Imprints of gravitational-wave polarizations on projected tidal tensor in three dimensions

Gravitational waves (GWs) distort galaxy shapes through the tidal effect, offering a novel avenue to probe the nature of gravity. In this paper, we investigate how extra GW polarizations beyond those predicted by general relativity imprint observable signatures on galaxy shapes. Since galaxy shapes are measured as two-dimensional images projected onto the celestial sphere, we present three-dimensional statistical quantities of the projected tidal tensor sourced by the tensor perturbation. We show that the presence of extra polarization modes modifies both the amplitude and angular dependence of the correlation functions. Furthermore, we identify a distinct observational channel for probing parity violation in helicity-two and helicity-one modes. In particular, we show that if they propagate at different speeds, galaxy surveys can disentangle the source of parity violation. Our findings establish a theoretical framework for using upcoming large-scale galaxy surveys to test modified gravity theories through the polarization content of GWs.

💡 Research Summary

This paper develops a comprehensive theoretical framework for probing the polarization content of a stochastic gravitational‑wave (GW) background through its tidal imprint on galaxy shapes. While intrinsic alignments (IA) of galaxies have traditionally been modeled as a response to the scalar gravitational potential, the authors emphasize that the time‑derivative of the tensor perturbation itself generates a traceless tidal tensor that directly distorts galaxy ellipticities.

In Section 2 the authors introduce a general decomposition of the metric perturbation h_{ij} in the synchronous gauge into six helicity states: the two GR‑allowed helicity‑2 modes (λ=±2), two helicity‑1 vector modes (λ=±1), a breathing scalar (λ=b) and a longitudinal scalar (λ=ℓ). Each mode is assigned an independent propagation speed c_{λ} and an initial power spectrum P^{(λ)}{ini}(k). The stochastic background is assumed isotropic, so cross‑spectra between different λ vanish. A chirality parameter χ^{(T)} (for helicity‑2) and χ^{(V)} (for helicity‑1) quantifies any left‑right asymmetry. The sub‑horizon solution of the wave equation yields h{λ}(η,k)≈a(η)^{-1}cos