Large-scale clustering of galaxies in general relativity

TL;DR

This work develops a gauge-invariant framework for the observed clustering of galaxies in general relativity, deriving the observed density contrast in synchronous-comoving gauge and showing how a gauge-invariant bias expansion can be formulated. It demonstrates that the large-scale galaxy power spectrum carries relativistic corrections that depend on the gauge choice for bias and includes volume and magnification effects, ultimately connecting these corrections to an effective local $f_{\rm NL}^{\rm eff}$ on horizon scales. The analysis confirms that, while the relativistic corrections can mimic a small local-type non-Gaussian signal (with $f_{\rm NL}^{\rm eff}$ typically around 0.2–0.5), their angular dependence and scale behavior differ from primordial NG, enabling discrimination with careful modeling. These results provide essential guidance for interpreting future surveys probing modes near the comoving horizon and establish a robust, gauge-consistent basis for large-scale structure analyses in a relativistic framework.

Abstract

Several recent studies have shown how to properly calculate the observed clustering of galaxies in a relativistic context, and uncovered corrections to the Newtonian calculation that become significant on scales near the horizon. Here, we retrace these calculations and show that, on scales approaching the horizon, the observed galaxy power spectrum depends strongly on which gauge is assumed to relate the intrinsic fluctuations in galaxy density to matter perturbations through a linear bias relation. Starting from simple physical assumptions, we derive a gauge-invariant expression relating galaxy density perturbations to matter density perturbations on large scales, and show that it reduces to a linear bias relation in synchronous-comoving gauge, corroborating an assumption made in several recent papers. We evaluate the resulting observed galaxy power spectrum, and show that it leads to corrections similar to an effective non-Gaussian bias corresponding to a local (effective) fNL < 0.5. This number can serve as a guideline as to which surveys need to take into account relativistic effects. We also discuss the scale-dependent bias induced by primordial non-Gaussianity in the relativistic context, which again is simplest in synchronous-comoving gauge.

Figures (8)

• Figure 1: Sketch of perturbed photon geodesics illustrating our notation. The observer is located at the bottom. The solid line indicates the actual photon geodesic tracing back to the source indicated by a star. The dashed line shows the apparent background photon geodesic tracing back to an inferred source position indicated by a circle.
• Figure 2: Coefficients of new terms in observed galaxy overdensity in synchronous comoving gauge Eq. (\ref{['eq:dgk']}) introduced by relativistic volume effect and bias. Here, we plot for $b=1.5$ (solid), $b=2$ (dotted), $b=3$ (dashed) and $b=4$ (dot-dashed) cases, and $b_e$ is calculated assuming the universality of the mass function [Eq. (\ref{['eq:btauU']})]. We ignore the magnification effect by setting $\mathcal{Q}=0$.
• Figure 3: Same as Fig. \ref{['fig:coeffs']}, but for fixed bias ($b=2$) and varying magnification with $\mathcal{Q}=-1$ (solid), $0.5$ (dotted), $1$ (dashed), $2$ (dot-dashed). For diffuse backgrounds, $\mathcal{Q}=1$, and the magnification effect cancels almost all volume distortions.
• Figure 4: Three different theoretical predictions of the observed galaxy power spectrum on large scales for galaxies with linear bias $b=2$ at redshift $z=0.5$, and assuming $\mathcal{Q}=0$: Newtonian linear theory kaiser:1987 (dotted line), relativistic linear theory with linear bias in uniform redshift gauge yoo/etal:2009 (dashed line), and relativistic linear theory with linear bias in synchronous comoving gauge (this work, solid line). We show both line-of-sight directional power spectrum ($\mu=1$, thick lines) and perpendicular directional power spectrum ($\mu=0$, thin lines). The vertical solid line indicates $k = aH$ at $z=0.5$.
• Figure 5: Observed galaxy power spectrum (Eq. (\ref{['eq:Pkg']})) as function of $k_{\parallel}$ and $k_\perp$. The color contours and dashed lines show the Newtonian result from Kaiser kaiser:1987, while the solid lines show the result including relativistic corrections. Here we have set $\mathcal{Q}=0$. For reference, we also show the real space power spectrum contours as white dotted lines.