Host Catalogs, Peculiar Velocities, and Lensing

A standard siren is only as accurate as the joint model connecting gravitational-wave amplitude, source population, galaxy survey, redshift, and propagation. At low redshift, host peculiar velocity can dominate the redshift-side error. At larger distance, catalog selection, photometric-redshift tails, distance–inclination structure, population selection, and weak lensing become increasingly important. None is a universal additive error bar: each changes a particular part of the generative likelihood.

Helpful background. Gravitational-Wave Astronomy supplies calibration, waveform, and detected-population selection models. Relativistic Cosmology supplies peculiar-velocity, lensing, and perturbed distance–redshift effects. Black-Hole Astrophysics supplies merger-rate evolution and host-galaxy weighting models.

Host support beyond a catalog list

Let $x=(z,\Omega,\gamma)$ describe redshift, sky position, and relevant galaxy properties, and let $p_0(x\mid\Lambda,\eta)$ be the normalized underlying host density. A survey does not reveal a complete set of possible hosts. If $C(x)$ is the probability that a host with properties $x$ enters the usable catalog, define

$$P_{\rm cat} =\int C(x)p_0(x)\,dx, \qquad p_{\rm cat}(x) =\frac{C(x)p_0(x)}{P_{\rm cat}},$$

and

$$p_{\rm out}(x) =\frac{[1-C(x)]p_0(x)}{1-P_{\rm cat}}, \qquad p_0(x) =P_{\rm cat}p_{\rm cat}(x) +(1-P_{\rm cat})p_{\rm out}(x).$$

Here $\Lambda$ denotes cosmology and $\eta$ denotes population and host-rate parameters; those conditionings are suppressed in the displayed equations. These densities condition on catalog membership before using the realized galaxy measurements. The in-catalog branch subsequently conditions on those measurements, giving the weighted galaxy sum in bright and dark standard sirens. Each branch and its scalar mixture weight are normalized. Substituting a position-dependent $C(x)$ directly as a mixture weight between two already normalized densities would not generally integrate to one. Usually $C$ is not one number: footprint, depth, dust, surface brightness, star–galaxy classification, redshift success, and quality cuts make it direction-, redshift-, and property-dependent.

For an observed catalog entry $g$, distinguish the redshift likelihood $p(d_{z,g}\mid z)$ from the normalized redshift posterior $p(z\mid d_{z,g})$. The latter includes the survey’s redshift prior. To reuse it with a different host prior, divide by the original prior and multiply by the target one wherever there is support; do not count the survey prior twice. Spectroscopic failures, photometric-redshift bias, catastrophic tails, duplicated objects, and cross-survey correlations must survive into the analysis. Host weights also require a physical hypothesis. A weight proportional to stellar mass, star-formation rate, or unity per galaxy gives a different merger environment and therefore a different redshift prior.

The normalization must follow the same catalog and merger model used in the numerator. Applying a completeness correction to the event likelihood but not to the detected-population selection factor is inconsistent. The galaxy-catalog formalism and internally consistent mock tests are developed in Gair et al. 2023, §§II–IV, PDF.

Peculiar velocity at low redshift

To first order in line-of-sight velocity,

$$1+z_{\rm obs} \simeq (1+z_{\rm cos}) \left(1+\frac{v_{\parallel}}{c}\right).$$

At sufficiently low redshift this becomes

$$cz_{\rm obs}\simeq H_0D_L+v_{\parallel}.$$

If $\delta v=v_{\rm adopted}-v_{\rm true}$ is the error in the adopted line-of-sight velocity correction, then

$$\frac{\delta H_0}{H_0} \simeq -\frac{\delta v}{H_0D_L}.$$

The sign depends on defining positive velocity as recession; the magnitude shows the essential scaling. The same velocity uncertainty is a small fraction of the Hubble flow for a distant host but a large fraction for a nearby one. A flow reconstruction adds information, yet its smoothing scale, tracer selection, reference frame, group assignment, and residual-velocity distribution become model inputs. Nearby hosts in a common large-scale environment can have correlated velocity errors, so combining events does not necessarily average them as independent noise. The perturbative origin and covariance of velocity-induced distance fluctuations are analyzed in Hui and Greene 2006, §§II–IV.

