From Theory Predictions to Tests of Gravity

A theory calculation becomes usable in an empirical test only when its parameters, branch, validity domain, and uncertainty are portable. A phase coefficient without the action that fixes it, a scalar charge without its stellar branch, or a propagation speed without its background is not yet a theory prediction. This page specifies the theoretical inputs to Tests of Gravity, where likelihoods, data products, and current numerical bounds are developed.

Helpful background. Einstein Equations defines the null theory; Gravitational Actions and Boundary Terms fixes couplings and matter variation; Formulations of General Relativity supplies branch and initial-data distinctions; Relativistic Two-Body Dynamics supplies invariant conservative outputs; Gravitational Radiation and Waveforms supplies detector-frame waveform variables; and Gravity Beyond General Relativity supplies the theory-family viability record.

The portable prediction packet

Begin with a formulation record, not a list of deviations. It must identify:

• the action, dynamical and nondynamical fields, gauge symmetries, physical matter metric, and field normalizations;
• the asymptotic or cosmological background, compact-object solution branch, equation of state or horizon conditions, and matter coupling;
• the approximation hierarchy—weak coupling, PN order, mass ratio, adiabaticity, screening regime, and numerical matching windows—and every relevant cutoff;
• source-frame masses, spins, charges, sensitivities, and frequencies together with the redshift map to detector-frame quantities; and
• conventions for Fourier phase, polarization basis, distance, and the sign of each exterior charge.

These axes cannot be collapsed. A theory can admit several backgrounds; one background can admit several strong-field branches; a branch can change under a different matter coupling; and every calculation occupies only a domain below its cutoff.

For a binary, a compact mathematical packet is a map

$$\mathcal P_{\mathcal T,\mathcal B}: \left(\lambda^I,m_A,S_A,\text{environment}\right) \longmapsto \left\{ E(\Omega),\mathcal F_a, h_A^{\rm src}(f_s), \omega_A(k,z),\nu_A(z),\mathcal M_{AB}(z) \right\},$$

where $\mathcal T$ is the theory and matter coupling, $\mathcal B$ is the background and compact-object branch, $\lambda^I$ are fundamental couplings, $\mathcal F_a$ are all open flux channels, and $\mathcal M_{AB}$ is any propagation-mixing matrix. The packet must also provide its domain $\mathcal D$ and an error decomposition, for example

$$\delta h=\delta h_{\rm PN}+\delta h_{\rm strong} +\delta h_{\rm num}+\delta h_{\rm EOS} +\delta h_{\rm prop},$$

with correlations retained rather than added as independent scalar error bars.

Portable deviation coordinates

Tests may use a generic frequency-domain coordinate system,

$$\widetilde h(f)=\widetilde h_{\rm GR}(f) \left[1+\alpha_{\rm ppE}v^a\right] \exp\!\left(i\beta_{\rm ppE}v^b\right), \qquad v=(\pi G M_z f)^{1/3},$$

where $M_z$ is the redshifted total mass and the amplitude bracket multiplies $\widetilde h_{\rm GR}$; it is not an independently added waveform. Berti, Yagi, and Yunes define this parameterized post-Einsteinian form in Berti, Yagi, and Yunes 2018, § 3.1, eq. (19), pp. 14–15. Generic coordinates are portable for null tests, but testing a specific theory requires the map

$$(\lambda^I,\mathcal B)\mapsto (\alpha_{\rm ppE},a,\beta_{\rm ppE},b)$$

and its domain. It must also carry amplitude ratios for additional polarizations, any frequency thresholds, dispersion coefficients, and $d_L^{\rm GW}/d_L^{\rm EM}$. If one coupling controls conservative dynamics, dipole generation, and propagation, those coefficients are correlated outputs—not independent theory parameters.

Results and open questions as of 2026-08-19

The following matrix records the durable calculation spine inspected through 2026-08-19. It reports no observational bounds.

