Mapping Theory Parameters Across Regimes

A cross-regime constraint must be written in the parameters of one specified theory, not assembled by identifying similarly named phenomenological coefficients. The same microscopic coupling can produce a PPN coefficient, a compact-body scalar charge, a gravitational-wave phase term, and a cosmological slip function through different solutions and approximations. Conversely, several couplings can project onto the same local observable.

Required background. Probability and Inference for Gravity supplies probability and statistical inference, likelihood transformations, covariance, and hierarchical modeling.

Helpful background. Weak Metric Perturbations and Gauge Freedom supplies weak-field gravity and gauge control. The Tests of Gravity overview supplies the regime map. Gravitational-Wave Astronomy supplies gravitational-wave data analysis and inference.

One theory, several effective coordinates

Let $\alpha$ denote parameters in an action together with boundary or initial data $b$. Experiment $j$ does not usually observe $\alpha$ directly. It measures data $d_j$ through effective quantities

$$\phi_j=F_j(\alpha,b;\mathcal E_j,\mathcal S_j,\mathcal R_j),$$

where $\mathcal E_j$ is the environment, $\mathcal S_j$ the source state, and $\mathcal R_j$ the approximation regime. The likelihood is

$$p(d_j\mid\alpha,b,\eta_j) =p\!\left(d_j\mid F_j(\alpha,b;\mathcal E_j,\mathcal S_j,\mathcal R_j),\eta_j\right),$$

with experiment-specific nuisance parameters $\eta_j$. Combining tests means evaluating these distinct maps at common theory parameters. It does not mean forcing $\phi_1=\phi_2$.

The boundary data can be physical. A cosmological scalar value may determine a local effective coupling; a halo potential can set whether a source is screened; a neutron-star equation of state can determine a strong-field sensitivity. If those quantities are uncertain, they belong in the joint inference rather than being fixed differently in separate translations.

Matching fields, units, and frames

A reliable map begins from an action and its matter coupling. Field redefinitions can move interactions between gravitational and matter sectors without changing operational observables, so coefficients must be compared in a common physical frame. Normalize Newton’s constant consistently: the bare coefficient in an action, the locally measured Cavendish constant, and the cosmological effective strength need not coincide.

Then solve the appropriate background and source problem. A weak-field PPN expansion assumes slowly moving sources and standardized potentials; compact bodies can acquire sensitivities not visible in that expansion; cosmological perturbations depend on scale and time as well as a background history. Will’s PPN construction makes the weak-field matching assumptions explicit Will 2014, §3.2, Box 2, and Table 2, while pulsar mappings show how body-dependent strong-field charges enter timing observables Freire and Wex 2024, §§3.4–5.

Finally, derive the observable in the convention used by each likelihood. Fourier sign, polarization basis, mass definition, redshift convention, and parameter normalization must agree. A sign or factor-of-two mismatch can masquerade as cross-regime tension.

Jacobians reveal what can be combined

Near a reference point, define the map Jacobian

$$J_{ja}=\frac{\partial F_j}{\partial\alpha_a}.$$

With nuisance inputs held fixed, the rank of the stacked Jacobian shows which theory directions change these observables to first order. Complementary regimes help when their rows point in different directions. If all tests constrain the same projection, a longer list of bounds improves precision but not theory-space coverage.

For example, two independent unit-variance measurements of $\phi_1=\alpha_1+\alpha_2$ and $\phi_2=\alpha_1-\alpha_2$ give $J^TJ=2I$ and constrain both directions. Repeating only $\phi_1$ leaves the direction $(1,-1)$ unconstrained however many measurements are added. For a Gaussian likelihood whose mean is $F(\alpha)$ and whose covariance $C$ is parameter-independent, the Fisher information with fixed nuisance inputs is $J^TC^{-1}J$. Fitting uncertain nuisance parameters can reduce this information through their correlations with $\alpha$.

