From Detected Mergers to Cosmic Populations: Gravitational-Wave Selection Effects and Hierarchical Inference

Catalogue and Population Papers Answer Different Questions

Catalogue studies organize candidates, search significance and event parameters; population studies further use these measurements to infer underlying distributions and occurrence rates. This article uses the 2023 published version, distinguishing it from potentially different values in early preprints or DCC versions. The probability that a candidate enters a catalogue, the probability that it is an astrophysical signal and the probability that it comes from a particular formation channel are distinct quantities that cannot be summarized by a single “confidence” value.[1] Population of Merging Compact Binaries Inferred Using G[2] GWTC-3: Compact Binary Coalescences Observed by LIGO an

Detectors Are Not Equally Sensitive to All Systems

Source mass, spin, orientation, distance and detector state all affect whether a signal exceeds the search threshold. Observing windows also differ in noise and effective operating time. Analyses should use a selection function consistent with the actual search, rather than simply drawing a boundary based on theoretical signal-to-noise ratio. If a class of systems has not been observed, one must first assess whether it is rare or difficult to detect; the two possibilities may be hard to distinguish with limited data.作者分析与复核建议 / Author analysis

Chirp Mass and the Early Inspiral

Chirp mass is a combination of the two component masses that affects frequency evolution in the inspiral waveform. Under a simplified approximation, detection distance can scale as a particular power of chirp mass, and observable volume amplifies the differences further. The figures in this article use only an idealized Euclidean, low-redshift, inspiral-dominated, fixed-noise relationship. High-mass systems enter the detector bandwidth differently, so this power law cannot be extrapolated indefinitely across the entire mass range.作者分析与复核建议 / Author analysis

An Analytical Counterexample Illustrating Selection Bias

Construct two source groups with equal underlying numbers and a factor-of-two difference in chirp mass. Under the idealized volume relationship above, the observable-volume ratio is 2 to the power 2.5, approximately 5.657, so the heavier group could account for about 85% of observations. This is not an actual black-hole fraction; it illustrates why a catalogue majority cannot directly be taken as a cosmic majority. Correction requires a selection model and its uncertainty, rather than simply plotting the true population structure from observed counts.作者分析与复核建议 / Author analysis

An Individual-Event Posterior Is Not a Likelihood

Public event parameters are often provided as posterior samples obtained under a particular prior. Treating these samples directly as a likelihood in population inference reuses the original prior or uses it incorrectly. Proper reweighting requires knowledge of the event-level prior and checks that the new population model has sufficient sample support. If the new model concentrates in a region barely sampled by the original posterior, importance weights may become unstable; increasing display precision cannot solve this.作者分析与复核建议 / Author analysis

Why Selection Normalization Cannot Be Omitted

A population likelihood conditional on the number of detected events requires division by the detection fraction predicted by the model. Otherwise, the model may favor sources that are more likely to appear in the catalogue rather than those that better represent the true population. The selection factor itself depends on population parameters and cannot be held fixed when comparing different mass distributions. This article provides the structural formula to aid verification; it neither reanalyzes real event samples nor estimates any new cosmic parameters.作者分析与复核建议 / Author analysis

Occurrence Rates and Distribution Shape

Estimating an occurrence rate per unit volume and time, rather than merely studying the composition of the detected sample, requires an effective spacetime volume and an observational count model. Different searches and mass ranges correspond to different observable volumes, so catalogue counts cannot simply be divided by calendar time. Source-frame time and observing time are also affected by redshift. Reports should specify the rate definition, mass range and cosmological assumptions to avoid directly comparing different estimands.作者分析与复核建议 / Author analysis

Parameter Degeneracies and the Source Frame

Connecting the redshifted mass measured from a waveform to the source-frame mass requires the redshift relation; distance is also correlated with factors such as orientation. Individual-event intervals can be broad, and population features must propagate these uncertainties. Reducing each event to a median and plotting a histogram discards degeneracies and measurement widths, potentially creating or erasing structure. A more reliable approach requires using the full event information and checking model specification.作者分析与复核建议 / Author analysis

Mass Peaks and Formation Channels

Local structure in a population may suggest a formation mechanism, but different astrophysical evolution, selection and measurement models can produce similar shapes. A statistical peak does not automatically establish a unique physical channel. Studies should compare alternative distributions, priors and simulation predictions, and clarify which features are genuinely identifiable. If a peak appears only in a particular parametric model without corresponding support from more flexible models, its interpretation should remain conditional.作者分析与复核建议 / Author analysis

Limitations of Spin Information

Spin magnitudes, orientations and mass ratio may be correlated, and measurement capability varies across individual events. The sensitivity of population posteriors to priors and parameterization must be shown. A single effective-spin parameter cannot be directly converted into the complete three-dimensional spins of all components, nor can a sign-related feature in a distribution establish a unique formation environment. Studies should explain which combination is actually constrained and which dimensions remain prior-dominated.作者分析与复核建议 / Author analysis

Injection Experiments and the Selection Function

Placing synthetic signals in actual or representative noise and running them through the search pipeline can estimate detection probabilities for different sources. However, finite injections also introduce variance, and the original injection distribution may not cover regions emphasized by a new model. Weights, threshold-passing criteria and effective-sample-size diagnostics must be retained. The selection function is not an error-free constant; its estimation error should enter population inference and robustness tests.作者分析与复核建议 / Author analysis

