Cross-Messenger Consistency Tests and Systematics

Cross-messenger consistency is not established by overlapping one-dimensional radius intervals, and “tension” is not established when two plotted contours look separated. Radio timing, X-ray pulse profiles, gravitational waves, nuclear experiments, and theory priors constrain different latent quantities with different selections and systematics. A defensible joint conclusion must preserve their likelihoods, expose shared assumptions, predict each held-out dataset, and survive alternative calibrations and EOS representations. This page supplies that decision procedure with evidence cutoff 2026-08-18.

Required background. Equivalence principles and gravitational measurement supplies the distinction between physical source parameters and messenger-dependent observables.

Helpful background. Stress–energy and conservation supplies the common material variables; the Einstein equation supplies the stellar map; relativistic matter and compact objects develops equilibrium, response, and dense-matter closure; gravitational radiation and waveforms supplies the wave likelihood; and computational relativity supplies reproducible forward-model tests.

When messenger likelihoods may be multiplied

Let $\phi$ denote the EOS, $\xi_k$ messenger-specific latent parameters, $\zeta$ shared nuisances or theory choices, and $d_k$ the data in messenger $k$. The factorization

$$p(\{d_k\}\mid\phi,\zeta,\{\xi_k\}) =\prod_k p(d_k\mid\phi,\zeta,\xi_k)$$

is valid only after conditioning on every cause of cross-dataset dependence. A radio mass likelihood used as a prior inside an X-ray pulse-profile posterior is one such dependence. Multiplying that X-ray posterior by the same radio likelihood again counts the timing data twice. A chiral-EFT ensemble calibrated to nuclear observables cannot later receive the same observables as an independent likelihood unless the calibration is removed or represented hierarchically. GW170817 strain and a kilonova model share binary masses, distance, inclination, merger physics, and sometimes posterior samples; they are not independent summaries merely because the photons and strain came from different instruments.

Published posteriors also carry interim priors. If samples approximate

$$p(\theta\mid d,\pi_0) \propto\mathcal L(d\mid\theta)\pi_0(\theta),$$

then a joint analysis reweights by $\mathcal L\propto p/\pi_0$, not by the posterior alone. Support lost because $\pi_0=0$ cannot be restored by reweighting. Exact sample provenance and effective sample size are therefore part of a consistency test.

At the cutoff, the named dataset versions are Australia Telescope National Facility 2026, ATNF v2.8.1 release-history row plus primary timing likelihoods; target-specific NICER/HEASARC ObsIDs with the exact CALDB, response, screening and background products for an explicitly declared subset of the five peer-reviewed targets J0030+0451, J0740+6620, J0437−4715, J1231−1411, and J0614−3329; the GWOSC GW170817 event release with current GWTC-5.0 catalog context; and the final PREX-II and CREX publications with a versioned chiral-EFT calculation. The J2124−3358 cutoff-date preprint remains a separately labeled optional input, as detailed in the NICER target inventory and analysis qualifications. An analysis must name the actual subset it uses. Raaijmakers et al. 2021, §§ 2–4 is a useful example because radio, NICER, gravitational-wave, and EFT contributions are turned on separately under two EOS representations rather than hidden in one synthetic constraint.

Predictive tests before a tension claim

Four tests answer different questions.

Prior prediction. Draw $\phi$ and nuisances from the complete prior, generate observables through each forward model, and compare the resulting range with the actual data. This reveals whether a narrow EOS or calibration prior excluded an observation before any likelihood was evaluated.

Posterior prediction. Fit a dataset or group $D_{-k}$ without messenger $k$, then evaluate

$$p(d_k\mid D_{-k}) =\int d\phi\,d\zeta\,d\xi_k\, p(d_k\mid\phi,\zeta,\xi_k) p(\xi_k\mid\phi,\zeta) p(\phi,\zeta\mid D_{-k}).$$

The held-out observation must be compared in its native count, strain, timing, or nuclear-observable space when possible, not only after both sides are compressed to $R_{1.4}$. Posterior-predictive checks and their graphical limitations are developed in Gelman et al. 2013, ch. 6.