Parameter-dependent covariance carries additional information that this mean-response formula omits. For example, $d\sim\mathcal N(0,\alpha^2)$ with $\alpha\gt0$ has a constant mean and hence $J=0$, yet its score is $\partial_\alpha\ln p=-1/\alpha+d^2/\alpha^3$ and its Fisher information is $2/\alpha^2$. Nonlinear mean maps require care too: at the null of $\phi=\alpha^2$, $J=0$ although a finite upper limit on $\phi$ still bounds $|\alpha|$. Rank loss at a turning point must not be mistaken for complete insensitivity.

Nonlinear maps require the full likelihood. A transformation that folds several theory branches onto one coefficient cannot be handled by translating interval endpoints. Branch weights, stability cuts, and Jacobians change the induced density. When published products contain only marginalized one-dimensional posteriors, an exact reconstruction may be impossible; the honest result is then a bounded approximation, not a synthetic joint constraint.

Renormalization or scale dependence adds another index. A Wilson coefficient quoted at laboratory momentum cannot be copied to binary or cosmological scales without its running and operator mixing. Likewise, a phenomenological graviton-dispersion coefficient is not automatically the same as a mass parameter in a nonlinear completion.

A mapping worksheet

Step	Required information	Independent check
Theory definition	Action, matter coupling, parameter scale, branch	Reproduce field equations and GR limit
Background and source	Boundary data, environment, compactness, equation of state	Solve a limiting analytic or independently computed case
Approximation	Expansion parameters, cutoff, matching region	Vary domain and recover adjacent regimes
Observable	Operational protocol and convention	Check invariance, dimensions, sign, and normalization
Statistical interface	Joint parameter vector, Jacobian, shared nuisance terms	Reproduce the experiment-level likelihood at a benchmark point

The output should be executable or at least machine-readable: for each theory point, return predicted experiment-level parameters plus a validity flag and theory uncertainty. A “no prediction” result caused by instability or cutoff is different from a very poor fit and should not be assigned an arbitrary likelihood.

Gravitational-wave frameworks make the distinction vivid. A complete theory can alter generation, propagation, polarization, and remnant spectra simultaneously; fitting those sectors independently and multiplying their marginalized constraints discards the predicted correlations Yunes, Siemens, and Yagi 2025, §§3–4.

Check your understanding

A toy mapping predicts a local coefficient $\delta_{\rm loc}=2\alpha^2$ and a binary dipole coefficient $B=\alpha^2(s_1-s_2)^2$, where the sensitivities $s_A$ depend on stellar structure. These stipulated relations illustrate parameter mapping; they are not the PPN relations of a specified scalar-tensor theory. Why can a bound on $\delta_{\rm loc}$ not be reused as a direct bound on $B$?

Solution

The two observables share $\alpha$, but the binary coefficient also contains strong-field sensitivities. Those sensitivities can depend on masses, equation of state, cosmological boundary value, and nonlinear scalarization; they are not determined by the weak-field PPN measurement. Therefore the correct combination evaluates both likelihoods at common $\alpha$ while solving or marginalizing the stellar sensitivities consistently. If $s_1=s_2$, dipole radiation can vanish even when the local coefficient is nonzero; if scalarization occurs, the weak-field expansion may fail. This counterexample shows why matching the microscopic parameter is necessary but not sufficient.

References

• Freire, P. C. C., and Wex, N. (2024), “Gravity experiments with radio pulsars,” Living Reviews in Relativity 27, 5. Freire and Wex 2024, §§3.4–5.
• Will, C. M. (2014), “The confrontation between general relativity and experiment,” Living Reviews in Relativity 17, 4. Will 2014, §3.2, Box 2, and Table 2.
• Yunes, N., Siemens, X., and Yagi, K. (2025), “Gravitational-wave tests of general relativity with ground-based detectors and pulsar-timing arrays,” Living Reviews in Relativity 28, 3. Yunes, Siemens, and Yagi 2025, §§3–4.