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Non-Parametric Reconstruction of the Hubble Parameter from the Fourth Gravitational Wave Transient Catalog and DESI Baryonic Acoustic Oscillations

Grégoire Pierra, Alberto Colombo, Simone Mastrogiovanni

TL;DR

The paper presents the first non-parametric reconstruction of the Hubble parameter $H(z)$ using gravitational-wave spectral sirens from GWTC-4.0, implemented with a spline-based representation of $\ln H(z)$ and a hierarchical Bayesian framework. It demonstrates that GW data alone provide meaningful constraints near $z\approx0.44$ (e.g., $H(z=0.44)=92.3_{-36.6}^{+29.9}$ km s$^{-1}$ Mpc$^{-1}$), but predictions at higher redshifts are strongly influenced by cosmological model priors; introducing BAO anchors from DESI improves constraints and identifies where GW data are most informative. The work introduces a constraining-power metric $\mathcal{C}_k$ to quantify data-driven redshift regions and shows that BAO anchoring shifts the inferred $H(z)$ shape toward the external measurements, highlighting the importance of combining GW data with independent probes to robustly recover the expansion history.

Abstract

The release of the fourth Gravitational Wave Transient Catalog (GWTC-4.0) by the LIGO-Virgo-KAGRA collaboration includes more than 200 compact binary coalescence (CBC) candidates that can be used to probe the cosmic expansion. The population of merging binary black holes has been used so far to provide a constraint on the Hubble constant and dark matter fraction under the hypothesis of a flat-$Λ$-Cold-Dark-Matter Universe. In this work, we provide the first non-parametric constrain on the Hubble parameter from 137 dark sirens reported in GWTC-4.0. We employ the relation between detector and source frame masses for detected GW signals, to obtain a statistical redshift evaluation for the population of binary black holes (BBHs). We model the Hubble parameter as a non-parametric autoregressive process in terms of the scale factor, using splines. In addition, we introduce two novel features: the use of \textit{anchor} points for $H(z)$ derived from an external probe - here, Baryon Acoustic Oscillations (BAOs) - and a constraining power coefficient that quantifies where the inference is most data-driven by GW detections. We highlight three key findings: (i) using GWs alone, the Hubble parameter determination is the most GW-data-driven around redshift $z = 0.44$, yielding to $H(0.44) = 92.3_{-36.6}^{+29.9}\rm\, km s^{-1} Mpc^{-1}$. Its value at $z = 0$, the Hubble constant, is therefore less constrained by the GW data. (ii) The Hubble parameter inferred from analyses assuming a flat-$Λ$CDM cosmological model is strongly affected by the cosmological model assumption. (iii) Introducing an anchor point for $H(z)$ enhances the inferred constraints and provides a clear visualization of the redshift range where GWs contribute most to the constraining power.

Non-Parametric Reconstruction of the Hubble Parameter from the Fourth Gravitational Wave Transient Catalog and DESI Baryonic Acoustic Oscillations

TL;DR

The paper presents the first non-parametric reconstruction of the Hubble parameter using gravitational-wave spectral sirens from GWTC-4.0, implemented with a spline-based representation of and a hierarchical Bayesian framework. It demonstrates that GW data alone provide meaningful constraints near (e.g., km s Mpc), but predictions at higher redshifts are strongly influenced by cosmological model priors; introducing BAO anchors from DESI improves constraints and identifies where GW data are most informative. The work introduces a constraining-power metric to quantify data-driven redshift regions and shows that BAO anchoring shifts the inferred shape toward the external measurements, highlighting the importance of combining GW data with independent probes to robustly recover the expansion history.

