Inferring, not just detecting: metrics for high-redshift sources observed with third-generation gravitational-wave detectors

TL;DR

This work tackles the challenge of not only detecting but also confidently locating high-redshift gravitational-wave sources with third-generation detectors. It introduces two metrics—the $z$-$z$ plot and the inference horizon—to quantify redshift inference using Fisher-matrix forecasts for networks like ET+CE, mapping luminosity distance to redshift under a flat $\Lambda$CDM cosmology. Analyzing both individual sources and populations, the study shows that $z_{90\%}$ can peak at an intermediate redshift and then decline, and that only a small fraction of high-$z$ events yield strong, credible redshift constraints; the inference horizon is typically well below the detection horizon. The findings suggest that while single-event claims of primordial origin are challenging, population analyses and stochastic background studies may offer more robust avenues for constraining high-redshift black-hole formation channels, and provide a framework to optimize 3G detector design for cosmological GW science.

Abstract

The detection of black-hole binaries at high redshifts is a cornerstone of the science case of third-generation gravitational-wave interferometers. The star-formation rate peaks at z~2 and decreases by orders of magnitude by z~10. Any confident detection of gravitational waves from such high redshifts would imply either the presence of stars formed from pristine material originating from cosmological nucleosynthesis (the so-called population III stars), or black holes that are the direct relics of quantum fluctuations in the early Universe (the so-called primordial black holes). Crucially, detecting sources at cosmological distances does not imply inferring that sources are located there, with the latter posing more stringent requirements. To this end, we present two figures of merit, which we refer to as "z-z plot" and "inference horizon", that quantify the largest redshift one can possibly claim a source to be beyond. We argue that such inference requirements, in addition to detection requirements, should be investigated when quantifying the scientific payoff of future gravitational-wave facilities.

Figures (4)

• Figure 1: Lower bound on the estimated redshift $z_{90\%}$ as a function of the true source redshift $z_{\rm true}$. We consider fiducial sources with different source-frame mass as reported on the color scale.
• Figure 2: $z$-$z$ plot for a GW150914-like source observed with a network of one ET and two CE instruments. A source at redshift $z_{\rm true}$ (bottom $x$-axis) can be placed at redshift larger than $z_{\rm c\%}$ ($y$-axis) at $c\%$ credible interval (color bar). Three representative values $c = 50, 90,$ and $99$ are indicated with dashed contours. The source SNR is reported on the top $x$-axis. The hatched region to the right corresponds to sources below the detectability threshold of $\rm SNR=8$.
• Figure 3: Population-averaged $z$-$z$ plots (bottom panels) for two populations of binary BH population, one where we extrapolate current LIGO/Virgo results (left panel) and one where we assume a BH-mass spectrum that is flat in log-space between $5~{\rm M_\odot}$ and $500~{\rm M_ \odot}$ (right panel). We consider a network of ET and two CE instruments; the resulting fractions of detected sources are reported in the top panels.
• Figure 4: Detection and inference horizons for a 3G detector network made of a triangular ET and two L-shaped CE instruments. The red curve indicates the larger redshift at which a source of a given mass $M$ is detectable, $z_{\rm hor}$, the solid blue curve indicates the true value of the redshift $z_{\rm peak}$ that provides the most stringent constraint, and the dashed blue curves indicates the corresponding lower bound $z_{\rm 90\%}$ on the redshift itself.