Tests of Gravity

A test of gravity is not a comparison between one equation and one measured number. It is a comparison between complete, operational predictions: a gravitational theory in a stated regime, a model for the source and its environment, a propagation law, an instrument and calibration model, a selection rule, a likelihood, and a decision criterion. This volume develops that chain from laboratory separations to cosmological distances. It explains what present observations establish, what remains conditional on nuisance models or parameterizations, and how apparently independent results may be combined without erasing their domains of validity. Empirical statements are current through 2026-08-19.

Helpful background. Weak Metric Perturbations and Gauge Freedom supplies the weak-field gravity and gauge reasoning used in the post-Newtonian, lensing, and cosmology chapters; Probability and Inference for Gravity supplies the probability, likelihood, prior, covariance, and model-comparison language used throughout.

From a theory to an empirical verdict

On a first visit, start with a familiar experiment: do two clocks at different heights tick at the rate ratio predicted by GR? The calculation predicts a proper-time difference; the apparatus records a frequency comparison; a transfer link, motion, and clock offsets connect the two. A discrepant ratio therefore prompts both a gravity question and a measurement question. Theory, Observable, and Measurement works through this example numerically before introducing the general inference machinery.

For a short first route, follow that clock example, Null Tests and Alternative-Model Tests, and the post-Keplerian timing example. They teach three reusable ideas: predict the recorded quantity, calibrate a discrepancy, and use an additional observable to test parameters inferred from the first measurements. The specialist chapters can then be read as applications of those ideas.

The reusable structure of a gravity test can be written schematically as

$$\begin{aligned} \mathcal T &\longrightarrow s_{\mathcal T}(\vartheta,\lambda;\mathcal E)\\ &\longrightarrow d\sim P_S\!\left(\,\cdot\mid R_{\kappa}[s_{\mathcal T}],\lambda,\kappa\right)\\ &\longrightarrow \mathcal L_S(\vartheta,\lambda,\kappa;d,\mathcal T) \xrightarrow{\;\pi,\mathcal R\;} \mathcal C. \end{aligned}$$

Here $\mathcal T$ is the theory or phenomenological framework, $\vartheta$ is the parameter of gravitational interest, and $\lambda$ collects source and environment parameters in the declared domain $\mathcal E$. The predicted signal $s_{\mathcal T}$ passes through the calibrated response $R_{\kappa}$ with nuisance parameters $\kappa$, and $P_S$ is the full sampling distribution under the declared selection mechanism $S$. The likelihood $\mathcal L_S$ is that sampling model viewed as a function of its parameters, including any selection normalization; $\pi$ is a prior measure when the procedure uses one, $\mathcal R$ is the calibrated interval or model-comparison rule, and $\mathcal C$ is the conclusion that rule licenses. Random noise is only one source of uncertainty. Imperfect source physics, calibration drift, population selection, correlated foregrounds, and approximation error can all imitate or hide a deviation.

This chain makes two distinctions non-negotiable. First, rejecting a particular GR-plus-source model is not automatically a discovery of modified gravity. Second, constraining a coefficient in a parameterized framework does not rule out every complete theory that contains additional fields: the mapping may depend on compactness, frequency, environment, screening, or cosmological history. The statistical vocabulary follows the distinctions among sampling distributions, likelihoods, priors, confidence procedures, and posterior model probabilities set out in Particle Data Group 2024, §§40.1–40.4, PDF.

Pages inherit metric signature $(-,+,+,+)$ and Fourier time dependence $e^{-i\omega t}$ unless they state a meaningful exception. Theoretical formulas normally use $c=\hbar=k_{\mathrm B}=1$ with $G$ explicit, while instrument specifications and measured quantities retain declared SI or astronomical units when those improve interpretation. A result involving an instrument, catalog, or simulation also states the relevant frame, data release, calibration, selection, and approximation regime. These declarations are part of the physical claim, not bookkeeping around it.

The volume in thirteen chapters

The chapters are ordered from test logic, through increasingly demanding physical regimes, to cross-regime synthesis. Each one asks a different question while preserving the same theory-to-data chain.

