Scalar Radiation and Cosmological Evolution

The scalar field measured by a compact binary is inherited from a cosmological history, then reshaped by each body’s strong self-gravity. The relevant theory-side outputs are the asymptotic scalar value, its drift, the body-dependent scalar charges, and the flux carried by radiative scalar modes. They are not fixed by the weak-field coupling alone. This page connects those scales and marks which conclusions are robust, branch-dependent, or still unresolved.

Required background. Einstein Equations supplies the FLRW and compact-binary backgrounds used to separate scalar and tensor dynamics.

Helpful background. Gravitational Actions and Boundary Terms supplies the scalar equation and its boundary data; Formulations of General Relativity supplies the gauge-invariant separation of radiative modes; Gravity Beyond General Relativity supplies the validity and branch checks needed before exporting a prediction; Relativistic Matter and Compact Objects supplies sensitivities, stellar structure, and binary multipoles.

From cosmological value to compact-body charge

Use the Einstein-frame normalization

$$\begin{aligned} S={}&\frac{1}{16\pi G_*}\int d^4x\sqrt{-g_*} \left[R_*-2(\nabla_*\phi)^2\right]\\ &+S_m\!\left[A^2(\phi)g^*_{ab},\psi\right], \qquad \alpha(\phi)=\frac{d\ln A}{d\phi}, \end{aligned}$$

and take the scalar to be massless for the moment. On a homogeneous FLRW background, dots denote Einstein-frame cosmic time and $H_*=\dot a_*/a_*$; the scale factor may be normalized to one at a chosen reference time. The scalar equation is

$$\ddot\phi+3H_*\dot\phi =-4\pi G_*\alpha(\phi)(\rho_*-3p_*).$$

Radiation has $\rho_*=3p_*$ and does not directly source the homogeneous mode, whereas nonrelativistic matter generally does. Hubble friction and the shape of $\alpha(\phi)$ can drive the scalar toward, away from, or around a weakly coupled point. Thus the local boundary data $(\phi_\infty,\dot\phi_\infty)$ are outputs of a cosmological branch, not arbitrary universal constants.

For a compact body $A$, integrate out its internal structure while holding its baryon number and entropy fixed. Its effective mass depends on the ambient scalar, and the dimensionless scalar charge is

$$\alpha_A(\phi_\infty) =\left.\frac{\partial\ln m_A(\phi_\infty)} {\partial\phi_\infty}\right|_{N_A,S_A}.$$

Weakly self-gravitating bodies have $\alpha_A\simeq\alpha(\phi_\infty)$. Strongly gravitating stars need not: their binding energy also varies with the ambient scalar, and a scalarized branch can have $\lvert\alpha_A\rvert\sim1$ even when $\lvert\alpha(\phi_\infty)\rvert\ll1$. The point-particle reduction and its strong-field body parameters are developed in Damour and Esposito-Farèse 1996, § II, equations (2.1)–(2.4).

The chain of dependence is therefore

$$\text{cosmological branch} \longrightarrow(\phi_\infty,\dot\phi_\infty) \longrightarrow m_A(\phi_\infty),\alpha_A \longrightarrow\text{near-zone dynamics and radiation}.$$

Changing the equation of state, scalar potential, cosmological initial data, or stellar solution branch can change the middle arrow without changing the weak-field action.

Dipole radiation and its discriminating limits

For a quasi-circular binary with total mass $M=m_A+m_B$, symmetric mass ratio $\eta=m_Am_B/M^2$, and orbital speed $v\ll1$, the leading scalar dipole flux scales as

$$\mathcal F_{\phi}^{\mathrm{dip}} \propto\frac{\eta^2}{G_*} (\alpha_A-\alpha_B)^2v^8.$$

The leading general-relativistic tensor quadrupole scales as

$$\mathcal F_{\mathrm{GR}}^{\mathrm{quad}} \propto\frac{\eta^2}{G_*}v^{10}.$$

Scalar dipole radiation is therefore enhanced by $v^{-2}$ relative to the quadrupole when the charge difference is nonzero. The invariant physical statement is the dependence on $(\alpha_A-\alpha_B)^2$ and the relative post-Newtonian order; numerical coefficients depend on the scalar normalization and orbital variables. Post-Newtonian scalar–tensor equations and flux terms are given in Mirshekari and Will 2013, § VI.D, equations (6.15)–(6.20), and § VII, equations (7.1)–(7.3).

