Binary Dynamics and Gravitational Radiation Beyond GR

Alternative-gravity binary predictions form a chain: the action and matter metric determine compact-body response; response determines conservative forces and radiative multipoles; those determine the emitted waveform; and the propagation equations map it to the detector. Breaking that chain produces apparently precise coefficients with no unique physical interpretation. This chapter develops every link while keeping empirical likelihoods and mutable bounds in Tests of Gravity.

Helpful background. Einstein Equations supplies the GR null limit; Gravitational Actions and Boundary Terms supplies field and matter variation; Formulations of General Relativity supplies constraints, gauges, and asymptotic data; Relativistic Two-Body Dynamics supplies PN and balance-law methods; and Gravitational Radiation and Waveforms supplies multipoles, waveforms, and detector response.

Enter this chapter

The recurring scalar example uses the Einstein-frame action

$$S=\frac{1}{16\pi G_*}\int d^4x\sqrt{-g_*} \left[R_*-2(\nabla\varphi)^2\right] -\sum_A\int m_A(\varphi)\,ds_A^* .$$

At fixed baryon number and other adiabatic invariants,

$$\alpha_A=\frac{\partial\ln m_A}{\partial\varphi_0}, \qquad q_A=m_A\alpha_A, \qquad \varphi=\varphi_0-\frac{G_*q_A}{r}+O(r^{-2}).$$

This sign and normalization remain fixed throughout. The effective Newtonian coupling is $G_{12}=G_*(1+\alpha_1\alpha_2)$, while the leading scalar dipole is controlled by $\Delta\alpha=\alpha_1-\alpha_2$. In Jordan variables the traditional sensitivity $s_A=\partial\ln M_A/\partial\ln\phi$ is translated at the interface rather than silently identified with $\alpha_A$. Damour and Esposito-Farèse define the strong-field mass derivatives and their stellar matching in Damour and Esposito-Farèse 1996, §§ II–III, eqs. (2.3)–(2.4) and (3.8)–(3.13).

The site uses signature $(-,+,+,+)$, explicit $G$, and Fourier convention $e^{-i\omega t}$. Source-frame frequency is converted to detector-frame frequency before applying cosmological propagation. A field redefinition may change intermediate charges or coefficients, but not a dimensionless detector observable when masses, clocks, matter coupling, and boundary data are transformed consistently.

The six-page route

1. Compact-Body Sensitivities and Charges turns an isolated object’s mass response into scalar or vector worldline data. It distinguishes the positive source convention $q_A=m_A\alpha_A$ from the negative exterior coefficient and requires fixed-baryon-number matching on a declared solution branch.
2. Post-Newtonian Dynamics Beyond GR derives the body-dependent Newtonian potential, organizes scalar and preferred-frame operators by PN order, and replaces gauge-dependent coordinate coefficients with invariant relations such as $E(\Omega)$.
3. Dipole and Additional-Field Radiation derives scalar multipoles from the retarded field. For a circular binary, $\boldsymbol D=\mu\Delta\alpha\,\boldsymbol r$, so dipole power is $-1$PN relative to tensor quadrupole power. Charge symmetry, mass thresholds, or absent healthy modes can remove that channel.
4. Polarizations, Speeds, and Propagation separates six possible metric tidal patterns from the physical eigenmodes selected by a theory’s principal symbol. It then derives group delay, dispersive phase, damping, and mode mixing.
5. Waveform Construction and Cross-Regime Consistency converts the accumulated source phase to Fourier phase and carries one coupling through generation and propagation in an executable four-refinement benchmark. It recovers the analytic GR chirp and demonstrates a finite phase discrepancy when the generation correction is counted twice.
6. From Theory Predictions to Tests of Gravity specifies the action, branch, charges, invariant dynamics, fluxes, propagation, cutoffs, and correlated uncertainty needed for null tests without importing empirical bounds.

