Instrumental Noise and Galactic Confusion

Instrument noise and unresolved Galactic binaries both contribute power to time-delay-interferometry channels, but they are not the same kind of uncertainty. Instrument terms propagate from proof masses, optical links, clocks, and residual laser noise through known transfer functions; the foreground is a population of physical signals whose resolvable membership changes with observing time and the global source fit. This page formulates their joint likelihood, explains the information that separates them, and validates a two-component spectral fit on a controlled injection.

Helpful background. Probability and Statistical Inference supplies likelihood, covariance, and Monte Carlo error. The gravitational-wave likelihood supplies multichannel Gaussian and Whittle models. Gravitational Radiation and Waveforms supplies compact-binary spectra, while Black-Hole Astrophysics supplies Galactic populations and selection effects.

Noise after time-delay interferometry

Let $\widetilde{\boldsymbol d}(f)$ contain selected TDI channels, for example $A,E,T$ under a declared normalization. A useful data model is

$$\widetilde{\boldsymbol d} =\sum_{s\in\mathcal R}\widetilde{\boldsymbol h}_s(\theta_s) +\widetilde{\boldsymbol g} +\widetilde{\boldsymbol n}_{\mathrm{inst}},$$

where $\mathcal R$ is the current set of resolved sources, $\boldsymbol g$ is the aggregate unresolved foreground, and $\boldsymbol n_\mathrm{inst}$ is instrument noise after TDI. In a locally stationary Gaussian approximation,

$$\mathbb E[ \widetilde{\boldsymbol r}(f) \widetilde{\boldsymbol r}^{\dagger}(f')] =\frac12\delta(f-f') \boldsymbol C(f), \qquad \boldsymbol C =\boldsymbol C_\mathrm{inst} +\boldsymbol C_\mathrm{conf}.$$

The factor one half is the one-sided-PSD convention for positive frequencies. The covariance need not be diagonal in the chosen channels when arms, link noises, or foreground anisotropy break the symmetries used to define orthogonal combinations.

Proof-mass acceleration and optical-path displacement have different link-frequency scaling. Schematically,

$$S_y^\mathrm{acc}(f) \propto\frac{S_a(f)}{(2\pi f c)^2}, \qquad S_y^\mathrm{opt}(f) \propto\frac{(2\pi f)^2S_x(f)}{c^2},$$

before multiplication by the channel’s ordered TDI transfer functions. Exact coefficients depend on which proof mass and optical bench enter each directed link. A strain-equivalent sensitivity curve folds in a response average and must not be substituted for the native TDI covariance without undoing that convention. Robson, Cornish, and Liu construct a widely used configuration-dependent sensitivity representation in Robson et al. 2019, §§2–3.

Confusion is a source-model remainder

The Galactic foreground is not defined by a fixed PSD alone. A global analysis proposes many compact binaries, fits their waveforms jointly with instrument noise and other source classes, and leaves a population-dependent unresolved remainder. Increasing observation time narrows frequency bins and accumulates modulation, so some sources move from “confusion” into the resolved model. Changing the detection or model-selection rule changes the same boundary.

At finite duration the foreground is anisotropic and seasonally modulated by constellation motion. It can also be non-Gaussian where only a few binaries occupy a resolution element. Treating it as stationary Gaussian colored noise can be a useful local approximation, but it discards population and modulation information that helps distinguish it from instrument noise.

The strongest supported strategy is therefore a joint or iterated global fit, not permanent subtraction of a universal foreground curve. Littenberg and Cornish demonstrate a prototype fit of Galactic binaries, massive-black-hole mergers, and an instrument-noise model on LISA Data Challenge 2b simulations Littenberg and Cornish 2023, §§II–V. It is a pipeline demonstration, not evidence of achieved flight separation.

Likelihood and identifiability

For an untapered segment of length $T$, define normalized positive-frequency residuals $\boldsymbol z_k=\sqrt{2/T}\,\widetilde{\boldsymbol r}(f_k)$. Their complex-bin covariance is the one-sided PSD matrix $\boldsymbol C_k$. Away from the real-valued DC and Nyquist bins, the approximately independent-bin Whittle log likelihood is

$$\ln\mathcal L =-\sum_k\left[ \ln\det\boldsymbol C_k +\boldsymbol z_k^\dagger \boldsymbol C_k^{-1} \boldsymbol z_k \right]+\text{constant}.$$

Instrument and foreground parameters are identifiable only through differences in spectral shape, cross-channel covariance, annual modulation, non-Gaussian structure, or informative external constraints. If two components have proportional covariance templates over the selected band, the likelihood measures only their sum. A tight posterior for each amplitude can then come only from priors or an omitted assumption.

Selection must be included in the model: observing duration and duty cycle, gaps, TDI generation, cadence, frequency band, resolved-source rule, source waveform family, population prior, and instrument-noise basis. Residuals after subtracting a point estimate understate source uncertainty; posterior source draws or joint marginalization propagate it.

