The effect of matter discreteness on gravitational wave propagation in post-geometrical optics

Technical Summary: The Effect of Matter Discreteness on Gravitational Wave Propagation in Post-Geometrical Optics

Problem Statement
The propagation of gravitational waves (GWs) is conventionally modeled using the geometrical optics approximation, where waves follow null geodesics. This approximation assumes the spacetime curvature radius is significantly larger than the GW wavelength. While this holds for macroscopic objects and the mean cosmic curvature, it potentially breaks down when considering the microscopic discreteness of matter. Individual particles (electrons, baryons, dark matter) generate sharp spikes in spacetime curvature at their locations. Although the mean curvature is negligible, the local curvature at a particle's position can be large, particularly if the particle is localized within its Compton wavelength.

Standard treatments often neglect the direct coupling of GWs to the Riemann tensor in the equation of motion, focusing instead on the background metric. However, for GWs—which have longer wavelengths and lower energies than electromagnetic waves—these curvature terms may induce significant effects on propagation, specifically on the angular diameter distance. Previous work on electromagnetic waves [8] suggested that matter discreteness could significantly alter the angular diameter distance to the Cosmic Microwave Background (CMB), potentially excluding certain dark matter mass ranges. This paper investigates whether similar effects apply to gravitational waves and whether the "post-geometrical optics" approximation remains valid in the presence of such curvature spikes.

Methodology
The authors apply the post-geometrical optics approximation, originally developed for electromagnetic waves in [8], to gravitational waves. The methodology involves:

1. Equation of Motion: Starting from the linearized Einstein equation, the authors derive the wave equation for the trace-reversed metric perturbation $\bar{h}_{\alpha\beta}$. Unlike the electromagnetic case, the GW equation includes a direct coupling to the Riemann tensor (specifically the Weyl tensor $C_{\alpha\mu\beta\nu}$ and Ricci tensor $R_{\alpha\beta}$), represented by the tensor $M_{\alpha\beta\mu\nu}$.
2. Post-Geometrical Expansion: The authors expand the solution in a series $\bar{h}_{\alpha\beta} = \sum \text{Re}(\bar{h}_{\alpha\beta}^n \epsilon^n e^{iS/\epsilon})$, where $\epsilon$ is the ratio of wavelength to other scales. Crucially, unlike standard geometrical optics, the curvature term $M_{\alpha\beta\mu\nu}$ is treated as order $\epsilon^{-2}$, meaning it is not assumed to be negligible compared to the leading kinetic terms.
3. Dispersion Relation: By analyzing the leading order ($\epsilon^{-2}$) terms, the authors derive a modified dispersion relation. Inside matter, the effective graviton mass squared becomes negative ($k^2 = -2\rho/3M_{Pl}^2$), implying that the wave vector becomes timelike and GWs slow down.
4. Angular Diameter Distance Calculation: The authors calculate the correction to the angular diameter distance ($D_A$) by integrating the area expansion rate of the ray bundle. This involves evaluating terms proportional to the square of the density gradients ($\partial_r \rho$) and the density itself, assuming a Gaussian density profile for particles with a width equal to the Compton wavelength.
5. Validity Checks: The authors verify the consistency of the local plane wave approximation by checking if the induced change in wavefront direction (deflection) remains small compared to the phase change, particularly in the presence of steep density gradients.

Key Contributions and Results

• Modified Dispersion and Mass: The study confirms that inside matter, GWs acquire an effective mass term proportional to the energy density $\rho$. The dispersion relation is $E^2 = |\vec{k}|^2 + 2\rho/3M_{Pl}^2$. This leads to non-geodesic motion driven by density gradients.
• Angular Diameter Distance Corrections: The correction to the angular diameter distance depends quadratically on matter density and its derivatives. For a single particle localized within its Compton wavelength, the correction term scales as $m^6 / (M_{Pl}^2 E^4)$.
• Regime of Validity and Caustic Formation:
  • The authors find that for gravitational waves detected by LIGO–Virgo–KAGRA (frequencies $\sim 10$ Hz), the approximation is valid for particle masses $m \lesssim 100$ MeV.
  • However, for electron masses ($m_e \approx 0.5$ MeV), the curvature effect is so large that it leads to the formation of caustics at extremely short distances ($\sim 0.1$ pc).
  • The formation of caustics implies a breakdown of the local plane wave approximation. Consequently, the post-geometrical optics approximation ceases to be applicable precisely in the regime where the curvature effects (and thus the distance corrections) become large.
• Comparison with Electromagnetic Waves:
  • Unlike photons, GWs interact with all matter, including charged particles (electrons and baryons), as they pass through them with negligible interaction.
  • While baryons fall outside the validity regime for current GW detectors, electrons do not. The cumulative effect of electrons leads to strong focusing and caustic formation on sub-cosmological scales.
  • The Weyl tensor, present in the GW equation but absent in the electromagnetic vector potential equation, contributes to the effective mass and polarization evolution. However, for the Gaussian profiles considered, the Weyl contribution is subdominant to the Ricci/density gradient terms.
• Polarization: The analysis indicates that the polarization vector is not parallel transported, exhibiting circular birefringence. However, the hierarchy of equations does not close for polarization at leading orders, preventing a quantitative evaluation of this effect within the current approximation.

Significance and Conclusions
The paper concludes that the "interesting regime" of the post-geometrical optics approximation—where curvature effects are significant enough to alter the angular diameter distance—is self-limiting. The approximation breaks down (via caustic formation) exactly when the curvature effects become large.

• Limitations: The study highlights that the current approximation cannot reliably evaluate the impact of curvature spikes from localized particles (like electrons) on GW propagation over cosmological distances. The assumption of a local plane wave fails when the induced graviton mass becomes comparable to the wave energy or when focusing becomes extreme.
• Future Needs: The authors emphasize that new methods are required to evaluate the effects of curvature spikes and to understand the localization scale of particles (determined by decoherence and interactions) more accurately. The localization scale is critical, as using the Compton wavelength (the assumption made here) maximizes the effect; if particles are less localized, the effect might be reduced, but if they are more localized, the breakdown of the approximation would be even more severe.
• Modest Claims: The paper does not claim to have ruled out specific dark matter models or to have found a new observational signature. Instead, it demonstrates that the theoretical framework used to predict such signatures for electromagnetic waves [8] encounters fundamental validity issues when applied to gravitational waves due to the formation of caustics. The authors stress that the regime of validity is limited and that the approximation is not applicable when curvature effects become large.