Consistency Requirements and Observational Tests

A low-acceleration theory becomes testable only after its action, matter coupling, solution branch, and regime of validity are fixed. Reproducing an algebraic galaxy scaling is not enough: the same model must conserve the appropriate charges, propagate healthy modes, recover general relativity in its declared limit, and define lensing, cosmology, and strong-field evolution. These are theory-admissibility questions. Likelihoods, survey systematics, parameter bounds, and empirical verdicts are handed off to Tests of Gravity.

Required background. Weak-Field Gravity supplies the Newtonian and post-Newtonian limits used as checks.

Helpful background. Einstein Equations supplies covariant conservation and metric dynamics; Gravitational Actions and Boundary Terms supplies the variational test; Formulations of General Relativity supplies constraint, hyperbolicity, and initial-data criteria; Gravity Beyond General Relativity supplies the common theory-space comparison; and Relativistic Matter and Compact Objects supplies matter and strong-field regimes that a galaxy-only model does not control.

Minimum specification of a theory

Before comparing predictions, record a tuple

$$\mathfrak T= \{S,\ \tilde g_{ab},\ \lambda^I,\ \mathcal B, \ \mathcal D_{\rm init},\ \mathcal C,\ \Lambda_{\rm ctrl}\}.$$

Here $S$ is the action; $\tilde g_{ab}$ is the metric or fields coupled to matter; $\lambda^I$ are parameters and free functions; $\mathcal B$ is the background and solution branch; $\mathcal D_{\rm init}$ gives initial and boundary data; $\mathcal C$ states constraints; and $\Lambda_{\rm ctrl}$ is the cutoff or other control scale. Two papers that differ in any of these entries need not be testing the same theory.

For modified inertia, $S$ must also state the trajectory kernel and its history prescription. For modified gravity, it must state the source equation and boundary conditions. For a relativistic completion, matter coupling cannot be inferred from the gravitational kinetic terms: it determines which rods, clocks, and null rays define observables.

Consistency filters before phenomenology

A useful ordering prevents an excellent fit from rescuing an ill-defined model.

1. Constraint and mode count. Identify primary and secondary constraints, then count propagating scalar, vector, and tensor modes on the relevant background.
2. Stability and characteristics. Require positive kinetic coefficients, acceptable gradient terms, and a hyperbolic initial-value problem in the branch used for prediction.
3. Conservation. Derive the field and matter equations from one invariant action, or otherwise demonstrate the corresponding stress-energy and Noether identities.
4. Limit recovery. Take the high-acceleration or decoupling limit inside the equations and verify the measured Newton constant, physical metric, and number of surviving modes.
5. Regime control. Check that gradients, curvature, frequency, and environmental scales remain below the stated cutoff and away from strong coupling.

The general logic of modes, weak limits, and theoretical viability is developed across Clifton et al. 2012, §§ 2–6. It applies equally to a low-acceleration proposal: a healthy Minkowski quadratic action is necessary evidence, but it does not prove nonlinear stability on a cosmological or compact-object background.

Cross-scale closure

After passing those filters, one fixed $\mathfrak T$ should connect five stages:

1. Specify admissible early-universe data.
2. Evolve their linear perturbations.
3. Follow the formation of nonlinear structure.
4. Derive the dynamics and lensing of galaxies and clusters.
5. Obtain consistent compact objects and radiative modes.

At every transition, the evolved fields must remain on the same admissible branch. The galaxy limit must produce its interpolation function rather than importing one; the physical metric must give both dynamics and lensing; and any unseen mass invoked for clusters must appear in the cosmological stress-energy budget. Compact-object boundary conditions and radiative modes then test whether the low-acceleration branch can be embedded in strong gravity.

No published construction cited here has a theorem closing this entire chain. TeVeS demonstrates a nonconformal relativistic route to dynamics and lensing (Bekenstein 2004, §§ III–VI); the Skordis–Zlosnik model demonstrates quasistatic and linear-cosmological mechanisms plus a quadratic stability analysis on a stated background (Skordis and Zlosnik 2021, equations (5)–(14)). Neither result alone controls every arrow above.

Results and open questions through 2026-08-19

The strongest conservative and permissive interpretations differ because they assign different weight to partial existence results and missing global control.

