Moving Constellations and Link Response

A space interferometer does not directly record a local strain at one beam splitter. Its elementary observable is a time-stamped comparison between light received from a distant spacecraft and a local laser. The gravitational wave enters through the metric difference at the two ends of that null path, while constellation motion changes the path, antenna pattern, and Doppler phase over the observation. This page derives the one-way fractional-frequency response and identifies exactly where the long-wavelength and rigid-array approximations fail.

Helpful background. Probability and Statistical Inference supplies likelihoods and covariance. The gravitational-wave likelihood supplies detector channels and one-sided-PSD weighting. Gravitational Radiation and Waveforms supplies tensor polarizations, while Black-Hole Astrophysics supplies the source populations interpreted later in the chapter.

A one-way light link

Use barycentric coordinate time $t$ and restore $c$ in instrument formulas. Light is emitted by spacecraft $i$ at $t_\mathrm e$ and received by spacecraft $j$ at $t$. Define the reception-time armlength and link direction by

$$t-t_\mathrm e=\frac{L_{ij}(t)}{c}, \qquad \widehat{\boldsymbol n}_{ij}(t) =\frac{\boldsymbol x_j(t)-\boldsymbol x_i(t_\mathrm e)} {L_{ij}(t)}.$$

The measured quantity $y_{ij}(t)=\delta\nu/\nu_0$ is a dimensionless fractional-frequency fluctuation at receiver $j$. In the weak-field, geometric-optics, linear-response limit, let a plane tensor wave be

$$h_{ab}(u) =\sum_{A=+,\times}e^A_{ab}(\widehat{\boldsymbol k},\psi) h_A(u), \qquad u=t-\frac{\widehat{\boldsymbol k}\cdot\boldsymbol x}{c},$$

where $\widehat{\boldsymbol k}$ points in the propagation direction. Holding the link geometry fixed during one light transit gives

$$y_{ij}^{\mathrm{GW}}(t) =\frac{\widehat n_{ij}^{a}\widehat n_{ij}^{b}} {2\left(1-\widehat{\boldsymbol k}\cdot\widehat{\boldsymbol n}_{ij}\right)} \left[ h_{ab}\!\left(t_\mathrm e,\boldsymbol x_i(t_\mathrm e)\right) -h_{ab}\!\left(t,\boldsymbol x_j(t)\right) \right].$$

This endpoint-difference response follows from the null propagation of the photon; Estabrook and Wahlquist derive the underlying Doppler observable in Estabrook and Wahlquist 1975, §§2–3. Reversing the link reverses its timestamped geometry, so $y_{ij}$ and $y_{ji}$ are not interchangeable.

The apparent denominator is regular when the wave and link become parallel because transverse polarization makes $\widehat n^a\widehat n^b h_{ab}$ vanish in the same limit. Numerically, the response should be evaluated with a stable sinc-like form rather than by subtracting two nearly equal endpoint values.

Transfer function and finite-arm scale

With the site convention $e^{-i\omega t}$, freeze $L_{ij}=L$ and write $\mu=\widehat{\boldsymbol k}\cdot\widehat{\boldsymbol n}_{ij}$. Relative to the receiving endpoint, the link contains the transfer factor

$$\mathcal T_y(\omega,\mu) =\frac{e^{i\omega L(1-\mu)/c}-1} {2(1-\mu)} =\frac{i\omega L}{2c} e^{i\omega L(1-\mu)/(2c)} \operatorname{sinc}\!\left[ \frac{\omega L(1-\mu)}{2c} \right].$$

Here $\operatorname{sinc}x=\sin x/x$. At low frequency, $|\mathcal T_y|\propto fL/c$: a fractional-frequency link measures the time derivative of the long-wavelength strain response. A phase or equivalent displacement observable integrates this factor and has a different normalization. Mixing link phase, pathlength, and fractional frequency is a dimensional error.

For the 2024 ESA LISA reference armlength $L=2.5\times10^9\,\mathrm m$, the transfer frequency

$$f_*=\frac{c}{2\pi L}\simeq19.1\,\mathrm{mHz}$$

marks where finite-arm structure becomes important. This number is configuration-specific, not a timeless property of every space interferometer. The full moving-response construction and its low-frequency limit are developed in Cornish and Rubbo 2003, §§II–IV.

Motion encodes the sky

The link response depends on source direction in three independent ways:

1. $\widehat n^a\widehat n^b e^A_{ab}$ projects polarization onto the instantaneous arm;
2. $u=t-\widehat{\boldsymbol k}\cdot\boldsymbol x/c$ carries a barycentric Doppler phase; and
3. $L_{ij}(t)$ changes the finite-arm transfer and emission timestamp.

