Time-Delay Interferometry

Time-delay interferometry forms virtual equal-optical-path interferometers from separately recorded one-way links. Correctly delayed copies of the same laser noise cancel even when the physical arms are unequal; when the arms flex, the delay operators no longer commute and their order becomes part of the observable. This page derives cancellation in a transparent two-arm model, identifies the flexing residual, and explains why the same combination retains a gravitational-wave response.

Helpful background. Probability and Statistical Inference supplies random processes and covariance. The gravitational-wave likelihood supplies multichannel covariance and PSD conventions. Gravitational Radiation and Waveforms supplies the incident polarizations, while Black-Hole Astrophysics supplies the source regimes for which TDI channels are later interpreted.

Delay operators and laser noise

For a time series $a(t)$, define a directed delay by

$$(\mathcal D_{ij}a)(t) =a\!\left(t-\frac{L_{ij}(t)}c\right).$$

The operator includes the link orientation, reception timestamp, and armlength evaluated according to a declared orbit convention. In a static array, the $L_{ij}$ are constant and all delays commute. In a moving array, $\mathcal D_{ij}\mathcal D_{kl}$ and $\mathcal D_{kl}\mathcal D_{ij}$ sample different retarded armlengths.

In a frozen array, a one-way fractional-frequency link contains a delayed transmitting-laser fluctuation and a local receiving-laser fluctuation,

$$y_{ij}(t) =\mathcal D_{ij}C_i(t)-C_j(t) +s_{ij}(t)+n_{ij}(t),$$

where $s_{ij}$ is the gravitational-wave term and $n_{ij}$ contains secondary instrument noises. The signs follow the fractional-frequency convention used on the preceding page. The same delay algebra applies to phase data with $C_i$ replaced by laser phase noise. For moving-arm frequency data, an additional Doppler factor is necessary, derived below; simply replacing constant lengths by $L_{ij}(t)$ in this frozen-frequency formula is insufficient.

A two-arm cancellation

First consider a pedagogical central-laser model in which synthesized round-trip measurements along arms 2 and 3 have laser parts

$$z_2=(\mathcal D_2^2-1)C+s_2, \qquad z_3=(\mathcal D_3^2-1)C+s_3.$$

The naive difference $z_2-z_3$ leaves unequal delayed copies of $C$. Instead form

$$X =(1-\mathcal D_3^2)z_2 -(1-\mathcal D_2^2)z_3.$$

For commuting delays, its laser term is

$$\begin{aligned} X_C &=(1-\mathcal D_3^2)(\mathcal D_2^2-1)C\\ &\quad-(1-\mathcal D_2^2)(\mathcal D_3^2-1)C\\ &=0. \end{aligned}$$

This cancellation is algebraic and does not require equal arms. The squares here represent a round trip only because each arm is static and reciprocal; a moving constellation requires distinct outward and return delays. Full Michelson TDI combines directed links and local measurements to remove independent spacecraft laser fluctuations. The construction is given in Tinto and Dhurandhar 2021, §§3–5.

Why the signal survives

The laser term is the same stochastic process $C$ entering with endpoint delays. The gravitational-wave terms $s_{ij}$ are different directional endpoint differences on different links. Substitution gives

$$X_\mathrm{GW} =(1-\mathcal D_3^2)s_2 -(1-\mathcal D_2^2)s_3,$$

which is generally nonzero. TDI cancels a particular common-noise algebra, not arbitrary time series. The signal can nevertheless have response zeros at frequencies and sky directions fixed by the transfer functions. Sensitivity must therefore be computed with the complete TDI response, not by assuming that laser-noise cancellation leaves unit strain response.

Cyclic relabeling gives Michelson-like $Y$ and $Z$. Linear combinations often denoted $A,E,T$ can diagonalize the instrumental-noise covariance under symmetric assumptions, but the exact transformation and normalization must be declared. A nominally null $T$ channel is only approximately insensitive to gravitational waves in the low-frequency, equal-arm limit.

Flexing arms and noncommuting paths

For two scalar armlengths $L_i(t)$ and $L_j(t)$,

$$\begin{aligned} \mathcal D_i\mathcal D_j a(t) =a\!\Bigl(&t-L_i(t)/c\\ &-L_j(t-L_i(t)/c)/c\Bigr). \end{aligned}$$

Expanding to first order in armlength rates gives

$$[\mathcal D_i,\mathcal D_j]a \simeq \frac{L_i\dot L_j-L_j\dot L_i}{c^2} \dot a\!\left(t-\frac{L_i+L_j}{c}\right),$$

where $[\mathcal D_i,\mathcal D_j]=\mathcal D_i\mathcal D_j-\mathcal D_j\mathcal D_i$. The expansion neglects quadratic armlength rates and armlength acceleration over a transit. A polynomial that cancels $C$ only by commuting delays therefore leaves a residual linear in the arm rates. Second-generation TDI uses longer ordered paths to cancel this leading mismatch; it is not exact cancellation for arbitrary accelerated motion Tinto and Dhurandhar 2021, §5.1.

The distinction is operational. Delay symbols cannot be sorted alphabetically in software, and an orbit interpolator must be evaluated at nested retarded times. Clock synchronization, ranging error, missing samples, and gap treatment can all convert nominal cancellation into residual laser noise.

