Where did Heavy Binaries Go? Gravitational-wave Populations Using Delaunay Triangulation with Optimized Complexity

TL;DR

The paper tackles how BBH mergers evolve with redshift by introducing a non-parametric joint-mass–redshift framework based on Delaunay triangulation and trans-dimensional Bayesian inference. By data-drivenly selecting the number and placement of triangulation nodes, it reconstructs the volumetric merger rate across $M_{tot}$ and $z$ using GWTC-4.0, requiring far fewer parameters than grid-based approaches. A key finding is a high-mass feature near $M_{tot} \sim 70\,M_\odot$ that appears at high redshift ($z \sim 1$) but fades by $z \sim 0.2$, with strong evidence that the rate at $z=1$ exceeds that at $z=0.2$ over relevant masses. The authors interpret this as signaling multiple BBH formation channels with different delay times, likely involving dense-environment channels for the heavy component and isolated-binary channels for the low-redshift background, and demonstrate a scalable, efficient method poised to handle higher-dimensional population analyses in GW astronomy.

Abstract

We investigate the joint mass-redshift evolution of the binary black-hole merger rate in the latest Gravitational-Wave Transient Catalog, GWTC-4.0. We present and apply a novel non-parametric framework for modeling multi-dimensional, correlated distributions based on Delaunay triangulation. Crucially, the complexity of the model -- namely, the number, positions, and weights of triangulation nodes -- is inferred directly from the data, resulting in a highly efficient approach that requires about one to two orders of magnitude fewer parameters and significantly less calibration than current state-of-the-art methods. We find no evidence for a peak at $M_{\mathrm{tot}} \sim 70\,\mathrm{M}_{\odot}$ at low redshifts ($z \sim 0.2$), where it would correspond to the $m_1 \sim 35\,\mathrm{M}_{\odot}$ feature reported in redshift-independent mass spectrum analyses, and we infer an increased merger rate at high redshifts ($z \sim 1$) around those masses, compatible with such a peak. When related to the time-delay distribution from progenitor formation to binary black-hole merger, our results suggest that sources contributing to the $m_1 \sim 35\,\mathrm{M}_{\odot}$ feature follow a steeper (shallower) time-delay distribution at high (low) redshifts. This hints at contributions from different formation channels -- for example dense environments and isolated binary evolution, respectively -- although firm identification of specific formation pathways will require further observations and analyses.

Figures (6)

• Figure 1: Delaunay triangulation to model the differential merger rate $\log_{10}\mathrm{d}_{\theta}N$ across two variables (on the horizontal and vertical axes, respectively). The central red dot represents a location $\theta$ where the rate needs to be computed. The highlighted area $S(\theta)$ represents the triangle (simplex in higher dimensions) containing $\theta$ whose vertices $v_i$ and weights $w_i$ are inferred from the data. The position of the four corners is fixed in advance, and their weights $W_{i}$ are inferred from the data. The rate $\log_{10}\mathrm{d}_{\theta}N$ is computed by interpolating the weights at the vertices of $S(\theta)$ using the barycentric coordinates $b_i(\theta)$ associated to $\theta$.
• Figure 2: Left: Posterior distribution of the volumetric differential rate reconstructed using Delaunay triangulation for two representative redshift values, $z=0.2$ (purple) and $z=1$ (orange). Solid curves indicate the medians while shaded regions encompass 90% symmetric credible intervals. The upper panel shows the Kullback-Leibler divergence between the posterior and prior volumetric differential rate distributions. Right: Posterior distribution of the ratio of the volumetric differential rate at the same two representative redshift values. The black solid curve denotes the posterior median. The shaded region contains the symmetric 90% credible interval. The horizontal red line corresponds to $\mathrm{d}_{\theta} \mathcal{R}(z=0.2)=\mathrm{d}_{\theta} \mathcal{R}(z=1)$. The dotted line corresponds to the $1.5\%$ credibility interval. The observed difference in rates is thus inconsistent with no evolution of the rate with $z$ at $\sim 98\%$ credibility, but remains consistent with a mass-uniform evolution (the envelope is consistent with a straight line).
• Figure 3: The color scale shows the median (50% quantile) posterior volumetric rate inferred using Delaunay triangulation on GWTC-4.0. The white contours indicate the relative uncertainty on the rate itself, defined as the ratio between the 95% quantile and the 5% quantile.
• Figure 4: Evolution of the volumetric differential merger rate as a function of cosmic time $t$ in log-log scale evaluated at $M_\mathrm{tot}=63\,\mathrm{M}_{\odot}$, i.e. a nominal location where evolution with redshift is tightly constrained. The solid curve indicates the median while the shaded region encloses the 90% symmetric credible interval. The dashed (dotted) line corresponds to a merger rate evolving as a power law with index $-1$$(-4)$ and an arbitrary normalization for reference.
• Figure 5: Left: Posterior predictive distribution of the mass-ratio distribution. The red (grey) solid line represent the median, while the shade (dotted) envelope contains the 90% credible interval assuming a broken (single) power-law distribution. Right: Posterior distribution of the power-law indices. The gray area represents the posterior probability on $(\alpha_1, \alpha_2)$ for the broken power-law model, while the dotted histogram represents the posterior probability on $\alpha_1$ for the single power-law model.