Evidence or check	Supported conclusion	Missing scope or adverse case
AQUAL/QUMOND actions	Nonrelativistic modified gravity can retain momentum and angular-momentum conservation while producing the low-acceleration law	Does not define light, causal propagation, cosmology, or compact objects
Conformal-scalar counterexample	Altering slow dynamics alone need not enhance lensing because null curves are conformally invariant	A disformal or otherwise enlarged matter coupling can evade this particular obstruction
TeVeS construction	A covariant scalar–vector–tensor theory can connect quasistatic dynamics and lensing	Viability depends on free functions, parameters, background, branch, and all extra modes
Skordis–Zlosnik calculation	One construction supplies a low-acceleration limit, dustlike linear cosmology, and conditions for healthy propagating modes	Its zero-frequency sector needs separate long-wavelength analysis; nonlinear well-posedness, all backgrounds, compact objects, and ultraviolet completion remain separate
Residual cluster requirements	A galaxy-scale algebraic law need not close the mass budget on a different scale	Quantitative significance depends on the full theory, baryonic model, added matter, and empirical systematics

The permissive position is that these mechanisms establish a viable research program: each former obstruction can be addressed in at least one controlled regime. The conservative position is that a patchwork of regimes is not yet a single predictive fundamental theory. The unresolved frontier is whether one fixed action, matter coupling, background history, and cutoff can pass the whole cross-scale chain without exchanging branches or retuning free functions.

The conclusions could change after a theorem changes the admissible theory class, an independent nonlinear calculation changes the stability of a phenomenological branch, a cutoff analysis invalidates a previously used regime, or a fixed prediction survives comparison with observations with its systematics exposed.

What crosses the empirical boundary

The theory side should deliver a versioned action, physical matter metric, parameter and free-function definitions, background and branch, initial and boundary data, cutoff conditions, and a reproducible map from fields to observables. It should also label invalid parameter regions before looking at data and state degeneracies with baryonic or dark-sector physics.

Tests of Gravity develops data selection, likelihood construction, nuisance parameters, calibration and selection systematics, inference, bounds, and comparison with competing models. This separation is not cosmetic. Allowing empirical preferences to silently redefine a free function, branch, or matter coupling would prevent falsification.

Diagnostic exercise

A proposed covariant model reproduces the deep circular scaling and a lensing potential, but on the same background its scalar quadratic action is

$$S^{(2)}_\pi=\frac12\int d^4x \left[-\dot\pi^2-c_s^2(\nabla\pi)^2\right].$$

Should this parameter point be sent to a galaxy likelihood analysis? Explain the independent consistency check that determines whether it is suitable.

Solution

No. With the displayed overall conventions, the coefficient of $\dot\pi^2$ is negative, so the scalar is a ghost on the very background used for the prediction. This is a theory-side exclusion independent of how well the circular and lensing formulas fit data. One should first verify that the sign is not an artifact of an uneliminated constraint or field redefinition by deriving the reduced quadratic action and checking its Hamiltonian. If the negative physical kinetic eigenvalue remains, that parameter point is not passed to likelihood inference. A different healthy branch would constitute a different specified input and must repeat the limit and observable mapping before empirical appraisal.

If $c_s^2\gt0$ in the displayed action, there is a second problem: variation gives $\ddot\pi+c_s^2\nabla^2\pi=0$, hence $\omega^2=-c_s^2 k^2$. The symbol $c_s^2$ in that expression is not a positive physical sound speed squared; the reversed time-kinetic sign makes the spatial gradient unstable as well.

References

• Bekenstein, Jacob D. (2004), “Relativistic Gravitation Theory for the MOND Paradigm,” Physical Review D 70, 083509; erratum 71, 069901. DOI. Open PDF.
• Clifton, Timothy, Pedro G. Ferreira, Antonio Padilla, and Constantinos Skordis (2012), “Modified Gravity and Cosmology,” Physics Reports 513, 1–189. DOI. Open PDF.
• Famaey, Benoît, and Stacy S. McGaugh (2012), “Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions,” Living Reviews in Relativity 15, 10. DOI. Open PDF.
• Skordis, Constantinos, and Tom Zlosnik (2021), “A New Relativistic Theory for Modified Newtonian Dynamics,” Physical Review Letters 127, 161302. DOI. Open PDF.