H₀ from gravitational-wave ×
galaxy cross-correlations


Cail Daley

CosmoStat, CEA Paris-Saclay


CosmoStat Journal Club

April 27, 2026

Standard sirens measure \(H_0\) without the distance ladder


Hubble’s law wants \(d_L\) and \(z\):

\[d_L(z) \;=\; (1+z)\,\frac{c}{H_0}\!\int_0^z\!\frac{dz'}{\sqrt{\Omega_m(1+z')^3 + \Omega_\Lambda}}\]

  • Traditional distance-ladder methods build \(d_L(z)\) by measuring redshifts of objects of known absolute luminosity, thus inferring \(d_L\) — with potentially compounding systematics.
  • The GW inspiral gives \(d_L\) directly from GR, without assumptions of intrinsic brightness or a multi-step ladder of propagating systematics.
  • But dark sirens have no optically identifiable host, and thus no spectrum / no \(z\).

A chirping waveform gives \(d_L\) absolutely — but never \(z\)

Amplitude alone is degenerate: \[h \;\propto\; \mathcal{M}_z^{5/3}\, f^{2/3}\,/\,d_L\]

Chirp rate breaks it — from phase, not amplitude: \[\dot f \;\propto\; \mathcal{M}_z^{5/3}\, f^{11/3} \;\;\Rightarrow\;\; \mathcal{M}_z\]

Plug \(\mathcal{M}_z\) back into \(h\) → \(d_L\) absolutely. GR-calibrated, no distance ladder.


What’s left: \(\mathcal{M}_z = (1+z)\,\mathcal{M}_\text{src}\). GW alone cannot separate mass from redshift → need external \(z\).

Siren H₀ landscape: precision driven by the one bright event

channel \(H_0\) (km/s/Mpc) precision
GW170817 only \(70^{+12}_{-8}\) ~14% Abbott+ 2017
GW170817 + jet \(\mathbf{68.9^{+4.7}_{-4.6}}\) ~7% Hotokezaka+ 2019
Dark (galaxy catalog, GWTC-4) \(76.6^{+13.0}_{-9.5}\) ~15% LVK 2025
Spectral (mass function, GWTC-4) broad ~20% LVK 2025


Today’s two papers:

  • Peak Sirens (2512.15380): \(67^{+18}_{-15}\) — new channel, first b_gw bound
  • Andrade-Oliveira+ (2601.04774): \(67.9^{+4.4}_{-4.3}\) — posterior-level fusion of spectral sirens + DES Y3 3×2pt + GW170817

With a counterpart (bright) or without (dark)

Bright — optical counterpart → host \(z\).

  • GW170817: the one clean event so far, ~1 per few years.

Dark — marginalize likelihood over every candidate host in the 3D localization volume.

  • Systematic floor: galaxy-catalog (in)completeness.

Spectral sirens: hierarchical modeling of the observed mass distribution


  • Observed and source-frame mass distributions \(p(m_\text{obs})\) and \(p(m_\text{src})\) are linked by the redshift distribution of the sources: \[m_z \;=\; (1+z)\,m_\text{src}\]

  • \(p(m_\text{src})\) has features — power-law, peak near \(\sim\!35\,M_\odot\), second peak / dip (top right).

  • Hierarchical Bayesian fit on \((d_L, m_\text{obs})\) per event → joint posterior on \((H_0, \Omega_m, \pi_\text{pop})\).

Paper 1

Andrade-Oliveira+ 2026 — spectral sirens × DES 3×2pt

Andrade-Oliveira+: break the \(\Omega_m\) degeneracy by multiplying in DES 3×2pt

Three independent \((H_0, \Omega_m)\) likelihoods:

  • Spectral sirens — 142 CBCs from GWTC-4.0 (LVK 2025 posterior)
  • DES Y3 3×2pt — gg + gκ + κκ, standard photometric cosmology
  • GW170817 + jet — superluminal-motion inclination prior (\(15^\circ\!<\theta_\text{jet}\!<\!29^\circ\))

Multiply posteriors:

  • Spectral sirens alone constrain both \(H_0\) and \(\Omega_m\) — weakly, and correlated
  • 3×2pt pins \(\Omega_m\) tightly → breaks the degeneracy → sharpens \(H_0\)
  • GW170817+jet is the precision booster (6.4% → 9.9% without jet)


Posterior-level only: DES galaxies are not used as dark-siren hosts — that would make the likelihoods correlated.

