Combining Short-Range Gravity Experiments

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Short-range experiments can be combined only after each one has been mapped into the same physical parameter space with its apparatus response and validity domain intact. Overlaying published exclusion curves is useful for orientation, but it is not a joint inference: contours can use different confidence constructions, composition charges, material models, priors, and environmental assumptions.

Required background. Weak-Field Gravity supplies the common Newtonian baseline; Probability and Statistical Inference supplies likelihood, prior-measure, covariance, and selection reasoning. Helpful background. Classical Field Theory supplies the mediator and environment maps that determine whether two experiments constrain the same parameter.

Put experiments in a common model

Let $\theta$ denote complete shared parameters: for a linear mediator these might include its mass, microscopic couplings to several matter charges, and any mixing angle. Experiment $i$ predicts an observable

\mathbf m_i = \mathcal R_i \left[ \mathbf s_i\bigl(f_i(\theta,E_i),\boldsymbol\nu_i\bigr) \right] +\mathbf b_i(\boldsymbol\nu_i,\boldsymbol\nu_{\rm sh}),

where $E_i$ specifies chamber, density, composition, geometry, and kinematic environment; $\mathcal R_i$ is the instrument response; $\boldsymbol\nu_i$ is experiment-specific; and $\boldsymbol\nu_{\rm sh}$ is shared. The map $f_i$ may reduce to $\alpha_{AB}$ for a linear fifth force, but it is a nonlinear field solution for screening. If two experiments use different test compositions, their quoted $\alpha$ values are not identical unless the charge model makes them so.

For conditionally independent data, a joint likelihood is

\mathcal L_{\rm joint}(\theta) = \int d\boldsymbol\nu_{\rm sh}\, \pi(\boldsymbol\nu_{\rm sh}) \prod_i \left[ \int d\boldsymbol\nu_i\, \mathcal L_i(\mathbf d_i\mid\theta,\boldsymbol\nu_i,\boldsymbol\nu_{\rm sh}) \pi_i(\boldsymbol\nu_i\mid\boldsymbol\nu_{\rm sh}) \right].

This is a likelihood marginalized with specified nuisance priors, not a profile likelihood. It represents a surface-potential calibration, common material density, or common $G$ uncertainty once through shared variables. It does not by itself fix reused measurements: overlapping raw data must be deduplicated or modeled with their joint sampling distribution before the conditional product is valid. If raw likelihoods are unavailable, publish sufficient statistics and covariance or keep the comparison qualitative. Multiplying $p$ -values, posterior densities computed under different priors, or points digitized from 95% contours does not reconstruct this likelihood. The distinctions are summarized in Particle Data Group 2024, §§40.2–40.4, PDF.

For example, two estimates $d_1=\alpha+b+\epsilon_1$ and $d_2=\alpha+b+\epsilon_2$ may share a zero-point error $b$ with variance $s_b^2$ . If independent $\epsilon_i$ each have variance $\sigma^2$ , the mean has variance $\sigma^2/2+s_b^2$ . Treating $b$ as independent between experiments incorrectly gives $(\sigma^2+s_b^2)/2$ . The common calibration floor persists no matter how many measurements reuse it.

Validity is not an exclusion

Each experiment has a theory-validity region $D_i$ . Outside it, an approximation may fail: a body is no longer pointlike, a single-mediator description fails, the chamber changes a screened solution, or the low-energy theory exceeds its cutoff. The correct contribution outside $D_i$ is “not evaluated” or a more complete calculation—not zero likelihood.

Prior transformations matter as well. If $m_\phi=\lambda^{-1}$ , then a prior uniform in $\log m_\phi$ is also uniform in $-\log\lambda$ , but a prior uniform in $m_\phi$ is not uniform in $\lambda$ . Likewise, a prior uniform in a microscopic coupling $g$ induces a nonuniform prior in $\alpha\propto g^2$ . Report the chosen base measure and repeat the combination with defensible alternatives.

Before combination, reproduce each experiment-level observable and contour from its versioned inputs. Recover the individual result when every other likelihood factor is removed. Then run leave-one-class-out combinations—torsion, Casimir, atoms, neutrons, resonators—and inspect whether one class dominates because of genuine information or an unexpectedly narrow nuisance prior. Profile-likelihood asymptotics require regularity and adequate sample size; their boundary behavior and finite-sample caveats are set out in Cowan et al. 2011, §§2.1–2.5 and §4.1.

Results and limitations through 2026-08-19

Position or input class	Evidence or result	Limitation and adversarial check
Torsion and macroscopic force data	Broad separation leverage and geometric harmonics constrain linear finite-range templates	Geometry, magnetic and electrostatic controls, and composition maps differ; recover each Newtonian calibration and perform apparatus-generation leave-out tests
Casimir and microscopic-force data	Short separations probe ranges inaccessible to larger bodies	Optical response, patches, distance zero, roughness, and background theory can dominate; broaden those shared nuisance models before combination
Atom, neutron, and screened-force data	Quantum phases and levels test distinct source, composition, and chamber configurations	Preparation, surfaces, detector response, and nonlinear screening prevent universal overlays; solve each chamber and remove any experiment outside its physical domain

The conservative conclusion is that a combination strengthens a constraint only within the intersection of model definition, apparatus calculation, and statistical compatibility. The competing practice of drawing the envelope of all exclusion contours answers a different question: which published curve is locally strongest. Negative or conflicting evidence includes a combined result that disappears under alternate priors, shared-background covariance, or one experiment’s removal.

An adversarial check is to reproduce every input likelihood, recover it as a special case, then perform leave-one-class-out, alternate-prior, and expanded-systematic fits while hiding a simulated common signal. The method should recover that injection without extending any factor beyond its domain. The remaining synthesis is unresolved or model-dependent whenever only contours, incomplete covariance, or inconsistent parameter definitions are available.

What could change the conclusion. The combined interpretation could change after a superseding calibrated dataset or likelihood product, a corrected common material or geometry input, a new physical parameter map, or an independent reproduction that changes an experiment-level likelihood. Merely redrawing an older contour at a newer date does not justify a changed joint conclusion.

Exercises

Experiment A reports a Gaussian likelihood for a coupling $g$ , while experiment B publishes a posterior for $\alpha=g^2$ obtained with a prior uniform in $\alpha\ge0$ . Explain why multiplying the two plotted curves is invalid and give a valid route to combination.

Solution

The posterior from B contains its prior. Recover $\mathcal L_B(\alpha)\propto p_B(\alpha\mid d_B)/\pi_B(\alpha)$ on the prior’s support, then compose the function with $\alpha=g^2$ : $\mathcal L_B(g)=\mathcal L_B(g^2)$ . A likelihood is a function of parameters, so this substitution introduces no Jacobian. Jacobians enter when transforming a prior or posterior density. B alone cannot distinguish the two signs of $g$ . The joint posterior is proportional to $\mathcal L_A(g)\mathcal L_B(g^2)\pi(g)$ with one declared prior on $g$ . In this special case B’s uniform prior makes its posterior shape proportional to its likelihood inside the stated support, but the support and sign treatment still matter; an arbitrary plotted curve does not supply them.

References

Cowan, G., Cranmer, K., Gross, E., and Vitells, O., “Asymptotic Formulae for Likelihood-Based Tests of New Physics,” European Physical Journal C 71 (2011) 1554, DOI.
Particle Data Group, “Statistics,” in Review of Particle Physics, Physical Review D 110 (2024) 030001, official review PDF.