Polarization and Propagation Frameworks

Gravitational-wave observations can test more than source dynamics. A theory may add radiative polarizations, alter the dispersion relation, damp amplitudes differently from electromagnetism, rotate polarization during propagation, or change the wave speed. These effects belong to distinct layers of the forward model; combining them into one undifferentiated “propagation test” obscures which data and assumptions constrain which phenomenon.

Required background. Weak Metric Perturbations and Gauge Freedom supplies weak-field gravity, linearized waves, gauge freedom, and tensor polarizations. Probability and Inference for Gravity supplies likelihoods, parameter degeneracy, and model comparison.

Helpful background. Relativistic Two-Body Dynamics supplies source generation and waveform phasing that must be separated from propagation effects.

Polarization as a detector-network problem

For a plane disturbance traveling along a fixed direction, a metric response can be decomposed into tensor plus and cross, two vector, scalar breathing, and scalar longitudinal patterns. This six-polarization classification is kinematic: a specified theory may propagate fewer modes, relate their amplitudes, make them massive, or invalidate the assumptions behind a null plane-wave classification Eardley et al. 1973, main classification.

Detector $I$ measures a linear response

$$h_I(t)=\sum_A F_I^A(\hat{\boldsymbol n},\psi)\,h_A(t-\Delta t_I),$$

where $A$ labels polarizations, $F_I^A$ is the antenna pattern, $\hat{\boldsymbol n}$ is source direction, $\psi$ denotes a polarization-basis angle, and $\Delta t_I$ is the geometric delay. Polarization inference is therefore a rank and conditioning problem for the network response matrix. Nearly aligned detectors can be sensitive to a signal yet poor at distinguishing its polarization content.

The source model matters. Testing pure tensor against pure scalar hypotheses is not the same as allowing a tensor signal plus a small scalar component. Compact-binary inclination, sky position, precession, higher harmonics, calibration phase, and detector glitches can mimic changes in relative response. The three-channel example and tensor-plane figure show exactly which combination removes a tensor signal and why that projection cannot identify every extra mode. Pulsar-timing arrays and astrometric correlations probe different angular response functions and supply complementary baselines.

Dispersion and arrival phase

A phenomenological dispersion relation can be written

$$E^2=p^2+A_\alpha p^\alpha,$$

in units with $c=1$. In a weak-deformation regime, the group velocity $v_g=\partial E/\partial p$ differs from unity by a frequency-dependent term. Propagation over distance then adds a waveform phase correction whose power of frequency is determined by $\alpha$, together with cosmological redshift factors for distant sources.

This parameterization requires a declared sign, normalization, energy range, and background evolution. For some $\alpha$, the phase scaling is degenerate with source-generation terms or arrival time. A massive graviton corresponds to a particular low-frequency dispersion, but a generic fitted $A_\alpha$ need not arise from a consistent massive-gravity theory.

Comparing gravitational and electromagnetic arrival times can constrain speed only after modeling intrinsic emission lag and identifying a common source. A near-luminal result at one frequency does not exclude frequency-dependent dispersion, extra polarizations, amplitude damping, or interactions confined to other environments.

Damping, friction, and birefringence

On a cosmological background, a tensor amplitude may obey schematically

$$h_A''+\bigl(2+\nu_A\bigr)\mathcal H h_A' +\left(c_{T,A}^2 k^2+a^2\mu_A^2\right)h_A=0,$$

where primes denote conformal-time derivatives, $\mathcal H$ is the conformal Hubble rate, $\nu_A$ modifies friction, $c_{T,A}$ is a propagation speed, and $\mu_A$ an effective mass. A nonzero friction modification changes the gravitational-wave luminosity distance relative to the electromagnetic one. That inference is correlated with binary inclination, calibration amplitude, lensing, host redshift, and cosmology.

Parity-violating propagation can make right- and left-handed components accumulate different phases or amplitudes. In a linear basis this appears as polarization rotation or distortion. It must be distinguished from source precession, detector calibration, and a parity-asymmetric selection function. A complete theory predicts how the effect changes with frequency, redshift, and source orientation; a free birefringence coefficient only tests that shape.

Yunes, Siemens, and Yagi separate generation, propagation, polarization, and residual frameworks and provide the relevant waveform mappings Yunes, Siemens, and Yagi 2025, §§3.1–3.5 and §§4.3.6–4.3.9. Preserving that separation prevents a propagation parameter from absorbing an incomplete source waveform.

Comparing predictions with measurements

Question	Minimal ingredients	Independent check
Are non-tensor modes present?	Network geometry, mixed-mode waveform, calibration covariance	Recover injected modes across changing detector subsets
Is propagation dispersive?	Source phasing, distance/redshift, signed dispersion convention	Verify predicted frequency and distance scaling
Is amplitude damping modified?	Distance ladder or host model, inclination, lensing, selection	Compare populations across redshift with shared calibration
Is propagation birefringent?	Polarization basis, precession model, handed detector response	Recover correlated phase/amplitude rotation in independent sources

The GR limit must be explicit: only tensor modes propagate, $A_\alpha=0$, $\nu_A=0$, $c_{T,A}=1$, $\mu_A=0$, and no helicity splitting remains. Each deformation should be switched on separately in injections before combinations are attempted. Multi-parameter fits are physically meaningful only when the data identify the response directions or a theory fixes their correlations.

Check your understanding

Why can two nearly coaligned interferometers detect a compact-binary signal strongly yet distinguish tensor and scalar polarizations poorly?

Solution

Detection asks whether a coherent signal lies in the network’s sensitive subspace. Polarization separation asks whether the columns $F_I^A$ of the response matrix are linearly independent and well conditioned. Nearly coaligned detectors have nearly proportional antenna responses, so different polarization mixtures can produce almost the same pair of strains after amplitude and inclination are adjusted. Therefore high signal-to-noise ratio does not guarantee mode identifiability. A differently oriented detector adds a response row and can increase rank. An external sky position instead reduces uncertainty in the existing matrix and arrival delays; it does not add a detector or automatically increase matrix rank. For a long signal, changing detector orientation can supply further independent responses when the source evolution is modeled consistently.

References

• Eardley, D. M., Lee, D. L., Lightman, A. P., Wagoner, R. V., and Will, C. M. (1973), “Gravitational-wave observations as a tool for testing relativistic gravity,” Physical Review Letters 30, 884–886. Eardley et al. 1973, main classification.
• 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, §§3.1–3.5 and §§4.1–4.4.

Further reading

• Nishizawa, A., Taruya, A., Hayama, K., Kawamura, S., and Sakagami, M.-A. (2009), “Probing non-tensorial polarizations of stochastic gravitational-wave backgrounds with ground-based laser interferometers,” Physical Review D 79, 082002. Nishizawa et al. 2009, §§II–III.