Estimation of the Hubble parameter from unedited compact object merger catalogues

TL;DR

The paper introduces a Bayesian hierarchical framework to infer cosmological and population parameters, notably the Hubble constant $H_0$, from compact-binary coalescence catalogues using only detection-level information. By modeling a mixture of astrophysical signals and background noise and leveraging single-candidate likelihoods $p(x|H_s,\lambda,Δ)$ and $p(x|H_n,Δ)$, the method naturally incorporates selection effects without per-event parameter estimation. A key contribution is the practical, detection-statistics-based estimator for the signal fraction $\bar{η}$ and its bias-corrected form, enabling informative population inferences even with marginal candidates. Proof-of-concept mock data analyses validate the approach, showing unbiased or near-unbiased recovery of $H_0$ and $\bar{η}$ under controlled conditions, while highlighting current limitations in signal-model convergence and computation that warrant further development for real data.

Abstract

In recent years, constraints on the Hubble parameter using multiple dark sirens have been made,relying on a galaxy catalogue, correlations between the mass and redshift distributions, or both. Those studies have typically used only significant gravitational wave candidates. In this work, we present a framework for cosmological inference that bypasses per-candidate parameter estimation, uses only detection-level information. This allows the population inference from a candidate list produced directly by a search pipeline, without additional selection cuts. Our method is particularly suited to extracting information from marginal candidates, which are essential for probing the distant universe.

Figures (10)

• Figure 1: Comparison between the unit-variance Gaussian distribution, ${\mathcal{N}}({\rho_\mathrm{obs}}|\sqrt{\nu+2}, 1)$, and the expected distribution of the observed SNR ${\rho_\mathrm{obs}}$, $p({\rho_\mathrm{obs}}|{\mathcal{H}_\text{s}}, \nu)$, derived from the non-central chi-squared distribution, conditioned by the non-centrality $\nu = ({\rho_\mathrm{opt}}{\mathcal{G}})^2 - 2$. Top panel: the two distributions are shown for different value of $\nu$; the colored solid lines represent the exact $p({\rho_\mathrm{obs}}|{\mathcal{H}_\text{s}}, \nu)$, while the gray dashed lines represent the unit-variance Gaussian distributions. The two distributions converge in the high-SNR limit. Bottom panel: the mean-squared error between $p({\rho_\mathrm{obs}}|{\mathcal{H}_\text{s}}, \nu)$ and the unit-variance Gaussian distribution as a function of $\sqrt{\nu + 2} = {\rho_\mathrm{opt}}{\mathcal{G}}$, showing an $\sim\nu^{-1}$ scaling.
• Figure 2: Expected SNR distribution assuming a given instrument and the population model described in Sec. \ref{['subsec:mda_pop_model']}. The gray lines, orange lines and blue lines correspond to Cosmic Explorer LIGO-P1600143, Advanced LIGO design sensitivity LIGO-T1800044, and S5 sensitivity, spanning GPS time 866736411--875232014, respectively.
• Figure 3: Detection statistic distribution models employed in the MDA. These PDF were generated based on data products provided by a previous study s5s6_2021_reanalysis. The ${H_0}$-dependence of this MDA signal model remains limited, since the LIGO detectors during the S5 run were not sensitive to very distant regions of the universe.
• Figure 4: Comparison between the distribution models, and the distribution of candidates' SNR and log-LR in an example MDA dataset.
• Figure 5: Posteriors marginalized over the signal fraction, ${\bar{\eta}}$, from MDA with different injected values of the Hubble constant ${H_0}$ and the signal fraction ${\bar{\eta}}$. Each row corresponds to a group of mock universes with different injected values of ${H_0}$, while each column corresponds to those with different injected values of ${\bar{\eta}}$. In each panel, the black solid line shows the posterior of ${H_0}$ marginalized over ${\bar{\eta}}$, whereas the colored lines shows the posteriors of ${H_0}$ calculated under different assumed values of ${\bar{\eta}}$. The curve corresponding to the true (injected) value of ${\bar{\eta}}$ for each mock universe is drawn with a thicker line, and the black dotted line indicates the true (injected) value of ${H_0}$. When ${\bar{\eta}}=0$ is assumed in the evaluation of the posteriors (in the case of blue lines in each panel), no constraint on ${H_0}$ is obtained. The marginalized posteriors show little sensitivity to the injected ${H_0}$ values, compared to their dependence on the injected ${\bar{\eta}}$ values. In contrast, the posteriors obtained using the true ${\bar{\eta}}$ tend to track the injected ${H_0}$ values reasonably well.