Cosmological and Radiative Predictions

Vector and preferred-foliation theories connect cosmology to radiation through the same action parameters, but not through one universal observable. In Einstein–æther gravity, different coupling combinations renormalize the cosmological and local Newton constants, while compact-body sensitivities control additional radiative channels. In generalized Proca theory, a temporal vector can modify the FLRW background even though its temporal component remains constrained. Whether a single nontrivial branch survives every background, compact-body, cutoff, and empirical test remains unresolved. These are branch-level predictions; empirical likelihoods and numerical bounds require an additional analysis of observations.

Required background. Frames, Tetrads, and Spinors supplies the comoving æther frame and polarization decomposition.

Helpful background. Einstein Equations supplies FLRW and wave equations; Gravitational Actions and Boundary Terms supplies action variation; Formulations of General Relativity separates constraints and radiation; and Gravity Beyond General Relativity supplies validity and theory-comparison criteria.

Cosmological versus local gravity in æther theory

Use signature $(-,+,+,+)$, minimally coupled matter, a spatially flat FLRW metric, and the isotropic æther branch

$$ds^2=-dt^2+a(t)^2\delta_{ij}dx^i dx^j, \qquad u^a=(1,0,0,0).$$

Spatial isotropy forces the æther to coincide with the comoving normal. Its stress tensor renormalizes the coefficient in the Friedmann equation, giving

$$G_{\mathrm{cosmo}}= \frac{G_{\mathrm{ae}}}{1+(c_{13}+3c_2)/2}.$$

The weak-field, static constraint instead gives

$$G_N=\frac{G_{\mathrm{ae}}}{1-c_{14}/2},$$

so

$$\frac{G_{\mathrm{cosmo}}}{G_N} =\frac{1-c_{14}/2}{1+(c_{13}+3c_2)/2}.$$

These relations are Jacobson 2008, §§2–3, Eqs. (2.1) and (3.2), PDF, translated from the source’s $(+,-,-,-)$ convention together with its unit-vector and action signs. Both denominators must be finite and have the sign required by their respective background equations. Equality of the two measured couplings occurs when $c_{13}+3c_2=-c_{14}$, but that single relation does not make perturbations or compact bodies identical to GR.

The result assumes the two-derivative action, homogeneous æther alignment, minimal matter coupling, and background inverse curvature radii and time-variation frequencies below the EFT cutoff. It is a background equation, so no subhorizon wavelength assumption is needed. Anisotropic vector backgrounds, a different matter metric, or higher-spatial-derivative Hořava operators define different cosmological calculations.

Weak-field radiation and extra channels

The flat-background spectrum contains spin-2, spin-1, and spin-0 waves with different speeds. For a slowly moving, weakly self-gravitating source, Foster’s leading post-Newtonian flux can be organized as

$$\dot E=-G_N\left\langle \frac{\mathcal A}{5}\dddot Q_{ij}\dddot Q_{ij} +\mathcal B\,(\dddot I)^2 +\mathcal C\,\dot\Sigma_i\dot\Sigma_i \right\rangle.$$

$Q_{ij}$ is the trace-free quadrupole, $I$ the trace moment, and $\Sigma_i$ a dipolar source. The dimensionless coefficients $\mathcal A$, $\mathcal B$, and $\mathcal C$ depend on the æther couplings and mode speeds. The derivation uses retarded Green functions for each wave sector and an energy-flux calculation at infinity Foster 2006, §§III–IV, Eqs. (1) and (71)–(79), PDF.

When the preferred-frame PPN parameters vanish and internal fields are weak, the monopole and dipole contributions can vanish at this order. An additional relation $\mathcal A=1$ then reproduces the GR quadrupole damping rate. This is a useful negative limit: one radiative observable can be GR-like on a nonzero-coupling curve even though extra modes still exist. Foster explicitly warns that the weak-field match does not establish viability for strongly self-gravitating bodies Foster 2006, §V, PDF.

