Population Timing and Theory Constraints

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Different pulsar binaries probe different parts of a gravity theory: double-neutron-star systems supply precise relativistic overdetermination, neutron-star–white-dwarf systems maximize charge asymmetry, triples test differential free fall, and different neutron-star masses sample possible nonperturbative branches. Combining them can constrain one common coupling more strongly than any system alone. The combination is valid only when each system keeps its own timing noise, selection, masses, environment, and effective charges, while shared theory and calibration parameters are represented once. The status described here is current through 2026-08-19.

Required background. Probability and Statistical Inference supplies hierarchical likelihoods, prior reweighting, selection normalization, and posterior-predictive checks.

Helpful background. Weak-Field Gravity supplies the asymptotic coupling regime; Relativistic Two-Body Dynamics supplies system-specific binary predictions; and Gravitational Radiation and Waveforms supplies the radiative observables being combined.

One theory, many compact systems

Let $\lambda$ denote a common theory coupling and $\mathcal E$ an equation-of-state hypermodel. System $j$ has local parameters $\theta_j$ : masses, orbital elements, distance, noise, and perhaps effective charges $\alpha_{A,j}(\lambda,\mathcal E)$ . Its likelihood is $p(d_j\mid\theta_j,\lambda,\mathcal E,\kappa)$ , where $\kappa$ contains shared clock, ephemeris, or calibration components. A fixed-count population likelihood can be written schematically as

p(\{d_j\}\mid\lambda,\Lambda,\mathcal E,\kappa,S) =\prod_{j=1}^{N} \frac{\displaystyle\int p(d_j\mid\theta_j,\lambda,\mathcal E,\kappa) p(\theta_j\mid\Lambda,\mathcal E)\,d\theta_j} {\xi(\lambda,\Lambda,\mathcal E,S)},

where $\Lambda$ describes the parent population. For one common discovery and follow-up rule,

\xi(\lambda,\Lambda,\mathcal E,S) =\int p(S\mid\theta,\lambda,\mathcal E)\, p(\theta\mid\Lambda,\mathcal E)\,d\theta.

The likelihood in the numerator is the ordinary data likelihood, evaluated for data that satisfy $S$ ; it must not already be conditioned on detection and then divided by $\xi$ a second time. Heterogeneous surveys and follow-up rules generally need separate efficiencies or an explicit survey-mixture model. A rate-inclusive likelihood has a different Poisson normalization; the convention must be declared.

This population model is not mandatory for every multi-pulsar gravity test. A targeted analysis can condition on a specified set of systems and their measured properties, sharing only theory parameters while treating each mass and distance as local nuisance variables. If selection adds no information about the common coupling after that conditioning, no inferred parent-population distribution or $\xi^{-N}$ correction is needed. Claims about which masses or systems are absent, or about how common a charge branch is, do require the discovery model. The conditioning question comes before choosing a hierarchy.

Published posterior samples cannot simply be multiplied. If samples for system $j$ were obtained under interim prior $\pi_j(\theta_j)$ , reweighting requires the factor $p(\theta_j\mid\Lambda,\mathcal E)/\pi_j(\theta_j)$ inside the integral. The selection-normalized construction and this posterior-recycling logic are derived in Mandel, Farr, and Gair 2019, §§2–4. Although that paper treats a general uncertain population, the same probability identity applies to a pulsar ensemble.

Theory coherence is equally important. A scalar-tensor analysis, for example, must solve the same stellar-structure model for every neutron star, propagate the corresponding charges into conservative and radiative timing parameters, and treat equation-of-state uncertainty consistently. A phenomenological dipole coefficient fitted independently to each system is useful for a null test but is not yet a constraint on one complete theory. Freire and Wex survey how systems with different companions and compactness close complementary theory regions in their 2024 review, §§4–5.

