Variance of gravitational-wave populations

Abstract

We quantify the impact of finite catalog size, or "catalog variance," on current gravitational-wave population analyses. The distribution of merging binary black holes is commonly reconstructed via hierarchical Bayesian inference, with uncertainties reported as credible intervals. Such intervals are conditioned on the specific realization of the observed events and are therefore themselves subject to variability arising from the finite size of the catalog. We estimate this "uncertainty on the uncertainty" using statistical bootstrapping applied to data segments containing both detected events and sensitivity injections. Applying this framework to GWTC-4, we find that the inferred population distributions exhibit substantially broader uncertainties than those obtained in a standard single-catalog analysis. In particular, the $\sim 35\,M_\odot$ peak in the primary-mass distribution is largely absorbed by statistical fluctuations once catalog variance is taken into account. Unlike other studies that rely on simulating catalogs by assuming an underlying population, this work provides the first data-driven assessment of the uncertainty intrinsic to the observed gravitational-wave catalog. Accounting for catalog variance is important for drawing robust astrophysical conclusions from gravitational-wave data, avoiding inferences driven by a particular finite realization rather than genuine population features.

Figures (4)

• Figure 1: Schematic representation of our resampling strategy. Detected events and sensitivity injections (grey arrows) are placed on a contiguous timeline, excising the breaks between observing runs (blue segments at the top). This timeline is then divided into $N$ equal-length segments (red, at the bottom), which are resampled with repetition to estimate the variance of the resulting population analysis.
• Figure 2: Reconstructed population distributions for BH binary parameters, showing the effect of catalog variance. The black dotted and solid curves, along with the shaded areas, represent median and 90% credibility regions obtained from the GWTC-4 data alone, reproducing the results of Ref. 2025arXiv250818083T. Red and blue shaded bands indicate the variability of the upper and lower bounds of these credibility regions across 700 bootstrap realizations of the catalog. In particular, red (blue) solid curves indicate the upper (lower) 90% confidence interval on the upper (lower) 90% credibility interval from GWTC-4, providing an estimate of the uncertainty associated with performing the population analysis on a single finite dataset. From left to right and from top to bottom, we show the differential merger rate as a function of primary mass $m_1$ at $z=0.2$, the differential merger rate as a function of mass ratio $q$ at $z=0.2$, the (normalized) distribution of spin magnitudes $\chi$, the (normalized) distribution of the cosine of the spin tilts $\theta$, and the merger rate as a function of redshift $z$ (these choices were made to facilitate direct comparison with Ref. 2025arXiv250818083T).
• Figure 3: As in Fig. \ref{['fig:catalog_variance']}, but using leave-one-out resampling instead of bootstraps.
• Figure 4: As in Fig. \ref{['fig:catalog_variance']}, but showing the effect of varying the number of bootstrap segments. We consider $N=15$ (dashed), $N=150$ (solid; the default value used in the main body of the paper), and $N=1500$ (dotted).