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.

The Laser Interferometer Space Antenna (LISA) will be the first space-based gravitational wave (GW) observatory. It will measure gravitational wave signals in the frequency regime from 0.1 mHz to 1 Hz. The success of these measurements will depend on the suppression of the various instrument noises. One important noise source in LISA will be tilt-to-length (TTL) coupling. Here, it is understood as the coupling of angular jitter, predominantly from the spacecraft, into the interferometric length readout. The current plan is to subtract this noise in-flight in post-processing as part of a noise minimization strategy. It is crucial to distinguish TTL coupling well from the GW signals in the same readout to ensure that the noise will be properly modeled. Furthermore, it is important that the subtraction of TTL noise will not degrade the GW signals. In the present manuscript, we show on simulated LISA data and for four different GW signal types that the GW responses have little effect on the quality of the TTL coupling fit and subtraction. Also, the GW signal characteristics were not altered by the TTL coupling subtraction.

The coupling of the angular jitter of the spacecraft and their sub-assemblies with the optical bench and the telescope into the interferometric length readout will be a major noise source in the LISA mission. We refer to this noise as tilt-to-length (TTL) coupling. It will be reduced directly by realignments, and the residual noise will then be subtracted in post-processing. The success of these mitigation strategies depends on an accurate computation of the TTL coupling coefficients. We present here a thorough analysis of the accuracy of the coefficient estimation under different jitter characteristics, angular readout noise levels, and gravitational wave sources. We analyze in which cases the estimates degrade using two estimators, the common least squares estimator and the instrumental variables estimator. Our investigations show that angular readout noise leads to a systematic bias of the least squares estimator, depending on the TTL coupling coefficients, jitter and readout noise level, while the instrumental variable estimator converges to an unbiased result as the data set length increases. We present an equation that predicts the estimation bias of the least squares method due to angular readout noise.

Tilts of certain elements within a laser interferometer can undesirably couple into measurements as a form of noise, known as tilt-to-length (TTL) coupling. This TTL coupling is anticipated to be one of the primary noise sources in the Laser Interferometer Space Antenna (LISA) mission, after Time Delay Interferometry (TDI) is applied. Despite the careful interferometer design and calibration on the ground, TTL is likely to require in-flight mitigation through post-processing subtraction to achieve the necessary sensitivity. Past research has demonstrated TTL subtraction in simulations through the estimation of 24 linear coupling coefficients using a noise minimization approach. This paper investigates an approach based on performing rotation maneuvers for estimating coupling coefficients with low uncertainties. In this study, we evaluate the feasibility and optimal configurations of such maneuvers to identify the most efficient solutions. We assess the efficacy of TTL calibration maneuvers by modulating either the spacecraft attitude or the Moving Optical Sub-Assembly (MOSA) yaw angle. We found that sinusoidal signals with amplitudes of around 30 nrad and frequencies near 43 mHz are practical and nearly optimal choices for such modulations. Employing different frequencies generates uncorrelated signals, allowing for multiple maneuvers to be executed simultaneously. Our simulations enable us to estimate the TTL coefficients with precision below 15 um/rad (1-sigma, in free space) after a total maneuver time of 20 minutes. The results are compared to the estimation uncertainties that can be achieved without using maneuvers.

A precise characterization of the magnetic properties of LISA Pathfinder free falling test-masses is of special interest for future gravitational wave observatory in space. Magnetic forces have an important impact on the instrument sensitivity in the low frequency regime below the millihertz. In this paper we report on the magnetic injection experiments performed throughout LISA Pathfinder operations. We show how these experiments allowed a high precision estimate of the instrument magnetic parameters. The remanent magnetic moment was found to have a modulus of (0.245 + 0.081) nAm2, the x-component of the background magnetic field within the test masses position was measured to be (414 + 74) nT and its gradient had a value of (-7.4 + 2.1) mu T/m. Finally, we also measured the test mass magnetic susceptibility at 5 mHz to be (-3.3723 + 0.0069) x 10-5. All results are in agreement with on-ground estimates.