Compact bodies and sensitivities

A compact object samples the ambient æther through its binding energy. Its long-distance worldline action is

$$S_A=-\int d\tau_A\,\widetilde m_A(\gamma_A), \qquad \gamma_A=-u_a v_A^a,$$

or, expanded about rest relative to the æther,

$$S_A=-m_A\int d\tau_A\left[ 1-\sigma_A(\gamma_A-1) +\frac{\sigma'_A}{2}(\gamma_A-1)^2+\cdots\right],$$

where

$$\sigma_A=-\left.\frac{d\ln\widetilde m_A}{d\ln\gamma_A} \right|_{\gamma_A=1}.$$

Here $m_A=\widetilde m_A(1)$ and

$$\sigma'_A=\sigma_A+\sigma_A^2+ \left.\frac{d^2\ln\widetilde m_A}{d(\ln\gamma_A)^2}\right|_1.$$

The prime labels the quadratic sensitivity coefficient; it is not a time derivative. The minus sign in the linear term follows directly from the definition of $\sigma_A$. As a check, in a local æther-rest frame $\gamma_A=1+v_A^2/2+O(v_A^4)$ and $d\tau_A=dt(1-v_A^2/2+O(v_A^4))$. Hence

$$L_A=-m_A+\frac12m_A(1+\sigma_A)v_A^2+O(v_A^4),$$

so the inertial coefficient is $m_A(1+\sigma_A)$, distinct from the rest parameter $m_A$.

This effective action and its matching interpretation are derived in Foster 2007, §II.A, p. 6, Eqs. (4)–(6), PDF. The sensitivities depend on the theory couplings, equation of state, compactness, and solution branch; they are not free universal constants. Unequal sensitivities give bodies different æther charges and can source dipole radiation, while common combinations also modify conservative dynamics and quadrupolar flux Foster 2007, §§III–IV, PDF.

The point-particle expansion assumes body size much smaller than orbital separation and background curvature radius, slow motion, adiabatic internal structure, and a successful match to a regular isolated-body solution. A sensitivity inferred for one stellar model cannot be transferred to a black hole or another equation of state without a new matching calculation.

Generalized Proca cosmology

For a flat FLRW background, isotropy permits a temporal vector

$$A^\mu=(\phi(t),0,0,0),\qquad X=\frac{\phi^2}{2},$$

so $F_{\mu\nu}=0$ on the background. Varying the generalized Proca action through $\mathcal L_5$ yields a vector equation with no $\dot\phi$,

$$\phi\left[G_{2,X}+3HG_{3,X}\phi+6H^2G_{4,X} +6H^2\phi^2G_{4,XX} -3H^3\phi G_{5,X}-H^3\phi^3G_{5,XX}\right]=0.$$

There is a trivial branch $\phi=0$ and a nontrivial algebraic branch on which the bracket vanishes and relates $\phi$ to $H$. Constant $H$ and $\phi$ can then define a de Sitter fixed point. The absence of $\dot\phi$ is the cosmological imprint of the constraint that removes the fourth vector polarization De Felice et al. 2016, §II, Eqs. (2.10)–(2.15), PDF.

Perturbations still contain two tensor, two transverse-vector, and one longitudinal-scalar gravitational/vector modes. After lapse, shift, and temporal-vector perturbations are eliminated, their high-momentum quadratic actions require

$$q_T\gt0,\quad q_V\gt0,\quad q_S\gt0, \qquad c_T^2\gt0,\quad c_V^2\gt0,\quad c_S^2\gt0.$$

The explicit background-dependent coefficients are derived in De Felice et al. 2016, §§III–IV, PDF. Because intrinsic vector interactions proportional to $F_{\mu\nu}$ vanish on the temporal background but enter $q_V$ and $c_V^2$, a viable expansion history does not imply viable perturbations.

Results and limitations through 2026-08-19

The evidence cutoff for this frontier assessment is 2026-08-19.

