Analysis of the residual force noise for the LISA Technology Package

The analysis of the noise sources perturbing a test mass (TM) geodesic motion is the main scientific objective of the LISA Technology Package experiment (LTP) on board of the LISA Pathfinder space mission. Information on force noise acting on TMs are obtained with a data reduction procedure involving system parameters. Such parameters can be estimated from dedicated experimental runs. Therefore the final estimation of force noise is affected by two sources of uncertainty. One is statistical and connected to the random nature of noisy signals. The other is connected to the uncertainties on the system parameters. The analysis of simulated LTP data is indicating that the major contribution to the force noise power spectral density uncertainties is coming from the statistical properties of the spectrum estimator.

💡 Research Summary

The paper presents a comprehensive study of the residual force noise acting on the test masses (TMs) of the LISA Technology Package (LTP) aboard the LISA Pathfinder mission, focusing on the quantification of uncertainties in the final force‑noise estimate. The LTP consists of two free‑falling TMs whose positions are measured by an interferometer. One TM serves as the drag‑free reference for the spacecraft (SC), while the second TM is kept centered by electro‑static suspension. The SC itself is controlled by micro‑Newton thrusters. The authors first write the equations of motion for TM1, TM2 and the SC in the Laplace domain, introducing the key dynamical parameters: the effective stiffnesses (ω₁², ω₂²), the drag‑free and suspension control gains (G_df, G_sus), interferometer cross‑talk (S₂₁), and various time delays (ΔT₁, ΔT_Δ).

Force‑noise power spectral densities (PSDs) are expressed for the two interferometer read‑out channels (the “o₁” channel measuring SC‑TM1 distance and the differential “o_Δ” channel measuring TM2‑TM1 separation). Equation (2) shows that the o₁ channel is dominated at low frequencies by spacecraft force noise (thruster noise, solar wind, asymmetric thermal emission) and that interferometer read‑out noise only becomes relevant at high frequencies. Conversely, the differential channel is dominated by the combined force noise on the two TMs, making it the optimal observable for estimating TM force noise in the mission‑relevant band (≈10⁻³ Hz to a few mHz).

System parameters are estimated through dedicated on‑flight experiments. In the first experiment, sine‑wave injections are applied to the drag‑free controller guidance; the responses in o₁ and o_Δ allow the extraction of G_df, ω₁², the thruster time constant τ_th, the o₁ delay ΔT₁, and, via the differential channel, the stiffness difference (ω₂²‑ω₁²) and interferometer cross‑talk S₂₁. In the second experiment, sine‑wave injections are applied to the suspension controller; the lack of response in o₁ confirms that differential motion does not leak into the first channel, while o_Δ provides G_sus, ω₂², the suspension time constant τ_sus, and the o_Δ delay ΔT_Δ. Because many physical parameters are highly correlated, the fitting matrix is rank‑deficient; the authors resolve this by singular‑value decomposition (SVD), fitting a set of linearly independent combinations, and then reconstructing the original physical parameters by solving the linear system that links SVD coefficients to the physical quantities. The resulting parameter values (Table 1) have very small relative uncertainties (e.g., G_df = 1.0813 ± 0.0005).

The conversion from measured displacements to effective forces per unit mass is performed offline using the calibrated dynamical model. The authors identify two distinct sources of uncertainty in the final force‑noise PSD: (1) propagation of the parameter‑fit uncertainties (σ_fit) and (2) statistical fluctuations inherent to the PSD estimator (σ_stat). To assess σ_fit, a Monte‑Carlo simulation is carried out: for each of N = 10⁴ realizations, the physical parameters are randomly drawn from normal distributions defined by their fitted means and standard deviations, while the displacement data remain fixed. For each realization the force time series is reconstructed, its PSD computed with a windowed averaged periodogram (8 averages, 50 % overlap, 4‑term Blackman‑Harris window), and the variance across realizations yields σ_fit(f).

σ_stat is derived from the known χ² statistics of the averaged periodogram. With eight averages the degrees of freedom are 16, and the 68 % confidence interval is obtained from the χ²(0.68, 16) distribution. The authors plot both uncertainty bands on the measured differential‑channel PSD (Figure 2). The σ_fit band (blue dots) lies essentially on top of the measured PSD, indicating that parameter‑fit errors contribute negligibly to the total error budget. In contrast, the σ_stat band (green dashes) is clearly separated from the PSD and dominates the overall uncertainty. This demonstrates that, provided the system‑identification experiments yield sufficiently accurate parameter estimates, the statistical properties of the PSD estimator are the limiting factor in the force‑noise determination.

In conclusion, the most accurate estimate of the residual force noise on LTP test masses is obtained from the PSD of the differential interferometer channel. The force‑noise contributions from the two TMs cannot be disentangled, but their sum can be measured with high fidelity. While the dynamical model parameters can be calibrated to sub‑percent precision using multi‑experiment SVD‑based fitting, the dominant source of uncertainty in the final force‑noise spectrum arises from the statistical variance of the spectral estimator itself. This insight has direct implications for the design of future space‑based gravitational‑wave observatories such as LISA, emphasizing the need for long integration times, sufficient averaging, and careful spectral‑analysis methodology to suppress σ_stat and achieve the required sensitivity.