LISA Pathfinder was a mission designed to test key technologies required for gravitational wave detection in space. Magnetically driven forces play a key role in the instrument sensitivity in the low-frequency regime, which corresponds to the measurement band of interest for future space-borne gravitational wave observatories. Magnetically induced forces couple to the test mass motion, introducing a contribution to the relative acceleration noise between the free-falling test masses. In this Letter we present the first complete estimate of this term of the instrument performance model. Our results set the magnetic-induced acceleration noise during the February 2017 noise run of 0.25_{-0.08}^{+0.15}  fm s^{-2}/sqrt[Hz] at 1 mHz and 1.01_{-0.24}^{+0.73}  fm s^{-2}/sqrt[Hz] at 0.1 mHz. We also discuss how the nonstationarities of the interplanetary magnetic field can affect these values during extreme space weather conditions.

The LISA Pathfinder (LPF) mission successfully demonstrated the feasibility of the technology needed for the future space borne gravitational wave observatory LISA. A key subsystem under study was the laser interferometer, which measured the changes in relative distance in between two test masses (TMs). It achieved a sensitivity of 32.0 thorn 2.4 -1.7 fm=ffiffiffiffiffiA/Hz, which was significantly better than the prelaunch tests. This improved performance allowed direct observation of the influence of laser frequency noise in the readout. The differences in optical path lengths between the measurement and reference beams in the individual interferometers of our setup determined the level of this undesired readout noise. Here, we discuss the dedicated experiments performed on LPF to measure these differences with high precision. We reached differences in path length difference between (368 +/- 5) mu m and (329.6 +/- 0.9) mu m which are significantly below the required level of 1 mm or 1000 mu m. These results are an important contribution to our understanding of the overall sensor performance. Moreover, we observed varying levels of laser frequency noise over the course of the mission. We provide evidence that these do not originate from the laser frequency stabilization scheme which worked as expected. Therefore, this frequency stabilization would be applicable to other missions with similar laser frequency stability requirements.

Electrostatic force actuation is a key component of the system of geodesic reference test masses (TM) for the LISA orbiting gravitational wave observatory and in particular for performance at low frequencies, below 1 mHz, where the observatory sensitivity is limited by stray force noise. The system needs to apply forces of order 10$^{-9}$ N while limiting fluctuations in the measurement band to levels approaching 10$^{-15}$ N/Hz$^{1/2}$. We present here the LISA actuation system design, based on audio-frequency voltage carrier signals, and results of its in-flight performance test with the LISA Pathfinder test mission. In LISA, TM force actuation is used to align the otherwise free-falling TM to the spacecraft-mounted optical metrology system, without any forcing along the critical gravitational wave-sensitive interferometry axes. In LISA Pathfinder, on the other hand, the actuation was used also to stabilize the TM along the critical $x$ axis joining the two TM, with the commanded actuation force entering directly into the mission's main differential acceleration science observable. The mission allowed demonstration of the full compatibility of the electrostatic actuation system with the LISA observatory requirements, including dedicated measurement campaigns to amplify, isolate, and quantify the two main force noise contributions from the actuation system, from actuator gain noise and from low frequency ``in band'' voltage fluctuations. These campaigns have shown actuation force noise to be a relevant, but not dominant, noise source in LISA Pathfinder and have allowed performance projections for the conditions expected in the LISA mission.

The tilt-to-length coupling during the LISA Pathfinder mission has been numerically and analytically modeled for particular time spans. In this work, we investigate the long-term stability of the coupling coefficients of this noise. We show that they drifted slowly (by 1 μm/rad and 6×10−6 in 100 days) and were strongly correlated to temperature changes within the satellite (8 μm/rad/K and 30×10−6/K). Based on analytical tilt-to-length coupling models, we attribute the temperature-driven coupling changes to rotations of the test masses and small distortions in the optical setup. Particularly, our findings lead to the conclusion that LISA Pathfinder’s optical baseplate was bent during the cooldown experiment, which started in late 2016 and lasted several months. Published by the American Physical Society 2024

