Impact of Spacecraft Orbit Uncertainties and Velocity Mismodeling on the LISA Gravitational-Wave Response

Technical Summary: Impact of Spacecraft Orbit Uncertainties and Velocity Mismodeling on the LISA Gravitational-Wave Response

Problem Statement
The Laser Interferometer Space Antenna (LISA) is designed to detect gravitational waves (GWs) in the millihertz frequency band using a constellation of three spacecraft. Accurate GW data analysis relies on precise knowledge of the detector's response function, which maps incoming GW strain to the measured fractional-frequency signal. This response depends critically on the spacecraft positions, velocities, and arm lengths. While previous studies have addressed instrumental calibration and trajectory simulation, two specific questions regarding the fidelity of the response model remained:

1. How do uncertainties in the knowledge of spacecraft orbits and arm lengths propagate into the GW response?
2. What is the impact of neglecting spacecraft velocity terms (Doppler boosting) in the response model, particularly for Galactic binary sources?

Methodology
The authors quantify these errors using the "mismatch" metric, which measures the fractional loss in signal-to-noise ratio (SNR) when an incorrect template is used. The study employs two complementary orbital scenarios:

1. Static Toy Model: A simplified, time-independent triangular configuration is used to isolate geometric effects. Three classes of perturbations are applied sequentially:

   • Arm-length variation: Independent Gaussian perturbations ($\sigma_L = 1$ m) to simulate high-precision onboard interferometric ranging.
   • Rigid-body rotation: Random rotations inducing positional uncertainties equivalent to $\sigma_{rot} = 50$ km.
   • Rigid-body translation: Global translations with $\sigma_{trans} = 50$ km to simulate absolute position uncertainties from ground-based radiometric tracking.

2. Realistic Evolving Scenario: A high-fidelity simulation using ESA's GODOT and MIDAS software packages. This involves:

   • Generating a "truth" trajectory with realistic perturbations (self-gravity, non-gravitational forces, clock noise).
   • Running an augmented Square-Root Information Filter (SRIF) to estimate the trajectory based on simulated observations (range, Doppler, pseudoranges).
   • Sampling from the resulting orbit covariance matrix to generate an ensemble of trajectory estimates.
   • Comparing the LISA response computed with these perturbed trajectories against a nominal response.

3. Response Mismodeling: The study compares the full Doppler-boosted response (including velocity terms $\dot{r}/c$) against a zeroth-order approximation where spacecraft velocities are neglected. This is evaluated using both a single-frequency monochromatic source and a fast Galactic binary waveform model (jaxgb).

4. Parameter Estimation: A Markov Chain Monte Carlo (MCMC) analysis is performed on a Galactic binary signal ($f = 10^{-4}$ Hz, SNR = 200) observed for one year to assess parameter biases resulting from velocity mismodeling.

Key Results

• Orbit Uncertainties:

  • In the realistic evolving scenario, orbit determination uncertainties produce negligible mismatches across the LISA band. The worst-case mismatch is below $10^{-7}$.
  • Errors are dominated by high-frequency effects ($>10$ mHz) where the response is most sensitive to absolute spacecraft positions.
  • Compared to the static toy model, the realistic simulation shows arm-length errors are 3–6 times smaller, position errors 8–20 times smaller, and angle errors ~300 times smaller.

• Velocity Mismodeling:

  • Neglecting spacecraft velocities leads to significantly larger errors, particularly at low frequencies.
  • At $\sim 10^{-4}$ Hz, mismatches are of order $10^{-4}$, with relative amplitude errors around $10^{-2}$ and phase errors reaching one radian.
  • The error scales approximately as $f^{-1}$ at low frequencies, decreasing to $\sim 10^{-9}$ at 100 mHz.
  • Velocity mismodeling dominates the error budget by approximately two orders of magnitude at 10 mHz and nine orders of magnitude at 0.1 mHz compared to orbit uncertainties.

• Parameter Estimation:

  • For a Galactic binary with $f = 10^{-4}$ Hz and SNR = 200, neglecting velocity terms in the response leads to parameter biases of less than 1$\sigma$.
  • The difference in mean log-likelihood between the full response and the velocity-neglected model is $\Delta \log \mathcal{L} \sim 10$.

Significance and Claims
This work provides the first characterization of how errors in the LISA gravitational wave response propagate from GW strain through the detector output to estimated parameters. The authors conclude that:

• Orbit Knowledge: Current orbit determination capabilities (with uncertainties of $\sim 50$ km) are sufficient, as the resulting mismatches are too small to affect LISA scientific exploitation.
• Velocity Terms: Neglecting spacecraft velocities is the leading source of modeling error for Galactic binary signals, particularly at low frequencies. However, even in the worst-case scenario tested (SNR=200), the resulting parameter biases remain below 1$\sigma$.
• Recommendation: For high-fidelity inference of low-frequency Galactic binaries, the full Doppler-boosted response should be adopted, as it introduces no additional computational cost compared to the zeroth-order approximation.
• Future Work: The authors suggest that the frequency-dependent zeros of the response could serve as a calibration diagnostic and that future studies should extend this analysis to Extreme Mass Ratio Inspirals (EMRIs), stellar-origin black hole binaries, and stochastic backgrounds.