Two adverse checks are immediate. If $\alpha_A=\alpha_B$, the center of scalar charge follows the center of mass and the dipole vanishes even though both bodies are charged. If the scalar has mass $m_s$, a Fourier harmonic of frequency $\omega$ propagates only when

$$k^2=\omega^2-m_s^2\gt0.$$

Below threshold it is evanescent and cannot be counted as flux at infinity. The massless scaling must also be modified near threshold by the group velocity and the appropriate multipole phase space.

Black holes and changing boundary data

On the stationary, asymptotically constant branch of the simplest massless scalar–tensor theories, regular black holes have no independent scalar charge. It is then consistent to set $\alpha_{\mathrm{BH}}=0$ in the point-particle description. This conclusion depends on the boundary condition.

Let the cosmological scalar vary adiabatically, $\phi_\infty(t)=\phi_0+\mu t$, with $\lvert\mu M\rvert\ll1$. Here $M=G_*M_{\mathrm{phys}}$ is the geometrized mass and $\mu=\dot\phi_\infty$ has inverse-time units. A regular Schwarzschild solution can inherit that time dependence. At leading order its far field contains

$$\phi=\phi_\infty(t)-\frac{4M^2\mu}{r}+O(r^{-2}),$$

up to the sign used to define the scalar charge. The charge is secondary and proportional to $M^2\dot\phi_\infty$; it disappears when the asymptotic drift is set to zero. The regular horizon construction is given in Jacobson 1999, §§ I–II, equations (1.1)–(2.7), and its binary interpretation and adiabatic domain are analyzed in Horbatsch and Burgess 2012, §§ 2.2–2.4.

This branch does not contradict a stationary no-hair theorem: the scalar does not inherit the timelike Killing symmetry, and the asymptotic boundary value is time dependent. Whether a realistic cosmological history produces a large enough $\dot\phi_\infty$, while remaining in the same low-energy regime, is a separate calculation.

Results and open questions through 2026-08-19

The evidence cutoff for this research bridge is 2026-08-19. The table separates established mechanisms from the assumptions that prevent them from becoming universal claims.

Evidence or process	Supported result or competing formulation	Scope, limitation, or unresolved status
Compact-star charges and binary flux	A body’s charge is a strong-field derivative of its mass; unequal charges generate a leading dipole term. Scalarized and weakly coupled stellar branches can therefore radiate differently. Damour and Esposito-Farèse 1996, §§ II and IV–V; Mirshekari and Will 2013, §§ VI.D–VII	Equal charges cancel the dipole, a scalar mass can close the radiative channel, and the charge depends on equation of state, boundary value, and solution branch. Generic nonlinear binary charges remain model dependent.
Black holes in massless scalar–tensor gravity	Constant asymptotic data support the conservative no-charge branch; slowly varying data permit a regular secondary charge proportional to $M^2\dot\phi_\infty$. Jacobson 1999, §§ I–II; Horbatsch and Burgess 2012, §§ 2.2–2.4	The permissive conclusion assumes an adiabatic, effectively massless scalar and controlled backreaction. It does not establish an independent stationary black-hole charge or a large cosmological drift.
Cosmological evolution and scalarized branches	Trace sourcing and Hubble friction select $\phi_\infty$; nonperturbative compact-star branches can amplify a locally small weak-field coupling. Doneva et al. 2024, §§ II.A–II.B and VI.A	Attraction toward a weakly coupled point is not universal. Potentials, phase transitions, initial data, matter content, and branch stability can alter the history; compatibility of all epochs and compact branches remains unresolved theory by theory.