The LISA Pathfinder (LPF) mission successfully demonstrated the feasibility of the technology needed for the future space borne gravitational wave observatory LISA. A key subsystem under study was the laser interferometer, which measured the changes in relative distance in between two test masses (TMs). It achieved a sensitivity of 32.0−1.7+2.4 fm/Hz, which was significantly better than the prelaunch tests. This improved performance allowed direct observation of the influence of laser frequency noise in the readout. The differences in optical path lengths between the measurement and reference beams in the individual interferometers of our setup determined the level of this undesired readout noise. Here, we discuss the dedicated experiments performed on LPF to measure these differences with high precision. We reached differences in path length difference between (368±5) μm and (329.6±0.9) μm which are significantly below the required level of 1 mm or 1000 μm. These results are an important contribution to our understanding of the overall sensor performance. Moreover, we observed varying levels of laser frequency noise over the course of the mission. We provide evidence that these do not originate from the laser frequency stabilization scheme which worked as expected. Therefore, this frequency stabilization would be applicable to other missions with similar laser frequency stability requirements.2 MoreReceived 2 October 2023Accepted 9 January 2024DOI:https://doi.org/10.1103/PhysRevD.109.042003Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasGravitational wave detectionMetrologyNoiseOptics & lasersPhysical SystemsLaser applicationsTechniquesData analysisGravitational wave detectorsOptical interferometrySatellite data analysisGravitation, Cosmology & AstrophysicsAtomic, Molecular & Optical

We present an in-depth analysis of the LISA Pathfinder differential acceleration performance over the entire course of its science operations, spanning approximately 500 days. We find: (1) The evolution of the Brownian noise that dominates the acceleration amplitude spectral density (ASD), for frequencies f greater than or similar to 1 mHz, is consistent with the decaying pressure due to the outgassing of a single gaseous species. (2) Between f = 36 mu Hz and 1 mHz, the acceleration ASD shows a 1/f tail in excess of the Brownian noise of almost constant amplitude, with similar or equal to 20% fluctuations over a period of a few days, with no particular time pattern over the course of the mission. (3) At the lowest considered frequency of f = 18 mu Hz, the ASD significantly deviates from the 1/f behavior, because of temperature fluctuations that appear to modulate a quasistatic pressure gradient, sustained by the asymmetries of the outgassing pattern. We also present the results of a projection of the observed acceleration noise on the potential sources for which we had either a direct correlation measurement or a quantitative estimate from dedicated experiments. These sources account for approximately 40% of the noise power in the 1/f tail. Finally, we analyze the possible sources of the remaining unexplained fraction and identify the possible measures that may be taken to keep those under control in LISA.

We present first-order models for tilt-to-length (TTL) coupling in LISA, both for the individual interferometers as well as in the time-delay interferometry (TDI) Michelson observables. These models include the noise contributions from angular and lateral jitter coupling of the six test masses, six movable optical subassemblies (MOSAs), and three spacecraft. We briefly discuss which terms are considered to be dominant and reduce the TTL model for the second-generation TDI Michelson X observable to these primary noise contributions to estimate the resulting noise level. We show that the expected TTL noise will initially violate the entire mission displacement noise budget, resulting in the known necessity to fit and subtract TTL noise in data post-processing. By comparing the noise levels for different assumptions prior to subtraction, we show why noise mitigation by realignment prior to subtraction is favorable. We then discuss that the TTL coupling in the individual interferometers will have noise contributions that will not be present in the TDI observables. Models for TTL coupling noise in TDI and in the individual interferometers are therefore different, and commonly made assumptions are valid as such only for TDI but not for the individual interferometers. Finally, we analyze what implications can be drawn from the presented models for the subsequent fit-and-subtraction in post-processing. We show that noise contributions from the test mass and inter-satellite interferometers are indistinguishable, such that only the combined coefficients can be fit and used for subtraction. However, a distinction is considered not necessary. Additionally, we show a correlation between coefficients for transmitter and receiver jitter couplings in each individual TDI Michelson observable. This full correlation can be resolved by using all three Michelson observables for fitting the TTL coefficients.