A controlled two-component fit

To test the spectral separation independently of a LISA pipeline, define dimensionless frequency $f\in[0.2,5]$ and PSD templates

$$u(f)=1+(f/1.8)^2, \qquad v(f)=\exp\!\left[ -\frac12\left(\frac{\ln(f/0.7)}{0.55}\right)^2 \right].$$

Inject

$$S(f)=1.3\,u(f)+0.7\,v(f)$$

and draw each periodogram ordinate from an exponential distribution with mean $S(f)$, as predicted for one complex Gaussian Fourier bin. Maximizing

$$-\ln\mathcal L =\sum_k\left[\ln S_k+\frac{I_k}{S_k}\right]$$

gives the following executed results. The quoted standard errors use the local Fisher matrix $F_{ij}=\sum_k t_i(f_k)t_j(f_k)/S_k^2$.

Frequency bins	Instrument amplitude	Foreground amplitude	Standard errors
2000	1.31643	0.67111	0.04622, 0.11670
10000	1.29667	0.71760	0.02043, 0.05224
50000	1.30976	0.68497	0.00921, 0.02333

The estimates fluctuate around $(1.3,0.7)$ and the local uncertainties contract approximately as the inverse square root of bin count. These controlled realizations demonstrate spectral separation for the declared shapes; a repeated-realization ensemble would be needed to measure bias and interval coverage. They do not validate a LISA noise budget, Galactic population, or global-fit pipeline.

Executable rerun

import numpy as np
from scipy.optimize import minimize

def run(size):
    rng = np.random.default_rng(13030)
    f = np.geomspace(0.2, 5.0, size)
    inst = 1.0 + (f / 1.8) ** 2
    gal = np.exp(-0.5 * (np.log(f / 0.7) / 0.55) ** 2)
    truth = 1.3 * inst + 0.7 * gal
    periodogram = truth * rng.exponential(size=size)

    templates = np.vstack([inst, gal])

    def negative_log_likelihood(log_amplitude):
        amplitude = np.exp(log_amplitude)
        spectrum = amplitude @ templates
        value = np.mean(np.log(spectrum) + periodogram / spectrum)
        gradient = amplitude * np.mean(
            templates * (1.0 / spectrum - periodogram / spectrum**2),
            axis=1)
        return value, gradient

    fit = minimize(negative_log_likelihood, np.log([1.0, 1.0]),
                   jac=True, method="BFGS", tol=1e-8)
    if not fit.success:
        raise RuntimeError(fit.message)
    amplitude = np.exp(fit.x)
    spectrum = amplitude @ templates
    fisher = np.einsum("ik,jk,k->ij", templates, templates,
                       1.0 / spectrum**2)
    error = np.sqrt(np.diag(np.linalg.inv(fisher)))
    print(size, *[f"{x:.5f}" for x in amplitude],
          *[f"{x:.5f}" for x in error])

for size in (2_000, 10_000, 50_000):
    run(size)

The run used macOS 26.6.1 on arm64, Python 3.9.6, NumPy==2.0.2, SciPy==1.13.1, double precision, and seed 13030. Save the code as two_component_whittle.py and run:

python3 -m venv .venv
. .venv/bin/activate
python3 -m pip install numpy==2.0.2 scipy==1.13.1
python3 two_component_whittle.py

Versioned simulation boundary

The configuration anchor is the ESA 2024 definition study ESA 2024, §§2–4. The named public simulation is LISA Data Challenge 2b “Spritz,” whose training products specify second-generation $X,Y,Z$ fractional-frequency channels at five-second cadence with simulated orbit, laser noise, secondary noise, gaps, artifacts, and an unresolved Galactic population LISA Data Challenge 2b, dataset description. No challenge file is analyzed on this page; the only computed dataset is the fully declared analytic injection above.

Every configuration, software, and forecast statement is frozen at 2026-08-18. A production rerun must add release and file hashes, orbit and clock products, link and TDI conventions, cadence and gaps, channel covariance, response code, source catalog, waveform versions, priors, resolved-source selection, sampler settings, effective samples, and posterior-predictive residuals. Systematic variants should alter arm and clock error, delay interpolation, residual laser noise, spectral basis, Galactic population, waveform accuracy, overlap model, duration, and source threshold. Refresh the scientific conclusions after a superseding ESA baseline or public challenge release.

Check your understanding

Let $F$ be the two-amplitude Fisher matrix in the benchmark. Define the template correlation $r=F_{12}/\sqrt{F_{11}F_{22}}$. What happens to component separation as $|r|\to1$?

Solution

The determinant is

$$\det F =F_{11}F_{22}-F_{12}^2 =F_{11}F_{22}(1-r^2).$$

As $|r|\to1$, $\det F\to0$ and the covariance $F^{-1}$ diverges along the difference between the two amplitudes. The data constrain only one linear combination, essentially their total power. More bins do not fix exact proportionality; separation requires a different band, channel covariance, modulation, non-Gaussian information, or an explicit prior.

References

• 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.
• Littenberg, Tyson B., and Neil J. Cornish. “Prototype Global Analysis of LISA Data with Multiple Source Types.” Physical Review D 107 (2023): 063004. DOI. Open PDF.
• Robson, Travis, Neil J. Cornish, and Chang Liu. “The Construction and Use of LISA Sensitivity Curves.” Classical and Quantum Gravity 36 (2019): 105011. DOI. Open PDF.