Result: \(H_0 = 67.9^{+4.4}_{-4.3}\), driven by the 3×2pt \(\Omega_m\) prior

  • Left: \((H_0, \Omega_m)\) 2D. 3×2pt (blue) × sirens (red) → tight joint contour (black).
  • Right: 1D \(H_0\). “All” (black) is the 6.4% result, competitive with Planck/SH0ES width. \(\Omega_m\) improved 22% over DES-only.

Paper 2

Santiago de Matos+ 2025 — “Peak Sirens”

Cross-correlation sirens: GW shell at \(d_L\) overlaps galaxy shell at \(z\) — only when \(H_0\) is right

  • 3D GW density field + galaxy field → cross-correlate. \(C_\ell^{gb}(z, d_L)\) peaks along the true \(d_L(z; H_0, \Omega_m)\); wrong \(H_0\) washes it out.
  • No host ID: catalog incompleteness only enters through field statistics, much less sensitive.

Inside the simulation machinery

A galaxy–BBH mock catalog pair for the ET+2CE network, built with GLASS. Left: galaxies. Center: true BBH positions. Right: BBH “event clouds” after convolving with the per-event distance and sky-localization uncertainties.

First detection: 5.9σ GW × galaxy cross-correlation


  • GWTC-3.0 (90 events) × GLADE+ galaxy catalog
  • Left: observed \(\sum_\ell w_\ell C_\ell^\text{obs}\) in \((z, d_L)\)
  • Right: theory at \(H_0=67.36\) — match along the magenta line

A new observable: GW bias \(b_\text{gw}\)

\[\delta_\text{gw}(\mathbf{x}) \;=\; b_\text{gw}\, \delta_m(\mathbf{x}) \qquad\Longrightarrow\qquad C_\ell^{gb} \;\propto\; b_g\, b_\text{gw}\, P_m(k;\,\Omega_m)\]


  • Galaxies trace matter with bias \(b_g\); so do BBH progenitors — but which hosts?
  • Massive / old populations? Dense environments?
  • Orthogonal to \(H_0\) (amplitude vs. shape) → co-fit
  • First observational constraint: \(b_\text{gw} < 4.3\) at 95% CI
  • Prior was \([0, 10]\) uniform
  • Forecasts: next-gen detectors (Einstein Telescope + Cosmic Explorer) will pin \(b_\text{gw}\) to order unity

Peak Sirens H₀ posterior: broad but detection is robust


  • Top: \(H_0\) marginal. With GW190814 bright bonus, peaks at 67.
  • Bottom-right: \(b_\text{gw}\) marginal. Consistent with 0, constrained \(<4.3\).
  • Contour: no degeneracy between \(H_0\) and \(b_\text{gw}\) — shape vs. amplitude.
  • LVK dark-sirens posterior (gray, dashed) shown for comparison.
  • \(H_0 = 67^{+18}_{-15}\) km/s/Mpc — wide, but first of its kind.

Weak-lensing magnification adds a tail at \(z_g < z_b\)


  • GWs from distant BBHs are magnified by intervening structure → counted in foreground galaxy shells they didn’t actually inhabit.
  • Lensing lifts the tail of \(C^{gb}(z_g, z_b)\) at \(z_g < z_b\) (right panel, dashed blue).
  • Peak is only slightly shifted → method is robust for LVK/O5, but next-gen precision will require modeling it.

Next-gen detectors push \(H_0\) into the sub-% regime


Ferri+ 2025 forecasts (ΛCDM, joint \((H_0, \Omega_m)\)):

network \(\sigma(H_0)/H_0\)
LVK O5 4% (with external \(\Omega_m\))
LVK + ET ~1%
ET + 2CE ~0.6%
  • Adding one next-generation detector (e.g. ET alone) gives an order-of-magnitude improvement in both \(H_0\) and \(\Omega_m\).
  • Competitive \(w_0 w_a\) constraints from the same data.