Frequency data also need Doppler factors

Let $p_i$ be laser phase noise, let $C_i=\dot p_i/\nu_0$ for a fixed reference frequency, and set $\tau_{ij}=L_{ij}/c$. Differentiating a delayed phase gives

$$\frac{1}{\nu_0}\frac{d}{dt} \bigl[\mathcal D_{ij}p_i-p_j\bigr] =(1-\dot\tau_{ij})\mathcal D_{ij}C_i-C_j.$$

The chain-rule factor $1-\dot\tau_{ij}$ accounts for the stretching of emission-time intervals relative to reception-time intervals. Define the frequency-data delay operator

$$\mathcal D_{ij}^{\nu}a \equiv(1-\dot\tau_{ij})\mathcal D_{ij}a.$$

Then differentiating an ordered phase-delay expression corresponds to using ordered $\mathcal D^\nu$ operators on frequency data. The index convention here is emitter first, receiver second; it is opposite to that of Bayle, Hartwig, and Staab 2021, §§II and IV. This correction and second-generation path ordering address different effects; a moving-arm fractional-frequency implementation needs both. The scalar-delay experiments below test path algebra, not a complete flight-frequency observable.

An independent cancellation control

For constant discrete delays of 17 and 23 samples, a length-8192 Gaussian laser series gives

$$\max|X_C|=8.88\times10^{-16}, \qquad \operatorname{rms}(X_C)=1.89\times10^{-16}$$

in double precision when the polynomial above is evaluated with circular shifts. The result checks unequal-arm cancellation to roundoff. It is independent of the laser spectrum because the identity holds sample by sample.

This exact roundoff result uses Python 3.9.6, NumPy 2.0.2, seed 1, and the parenthesized evaluation order shown below on macOS arm64:

import numpy as np

rng = np.random.default_rng(1)
C = rng.normal(size=8192)

def D(x, samples):
    return np.roll(x, samples)

z2 = D(D(C, 17), 17) - C
z3 = D(D(C, 23), 23) - C
X = (z2 - D(D(z2, 23), 23)) - (z3 - D(D(z3, 17), 17))

print(np.max(np.abs(X)))
print(np.sqrt(np.mean(X * X)))

It prints 8.881784197001252e-16 and 1.8944054239707402e-16. Algebraic cancellation is exact; the last digits are implementation- and evaluation-order-dependent floating-point artifacts, not physical residuals.

For a separate analytic flexing test, take

$$\begin{aligned} C(t)&=\sin(2\pi\,0.03\,t),\\ L_2(t)/c&=8+0.01\sin(2\pi t/1000),\\ L_3(t)/c&=9+0.012\cos(2\pi t/1000), \end{aligned}$$

with times in seconds. Directly composing the retarded times gives

$$\operatorname{rms}\!\left[ (\mathcal D_2\mathcal D_3-\mathcal D_3\mathcal D_2)C \right] =7.80375\times10^{-5}.$$

Sampling the periodic grid $0\le t\lt1000\ \mathrm{s}$ with 1000, 2000, 10000, and 50000 points changes the reported value by less than $3\times10^{-15}$. The nonzero limit is physical for this first-generation commutator, not interpolation error. A second-generation implementation should suppress the appropriate longer path commutator; a full verification also injects a link-level gravitational wave and confirms that it remains.

Versioned data and reproduction

The ESA 2024 definition study supplies the reference link and observatory architecture ESA 2024, §§2–4. As a public simulation example, LISA Data Challenge 2b “Spritz” provides second-generation $X,Y,Z$ fractional-frequency channels, simulated orbits, gaps, artifacts, and component training streams LISA Data Challenge 2b, dataset description. These descriptions are frozen at 2026-08-18 and are simulated products, not flight calibration or achieved performance.

A reproducible TDI result records the challenge release and file hashes or analytic orbit, link orientation and timestamp convention, cadence, gap policy, $L_{ij}(t)$ source, interpolation order, clock and ranging model, laser and secondary-noise spectra, exact ordered delay word, $X/Y/Z$ or $A/E/T$ normalization, code commit, runtime, and numerical precision. When source recovery or sensitivity is claimed, it also records observing span and duty, selected channels and band, waveform and source population, foreground, covariance, priors, statistic, and selection threshold. Validation compares an independent implementation, refines cadence and delay interpolation, checks constant equal and unequal arms, introduces controlled flexing, and measures both laser suppression and injected-signal transfer. Synthetic LISA demonstrates this end-to-end simulation philosophy in Vallisneri 2005, §§II–V.

Check your understanding

Substitute the laser parts of $z_2$ and $z_3$ into $X$ and prove cancellation for unequal constant arms. Which single algebraic step fails when the arms vary?

Solution

The laser contribution is

$$\begin{aligned} X_C &=(1-\mathcal D_3^2)(\mathcal D_2^2-1)C\\ &\quad-(1-\mathcal D_2^2)(\mathcal D_3^2-1)C. \end{aligned}$$

Since constant delays commute,

$$(1-\mathcal D_3^2)(\mathcal D_2^2-1) =-(\mathcal D_3^2-1)(\mathcal D_2^2-1)$$

and the second term is the negative of the same product, so $X_C=0$. For varying arms, exchanging $\mathcal D_2$ and $\mathcal D_3$ changes the nested retarded armlength. The factor ordering cannot be swapped, leaving a commutator proportional at first order to $L_2\dot L_3-L_3\dot L_2$.

References

• Bayle, Jean-Baptiste, Olaf Hartwig, and Martin Staab. “Adapting Time-Delay Interferometry for LISA Data in Frequency.” Physical Review D 104 (2021): 023006. DOI. Open PDF. Erratum.
• European Space Agency. LISA Definition Study Report. ESA-SCI-DIR-RP-002 (2024). Official PDF.
• LISA Data Challenge. “LISA Data Challenge 2b: Spritz.” Public simulation release, accessed 2026-08-18. Dataset description.
• Tinto, Massimo, and Sanjeev V. Dhurandhar. “Time-Delay Interferometry.” Living Reviews in Relativity 24 (2021): 1. DOI.
• Vallisneri, Michele. “Synthetic LISA: Simulating Time Delay Interferometry in a Model LISA.” Physical Review D 71 (2005): 022001. DOI. Open PDF.