As the constellation moves, these modulations separate sky position, polarization, inclination, phase, and amplitude that would be degenerate in a frozen long-wavelength detector. The information depends on how much modulation is observed before a signal ends. A short merger-dominated source and a nearly monochromatic source therefore use the same response but acquire different localization information.

A rigid adiabatic response evaluates a static link at the slowly changing spacecraft positions associated with each waveform time or frequency. The first neglected effects are motion during a light transit, arm flexing within composed delays, orbit and clock error, and any interpolation error used to evaluate a waveform at retarded times. These become consequential for time-delay interferometry rather than being absorbed into an antenna factor.

From links to calibrated data

One link also contains transmitting and receiving laser-frequency noise, proof-mass acceleration, optical-path noise, clock noise, and other instrument terms. A schematic measurement with frozen geometry is

$$y_{ij} =C_i\!\left(t-\frac{L_{ij}}c\right)-C_j(t) +y_{ij}^{\mathrm{GW}}+n_{ij},$$

where $C_i$ is fractional laser-frequency noise and $n_{ij}$ collects terms whose exact signs depend on the phasemeter convention. When $L_{ij}$ varies, the transmitting term becomes $(1-\dot L_{ij}/c)C_i(t-L_{ij}/c)$ after removal of the deterministic carrier beat. This follows by differentiating the received laser phase at its retarded emission time. Even a small factor matters because laser noise greatly exceeds the target signal; Time-Delay Interferometry derives the corresponding frequency-delay operator and cancellation.

The calibration record must name the one-way phase or fractional-frequency convention, emitting and receiving indices, timestamp, orbit and armlength product, clock convention, laser-noise sign, sampling cadence, delay interpolation, and TDI generation. A channel called $X$ without this information is not a reproducible data product.

Limits of the results and independent checks

The reference configuration comes from the LISA Definition Study Report, ESA-SCI-DIR-RP-002 (2024), especially its instrument, measurement, and data-analysis descriptions ESA 2024, §§2–4. All mission-dependent statements here are frozen at 2026-08-18. No flight data or achieved sensitivity exists in this chapter; reference-design requirements and simulations are not observations.

The exact public simulation anchor is LISA Data Challenge 2b “Spritz,” which provides simulated second-generation $X,Y,Z$ fractional-frequency TDI streams at 5 s cadence together with orbit, noise, gap, artifact, and foreground specifications LISA Data Challenge 2b, dataset description. These are downstream combinations rather than raw one-way links, so a link-level reproduction must regenerate the directed observables and demonstrate that their ordered TDI combination agrees with the release convention.

For a forecast, the response enters a declared TDI data model and full cross-channel covariance; selected channels, observing span and duty, cadence and band, source population and waveform, foreground, priors, and threshold define the probability measure and selection. A reproducible simulation specifies those quantities plus the ESA configuration, orbit file or analytic orbit, coordinate frame, $L_{ij}(t)$ convention, link timestamps, noise realization, random seed, and response software commit. Independent checks set $h_{ab}=0$, reverse a link with the corresponding timestamp change, verify the $\omega L/c\to0$ limit, compare the endpoint and sinc forms near $\mu=1$, and refine retarded-time interpolation. The next page checks that delayed links cancel laser noise while retaining their unequal directional signal terms.

Check your understanding

Take a frozen link perpendicular to the propagation direction, so $\mu=0$, and a monochromatic projected strain amplitude $H$. Show that the link’s transfer magnitude is $|H\sin(\omega L/2c)|$. What is its low-frequency limit?

Solution

For $\mu=0$, the endpoint factor multiplying the projected amplitude is

$$\frac{e^{i\omega L/c}-1}{2} =i\,e^{i\omega L/(2c)} \sin\!\left(\frac{\omega L}{2c}\right).$$

Its magnitude is therefore

$$\left\lvert y^{\mathrm{GW}}\right\rvert =|H|\left\lvert \sin\!\left(\frac{\omega L}{2c}\right) \right\rvert.$$

When $\omega L/c\ll1$,

$$\left\lvert y^{\mathrm{GW}}\right\rvert \simeq |H|\frac{\omega L}{2c}.$$

The response vanishes linearly with frequency because $y$ is fractional frequency. A strain-like phase channel divides out or integrates a different transfer factor.

References

• Cornish, Neil J., and Louis J. Rubbo. “The LISA Response Function.” Physical Review D 67 (2003): 022001. DOI. Open PDF.
• Estabrook, Frank B., and Hugo D. Wahlquist. “Response of Doppler Spacecraft Tracking to Gravitational Radiation.” General Relativity and Gravitation 6 (1975): 439–447. DOI.
• European Space Agency. LISA Definition Study Report. ESA-SCI-DIR-RP-002 (2024). Official PDF.
• LISA Data Challenge. “Challenge 2b: Spritz.” Public simulated dataset and documentation. Dataset.