. Wanner, Postprocessing subtraction of tilt-tolength noise in LISA in the presence of gravitational wave signals, Physical Review D 111, 043048 (2025).
  • [32] N. Houba, S. Delchambre, T. Ziegler, G. Hechenblaikner, and W. Fichter, LISA spacecraft maneuver design to estimate tilt-to-length noise during gravitational wave events, Phys. Rev. D 106, 022004 (2022).
  • [33] N. Houba, S. Delchambre, T. Ziegler, and W. Fichter, Optimal estimation of tilt-to-length noise for spaceborne gravitational-wave observatories, Journal of Guidance, Control, and Dynamics 45, 1078 (2022).
  • [34] H. Inchauspé, M. Hewitson, O. Sauter, and P. Wass, New LISA dynamics feedback control scheme: Common-mode isolation of test mass control and probes of test-mass acceleration, Physical Review D 106, 022006 (2022).
  • [35] S. Vidano, C. Novara, L. Colangelo, and J. Grzymisch, The LISA DFACS: A nonlinear model for the spacecraft dynamics, Aerospace Science and Technology 107, 106313 (2020).
  • [36] L. Heisenberg, H. Inchauspé, D. Q. Nam, O. Sauter, R. Waibel, and P. Wass, LISA dynamics and control: Closed-loop simulation and numerical demonstration of time delay interferometry, Physical Review D 108, 122007 (2023).
  • [37] C. Novara, S. Vidano, M. Pagone, J. Grzymisch, and V. Preda, LISA drag-free attitude control system: Robust stability and performance analysis, Acta Astronautica 240, 115 (2026).
  • [38] M. Armano, H. Audley, J. Baird, P. Binetruy, M. Born, D. Bortoluzzi, E. Castelli, A. Cavalleri, A. Cesarini, A. Cruise, K. Danzmann, M. de Deus Silva, I. Diepholz, G. Dixon, R. Dolesi, et al., Calibrating the system dynamics of LISA Pathfinder, Physical Review D 97, 122002 (2018).
  • [39] LISA Pathfinder Collaboration, M. Armano, H. Audley, J. Baird, P. Binetruy, M. Born, D. Bortoluzzi, E. Castelli, A. Cavalleri, A. Cesarini, A. Cruise, K. Danzmann, M. de Deus Silva, I. Diepholz, G. Dixon, et al., LISA Pathfinder platform stability and drag-free performance, Physical Review D 99, 082001 (2019).
  • [40] G. H. Kaplan, The iau resolutions on astronomical reference systems, time scales, and earth rotation models, arXiv preprint astro-ph/0602086 (2006).
  • [41] J. Diebel, Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors, Matrix 58 (2006).
  • [42] J.-B. Bayle, O. Hartwig, A. Petiteau, and M. Lilley, LISANode (1.4) (2022).
  • [43] J.-B. Bayle and O. Hartwig, Unified model for the LISA measurements and instrument simulations, Physical Review D 107, 083019 (2023).
  • [44] J.-B. Bayle, Simulation and Data Analysis for LISA (Instrumental Modeling, Time-Delay Interferometry, NoiseReduction Performance Study, and Discrimination of Transient Gravitational Signals), Theses, Université de Paris ; Université Paris Diderot ; Laboratoire Astroparticules et Cosmologie (2019).
  • [44] J.-B. Bayle, Simulation and Data Analysis for LISA (Instrumental Modeling, Time-Delay Interferometry, NoiseReduction Performance Study, and Discrimination of Transient Gravitational Signals), Theses, Université de Paris ; Université Paris Diderot ; Laboratoire Astroparticules et Cosmologie (2019).
  • [45] M. Otto, Time-delay interferometry simulations for the laser interferometer space antenna, Ph.D. thesis, Leibniz University, Hannover, Germany (2016).
  • [46] W. Martens and E. Joffre, Trajectory design for the ESA LISA mission, The Journal of the Astronautical Sciences 68, 402 (2021).
  • [47] M. Staab, J.-B. Bayle, and O. Hartwig, PyTDI (1.4.0) (2025).
  • [48] J.-B. Bayle, O. Hartwig, A. Hees, W. Kastaun, M. Staab, S. Paczkowski, and G. Wanner, LISA Instrument v2.0.0 documentation (2025), last accessed 12 January 2026.
  • [49] A. N. Heinz Werner Engl, Martin Hanke, Regularization of Inverse Problems, 1st ed., Mathematics and Its Applications (Springer Dordrecht, 1996).
  • [50] T. A. Prince, M. Tinto, S. L. Larson, and J. W. Armstrong, LISA optimal sensitivity, Phys. Rev. D 66, 122002 (2002).
  • [51] G. Heinzel, J. Álvarez-Vizoso, M. Dovale-Álvarez, and K. Wiesner, Frequency planning for LISA, Phys. Rev. D 110, 042002 (2024).
  • [52] M. Hewitson, Tech. Rep. LISA-LCST-INST-DD-003 (Max-Planck Institute for Gravitational Physics (AEI), 2021).
  • [53] S. Babak, M. Hewitson, and A. Petiteau, LISA sensitivity and SNR calculations (2021), arXiv:2108.01167 [astroph.IM].
  • [54] D. Gomes de Pinho Zanco, L. Szczecinski, and J. Benesty, Automatic regularization for linear mmse filters, Signal Processing 230, 109820 (2025).
  • [55] M. Muratore, O. Hartwig, D. Vetrugno, S. Vitale, and W. J. Weber, Effectiveness of null time-delay interferometry channels as instrument noise monitors in LISA, Phys. Rev. D 107, 082004 (2023).
  • [56] J.-B. Bayle, A. Hees, M. Lilley, C. Le Poncin-Lafitte, W. Martens, and E. Joffre, LISA Orbits (2.4.2) (2024).
  • [57] G. Heinzel, LISA Frequency Planning, Tech. Rep. LISAAEI-INST-TN-002 (AEI Hannover, 2018).
  • [58] J.-B. Bayle, psd (1.5.2) (2020).
  • [59] P. Welch, The use of fast fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms, IEEE Transactions on audio and electroacoustics 15, 70 (2003).
  • [60] G. Heinzel, A. Rüdiger, and R. Schilling, Spectrum and spectral density estimation by the discrete fourier transform (dft), including a comprehensive list of window functions and some new at-top windows (2002), unpublished.
  • [61] A. Nuttall, Some windows with very good sidelobe behavior, IEEE Transactions on Acoustics, Speech, and Signal Processing 29, 84 (2003).
  • [62] C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Rı́o, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, and T. E. Oliphant, Array programming with NumPy, Nature 585, 357 (2020).
  • [63] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, and P. van Mulbregt, SciPy 1.0: fundamental algorithms for scientific computing in Python, Nature Methods 17, 261 (2020).
  • [64] J. D. Hunter, Matplotlib: A 2d graphics environment, Computing in Science & Engineering 9, 90 (2007).
  • [65] Wolfram Research Inc., Mathematica, Version 14.2 (2025), champaign, IL, 2025.
  • [66] G. Bianchi and R. Sorrentino, Electronic Filter Simulation & Design, 1st ed. (McGraw-Hill Professional, 2007).