GW170817 illustrates the correct boundary: its standard-siren likelihood used a measured host recessional velocity together with an explicit peculiar-velocity estimate and uncertainty LIGO/Virgo and partners 2017, Methods. That treatment is evidence for a concrete event model, not a universally adequate velocity prescription.

Weak lensing and propagation

In geometric optics, magnification $\mu$ multiplies flux and scales the gravitational-wave amplitude by $\sqrt{\mu}$. If an analysis assumes $\mu=1$, it infers

$$D_L^{\rm inf}=\frac{D_L}{\sqrt{\mu}}.$$

Weak lensing therefore broadens and skews the distance distribution at cosmological redshift. Its ensemble mean, median, and mode need not coincide, and a detection threshold preferentially admits magnified sources near threshold. Replacing the magnification distribution with a symmetric Gaussian distance error can lose both the tail and magnification selection. Forecasts showing lensing as an eventual precision limit depend on source redshift, angular resolution, selection, matter modeling, and any delensing tracer Holz and Hughes 2005, §§2–3.

Lensing is not the only distance-side effect. Amplitude calibration, waveform mismatch, neglected precession or higher modes, and detector-network polarization coverage couple to inclination and distance. Modified gravitational-wave propagation would change the relation between gravitational-wave and electromagnetic luminosity distances, but it must not be invoked as an explanation until detector, waveform, population, catalog, and lensing alternatives have been tested.

Coupled population and selection effects

Detector-frame masses satisfy $m_{\rm det}=(1+z)m_{\rm src}$. Consequently, a compact-binary mass model can inform redshift while cosmology changes the inferred source-frame population. Merger-rate evolution, mass-spectrum features, spin assumptions, event threshold, and posterior support can all couple to $H_0$. A galaxy catalog adds another route to redshift, but it does not remove population selection.

A useful stress-test sequence is:

1. recover injected cosmology from a complete mock catalog with exact redshifts and a matching source population;
2. impose the real angular mask and magnitude or quality cuts, retaining an out-of-catalog term;
3. replace exact redshifts with the calibrated spectroscopic or photometric error model;
4. add correlated peculiar velocities and a non-Gaussian magnification distribution;
5. perturb calibration, waveform, host weights, population hyperpriors, and detection selection one at a time;
6. refine event count, catalog integration, selection-injection count, sky resolution, and posterior-sample effective size, then test coverage over many independent realizations.

One mock need not peak exactly at its injected value. Check the full prior-predictive ensemble for nominal Bayesian coverage, and separately use fixed-cosmology ensembles to measure bias and frequentist coverage; a proper Bayesian posterior need not have nominal fixed-parameter coverage. A failed complete-catalog baseline can reveal likelihood or selection errors; a new shift after adding the survey mask motivates catalog-model checks; a redshift-dependent shift after lensing is added motivates magnification and selection checks.

Evidence as of 2026-08-18

Evidence	Supported reading	Limitation or alternative
The official GWTC-5.0 cosmology analysis combines 236 gravitational-wave sources, GW170817, source-population information, and DES-Y6 galaxy information and reports a broad $H_0$ constraint LIGO–Virgo–KAGRA 2026, abstract and §§3–5.	Current catalogs carry cosmological information when population, host, and selection models are combined.	The quoted posterior is conditional on catalog, mass-population, cosmological, calibration, waveform, and selection choices; it is not a systematics-free distance ladder.
Internally consistent galaxy-catalog mocks recover an unbiased cosmology, while mismatched data generation or selection can create a spurious bias Gair et al. 2023, §§III–IV, PDF.	Dark-siren inference is statistically sound under its generative assumptions.	A synthetic success does not establish the completeness or host-rate model of a real survey.
GW170817 combines a direct distance likelihood with a host velocity model LIGO/Virgo and partners 2017, Methods, PDF.	A bright siren can isolate a host and make the low-redshift velocity term explicit.	One nearby event cannot calibrate population, catalog, or high-redshift lensing systematics.
Perturbation theory predicts a peculiar-velocity contribution that is strongest nearby and a lensing contribution that grows in importance with distance Hui and Greene 2006, PDF.	Different redshift regimes have different dominant propagation systematics.	The scaling does not supply a catalog-specific covariance or a validated delensing correction.