Results and limitations through 2026-08-19

Position or evidence	Population-level result	Limitation or counterevidence
Multiple precisely timed systems are individually compatible with their GR timing maps	A common deviation is not required across the sampled masses, companions, periods, and environments	The sample is targeted, heterogeneous, and incomplete; null sensitivity differs sharply by system
Asymmetric neutron-star–white-dwarf binaries constrain dipole channels while double-neutron-star binaries constrain other combinations	Complementary systems close theory regions that one binary leaves open	Charge maps depend on equation of state, scalar mass, screening, and coupling branch
Leave-one-system-out results can reveal domination by a single high-precision binary	Stable combinations support a genuinely shared constraint	Instability may reflect either an outlier, shared calibration, or legitimate mass-dependent scalarization rather than bad data

The strongest conservative conclusion is that no common departure is needed in the well-studied timing sample under the theory maps that have been evaluated. A competing interpretation is that the sampled masses, companion types, and environments do not span every viable activation or screening window. The latter is a limitation, not positive evidence for an extra field. It remains unresolved how completely current systems cover abrupt, equation-of-state-dependent charge branches and massive-field radiation thresholds.

Versioned inputs, selection, and covariance

Every contributing system needs a primary timing dataset or precisely identified observing release: TOAs, templates and backend metadata, clock and ephemeris corrections, dispersion treatment, fitted parameters and covariance, software version, and retrieval date. Calibration changes can correlate systems observed with the same backend. A common Solar-System ephemeris or time standard creates off-diagonal covariance; treating those errors as independent artificially sharpens the product likelihood.

The selection model records survey discovery thresholds, sky and frequency coverage, acceleration-search sensitivity, source classification, availability of optical or astrometric follow-up, publication choices, timing-baseline cuts, and the criterion for entering the gravity sample. A population of “systems with published useful limits” is not complete in a well-defined physical volume. It can still support conditional inference, provided the conditioning is stated and the claim does not silently extend to unobserved configurations.

Systematics include red timing noise, dispersion and scattering, heterogeneous distances, Galactic acceleration, mass covariance, companion classification, equation of state, system-specific charge branches, and shared clock or ephemeris error. The normalized likelihood, priors on $\lambda$ , $\Lambda$ , and $\mathcal E$ , interval convention, sampler diagnostics, and posterior-predictive residuals must accompany any numerical population result. A machine-readable data product and rerun pipeline are required to reproduce it; summary upper limits are insufficient inputs.

Adversarial population checks

An independent check injects a common coupling into mock systems while calculating object-dependent charges, then verifies unbiased recovery after applying the same detection and follow-up selection. An adversarial analysis repeats the inference after removing each system in turn, changing the equation-of-state hypermodel, perturbing the selection efficiency, and allowing shared calibration covariance. It should also demonstrate the bias produced by naïvely multiplying posterior samples without undoing interim priors.

A useful negative control injects no common coupling but gives one binary an unmodeled red-noise component. A robust hierarchy should identify posterior-predictive failure in that system rather than interpret it as mass-dependent gravity. Conversely, a genuine scalarization threshold should appear coherently in systems near the predicted compactness and should survive alternative timing-noise models.

What could change the conclusion. The population conclusion could change after a new system fills an unsampled mass or companion regime, a superseding timing or astrometric release changes an influential likelihood, an equation-of-state calculation moves a charge branch, or a selection-aware blinded injection produces a stable common coupling under leave-one-out and shared-calibration tests. Any of these changes can alter the mapped theory coverage without invalidating earlier system-specific measurements.

Exercise

Two analyses of the same three binaries use identical likelihoods but different interim mass priors. Why can multiplying their published posterior densities produce different “combined constraints” even though the data are unchanged?

Solution

Each posterior is proportional to its likelihood times its interim prior. Multiplying posteriors therefore multiplies those priors as well as the likelihoods, effectively counting prior information once per system. If the interim mass priors differ, the accidental factor also differs. The correct hierarchical integral divides each posterior representation by its interim prior and inserts the intended common population measure. Therefore the reconstructed likelihood contribution, not the raw posterior density, is what may be coherently combined.

References

Freire, P. C. C., and Wex, N. (2024), “Gravity Experiments with Radio Pulsars,” Living Reviews in Relativity 27, article 5. DOI.
Mandel, I., Farr, W. M., and Gair, J. R. (2019), “Extracting distribution parameters from multiple uncertain observations with selection biases,” Monthly Notices of the Royal Astronomical Society 486, 1086–1093. DOI.