Theory ingredient	Result and source	Inputs to observational tests	Scope or limitation
compact-body response	$m_A(\varphi_0)$ defines $\alpha_A$ and $\beta_A$; isolated-star exterior matching supplies mass and charge in Damour and Esposito-Farèse 1996, § II, eqs. (2.3)–(2.4); § III, eqs. (3.8)–(3.13)	tabulated $m_A,\alpha_A,\beta_A$ versus asymptotic field, baryon mass, spin, EOS, and branch, with interpolation error	one-scalar model and selected stellar construction; weak-field coupling does not replace a nonperturbative charge
conservative binary dynamics	body-dependent Newtonian coupling and 1PN/2PN equations, Lagrangian, energy, and relative motion in Mirshekari and Will 2013, Table I and eqs. (1.3)–(1.8), pp. 3–5; § VI A–C, eqs. (6.1)–(6.13), pp. 21–26	invariant $E(\Omega)$ or equivalent Hamiltonian, declared PN gauge, charge convention, and truncation estimate	massless scalar–tensor, nonspinning compact bodies, adiabatic point-particle sensitivities, through stated PN order
dissipative generation	leading dipole is $-1$PN relative to tensor quadrupole; balance-law coefficients are derived in Mirshekari and Will 2013, § VI D, eqs. (6.14)–(6.20), pp. 26–27	each open flux, harmonic threshold, absorption term, and the common charges entering conservative motion	near/far-zone PN matching; includes eccentric general motion at that PN order, but is not a generic massive-field, nonadiabatically scalarizing, or merger flux
generic waveform coordinates	amplitude and phase deformations and the separation of generation from propagation in Berti, Yagi, and Yunes 2018, § 3.1, eqs. (18)–(19), pp. 13–15	coefficient map, source/detector-frame definition, parameter correlations, prior-free validity interval	different mechanisms can share the same exponent; a generic coefficient is not a unique fundamental coupling
propagation	modified dispersion and group velocity in Berti, Yagi, and Yunes 2018, § 3.2, eqs. (20)–(22), pp. 17–18; modified damping distance in Belgacem et al. 2018, § II, eqs. (9)–(13), pp. 1–2	polarization-resolved dispersion, damping, mixing, cosmological background, redshift convention, and WKB domain	a speed, mass, or friction law on one background need not apply to another; mixing invalidates a single-distance description
polarization content	six possible metric tidal patterns and detector-response logic in Eardley et al. 1973, pp. 884–886	normalized polarization tensors, antenna response, source excitation, and correlations among modes	kinematic classification does not prove a mode is healthy, radiative, independently excited, or distinguishable
strong-field completion	analytical and numerical scalar–tensor merger results and their open modeling questions are surveyed in Berti, Yagi, and Yunes 2018, §§ 4.1–4.3 and § 5, pp. 25–32	matching window, branch continuity, numerical resolution/error, GR recovery, and an explicit termination if merger is uncontrolled	no universal inspiral-to-ringdown completion follows from a PN coefficient; theory and branch coverage remain incomplete

Competing uses of generic parameters

There are three defensible but different interpretations.

Theory-first interpretation. Only the correlated image of $\mathcal P_{\mathcal T,\mathcal B}$ is called a test of that theory. This preserves common couplings and strong-field branches, but it can withhold a merger prediction when the calculation is incomplete.

Phenomenological interpretation. Independent amplitude, phase, polarization, and propagation coefficients are useful coordinates for finding disagreement with GR. This is broader and can reveal an anomaly, but a nonzero coefficient is not uniquely attributable to one action or coupling.

Strong-field-agnostic interpretation. Treat object charges or sensitivities as independent parameters. This avoids committing to an equation of state or scalarization branch, but it tests a compact-body effective description rather than the underlying microscopic parameter map.

An observational test should state which interpretation is being used. It should never quote a phenomenological coefficient as a theory coupling without the intervening map.

