Large-Scale Structure with Gravitational Waves I: Galaxy Clustering

TL;DR

The study derives the linear-order effects of a stochastic gravitational-wave background on observed galaxy clustering, showing that intrinsic densities are unaltered by tensor perturbations at this order and that observable signals arise from projection effects on light propagation. Using a geodesic-based formalism with a TT tensor perturbation, the authors decompose the GW impact into redshift perturbations, volume distortions, and magnification biases, and present a comprehensive expression for the tensor contribution to the observed density. They compute the angular power spectrum C_l, highlighting the importance of the observer term for the quadrupole and demonstrating the breakdown of the Limber approximation for tensor modes. Across parameter studies, tensor effects are found to be highly suppressed relative to scalar contributions (especially for current limits on the inflationary GW amplitude), with most sensitivity confined to the very largest scales; the most promising signal would come from higher-order, four-point correlations via h_ij δ_g and from dedicated cross-correlation strategies, motivating future work on non-Gaussian and standard-ruler-based approaches. Overall, the work clarifies that, while linear GW imprints on galaxy clustering are theoretically well-defined, their practical detectability with current or near-future surveys is limited, emphasizing alternative observational strategies for GW backgrounds from inflation.

Abstract

Observed angular positions and redshifts of large-scale structure tracers such as galaxies are affected by gravitational waves through volume distortion and magnification effects. Thus, a gravitational wave background can in principle be probed through clustering statistics of large-scale structure. We calculate the observed angular clustering of galaxies in the presence of a gravitational wave background at linear order including all relativistic effects. For a scale-invariant spectrum of gravitational waves, the effects are most significant at the smallest multipoles (2 <= l <= 5), but typically suppressed by six or more orders of magnitude with respect to scalar contributions for currently allowed amplitudes of the inflationary gravitational wave background. We also discuss the most relevant second-order terms, corresponding to the distortion of tracer correlation functions by gravitational waves. These provide a natural application of the approach recently developed in arXiv:1204.3625.

Figures (7)

• Figure 1: Contributions to the observed galaxy angular power spectrum from inflationary gravitational waves, for a sharp source galaxy redshift of $\tilde{z} = 2$, and using the tensor mode power spectrum defined in § \ref{['sec:prelim']}. The black solid line shows the total contribution, while the colored lines show contributions proportional to line-of-sight integrals of $h_\parallel$ (blue dotted), $h_\parallel'$ (green short-dashed) and $\nabla_\Omega^2 h_{\parallel}$ (magenta long-dashed). Here, we have assumed $b_e = 2.5,\; Q = 1.5$. The black square at $l=2$ indicates the result for $l=2$ if the observer term is neglected (see § \ref{['sec:quad']}).
• Figure 2: Contributions to the kernel $F_l(k)$ for $l=2$ (thick) and $l=20$ (thin, only total contribution shown, scaled by $10^4$), for a sharp source redshift $\tilde{z}=2$ and the same parameters as in Fig. \ref{['fig:Cl_TT']}. Note that the separate contributions have non-zero weight for $k\to 0$, while the total $F^g_l$ (black solid) is only non-zero for $k\gtrsim 10^{-4}\:h/{\rm Mpc}$, as required by causality.
• Figure 3: Contributions of tensor modes to the angular galaxy power spectrum for a broad redshift distribution (Eq. (\ref{['eq:dNdz']})) expected for LSST, separated into different contributions as in Fig. \ref{['fig:Cl_TT']} (again using $b_e=2.5,\;Q=1.5$). The thick lines show the exact calculation, while the thin lines show the Limber approximation using Eq. (\ref{['eq:CABLimber']}).
• Figure 4: Total tensor mode contribution to the angular galaxy power spectrum as a function of source redshift, for a Gaussian redshift distribution centered on $\tilde{z}$ with an RMS width of $0.03(1+\tilde{z})$. Here, $b_e=2.5,\;Q=1.5$ fixed.
• Figure 5: Relative impact on the total tensor mode contribution to galaxy clustering when changing $b_e$ from 2.5 to 4 and 1 (thick and thin green long-dashed, respectively), and when changing $\mathcal{Q}$ from 1.5 to 3 and 0 (thick and thin red dot-dashed, respectively). Here, a Gaussian redshift distribution centered on $\tilde{z}=2$ with RMS width $0.03(1+\tilde{z})$ was assumed.