Chapter	Central task	Boundary that must remain visible
The Logic of Testing Gravity	Turn a hypothesis into an observable, likelihood, and calibrated conclusion.	A significance or Bayes factor tests a declared composite model, not “gravity” in isolation.
Parameterized Frameworks and Null Tests	Build PPN, post-Keplerian, waveform, ringdown, polarization, and residual tests.	A deformation basis is a controlled language for departures, not automatically a covariant or viable theory.
Equivalence Principles and Universality of Free Fall	Compare composition, antimatter, quantum probes, and compact self-gravity.	Weak, Einstein, and strong equivalence principles are logically distinct, and weak-field agreement need not fix compact-body charges.
Clocks, Redshift, and Local Symmetries	Use frequency ratios to test redshift, Lorentz symmetry, position invariance, and varying couplings.	Clock realization, transfer links, geopotential models, and coefficient conventions enter the observable.
Laboratory and Short-Range Gravity	Search for inverse-square violations, fifth forces, and screened scalars with force, torque, atom, and neutron probes.	Geometry, material response, electrostatic and Casimir backgrounds, and chamber-dependent screening can dominate the inference.
Solar-System and Spacecraft Tests	Test orbital motion, light propagation, ranging, radio tracking, and frame dragging.	Ephemerides, reference frames, solar plasma, nongravitational forces, and global covariance prevent a residual from being read as a standalone bound.
Binary Pulsars and Relativistic Timing	Overdetermine relativistic binaries with post-Keplerian parameters, damping, and strong-equivalence tests.	Clock, ephemeris, dispersion, kinematic, Galactic, stellar-structure, and theory-specific charge corrections must be propagated together.
Galactic and Astrophysical Dynamics	Infer gravitational fields from stellar, cluster, galactic, and wide-binary motions.	Baryons, dark matter, anisotropy, disequilibrium, multiplicity, and selection compete with a change in the force law.
Gravitational Lensing Tests	Compare photon propagation, dynamical mass, time delays, and growth through the two metric potentials.	Mass-sheet and source-position transformations, line-of-sight structure, redshift calibration, and source modeling limit uniqueness.
Electromagnetic Strong-Field Tests	Reconstruct strong-field geometry from orbits, spectra, timing, horizon-scale imaging, polarization, and disruptions.	Accretion, plasma, emissivity, variability, distance, and selection are inseparable from the metric comparison.
Gravitational-Wave Tests	Test generation, propagation, polarization, remnant response, residuals, and population consistency.	Waveform error, calibration, event selection, priors, detector geometry, and combination rules condition every bound.
Cosmological Tests of Gravity	Compare expansion, growth, lensing, clustering, screening, and standard-siren propagation.	Gauge and function conventions, bias, nonlinear and baryonic physics, neutrinos, selection, and cross-probe covariance can mimic gravitational slip or altered growth.
Cross-Regime Inference and Evidence Synthesis	Combine experiments only after mapping their parameters, likelihoods, selections, and validity domains.	A universal-parameter overlay or multiplication of overlapping likelihoods can manufacture precision and conceal screening or shared data.

The table also connects to the underlying theory. General Relativity develops the baseline field equations, equivalence principles, and weak-field expansion. Relativistic Two-Body Dynamics, Gravitational Radiation, and Relativistic Cosmology develop the corresponding dynamical, waveform, and cosmological derivations. Gravitational-Wave Astronomy explains detector and catalog inference, while Black-Hole Geometry, Black-Hole Perturbation Theory, and Black-Hole Astrophysics develop the strong-field geometry, response, and source models. The construction and consistency of alternative actions belong to Gravity Beyond General Relativity. The Black Hole Atlas gives object-first orientation for the compact systems used in strong-field tests, the Research Atlas carries dated result summaries, Computational Relativity develops numerical methods and reproducible execution practice, and the Reference carries reusable definitions and conventions. This volume develops the operational comparison: which prediction was tested, with which data and nuisance assumptions, and how strong a conclusion follows.

Three routes through the material

Design a test. Begin with the logic chapter, choose the narrowest useful parameterized or theory-specific framework, then study the relevant physical regime. Finish with cross-regime synthesis only after the single-regime likelihood and failure modes are explicit. This route is best for a researcher turning a proposed observable into an analysis.

Follow increasing field strength and scale. Move from equivalence and clock experiments through short-range and Solar-System tests, then to pulsars, galactic dynamics, lensing, electromagnetic compact-object observations, gravitational waves, and cosmology. Field strength, curvature, potential, acceleration, compactness, and distance do not order all experiments in the same way; the route is therefore comparative rather than a single linear “strength of gravity” axis.

Interrogate an apparent anomaly. Read the relevant domain chapter together with nuisance-systematic analysis, residual tests, and the final synthesis. Ask whether the effect survives a held-back data split, alternative calibration and source models, a coverage or injection study, and a parameter mapping to a second regime. An anomaly that disappears under one of these checks is still scientifically useful: it has located a failure in the composite prediction.

What present evidence allows us to say

The evidence inspected through 2026-08-19 supports a strong but deliberately conditional statement: GR remains compatible with the high-precision tests covered here inside their stated theory, source, instrument, and statistical domains. Local, Solar-System, and classical weak-field tests provide stringent checks of equivalence, post-Newtonian motion, and propagation Will 2014, §§2–6. Relativistic pulsars extend the comparison to compact self-gravity and radiative damping Freire and Wex 2024, §§3–5. Gravitational-wave observations test dynamical generation, propagation, polarization, and remnant response, with conclusions limited by waveform, detector, and population assumptions Yunes, Siemens, and Yagi 2025, §§3–4.