The general inspiral–merger–propagation completion remains unresolved: no single calculation establishes it for every action, background, compact-object branch, matter coupling, and frequency range. This is a boundary on the claim, not a universal negative verdict about alternative-gravity waveforms.

Adversarial and null checks

The independent null check sends all extra charges, response derivatives, non-GR conservative coefficients, extra fluxes, mode masses, speed shifts, damping, and mixing to zero. The result must be a two-tensor GR waveform with $d_L^{\rm GW}=d_L^{\rm EM}$. A decoupling limit should agree whether taken in the action, compact-body solver, PN dynamics, and propagation equation.

Several adverse cases prevent false inference:

• $\alpha_1=\alpha_2\ne0$ removes scalar dipole radiation but not the conservative scalar force;
• a kinematically allowed polarization may have zero source projection or an unhealthy kinetic term;
• GR-like generation can coexist with non-GR cosmological damping;
• scalarized and unscalarized stars at the same fundamental couplings can produce different charges; and
• two mechanisms can generate the same power $v^b$, making only a combination identifiable over a finite band.

These counterexamples are formulation diagnostics. They show why a “no dipole,” “two-polarization,” or “GR-like inspiral” result is not by itself a full GR limit.

What could change these conclusions

The conclusions could change after any primary calculation supplies a controlled alternative-theory merger across a previously missing branch, changes the common generation–propagation parameter map, proves or disproves a mode’s hyperbolicity, or revises an inspected source through a correction, retraction, or convention-changing result. A new likelihood or numerical bound alone changes the observational assessment in Tests of Gravity, without changing this theoretical map.

Exercises

Choose the formulation represented by a result. An analysis varies two independent coefficients, $\beta_{-7}$ for a $v^{-7}$ phase term and $\gamma_{\rm prop}$ for dispersion. A scalar theory instead predicts $\beta_{-7}=C_{\rm gen}\lambda(\alpha_1-\alpha_2)^2$ and $\gamma_{\rm prop}=C_{\rm prop}\lambda$. What must be supplied before the result can be interpreted as a constraint on $\lambda$?

Solution

The two-coefficient analysis is initially a phenomenological null test. A theory interpretation requires the constants $C_{\rm gen}$ and $C_{\rm prop}$ in one normalization; the compact-body branch and charge map $\alpha_A(\lambda,\varphi_0,\mathrm{EOS},m_A)$; source-to-detector frequency conversion; the common validity band; covariance between the two waveform shapes; and evidence that neither correction is already included in the baseline. The likelihood must then be restricted or reweighted onto the one-parameter curve $(\beta_{-7}(\lambda),\gamma_{\rm prop}(\lambda))$. Fitting the coefficients independently and converting each to a separate “bound on $\lambda$” would discard the theory relation and can count the same coupling twice.

References

• Belgacem, Enis, Yves Dirian, Stefano Foffa, and Michele Maggiore. “The Gravitational-Wave Luminosity Distance in Modified Gravity Theories.” Physical Review D 97 (2018): 104066. DOI; Open PDF.
• Berti, Emanuele, Kent Yagi, and Nicolás Yunes. “Extreme Gravity Tests with Gravitational Waves from Compact Binary Coalescences: (I) Inspiral–Merger.” General Relativity and Gravitation 50 (2018): 46. DOI; Open PDF.
• Damour, Thibault, and Gilles Esposito-Farèse. “Tensor-Scalar Gravity and Binary-Pulsar Experiments.” Physical Review D 54 (1996): 1474–1491. DOI; Open PDF.
• Eardley, Douglas M., David L. Lee, Alan P. Lightman, Robert V. Wagoner, and Clifford M. Will. “Gravitational-Wave Observations as a Tool for Testing Relativistic Gravity.” Physical Review Letters 30 (1973): 884–886. DOI.
• Mirshekari, Saeed, and Clifford M. Will. “Compact Binary Systems in Scalar-Tensor Gravity: Equations of Motion to 2.5 Post-Newtonian Order.” Physical Review D 87 (2013): 084070. DOI; Open PDF.