Abstract

The release of the fourth Gravitational Wave Transient Catalog (GWTC-4.0) by the LIGO-Virgo-KAGRA collaboration includes more than 200 compact binary coalescence (CBC) candidates that can be used to probe the cosmic expansion. The population of merging binary black holes has been used so far to provide a constraint on the Hubble constant and dark matter fraction under the hypothesis of a flat--Cold-Dark-Matter Universe. In this work, we provide the first non-parametric constrain on the Hubble parameter from 137 dark sirens reported in GWTC-4.0. We employ the relation between detector and source frame masses for detected GW signals, to obtain a statistical redshift evaluation for the population of binary black holes (BBHs). We model the Hubble parameter as a non-parametric autoregressive process in terms of the scale factor, using splines. In addition, we introduce two novel features: the use of \textit{anchor} points for derived from an external probe - here, Baryon Acoustic Oscillations (BAOs) - and a constraining power coefficient that quantifies where the inference is most data-driven by GW detections. We highlight three key findings: (i) using GWs alone, the Hubble parameter determination is the most GW-data-driven around redshift , yielding to . Its value at , the Hubble constant, is therefore less constrained by the GW data. (ii) The Hubble parameter inferred from analyses assuming a flat-CDM cosmological model is strongly affected by the cosmological model assumption. (iii) Introducing an anchor point for enhances the inferred constraints and provides a clear visualization of the redshift range where GWs contribute most to the constraining power.

Paper Structure

This paper contains 12 sections, 20 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. A non-parametric cosmological model
  3. The spectral siren inference framework
  4. Hierarchical likelihood and constraint estimators
  5. Population and cosmological models
  6. Data: GW catalog, injections, and BAO measurements
  7. Results
  8. The Hubble parameter with GWs
  9. Anchoring H(z) with BAOs
  10. Quantifying the GW constraining power
  11. Conclusions
  12. Acknowledgments

Figures (5)

  • Figure 1: Graphical representation of the evolution of the mass spectrum in the detector frame as observed at increasing luminosity distances, assuming different values of the Hubble constant ($H_0$) and the matter-density parameter ($\Omega_{\rm m,0}$) within a flat-$\Lambda$CDM cosmological model. The figure illustrates how changes in cosmological parameters affect the observed shape of mass features at larger distances.
  • Figure 2: Prior and posterior predictive distributions of the Hubble parameter $H(z)$ as a function of cosmological redshift in $\rm log_{10}$ space, from a spectral siren inference using 137 GW detections with FAR$< 0.25\,yr^{-1}$, from the GWTC-4.0 catalog. Left: Non-parametric inference with the spline $H(z)$ model. Binned constraints are plot as continuous to improve readability. Right: Parametric inference of the Hubble parameter $H(z)$ assuming a flat-$\Lambda$CDM cosmological model. The dark colored contours represent the 68.3% highest density intervals (HDI) around the maximum a posteriori (MAP) estimation.
  • Figure 3: Reconstruction of the inferred Hubble parameter $H(z)$ using our non-parametric model with splines, for GW detections only (blue), BAOs only (purple), and combining GW and BAO (yellow). The BAO data points estimated by DESI are also shown in dark purple on this figure. For each posterior predictive check shown here, the MAP is represented by the solid colored line and the 68.3% HDI are shown with the contours.
  • Figure 4: Evolution of the constraining-power coefficient $\mathcal{C}_{k}$ at each redshift nodes $z_{k}$, normalized with respect to their maximum. The lower the size marker, the more data-driven is the constraint at a given node. The left panel shows the constraining power on the Hubble parameter $H(z)$, while the right panel shows the constraining power on the luminosity distance $d_{\rm L}(z)$. Results are obtained with the non-parametric cosmological model with splines, using GWs only (blue), the combination of GWs with BAO (yellow); and with the parametric standard flat-$\Lambda$CDM cosmological model (pink).
  • Figure 5: Posterior distributions of the Hubble parameter estimated at $z=0.44$, being the redshift bin with the strongest constraints coming from the GW data. We show results for the three variants of our non-parametric approach with splines: GW, BAO, and GW+BAO. In red, the standard spectral siren approach is used, assuming a flat-$\Lambda$CDM cosmological model. The reference values of $H(z=0.44)$ from SH0ES and Planck are indicated in pink and green, respectively.