The strongest conservative position is that portable predictions should be computed on a cosmological branch already shown to be regular, stable, and within its cutoff, using charges obtained from the matching compact-object solutions. The strongest permissive position is that nonperturbative scalarization or slowly changing boundary data can create substantial local charges even when weak-field coupling is small. These positions are not logical opposites: they emphasize different solution branches and data classes. The evidence does not establish that either mechanism is generic across scalar–tensor theory space.

The conclusions could change after a nonlinear binary calculation changes the charge or radiative scaling, a cosmological analysis moves the admissible solution branch, a stability or cutoff result excludes a mechanism, or an independent derivation establishes scalar radiation in a parameter regime presently listed as unresolved.

From theory predictions to observational tests

A reusable theory result should export $\{\alpha_A,\alpha_B,m_s,\phi_\infty,\dot\phi_\infty\}$, the scalar normalization, polarization content, flux formula, frequency range, solution branch, and cutoff. Detector response, population assumptions, environmental systematics, likelihoods, and numerical bounds belong to Tests of Gravity. A bound obtained for constant charges cannot be applied without qualification to dynamically scalarizing charges, and a massless template cannot constrain a mode that is evanescent over the observed band.

Common pitfalls

Use the weak-field coupling as every body’s charge. Strong binding energy and branch selection make $\alpha_A$ body dependent.

Infer dipole radiation from nonzero individual charges. The leading dipole depends on their difference and vanishes for equal charges.

Treat a cosmological scalar as constant and time dependent simultaneously. The stationary no-charge branch and the adiabatically induced black-hole charge impose different boundary data.

Exercises

Classify the leading scalar radiation for four binaries: equal massless charges; unequal massless charges; a star with charge $\alpha_A$ orbiting a stationary no-hair black hole; and the same system when the scalar frequency lies below a nonzero mass threshold. State which assumption decides each case.

Solution

For equal charges,

$$\alpha_A-\alpha_B=0,$$

so the leading dipole vanishes even though higher scalar multipoles can remain. For unequal massless charges,

$$\mathcal F_{\phi}^{\mathrm{dip}} \propto(\alpha_A-\alpha_B)^2v^8,$$

so dipole radiation is present and is one post-Newtonian order earlier than the tensor quadrupole.

On the stationary, constant-boundary black-hole branch, $\alpha_{\mathrm{BH}}=0$. A charged star then has $\Delta\alpha=\alpha_A$ and radiates a scalar dipole. With slowly changing boundary data this assignment must be replaced by the induced, convention- dependent black-hole charge proportional to $M_{\mathrm{BH}}^2 \dot\phi_\infty$; the charge difference must be recomputed.

Finally, if every relevant harmonic satisfies $\omega\lt m_s$, then $k^2=\omega^2-m_s^2\lt0$. The scalar is evanescent, so the massless dipole formula does not represent flux at infinity regardless of $\Delta\alpha$. The deciding assumptions are, respectively, charge difference, compact-object branch and boundary data, and propagation above the mass threshold.

References

• Damour, Thibault, and Gilles Esposito-Farèse (1996), “Tensor-Scalar Gravity and Binary-Pulsar Experiments,” Physical Review D 54, 1474–1491. DOI; Open PDF.
• Doneva, Daniela D., Fethi M. Ramazanoğlu, Hector O. Silva, Thomas P. Sotiriou, and Stoytcho S. Yazadjiev (2024), “Spontaneous Scalarization,” Reviews of Modern Physics 96, 015004. DOI; Open PDF.
• Horbatsch, Michael, and C. P. Burgess (2012), “Cosmic Black-Hole Hair Growth and Quasar OJ287,” Journal of Cosmology and Astroparticle Physics 2012(05), 010. DOI; Open PDF.
• Jacobson, Ted (1999), “Primordial Black Hole Evolution in Tensor-Scalar Cosmology,” Physical Review Letters 83, 2699–2702. DOI; Open PDF.
• Mirshekari, Saeed, and Clifford M. Will (2013), “Compact Binary Systems in Scalar-Tensor Gravity: Equations of Motion to 2.5 Post-Newtonian Order,” Physical Review D 87, 084070. DOI; Open PDF.