We present a study of the tilt-to-length coupling noise during the LISA Pathfinder mission and how it depended on the system's alignment. Tilt-to-length coupling noise is the unwanted coupling of angular and lateral spacecraft or test mass motion into the primary interferometric displacement readout. It was one of the major noise sources in the LISA Pathfinder mission and is likewise expected to be a primary noise source in LISA. We demonstrate here that a recently derived and published analytical model describes the dependency of the LISA Pathfinder tilt-to-length coupling noise on the alignment of the two freely falling test masses. This was verified with the data taken before and after the realignments performed in March (engineering days) and June 2016, and during a two-day experiment in February 2017 (long cross-talk experiment). The latter was performed with the explicit goal of testing the tilt-to-length coupling noise dependency on the test mass alignment. Using the analytical model, we show that all realignments performed during the mission were only partially successful and explain the reasons why. In addition to the analytical model, we computed another physical tilt-to-length coupling model via a minimizing routine making use of the long cross-talk experiment data. A similar approach could prove useful for the LISA mission.

A comprehensive summary of the measurements made to characterize test mass charging due to the space environment during the LISA Pathfinder mission is presented. Measurements of the residual charge of the test mass after release by the grabbing and positioning mechanism, show that the initial charge of the test masses was negative after all releases, leaving the test mass with a potential in the range $-12$ mV to $-512$ mV. Variations in the neutral test mass charging rate between $21.7$ e s$^{-1}$ and $30.7$ e s$^{-1}$ were observed over the course of the 17-month science operations produced by cosmic ray flux changes including a Forbush decrease associated with a small solar energetic particle event. A dependence of the cosmic ray charging rate on the test mass potential between $-30.2$ e s$^{-1}$ V$^{-1}$ and $-40.3$ e s$^{-1}$ V$^{-1}$ was observed and this is attributed to a contribution to charging from low-energy electrons emitted from the gold surfaces of the gravitational reference sensor. Data from the on-board particle detector show a reliable correlation with the charging rate and with other environmental monitors of the cosmic ray flux. This correlation is exploited to extrapolate test mass charging rates to a 20-year period giving useful insight into the expected range of charging rate that may be observed in the LISA mission.

In a previous article [1], we have reported on the first subpicometer interferometer flown in space as part of ESA's LISA Pathfinder mission, and have shown the residual sensor noise to be on the level of 32.0 thorn 2.4 -1.7 fm/ . This review provides a deeper and more complete overview of the full system and its ffiffiffiffiffiHz interferometric mission performance under varying operational conditions, allowing a much more detailed view on the noise model. We also include the optical measurements of rotations through differential wave ffiffiffiffiffifront sensing (DWS), which reached a sensitivity of as good as 100 prad/ Hz. We present more evidence for the long-term stability of the interferometric performance and components. This proves a solid foundation for future interferometry in space such as the LISA mission.

We present an in depth analysis of the transient events, or glitches, detected at a rate of about one per day in the differential acceleration data of LISA Pathfinder. We show that these glitches fall in two rather distinct categories: fast transients in the interferometric motion readout on one side, and true force transient events on the other. The former are fast and rare in ordinary conditions. The second may last from seconds to hours and constitute the majority of the glitches. We present an analysis of the physical and statistical properties of both categories, including a cross-analysis with other time series like magnetic fields, temperature, and other dynamical variables. Based on these analyses we discuss the possible sources of the force glitches and identify the most likely, among which the outgassing environment surrounding the test-masses stands out. We discuss the impact of these findings on the LISA design and operation, and some risk mitigation measures, including experimental studies that may be conducted on the ground, aimed at clarifying some of the questions left open by our analysis.

