Gravitational waves in Palatini gravity for a non-minimal geometry-matter coupling

We discuss the propagation of gravitational waves over a non-Riemannian spacetime, when a non-minimal coupling between the geometry and matter is considered in the form of contractions of the energy momentum tensor with the Ricci and co-Ricci curvature tensors. We focus our analysis on perturbations on a Minkowski background, elucidating how derivatives of the energy momentum tensor can sustain non-trivial torsion and non-metricity excitations, eventually resulting in additional source terms for the metric field. These can be reorganized in the form of D’Alembert operator acting on the energy momentum tensor and the equivalence principle can be reinforced at the linear level by a suitable choice of the parameters of the model. We show how tensor polarizations can exhibit a subluminal phase velocity in matter, evading the constraints found in General Relativity, and how this allows for the kinematic damping in specific configurations of the medium and of the geometry-matter coupling. These in turn define regions in the wavenumber space where propagation is forbidden, leading to the appearance of typical cut-off scale in the frequency spectrum.

💡 Research Summary

This paper presents a detailed investigation into the propagation of gravitational waves (GWs) within a modified theory of gravity formulated in the Palatini approach, featuring a non-minimal coupling between geometry and matter. The primary goal is to understand how such couplings alter GW propagation through material media, potentially enabling phenomena like subluminal phase velocities and Landau damping, which are forbidden in standard General Relativity (GR) for tensor modes in flat spacetime.

The authors construct a general action within the metric-affine formalism, where the connection is independent of the metric. The gravitational Lagrangian is a function f(R, X, Y), where R is the Ricci scalar, and X and Y are two specific scalar contractions between the curvature tensors and the energy-momentum tensor T_μν: X = R_μν T^μν and Y = g^μν R^ρ_μσν T^σ_ρ (the contraction with the co-Ricci tensor). These couplings are chosen because they preserve projective invariance, a symmetry relevant for stability in metric-affine theories. The model is recast into a dynamically equivalent multi-scalar-tensor theory for analytical convenience.

The core technical achievement lies in solving the system of equations. Varying the action with respect to the independent connection yields an algebraic (not differential) equation linking the torsion and non-metricity components linearly to first derivatives of T_μν. In the linearized regime around a Minkowski background, this equation can be solved exactly. Substituting this solution into the linearized field equations for the metric perturbation h_μν results in an effective GW equation.

The key result is that the non-minimal coupling, particularly through the Y-term (co-Ricci coupling), modifies the linearized GW equation in a significant yet structured way. The final equation for the tensor modes takes the form: ◻h_μν = standard GR source + β ◻T_μν, where β is a parameter determined by the coupling constant. This implies that derivatives of the matter distribution itself act as an additional dynamic source for GWs. The parameter associated with the X-term (Ricci coupling) can be set to zero to better recover the equivalence principle at the linear level.

To explore the physical consequences, the authors analyze GW propagation through a collisionless medium using kinetic theory. They couple the modified GW equation to the Vlasov equation for the particle distribution function. Analyzing plane-wave solutions reveals a modified dispersion relation ω(k). Two major phenomenological consequences emerge:

1. Cut-off Scale: A critical wavenumber k_c exists below which no real-frequency solution for the GW exists, implying complete suppression of propagation for long wavelengths (low frequencies). This cut-off scale k_c is inversely proportional to the coupling parameter β and the mean thermal speed of the medium particles.
2. Subluminal Propagation and Landau Damping: For wavenumbers above k_c, the phase velocity of the GW can become subluminal (less than the speed of light) depending on the medium’s velocity distribution. This condition opens the door for Landau damping – a kinematic damping mechanism where the wave transfers energy to particles in the medium via resonance. This is a striking result, as such damping is strictly prohibited for tensor GWs in GR on a flat background with an isotropic medium.

The paper also discusses phenomenological constraints. The cut-off scale k_c influences the static limit of the theory, leading to a modified Poisson equation for the Newtonian potential. Using precision solar system tests (e.g., from the Cassini mission), a lower bound on k_c (and thus an upper bound on the coupling strength) is estimated. These predictions suggest that future third-generation GW observatories (like Einstein Telescope and Cosmic Explorer), with their enhanced sensitivity across a broad frequency band, could potentially detect signatures of such modified propagation, such as frequency-dependent attenuation or band-stop filtering effects in the GW signal from distant sources. The work highlights how Palatini formulations of geometry-matter coupling can lead to distinct, testable signatures in GW astronomy that differ from their metric counterparts.