Looking To The Horizon: Probing Evolution in the Black Hole Spectrum With Gravitational Wave Catalogs

TL;DR

This work develops a non-parametric, iterative reweighted KDE framework with optimized multi-dimensional bandwidths to reconstruct the BBH merger rate as a function of masses and distance, explicitly accounting for selection effects. Applying this method to GWTC-3 data, the authors find persistent features in the BBH mass spectrum near $10\,M_\odot$ and $35\,M_\odot$, but no statistically robust evidence for evolution with redshift within the current detectability horizon and sample size. The approach carefully handles measurement uncertainties, selection biases, and censoring, providing a conservative assessment of mass–redshift correlations while highlighting the limitations imposed by sparse high-mass statistics. The work sets the stage for applying the framework to future, larger GW catalogs (e.g., from O4 and beyond) to robustly test formation-channel predictions through mass–redshift evolution.

Abstract

The population of black holes observed via gravitational waves currently covers the local universe up to a redshift $z\lesssim 1$, for the most massive merging binaries, or $z\lesssim 0.25$ for low-mass BH binaries (BBH). Evolution of the BBH mass spectrum over cosmic time will be a significant probe of formation channels and environments. We demonstrate a reconstruction of the BBH merger rate, allowing for general dependence on binary masses and luminosity distance or redshift and accounting for selection effects, via iterative kernel density estimation (KDE) with optimized multidimensional bandwidths. Performing such reconstructions under a range of detailed assumptions, we see no significant evidence for the evolution of BBH masses with redshift, over the range where detected events are available. At most, possible trends towards increasing merger rate with redshift for primary masses $m_1\gtrsim 50\,M_\odot$, or towards decreasing merger rate with redshift for primary masses $m_1 \lesssim 40 M\odot$ may be supported. We compare these findings with previous investigations and caution against over-interpreting the current, sparse, data. Significantly upgraded detectors and/or facilities, and longer observing times, are required to harness any correlations of the BBH mass distribution with redshift.

Figures (15)

• Figure 1: Verification of the multidimensional bandwidth optimization using mock data consisting of a mixture of two Gaussian components, represented by red $+$ symbols: contours and blue shading show the KDE in each case. Top: Adaptive KDE with an isotropic kernel (in standardized coordinates): the single bandwidth parameter $h$ and adaptive parameter $\alpha$ optimize the cross-validated likelihood. Bottom: Adaptive KDE with different optimized bandwidths $h_{1,2}$ for each coordinate.
• Figure 2: Detection probability $p_\mathrm{det}$ as a function of the primary mass $m_1$ and luminosity distance $d_L$ for PE samples from 69 BBH events in GWTC-3. The top panel assumes a power-law distribution for the mass ratio, $p(q) \propto q^{1.26}$, while the bottom panel uses posterior sample values of secondary mass $m_2$.
• Figure 3: Rate density as a function of primary mass $m_1$ and luminosity distance $d_L$, assuming a power-law distribution for $m_2$. Red $+$ symbols represent the medians of PE samples for GWTC-3 BBH events, while the orange contours correspond to $p_\mathrm{det}$ levels. One event, GW190805_211137, has a median $d_L \sim 7500$ Mpc outside the range of the plot.
• Figure 4: Top panel: Merger rate density (median estimate) as a function of the primary mass $m_1$, with different curves representing redshift values indicated by color, assuming a power law distribution of mass ratio. Estimates at low mass are increasingly affected by censorship towards high redshift: corresponding regions with $p_\mathrm{det}<0.1$ are shown as dotted curves. Bottom panel: Rate density for a subset of redshift values, with shaded 90% uncertainty intervals. We omit values with excessively high uncertainty for clarity (as also in Fig. \ref{['fig:offset-m1-rate-powerlaw-m2']}).
• Figure 5: Merger rate density as a function of primary mass with symmetric 90% confidence intervals, corresponding to the results shown in Fig. \ref{['fig:median-m1-rate-powerlaw-m2']}. Each curve shows a constant distance or redshift, shown by the color scale, with constant offsets applied for clarity. We only plot values where both (1) the 5th percentile is $>10^{-3}\times$ median, and (2) the 95th percentile is $\leq 200\times$ the median.