The coupling of an angular jitter into the interferometric phase readout is summarized under the term tilt-to-length (TTL) coupling. This noise is expected to be a major noise source in the intersatellite interferometry for the Laser Interferometer Space Antenna (LISA) space mission. Despite efforts to reduce it by satellite construction, some remaining TTL noise will need to be removed in postprocessing on Earth. Therefore, such a procedure needs to be developed and validated to ensure the success of the LISA mission. This paper shows a method to calibrate and subtract TTL noise that has no impact on LISA science operations. This solution relies on noise minimization and uses the differential wavefront sensing (DWS) measurements to estimate the TTL contribution. Our technique is applied after the laser frequency noise is suppressed via the time-delay interferometry (TDI) postprocessing algorithm. We use a simulation to show as a proof-of-principle that we can estimate the TTL coefficients to the required accuracy level based on the current design configuration of LISA. We then use these estimates to subtract the TTL noise, ensuring that any remaining TTL noise is below the current estimate of the other noise sources. We validate the procedure on simulated data for different operating scenarios. Our work shows that it is indeed possible to estimate the effect of TTL coupling and subtract it a posteriori from the TDI data streams.

LISA Pathfinder is a technology demonstrator space mission, aimed at testing key technologies for detecting gravitational waves in space. The mission is the precursor of LISA, the first space gravitational waves observatory, whose launch is scheduled for 2034. The LISA Pathfinder scientific payload includes two gravitational reference sensors (GRSs), each one containing a test mass (TM), which is the sensing body of the experiment. A mission critical task is to set each TM into a pure geodesic motion, i.e. guaranteeing an extremely low acceleration noise in the sub-Hertz frequency bandwidth. The grabbing positioning and release mechanism (GPRM), responsible for the injection of the TM into a geodesic trajectory, was widely tested on ground, with the limitations imposed by the 1-g environment. The experiments showed that the mechanism, working in its nominal conditions, is capable of releasing the TM into free-fall fulfilling the very strict constraint imposed on the TM residual velocity, in order to allow its capture on behalf of the electrostatic actuation. However, the first in-flight releases produced unexpected residual velocity components, for both the TMs. Moreover, all the residual velocity components were greater than maximum value set by the requirements. The main suspect is that unexpected contacts took place between the TM and the surroundings bodies. As a consequence, ad hoc manual release procedures had to be adopted for the few following injections performed during the nominal mission. These procedures still resulted in non compliant TM states which were captured only after impacts. However, such procedures seem not practicable for LISA, both for the limited repeatability of the system and for the unmanageable time lag of the telemetry/telecommand signals (about 4400 s). For this reason, at the end of the mission, the GPRM was deeply tested in-flight, performing a large number of releases, according to different strategies. The tests were carried out in order to understand the unexpected dynamics and limit its effects on the final injection. Some risk mitigation maneuvers have been tested aimed at minimizing the vibration of the system at the release and improving the alignment between the mechanism and the TM. However, no overall optimal release strategy to be implemented in LISA could be found, because the two GPRMs behaved differently. (C) 2020 COSPAR. Published by Elsevier Ltd. All rights reserved.

We report on the first subpicometer interferometer flown in space. It was part of ESA's Laser Interferometer Space Antenna (LISA) Pathfinder mission and performed the fundamental measurement of the positional and angular motion of two free-falling test masses. The interferometer worked immediately, stably, and reliably from switch on until the end of the mission with exceptionally low residual noise of 32.0_{-1.7}^{+2.4}  fm/sqrt[Hz], significantly better than required. We present an upper limit for the sensor performance at millihertz frequencies and a model for the measured sensitivity above 200 mHz.

Electrostatic actuation of a free-floating test-mass was tested in the Laser Interferometer Space Antenna (LISA) Pathfinder mission, and it will be integrated into the LISA. We have investigated the LISA Pathfinder actuation accuracy with respect to the precision of fractional digits in the field programmable gate array (FPGA) code of actuation electronics. The LISA Pathfinder data showed that the rounding errors in the FPGA code result in an erroneous force that contaminated the main mission observable, and this error was compensated in the post-processing of the LISA Pathfinder data. To avoid a similar issue for the LISA, the LISA actuation accuracy can be improved by increasing the number of fractional digits in the FPGA code. However, this is restricted by some hardware limitations. In this paper, we investigate the necessary enlargement of the FPGA to fulfill the LISA acceleration requirements and propose a design optimization for LISA actuation electronics.