Leave-one-messenger-out stability. Repeat the EOS inference omitting radio, X-ray, gravitational-wave, laboratory, and theory inputs in turn. Track pressure by density, sound speed, radius, maximum mass, and phase-transition probability. A result driven by one channel is not invalid, but it must be labeled as such.

Prior- and parameterization-aware discrepancy. Evidence ratios depend on prior volume. “Suspiciousness” and related information-corrected measures can be useful summaries, but only with their dimensionality, shared parameters, correlations, and calibration established; Handley and Lemos 2019, §§ III.A–C and IV.B–C, Eqs. (6)–(10) and (22)–(26) explains why a raw evidence ratio is not a prior-independent tension scale. No scalar replaces inspection of which native residuals fail.

Synthetic injection is the calibration of these tests. Generate mutually consistent mock radio, X-ray, wave, and nuclear data from one EOS, pass them through the actual posterior-reuse pipeline, and verify that the method does not manufacture a conflict. Then inject a known channel bias or an EOS outside one representation family and measure detection power.

Systematic alternatives by messenger

Messenger	Calibration and selection	Forward-model uncertainty	Minimum stress test
Radio timing	Clock, ephemeris, dispersion, kinematic correction; survey and follow-up selection	Timing model, companion and population model	Alternative ephemerides and population families; leave out each influential mass
X-ray pulse profile	NICER CALDB/response, detector and good-time cuts, clock and background	Atmosphere, hot-region topology, exterior spacetime, mass/distance priors	Perturbed response and background; independent ray tracer; alternate spot and atmosphere families
Gravitational waves	Strain calibration, data-quality segments, gating, PSD and search threshold	Point-particle/tidal waveform, spins, merger cutoff, common-EOS relation	Independent waveform, frequency band, spin prior and calibration realization; event injection
Nuclear experiment	Beam polarization, spectrometer response, kinematic and centrality cuts	Finite-nucleus functional, transport model, electroweak and in-medium corrections	Joint PREX-II/CREX fits; alternate functionals and transport codes with covariance
Nuclear theory	Interaction order, regulator, low-energy constants and calibration set	EFT and many-body truncation, emulator and breakdown density	Correlated truncation model; omit highest-density points; alternate interaction families
Transient electromagnetic counterpart	Photometric calibration, cadence, band selection and host distance	Ejecta geometry, opacity, heating, thermalization and remnant mapping	Competing radiative-transfer grids; omit counterpart or broaden model discrepancy

Systematic uncertainty is propagated as nuisance structure or model averaging, not added afterward until intervals overlap. Conversely, unlimited discrepancy parameters can make every dataset “consistent” by removing its information. Their priors require external calibration or synthetic recovery.

Selection is messenger-specific. A radio catalog is filtered by discovery and follow-up; NICER targets are intentionally selected; gravitational-wave events pass a search threshold; nuclear experiments choose nuclei and kinematics. Population-normalized likelihoods include the appropriate detection denominator, while a targeted single-object likelihood conditions on its selection. Mixing those cases creates a false EOS preference.

Pang et al. 2021, §§ 2–4 demonstrates how adding counterpart models changes a multimessenger combination and why model-dependent transient inputs should be removable. Agreement between that analysis and another is meaningful only after common events, priors, nuclear inputs, and EOS representations have been accounted for.

Reproduction package and decision rule

A reproducible joint analysis archives each native data product or lawful retrieval recipe, checksum, calibration and response, selection table, likelihood implementation, interim prior, nuisance prior, EOS map, software environment, sampler settings, random seeds, and convergence output. The posterior-sample and plotting package accompanying Raaijmakers et al. 2021, Zenodo version 1 is an example of a citable data-product boundary; reproducing a plot still does not verify every upstream likelihood.