Theory, branch, and regime	Calculation or result	What is established	Limitation and exact locator
Einstein–æther; homogeneous isotropic background and weak-field static branch	Distinct $G_{\mathrm{cosmo}}$ and $G_N$ are fixed by different coupling combinations	A direct theory-side link between background expansion and local gravity	Equality can occur on a non-GR coupling surface; Jacobson 2008, §§2–3, PDF
Einstein–æther; weak source and leading post-Newtonian radiation	Spin-2, spin-1, and spin-0 flux reduces to quadrupole, monopole, and dipole source terms	Extra channels and a GR-like damping subspace are analytically identified	Neglects strong internal fields; Foster 2006, §§III–V, PDF
Einstein–æther; compact objects represented by matched worldlines	Body-dependent sensitivities modify conservative dynamics and radiative multipoles	Strong-field structure enters through explicit matching coefficients	Sensitivities require object- and branch-specific calculations; Foster 2007, §§II–IV, PDF
Generalized Proca; temporal-vector FLRW branch with perfect-fluid matter	Algebraic background branch admits de Sitter solutions; quadratic tensor, vector, and scalar conditions are derived	Existence and linear stability criteria for specified functions and histories	Not a universal attractor or empirical fit for arbitrary $G_i$; De Felice et al. 2016, §§II–IV, PDF

Competing interpretations and observational tests

The strongest permissive interpretation is that shared couplings generate cross-regime consistency tests: the same æther parameters enter $G_{\mathrm{cosmo}}/G_N$, mode propagation, and radiation, while generalized Proca functions control both the background branch and perturbation kinetics. A model surviving all of these calculations becomes sharply predictive.

The strongest conservative interpretation is that no single “vector signature” exists. Background choice can hide intrinsic vector operators, weak-field radiation can lie on a GR-like degeneracy, compact sensitivities are body dependent, and a different matter coupling changes every observable map. Agreement in one regime cannot be promoted across the others without explicit matching.

To compare these theories with data, begin with the background equations, flux structures, sensitivities, polarizations, and validity regimes developed here. Population assumptions, waveform systematics, cosmological nuisance parameters, likelihoods, and current observational bounds are developed in Tests of Gravity. Microscopic completions and radiative naturalness are studied in Quantum Gravity.

The conclusions could change after a new isolated-body or merger calculation changes sensitivities; a nonlinear stability result changes an FLRW branch; a cutoff or radiative correction invalidates the classical functions used here; or an empirically tested joint cosmology-and-radiation prediction survives the full Tests-of-Gravity systematics analysis.

Common pitfalls

Using $G_N$ in the Friedmann equation by definition. The æther stress renormalizes the cosmological coefficient differently. Derive both constants before comparing them.

Assuming a temporal vector has no perturbative vector physics. $F_{\mu\nu}=0$ on the background, but intrinsic vector operators reappear at quadratic order.

Exercises

Find the æther coupling relation for which $G_{\mathrm{cosmo}}=G_N$. Then explain why a binary with unequal sensitivities can still differ radiatively from GR on that surface.

Solution

Equating the denominators gives

$$1+\frac{c_{13}+3c_2}{2}=1-\frac{c_{14}}2,$$

and therefore

$$c_{13}+3c_2=-c_{14}.$$

This makes the homogeneous expansion and weak-field static experiments use the same effective Newton constant, but it does not set the individual mode couplings or compact charges to their GR values. If $\sigma_1\ne\sigma_2$, the two bodies respond differently to the ambient æther. Their charge contrast supplies a nonzero dipolar source $\Sigma_i$, so the $\mathcal C\,\dot\Sigma_i\dot\Sigma_i$ term can remain even though $G_{\mathrm{cosmo}}/G_N=1$. Background degeneracy is therefore not radiative equivalence.

References

• De Felice, A., L. Heisenberg, R. Kase, S. Mukohyama, S. Tsujikawa, and Y.-L. Zhang. “Cosmology in Generalized Proca Theories.” Journal of Cosmology and Astroparticle Physics 06 (2016) 048. DOI. Open PDF.
• Foster, B. Z. “Radiation Damping in Einstein-Aether Theory.” Physical Review D 73 (2006) 104012. DOI. Open PDF.
• Foster, B. Z. “Strong Field Effects on Binary Systems in Einstein–Aether Theory.” Physical Review D 76 (2007) 084033. DOI. Open PDF.
• Jacobson, T. “Einstein-æther gravity: a status report.” Proceedings of Science QG-PH (2007) 020. DOI. Open PDF.