Are all Binary Black Holes Detected by LIGO-Virgo-KAGRA Following the Universal Time-Delay Distributions? Probably Not

TL;DR

The paper addresses whether binary black hole mergers detected by LVK follow a universal delay-time distribution (DTD) or exhibit mass-dependent evolution. It introduces a grid-based, non-parametric framework to reconstruct the mass-dependent DTD $p_t(t_d|m)$ by exploring causality-constrained trajectories on a $(t_d, \log p_t)$ grid and convolving with the cosmic star-formation rate $R_{ m SFR}(z_f)$, while accounting for detector selection. Applying this method to GWTC-3 and GWTC-4, the authors find that low-mass BBHs in the range $20$--$40\,M_\odot$ have a broad, nearly scale-invariant DTD across $\sim1$--$10$ Gyr, whereas high-mass BBHs in $40$--$100\,M_\odot$ preferentially merge on shorter delays around $2$--$6$ Gyr, with a higher confidence that the DTD deviates from a simple power law. They also infer distinct local merger rates $R_0$ for the two mass bins and demonstrate that mass-dependent channels play a crucial role in BBH formation, highlighting the value of non-parametric methods for GW population studies.

Abstract

The delay time distribution (DTD) of binary black hole (BBH) mergers encodes the evolutionary link between the formation history and gravitational-wave (GW) emission. We present a non-parametric reconstruction of the mass-dependent DTD using the BBHs from the GWTC-4 that avoids restrictive assumptions of only power-law forms. Our analysis reveals for the first time the signature for mass-dependent evolutionary pathways: lower-mass systems ($20$-$40\,M_\odot$) are consistent with a scale-invariant DTD, whereas higher-mass BBHs ($40$-$100\,M_\odot$) provide the first direct tentative evidence of DTD that deviate from simple power laws, with a pronounced preference for rapid mergers around $2-6$ Gyrs. These findings reveal the advantage of the non-parametric technique in reconstructing the mass-dependent DTD and discovering for the first-time the presence of a potential time-scale associated with high-mass GW events.

Figures (5)

• Figure 1: Flowchart summarizing the non-parametric grid-based framework for inferring the BBHs DTD and merger rate. The framework begins with the construction of a delay time-probability grid, followed by the enumeration of all possible trajectories and the selection of causality-constrained ones. These trajectories are convolved with the cosmic star formation rate and cosmological model to obtain the merger rate evolution. Incorporating the BBH population model and detector selection function yields the detectable BBH population, which is then compared with the observed GW catalog. Bayesian evidence evaluation across trajectories enables inference of both the DTD shape and the local merger rate.
• Figure 2: Normalized redshift distributions of BBHs from the combined GWTC-3 and GWTC-4 catalogs, binned by source-frame primary mass: $20$–$40M_\odot$ (blue) and $40$–$100M_\odot$ (red). Solid curves represent the observed distributions, while dashed curves show the reconstructed trajectories from the best-fit delay–time distribution model obtained using our grid-based technique (see Sec. \ref{['sec:result']} for more details on this.). The lower-mass systems peak at $z \sim 0.2$, reflecting more recent mergers with longer delay times, whereas higher-mass systems peak at $z \sim 0.5$ and exhibit broader distributions extending to earlier cosmic epochs, consistent with shorter delay times. The close agreement between the reconstructed trajectories and the observations supports the inferred form of the underlying time-delay function governing BBH formation channels.
• Figure 3: Grid-based exploration of DTDs in $(t_d, \log_{10} p_t)$ space. The $5 \times 5$ grid spans delay times from 0.5 to 13 Gyr and probability densities over six orders of magnitude. Solid lines show three example valid trajectories that satisfy the causality constraint $t_d^{(i)} \leq t_d^{(i+1)}$, while dashed lines illustrate invalid trajectories that violate causality by decreasing in delay time (marked with ×). Arrows indicate the direction of temporal evolution. Each trajectory through the grid defines a unique DTD via cubic spline interpolation between grid points, enabling systematic exploration of all physically allowed evolutionary pathways without assuming specific parametric forms.
• Figure 4: Envelope of the top 50 highest-evidence delay–time distribution trajectories reconstructed from the GWTC-3 and GWTC-4 catalogs. The shaded regions show the allowed range of $\log P(t_d)$ as a function of delay time for binaries in two mass intervals: $20$–$40,M_\odot$ (pink) and $40$–$100,M_\odot$ (gold), while the dark solid curves denote the single best-evidence trajectory in each mass bin. Lower-mass systems exhibit broadly flat distributions across the delay-time range, whereas higher-mass binaries show a pronounced concentration toward short delays ($\lesssim 5$ Gyr), providing direct non–power-law evidence for merger timescales.
• Figure 5: Posterior distributions of the local binary-black-hole merger rate $R_0$ for two mass bins, $20$-$40\,M_\odot$ and $40$-$100\,M_\odot$. Solid curves show the marginalized one-dimensional posteriors; vertical dashed lines mark the posterior medians and dotted lines indicate the 68% credible bounds.