The finite test sequence is:

1. recover analytic Gaussian and stellar-polytrope benchmarks;
2. reproduce each messenger likelihood separately from its named version;
3. verify interim-prior removal and selection normalization on synthetic populations;
4. run the joint baseline under pinned software and convergence thresholds;
5. produce prior-predictive, posterior-predictive, and leave-one-messenger-out results;
6. repeat the high-impact calibration and model alternatives in the table; and
7. compare at least two EOS representations, including one that admits sharp transitions.

A “consistent” conclusion means that held-out native data are plausible under the joint model and that the main material conclusion survives declared alternatives. A “model-dependent discrepancy” means the result changes with an admissible calibration, likelihood, prior, or EOS family. A “persistent tension” is reserved for a discrepancy that survives those changes and synthetic calibration with a stated false-alarm or predictive tail probability. Only then is new matter physics one candidate explanation; unmodeled instrument, population, atmosphere, waveform, nuclear, or transient effects remain alternatives.

This page specifies the procedure but does not claim a new joint fit or a current dense-matter tension.

Check your understanding

Two analyses of the same derived radius scale give $12.0\pm0.5\,\mathrm{km}$ and $13.2\pm0.6\,\mathrm{km}$ with independent Gaussian statistical errors. Compute the naive standardized difference. Then compare two explicitly hypothetical systematic models: independent zero-mean Gaussian offsets of standard deviation $0.5\,\mathrm{km}$ in each analysis, or one common offset of that size added to both.

Solution

Without the additional systematics,

$$z=\frac{|13.2-12.0|}{\sqrt{0.5^2+0.6^2}} =\frac{1.2}{\sqrt{0.61}} \simeq1.54.$$

Adding independent $0.5\,\mathrm{km}$ systematic variances gives

$$z_{\rm sys} =\frac{1.2}{\sqrt{0.5^2+0.6^2+0.5^2+0.5^2}} =\frac{1.2}{\sqrt{1.11}} \simeq1.14.$$

For one common offset $b$, write $R_1=R+e_1+b$ and $R_2=R+e_2+b$. It cancels in $R_2-R_1$. Equivalently,

$$\operatorname{Var}(R_2-R_1) =\operatorname{Var}(R_2)+\operatorname{Var}(R_1) -2\operatorname{Cov}(R_1,R_2),$$

so the two added $0.25\,\mathrm{km}^2$ variances cancel the covariance contribution $2(0.25\,\mathrm{km}^2)$. The discrepancy remains $z\simeq1.54$, although uncertainty in the common absolute radius increases. Shared uncertainty does not necessarily make a discrepancy smaller.

Neither number alone establishes consistency or tension: the estimates may share data or priors, the derived radius may hide different native observables, and the systematic distributions require calibration rather than arbitrary inflation. The calculation shows why covariance and model structure precede a scalar discrepancy label.

References

• Australia Telescope National Facility. “ATNF Pulsar Catalogue.” PSRCAT version 2.8.1, released 18 June 2026. Official version history.
• Gelman, Andrew, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. Bayesian Data Analysis. 3rd ed. Boca Raton: CRC Press, 2013. DOI.
• Handley, Will, and Pablo Lemos. “Quantifying Tensions in Cosmological Parameters: Interpreting the DES Evidence Ratio.” Physical Review D 100 (2019): 043504. DOI. Open PDF, corrected v5.
• Pang, Peter T. H., et al. “Nuclear Physics Multimessenger Astrophysics Constraints on the Neutron-Star Equation of State: Adding NICER’s PSR J0740+6620 Measurement.” Astrophysical Journal 922 (2021): 14. DOI.
• Raaijmakers, Geert, et al. “Constraints on the Dense Matter Equation of State and Neutron Star Properties from NICER’s Mass-Radius Estimate of PSR J0740+6620 and Multimessenger Observations: Posterior Samples and Scripts for Generating Plots.” Zenodo dataset, version 1, 16 April 2021. DOI.
• Raaijmakers, Geert, et al. “Constraints on the Dense Matter Equation of State and Neutron Star Properties from NICER’s Mass–Radius Estimate of PSR J0740+6620 and Multimessenger Observations.” Astrophysical Journal Letters 918 (2021): L29. DOI.