Redshift Evolution and Cosmic History

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Gravitational-wave catalogs can constrain how the merger-rate density changes across cosmic time, but they do not provide spectroscopic redshifts for ordinary binary black holes. The inference comes from luminosity distance, redshifted masses, an assumed cosmology, and a rapidly changing selection function. A rising detected count with distance is therefore not by itself evidence for an evolving source rate. As of 2026-08-18, current catalogs favor positive low-redshift evolution for binary-black-hole mergers, while detailed cosmic-history and formation-delay interpretations remain model dependent.

Required background. Probability and Statistical Inference supplies hierarchical and point-process models. Helpful background. Parameter Degeneracies explains event distance–inclination uncertainty; Merger Rates and Sensitive Spacetime Volume supplies the selection-weighted count measure.

The observable redshift distribution

For source-frame merger-rate density $\mathcal R(z,\lambda)$ and intrinsic parameters $\lambda$ , the detected differential count is

\frac{dN_{\rm det}}{dz} =T_{\rm obs}\frac{dV_c}{dz}\frac{1}{1+z} \int d\lambda\, \mathcal R(z,\lambda) p_{\rm det}(z,\lambda).

Three factors can make detected counts change with redshift even if the source population does not evolve: the comoving shell volume $dV_c/dz$ , source-to-detector time dilation $(1+z)^{-1}$ , and detection probability. Conversely, a declining efficiency can hide an intrinsically increasing rate. Only their joint forward model is identifiable from the catalog.

For an event without an electromagnetic redshift, the waveform measures detector-frame masses and luminosity distance:

m^{\rm det}=(1+z)m^{\rm src}, \qquad D_L=D_L(z;H_0,\Omega_m,\ldots).

The distance–inclination degeneracy broadens $D_L$ ; weak lensing and calibration add uncertainty; and a cosmological model maps that distance to $z$ . A mass model can sharpen redshift by insisting that source masses lie in a narrow range, but then the apparent redshift evolution inherits that mass model. Joint mass–redshift inference is essential whenever either may evolve.

Useful rate models

A local phenomenological model is

\mathcal R(z)=\mathcal R_0(1+z)^\kappa.

It gives an interpretable low-redshift slope but cannot represent a turnover across the full history of star formation. A broken-power-law or Madau–Dickinson-like form adds rise, turnover, and decline parameters. Splines, Gaussian processes, or autoregressive processes allow more freedom but need smoothing priors and adequate high-redshift sensitivity.

A formation model instead convolves the progenitor formation rate $\mathcal F$ with a delay-time distribution:

\mathcal R(t) =\int_{t_{d,\min}}^{\min(t,t_{d,\max})} d t_d\,p(t_d\mid\Lambda_d) \mathcal F(t-t_d\mid Z,\Lambda_f),

where metallicity $Z$ and binary efficiency may also evolve. Many combinations of $\mathcal F$ , $p(t_d)$ , metallicity dependence, and channel mixture produce the same low-redshift $\mathcal R(z)$ . Measuring $\kappa$ is therefore not the same as measuring a delay-time law.

Selection and correlated evolution

The selection factor for hyperparameters is

\alpha(\Lambda)= \int dz\,d\lambda\, p_{\rm det}(z,\lambda) p(z,\lambda\mid\Lambda).

If high-mass systems are visible farther away, an apparent trend of mass with redshift can arise from selection. If the mass spectrum genuinely changes with redshift but the model forces independence, the fitted rate evolution can absorb that change. Spin-dependent sensitivity and waveform duration provide additional couplings.

Catalog-specific injections must span source masses, spins, redshift, sky position, and observing epochs. The proposal density and recovered weights should be checked in each redshift region. High-redshift bins with a tiny injection effective sample size are prior extrapolations even if the hyperposterior sampler itself converges.

Current evidence and competing readings

The GWTC-5.0 population analysis fits parametric and weakly parameterized redshift models with the official event and injection products. It favors a binary-black-hole merger-rate density increasing with redshift over the catalog’s informative range and finds broad consistency with a star-formation-like rise LVK 2026, §4.4.4 and Appendix B.5. The statement is conditional on the catalog selection, cosmology, mass–spin model, event treatment, and the limited redshift lever arm.

Analysis direction	Supported conclusion	Remaining ambiguity
Official parametric rate evolution	A constant source-frame rate is not the best current description over the analyzed range	The exact functional form and turnover at higher redshift
Weakly parameterized reconstruction	The broad rise is not solely a single power-law ansatz	Fine structure depends on smoothing and sparse high-redshift support
Joint mass–redshift models	Some apparent evolution can trade against mass-spectrum evolution	Whether the mass distribution itself evolves and by which mechanism
Formation-delay convolution	Current rate evolution constrains combinations of formation and delay histories	A unique isolated, dynamical, or hierarchical channel

Earlier work already demonstrated why mass and redshift must be fitted together Fishbach, Holz, and Farr 2018, §§ 2–4. Flexible analyses can expose structure omitted by parametric forms, but their broader uncertainty is part of the result, not a defect Callister and Farr 2024, §§ II and IV.

Independent checks

Non-evolving mock. Draw a constant comoving source-frame rate, apply realistic detector selection, and verify that the pipeline does not infer positive $\kappa$ from geometric volume alone.
Correlated mock. Generate a mass spectrum that changes with redshift. Fit both independent and joint models; the difference shows how misspecification migrates into $\mathcal R(z)$ .
Cosmology stress test. Vary the distance–redshift relation consistently in event likelihoods, rate measure, and selection injections. Changing it in only one place is not a sensitivity analysis.
High-redshift support. Plot recovered-injection effective sample size and posterior predictive detected counts versus redshift.
Leave-loud-events-out. Refit after removing individually influential distant events; report whether the qualitative rise persists.

The result should be reviewed after a superseding catalog or injection release, materially changed distance calibration or waveform likelihoods, or an independent joint mass–redshift analysis that alters the evidence ordering. Claiming a formation history additionally requires predictive agreement in mass, spin, mass ratio, and rates, not only a redshift curve.

Exercises

1. Time dilation. A source population produces $R_s$ mergers per unit source proper time in a fixed comoving volume at redshift $z$ . How many occur per detector time?

Solution

Cosmological time dilation gives $dt_{\rm det}=(1+z)dt_{\rm src}$ . The detector-frame rate is therefore $R_s/(1+z)$ . This is the origin of the factor in the detected-count integral; omitting it overpredicts the contribution of distant sources.

2. Why mass evolution can mimic rate evolution. Suppose high-mass binaries become more common at larger redshift but the analysis assumes one redshift-independent mass distribution. Explain the possible bias.