The strongest supported interpretation at the cutoff is that present standard-siren constraints are informative and improving, while remaining conditional on explicit population and survey models. The conservative interpretation is not that the method is invalid; it is that no single combined posterior uniquely identifies which catalog, population, velocity, lensing, or detector assumption would dominate a future tension. Those positions have unequal scope, not equal evidential weight.

Every result should freeze the event and calibration release, sky maps and posterior priors, host catalog version and mask, redshift and flow products, completeness model, selection injections, waveform, population model, software environment, and file hashes. Reassess after a superseding official catalog or cosmology release, a corrected selection set, a material redshift/catalog update, or an independent mock that changes the bias ranking. No public-event or cosmological fit was rerun for this page.

Common pitfalls

Treating a catalog as the host population. It is a selected observation of that population. Missing support needs a model, and the model must enter both event and selection terms.

Adding all systematics in quadrature. Several effects shift or reshape the likelihood and share latent parameters. A scalar error budget cannot represent correlated peculiar velocities, skewed lensing, or population–cosmology degeneracy.

Calling a broad posterior robust because variants overlap. Variants can share the same omitted support or stale selection function. Validation needs adversarial mocks and coverage, not only agreement among related pipelines.

Exercises

1. Size of a velocity systematic. For a host at $D_L=100\,\mathrm{Mpc}$ in a model with $H_0=70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, estimate the magnitude of the fractional $H_0$ shift caused by a $300\,\mathrm{km\,s^{-1}}$ error in the peculiar-velocity correction.

Solution

The Hubble-flow velocity is

$$H_0D_L=7000\,\mathrm{km\,s^{-1}}.$$

Therefore

$$\left|\frac{\delta H_0}{H_0}\right| \simeq\frac{300}{7000} \simeq0.043.$$

The magnitude is about $4.3\%$. With positive recession velocity, overcorrecting the peculiar velocity by $300\,\mathrm{km\,s^{-1}}$ lowers the inferred $H_0$ by that fraction. The exercise isolates one term; a real result also propagates distance uncertainty and velocity covariance.

2. Catalog incompleteness changes odds. At two candidate cosmologies $A$ and $B$, an in-catalog likelihood is $(0.9,0.3)$ and an out-of-catalog likelihood is $(0.2,0.6)$, in the same arbitrary likelihood units. Take the integrated catalog-membership probability to be $P_{\rm cat}=0.7$ for both cosmologies and assume equal detection fractions $\alpha(A)=\alpha(B)$. Compute the likelihood ratio $\mathcal L(A)/\mathcal L(B)$ and compare it with the ratio obtained by dropping the out-of-catalog term.

Solution

The mixture gives

$$\mathcal L(A)=0.7(0.9)+0.3(0.2)=0.69,$$

and

$$\mathcal L(B)=0.7(0.3)+0.3(0.6)=0.39.$$

Thus $\mathcal L(A)/\mathcal L(B)\simeq1.77$. The equal selection factors cancel here. Dropping missing hosts gives $0.9/0.3=3$. The incomplete analysis more strongly favors $A$ even though no new data were added. If the detection fractions differ, the conditioned single-event likelihood ratio also includes $\alpha(B)/\alpha(A)$.

References

• Gair, Jonathan R., et al. “The Hitchhiker’s Guide to the Galaxy Catalog Approach for Dark Siren Gravitational-Wave Cosmology.” Astronomical Journal 166 (2023): 22. DOI. Open PDF.
• Holz, Daniel E., and Scott A. Hughes. “Using Gravitational-Wave Standard Sirens.” Astrophysical Journal 629 (2005): 15–22. DOI. Open PDF.
• Hui, Lam, and Patrick B. Greene. “Correlated Fluctuations in Luminosity Distance and the Importance of Peculiar Motion in Supernova Surveys.” Physical Review D 73 (2006): 123526. DOI. Open PDF.
• LIGO Scientific Collaboration, Virgo Collaboration, and partners. “A Gravitational-Wave Standard Siren Measurement of the Hubble Constant.” Nature 551 (2017): 85–88. DOI. Open PDF.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. “GWTC-5.0: Constraints on the Cosmic Expansion Rate and Modified Gravitational-Wave Propagation.” arXiv:2605.27227v2 (2026). Preprint. Open PDF.