Galactic and cosmological observations do not test field equations in isolation. Their interpretation requires source and matter-sector models, and cosmological probes additionally require a background and perturbation model. They reveal phenomena that restricted baryons-only models do not describe, but do not by themselves decide between new gravitating components, modified field equations, or inadequacies in the astrophysical model. Growth, lensing, clustering, and expansion become a gravity test only when the background, perturbation, bias, neutrino, baryonic, and nonlinear sectors are specified together Ferreira 2019, §§2–6.

Several conclusions therefore coexist without contradiction:

• Agreement with GR in one regime constrains the alternatives active there; it does not prove that all deviations vanish at every scale or environment.
• A null result is informative only relative to its sensitivity. An alternative can evade a test through a weak coupling, a massive excitation, a restricted activation window, or screening.
• An observational residual belongs first to the complete forward model. New gravity becomes a credible explanation only after calibration, source, environment, selection, and statistical alternatives pass discriminating checks.
• A preference between complete theories requires a specified joint analysis. The regime-by-regime examples in this volume do not themselves perform an exhaustive ranking of alternatives or establish a universal cross-regime likelihood.

The agreement summarized here is not a theorem of uniqueness. A new release, a revised calibration, an independently replicated anomaly, or a viable theory with successful held-back predictions can change it. Numerical bounds and catalog membership should always be read from the dated page and named release rather than remembered as permanent constants.

A reusable test-design checklist

Before interpreting a constraint, make the following seven objects explicit.

1. Claim. State the null and alternative, including which parameters are fixed, fitted, or marginalized and what observation would distinguish them.
2. Domain. Specify the field, scale, frequency, environment, matter content, and approximation regime. Check the GR or null limit and at least one other discriminating limit.
3. Forward model. Propagate the theory through source dynamics, radiation or light transport, detector response, and data reduction. Keep convention translations and dimensions visible.
4. Data contract. Name the release, calibration or response version, selections, masks, time span, and any mutable auxiliary product. Record what is actually public enough to reproduce.
5. Uncertainty model. Include nuisance parameters, covariance, foregrounds, population assumptions, waveform or simulation error, and competing source explanations.
6. Calibrated inference. Declare likelihood, priors, interval or decision rule, trials, and combination method. Use injections, mock catalogs, alternative pipelines, or known limits to check coverage and bias.
7. Conclusion and trigger. Say exactly what the result establishes and what remains model-dependent. Name the release, calculation, or observation that would require the conclusion to be reviewed.

This checklist prevents a common asymmetry: demanding a complete theory for a deviation while allowing the reference analysis to hide an incomplete source or instrument model. Both sides of the comparison must meet the same operational standard.

Review the volume

An analysis reports a population-level $3\sigma$ departure in a gravitational-wave deformation coefficient. A Solar-System analysis gives a tight constraint on a parameter with the same symbol. Explain why multiplying the two likelihoods is not yet justified, and list the checks needed before a cross-regime conclusion.

Solution

The shared symbol does not establish a shared physical parameter. First derive both coefficients from one complete theory, including units, normalization, field basis, scale and frequency dependence, compact-body sensitivities, and any screening or environmental branch. Verify that the complete theory is valid in both experimental domains and derive the map into each coefficient; the two regimes need not overlap physically. Next recover the two likelihood factors with their original priors and measures. Where a posterior density is available, dividing by its interim prior recovers a likelihood only on the supported domain and only up to the normalization needed for model evidence. Check whether the analyses reuse events, external data, or uncertain calibration inputs; these create statistical dependence and must appear once or through a joint covariance. Apply population hyperpriors and common theory assumptions only once in the common hierarchy so that they are not double-counted. For the gravitational-wave result, identify the statistic, null calibration, and trials factor behind the reported $3\sigma$, then test waveform alternatives, event selection, posterior predictive behavior, catalog jackknifes, and injection coverage. For the Solar-System result, retain the global ephemeris, propagation, plasma, and spacecraft-force covariance. Only after these steps can one write a joint likelihood with a common theory parameter and correctly represented shared nuisance variables. Otherwise the apparent conflict could be a basis mismatch, screening effect, prior artifact, selection bias, or source-model failure rather than evidence that one regime has falsified the other.

References

• Ferreira, Pedro G. (2019), “Cosmological Tests of Gravity,” Annual Review of Astronomy and Astrophysics 57, 335–374. DOI.
• Freire, Paulo C. C., and Norbert Wex (2024), “Gravity Experiments with Radio Pulsars,” Living Reviews in Relativity 27, article 5. DOI.
• Particle Data Group (2024), “Statistics,” in Review of Particle Physics, Physical Review D 110, 030001. Official review PDF.
• Will, Clifford M. (2014), “The Confrontation between General Relativity and Experiment,” Living Reviews in Relativity 17, article 4. DOI.
• Yunes, Nicolás, Xavier Siemens, and Kent Yagi (2025), “Gravitational-Wave Tests of General Relativity with Ground-Based Detectors and Pulsar-Timing Arrays,” Living Reviews in Relativity 28, article 3. DOI.