Mirshekari and Will provide the massless scalar–tensor calculation from sensitivities through 2.5PN motion and balance laws in Mirshekari and Will 2013, §§ II–VII, pp. 6–31. Berti, Yagi, and Yunes explain how theory calculations map to generic generation and propagation coordinates, and why inspiral-to-merger completion is theory dependent, in Berti, Yagi, and Yunes 2018, §§ 3–5, pp. 12–32.

From conservative motion to detector strain

Once isolated-body matching has supplied $m_A$, charges, and response derivatives, the conservative calculation produces an invariant binding energy $E(\Omega)$. Radiation generation supplies a sum over all healthy open channels,

$$\mathcal F=\mathcal F_T+\mathcal F_\varphi +\mathcal F_V+\cdots,$$

and adiabatic balance gives

$$\dot\Omega=-\frac{\mathcal F}{dE/d\Omega}.$$

Source multipoles then determine polarization-resolved amplitudes and phases. A propagation operator uses the same theory parameters and background to apply dispersion, damping, and mixing:

$$\{h_A^{\rm src}(f_s)\} \xrightarrow[ f_s=(1+z)f\ ]{\text{propagation}} \{h_A^{\rm det}(f)\}.$$

Finally a detector tensor projects these fields into measured strain. The possible six metric tidal patterns were identified by Eardley et al. 1973, pp. 884–886, but a kinematic basis does not establish that every pattern is dynamically healthy, sourced, or independently measurable.

Four separations keep the calculation honest:

• generation versus propagation: the former changes orbital evolution and emitted fields; the latter changes waves after emission;
• coordinate coefficients versus invariants: an acceleration depends on PN gauge, while $E(\Omega)$ and detector strain are portable;
• possible polarization versus physical mode: constraints, kinetic signs, hyperbolicity, and matter coupling must be checked before interpretation; and
• theory prediction versus empirical test: this chapter supplies maps and domains, while Tests of Gravity develops likelihoods, systematics, current bounds, and empirical conclusions.

Validity boundaries

No equation in this chapter is universal across all alternatives to GR. The scalar derivations assume a canonical long-range mode and adiabatic compact-body charges. The PN expansion requires $v\ll1$ and gradients below the field-theory cutoff. Massive fields introduce harmonic thresholds; scalarization can make charges time dependent; preferred-vector theories require a declared æther frame; and cosmological damping requires a specified background. A controlled inspiral calculation does not automatically extend through merger and ringdown. Where that extension is missing, the waveform must terminate with an explicit validity statement.

The decisive GR check is simultaneous: extra charges and response derivatives vanish, $G_{12}\to G$, additional fluxes close, only two tensor polarizations remain, their speed is one, and propagation damping reduces to the electromagnetic luminosity distance. Equal charges alone remove dipole radiation but do not satisfy this full limit.

Review the chapter

Diagnose a proposed waveform. A model uses weak-field scalar coupling $\alpha_0$ for both neutron stars, sets the dipole term to zero because the charges are equal, imports a merger correction calibrated on a scalarized branch, and adds a propagation phase with an independent coefficient. Identify the required repairs before this is a theory prediction.

Solution

First compute each star’s strong-field $\alpha_A$ and $\beta_A$ at fixed baryon number, asymptotic field, EOS, mass, and branch; weak-field $\alpha_0$ need not equal either charge. Second use those same charges in $G_{12}$, $E(\Omega)$, and every flux. Equal charges can consistently remove the leading dipole, but they do not remove conservative scalar effects or higher multipoles. Third, the merger model must lie on the same scalarized or unscalarized branch and overlap a controlled inspiral window; otherwise terminate the waveform before merger. Fourth, derive the propagation coefficient from the same fundamental coupling and cosmological background, retain its correlation with generation, and convert source to detector frequency. Finally verify the simultaneous GR limit, PN and EFT cutoffs, numerical convergence, and absence of duplicated corrections before handing the model to Tests of Gravity.

References

• 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.