Candidate Purity and Thresholds

Relaxing a catalogue threshold can add potential signals as well as background candidates. Using only a hard threshold discards information near the boundary, whereas mixture models depend on background calibration. Studies should disclose candidate inclusion rules and astrophysical probabilities, and test sensitivity to thresholds. The most favorable threshold must not be selected after the fact to make population features more significant, and not all candidates should be treated as confirmed events.作者分析与复核建议 / Author analysis

Cosmology and Population Evolution

Greater distances correspond to an earlier Universe, and source occurrence rates and mass distributions may evolve with redshift. If a model assumes no evolution, an observed mass–distance relationship may be misinterpreted. Detection efficiency and distance errors can also create correlations. Analyses should compare identifiable evolution models and clarify which conclusions depend on the cosmological background; limited observations cannot support arbitrarily complex evolution functions.作者分析与复核建议 / Author analysis

Priors and Model Checks

When population samples are relatively limited, priors on smoothing, peak widths or boundaries may affect results. Posterior predictive checks should assess whether the model can reproduce distributions at the observational level, rather than merely examining whether parameter posteriors are narrow. Predictions must pass through the selection and measurement processes again; directly comparing the true population curve with a biased catalogue histogram confuses the model and observational levels.作者分析与复核建议 / Author analysis

The Role of Fast AI Inference

Neural density estimation and other fast methods can reduce computation, but require calibration in parameter tails, under different noise conditions and for unseen waveforms. Greater speed does not imply a correct posterior, especially for rare events and out-of-distribution signals. Models should also retain their training simulations and waveform versions. This article provides no new evaluation of any AI inference engine; it only explains that the same selection, prior and coverage checks are required.作者分析与复核建议 / Author analysis

Version Updates and Duplicate Events

Catalogue updates may revise the parameters, significance or candidate status of the same event. Combining versions requires deduplication by event identity while preserving provenance and revisions. Samples from old and new versions must not be treated as additional independent events, nor should new posteriors silently rewrite old conclusions. Population results should be tied to precise catalogue and data-release versions so that later researchers can reconstruct the input set used at the time.作者分析与复核建议 / Author analysis

Limits of Recomputing the Analytical Figures

The mass–volume example can be verified directly by evaluating the power law and the two-group counts; chirp mass can be checked for exchange symmetry and mass scaling. The figure contains no real detector noise, injections or cosmological volume and therefore cannot estimate actual source fractions. It demonstrates that selection alone is sufficient to alter the observed composition; real correction requires a complete search and measurement model.作者分析与复核建议 / Author analysis

Falsifiable Physical Interpretations

A formation-channel interpretation should make testable predictions for future joint distributions of mass, spin and redshift, passed through the same selection pipeline. If new events can only be explained after readjusting the model, the failure of the original prediction should be reported. Retaining alternative models and observation-level predictions in advance helps distinguish genuine physical information from parametric flexibility and allows new observations to change our understanding explicitly.作者分析与复核建议 / Author analysis

Conclusions and Research Package

Verifiable materials should connect the catalogue, event priors, posteriors, search thresholds, injection-based selection, effective time and population model. The strength of gravitational-wave population studies lies in inferring cosmic structure from limited, selected events while faithfully propagating uncertainty at every level. Separating catalogue, underlying population and formation mechanism enables new detections to test physics rather than continually embellish a biased histogram.作者分析与复核建议 / Author analysis

Core Equations and Derivations

Methods and Evidence Workflow

Analytical Figures and Numerical Examples

Recomputable worked example

A chirp-mass ratio of 2 gives an ideal volume ratio of 2^2.5=5.656854; with equal underlying group fractions, the heavier group's catalogue fraction is 0.849779. No real events were reanalyzed.

Comparison of Methods

Verification and Experimental Plan

1. Fix the catalogue and data-release versions and remove duplicates.
2. Retain event priors and full posterior information.
3. State candidate thresholds and background treatment.
4. Use injection-based selection matched to the search.
5. Check weights and effective sample coverage.
6. Distinguish distribution-shape estimation from occurrence-rate estimation.
7. Perform posterior prediction and prior-sensitivity checks at the observational level.
8. Release code, failed models and falsifiable future predictions.

Scope and Limitations

• No new gravitational-wave detections or reanalysis of real events.
• GWTC-3 is a methodological case, not a claim to use the latest catalogue.
• The volume power law applies only under the explicitly stated idealized approximations.
• Synthetic group fractions are not cosmic black-hole fractions.
• Statistical population structure does not uniquely determine a formation channel.

Evidence from the Original Research and Its Scope

Research Questions Worth Testing

1. When does selection-function error dominate inferred population features?
2. Can model peaks persist across priors and parameterizations?
3. Which future joint distributions can best distinguish formation channels?

Mapping Primary Sources to Claims

1. Population of Merging Compact Binaries Inferred Using Gravitational Waves through GWTC-3R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, N. Adhikari, et al. · 2023-03-29Abstract; population likelihood and selection effects
2. GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing RunR. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, N. Adhikari, et al. · 2023-12-04Abstract; search and parameter-estimation methods