Takeaway: siren cosmology is fanning out into three complementary channels

Three paths to \(z\):

  • Counterparts — precision, but luck-limited
  • Catalogs + mass function — per-event, systematics from completeness / population priors
  • Cross-correlation — field-level, robust to incompleteness, gives \(b_\text{gw}\) as a bonus

Both papers, same 3 months:

  • Peak Sirens: first detection of a new channel
  • Andrade-Oliveira+: matches the bright-siren benchmark by stacking independent probes

Next: O5 will triple the event count. ET+CE forecasts promise 1% \(H_0\).

Acknowledgements & further reading

Primary papers (today’s JC):

Methods companion:

Pedagogical inspiration:

Foundational & landscape:

  • Schutz 1986 — original standard-siren proposal
  • Chernoff & Finn 1993; Farr+ 2019 — spectral sirens
  • Mastrogiovanni+ 2021, 2024 — LVK O4a gwcosmo
  • Hotokezaka+ 2019 — GW170817 + superluminal jet
  • LVK 2025 — GWTC-4 cosmology
  • Scelfo+ 2018, Libanore+ 2022 — GW bias theory

Q: what redshifts are the GWTC-3 events actually at?


  • LVK O5 forecast BBHs live almost entirely at \(z \lesssim 1\), concentrated in \(z \sim 0.1\text{--}0.5\).
  • GWTC-3 (what Peak Sirens actually uses) is tighter still — most events at \(z \lesssim 0.3\).
  • 3G pushes the accessible range to \(z \sim 2\) in the cross-correlation analysis (even though ET alone detects BBHs to \(z \sim 10\)).

Q: what’s actually in GLADE+ — and why not DESI?

GLADE+ = a purpose-built merger of six all-sky catalogs (Dálya+ 2022):

  • 2MASS XSC — NIR extended sources
  • 2MPZ — 2MASS photo-z’s
  • WISE × SuperCOSMOS — mid-IR photo-z’s
  • HyperLEDA — morphologies, velocities
  • GWGC — original GW catalog
  • SDSS-DR16Q — quasars

~23M galaxies, all-sky, heterogeneous \(z\) quality.

Why this, not DESI?

  • GW events can land anywhere on the sky. DESI spectroscopic covers ~17 k deg² (~40% of sky) — you’d lose half your events.
  • GLADE+ is the heritage catalog for GW cosmology → direct comparability with prior gwcosmo results.
  • Authors flag shallowness as the main limitation; their stated next step is DESI Legacy Imaging (photo-z’s, wider, deeper than GLADE+). Real all-sky+deep waits for Euclid / LSST.

Q: how complete is the GLADE+ galaxy catalog?

GLADE+ stitches together 2MASS, GWGC, SDSS, HyperLEDA, WISE — ~23M galaxies.

  • Essentially complete at \(z \lesssim 0.03\) (~130 Mpc).
  • Usable, but increasingly sparse, to \(z \sim 0.1\)–0.3.
  • Beyond \(z \sim 0.3\), incompleteness dominates.
  • Most GWTC-3 BBHs live where GLADE+ is already thinning out.
  • Catalog-dark analyses have to marginalize over the missing population → the systematic floor.
  • Cross-correlation sirens only feel this through field statistics — much gentler dependence.
  • Deeper surveys (DESI, Euclid, LSST) are the obvious upgrade.

Q: could you use weak lensing or CMB lensing instead of galaxies?

Weak lensing (\(\kappa_\text{gal}\))

  • Weighs all matter along the line of sight — clean on galaxy-catalog incompleteness.
  • Paper 2 already uses DES 3×2pt, but at the posterior level, not as a cross-correlation field.
  • A \(C_\ell^{g\kappa b}\) field-level analysis is the natural next step.

CMB lensing (\(\kappa_\text{CMB}\))

  • Kernel peaks at \(z \sim 2\) — perfectly matched to 3G BBHs.
  • No catalog needed at all; use existing Planck / ACT / SO \(\kappa\) maps.
  • Cross-correlating GW density with \(\kappa_\text{CMB}\) is completely incompleteness-free — and as yet unmeasured.