Cross-Correlation for Stochastic Backgrounds

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A stochastic gravitational-wave background can lie below the noise of every individual detector. Cross-correlation exposes it because an astrophysical field produces a calculable common response, whereas independent instrumental noises average toward zero. The method succeeds only to the extent that correlated environmental or instrumental noise is measured, modeled, or vetoed.

Helpful background. Power Spectral Density and Nonstationarity supplies the power-spectrum conventions and stationarity diagnostics used below.

Cross power from a common random field

Write the Fourier data from detectors $1$ and $2$ as

\widetilde s_I(f) = \widetilde h_I(f)+\widetilde n_I(f).

For an isotropic, stationary, Gaussian, unpolarized tensor background, define the one-sided detector-response-averaged strain spectrum by

\left\langle \widetilde h_1^*(f)\widetilde h_2(f') \right\rangle = \frac12\delta(f-f')\, \gamma_{12}(f)S_h(f).

If the detector noises are mutually uncorrelated,

\left\langle \widetilde n_1^*(f)\widetilde n_2(f') \right\rangle=0,

then the expectation of cross power contains the background even when $S_h\ll P_1,P_2$ . Each finite segment remains noisy; averaging many statistically independent segments reduces the standard error approximately as their number to the power $-1/2$ .

For two right-angle interferometers, the normalized tensor overlap reduction function is

\gamma_{12}(f) = \frac{5}{8\pi} \sum_{A=+,\times} \int_{S^2}d\boldsymbol\Omega\, F_1^A(\boldsymbol\Omega) F_2^A(\boldsymbol\Omega) e^{2\pi i f\boldsymbol\Omega\cdot\Delta\boldsymbol x/c}.

Here $\widetilde s(f)=\int s(t)e^{+2\pi ift}dt$ , $\boldsymbol\Omega$ points along wave propagation, and $\Delta\boldsymbol x=\boldsymbol x_2-\boldsymbol x_1$ . The antenna factors encode relative orientation; the positive phase follows from $h(t,\boldsymbol x)\propto e^{-2\pi if(t-\boldsymbol\Omega\cdot\boldsymbol x/c)}$ . The normalization gives $\gamma(0)=1$ for colocated, coaligned ideal detectors. For real long-wavelength antenna factors and an isotropic unpolarized sky, opposite directions make $\gamma$ real. Separated or misaligned detectors have frequency-dependent response, including zeros where sky contributions cancel Allen and Romano 1999, §III, “Overlap reduction function”.

The optimal filter

Consider the real cross-correlation statistic

Y = 2\operatorname{Re} \int_0^\infty df\, \widetilde s_1^*(f)\widetilde s_2(f)\, \widetilde Q(f).

For an untapered long segment of duration $T$ , the finite-time delta function has $\delta_T(0)=T$ . In the weak-signal approximation and for a real filter, the mean and variance are

\langle Y\rangle = T\int_0^\infty df\, \gamma(f)S_h(f)\widetilde Q(f),

\sigma_Y^2 =\frac12 T\int_0^\infty df\, P_1(f)P_2(f) \lvert\widetilde Q(f)\rvert^2.

Maximizing $\langle Y\rangle/\sigma_Y$ with Cauchy–Schwarz gives

\widetilde Q(f) \propto \frac{\gamma(f)S_h(f)} {P_1(f)P_2(f)}.

Thus frequencies with strong common response and low noise receive greatest weight; a noisy band or a zero of $\gamma$ contributes little. For a chosen spectral shape $S_h(f)=A\,H(f)$ , an unbiased estimator is obtained by taking

\widetilde Q(f)=\mathcal N\frac{\gamma(f)H(f)}{P_1(f)P_2(f)}, \qquad \mathcal N^{-1}=T\int_0^\infty \frac{\gamma^2(f)H^2(f)}{P_1(f)P_2(f)}df.

Then $\langle Y\rangle=A$ and $\sigma_A^{-2}=2T\int_0^\infty\gamma^2H^2/(P_1P_2)\,df$ . The units and the factor two follow from using independent positive-frequency bins; they must be recalculated for the windowed estimator actually implemented.

The dimensionless energy-density spectrum is

\Omega_{\rm GW}(f) = \frac{1}{\rho_c} \frac{d\rho_{\rm GW}}{d\ln f}, \qquad S_h(f) = \frac{3H_0^2}{10\pi^2} \frac{\Omega_{\rm GW}(f)}{f^3},

This $S_h$ includes the sky/polarization average of a right-angle detector. The total metric-strain convention often used with characteristic strain instead has $S_h^{\rm metric}=5S_h$ and $h_c^2=fS_h^{\rm metric}$ , giving $\Omega_{\rm GW}=2\pi^2f^2h_c^2/(3H_0^2)$ . Keeping this factor five is essential when comparing interferometer and PTA plots. A power-law search specifies $\Omega_{\rm GW}(f)=\Omega_\alpha(f/f_{\rm ref})^\alpha$ ; its result is conditional on $\alpha$ , $f_{\rm ref}$ , the analyzed band, and $H_0$ convention.

From ideal derivation to real data

Data are divided into segments, windowed, Fourier transformed, and combined with inverse-variance weights. Window overlap and spectral leakage correlate neighboring estimates, so the effective number of averages is not simply the raw segment count. Nonstationary segments and notched lines must be selected by rules that do not bias the cross power.

The central failure mode is correlated noise:

\left\langle \widetilde n_1^*(f)\widetilde n_2(f) \right\rangle\neq0.

Global magnetic fields, timing artifacts, shared electronics, and narrow spectral lines can mimic a common signal. Witness magnetometers, coherence studies, alternative baselines, and frequency-resolved residuals are therefore part of the inference. Time shifts provide a complementary control only when they decorrelate the analyzed signal: shifting by more than the intersite light-travel time alone does not guarantee this for a narrow-band stationary process. A shift multiplies cross power by a frequency-dependent phase, and suppression depends on the weighted bandwidth. It can also destroy correlated instrumental noise, so a null shifted result does not certify clean zero-lag data.

Reproduction and null checks

A reproducible calculation records segment duration and overlap, window normalization, coarse-graining, analyzed and notched bands, power-spectrum estimator, detector calibration, overlap function implementation, cosmological convention, and likelihood.

The minimal benchmark generates two independent Gaussian noises and a correlated Gaussian field with known $S_h$ . Refine segment length and frequency binning until recovered amplitude and variance stabilize; confirm that window normalization preserves injected cross power. Then perform:

zero-signal trials to test mean, variance, and upper-limit coverage;
time shifts verified to suppress the target cross power, without assuming that they preserve zero-lag environmental correlations;
sign or phase injections that traverse zeros of $\gamma(f)$ ;
independent-baseline and environmental-witness comparisons;
loud software and, where available, hardware injection recovery.

Observational status through 2026-08-18

The official O4a isotropic analysis reports no statistically significant stochastic-background detection and sets upper limits for stated spectral models LVK 2025. Those bounds describe the recovery and likelihood of the released detector network, data-quality cuts, frequency band, overlap functions, calibration, and assumed spectral shapes. They are not measurements of a positive background amplitude and should not be plotted as detections.

The release must be refreshed when additional observing data, calibration, correlated-noise subtraction, or an official stochastic analysis supersedes it. The cited paper and data products, rather than mutable summary values, are the numerical authority.

Exercises

Show that $\gamma(0)=1$ for ideal colocated, coaligned right-angle detectors using

\sum_A\int d\boldsymbol\Omega\, \left[F^A(\boldsymbol\Omega)\right]^2 = \frac{8\pi}{5}.

Solution

Colocation makes the phase factor unity and coalignment gives $F_1^A=F_2^A=F^A$ . Therefore

\gamma(0) = \frac{5}{8\pi} \sum_A\int d\boldsymbol\Omega\, \left[F^A\right]^2 = \frac{5}{8\pi}\frac{8\pi}{5} = 1.

Equal-variance independent segment estimates $Y_a$ have variance $\sigma^2$ . Find the variance of their average over $N$ segments, and state why overlapping windows can spoil the result.