Going beyond the Kaiser redshift-space distortion formula: a full general relativistic account of the effects and their detectability in galaxy clustering

TL;DR

The paper develops a full general relativistic account of galaxy clustering beyond the Kaiser redshift-space distortion formula, identifying two horizon-scale GR terms linked to velocity and gravitational potential that survive on large scales. It shows Newtonian redshift-space distortions recover the Kaiser form for small scales but miss crucial GR effects such as luminosity-distance fluctuations, which become relevant only at large scales. Using a Fisher matrix approach, the authors demonstrate that single-tracer surveys are cosmic-variance limited and cannot detect these GR terms, while a multi-tracer, shot-noise-cancelling strategy can boost detectability, potentially yielding significant measurements of the velocity term $\\mathcal{R}$ (and, more weakly, the potential term $\\mathcal{P}$). They further incorporate primordial non-Gaussianity via $f_{\\rm NL}$ and show that GR effects do not strongly degrade $f_{\\rm NL}$ constraints, with multi-tracer techniques helping to disentangle the two. Overall, the work provides a practical framework for horizon-scale tests of general relativity and the impact of GR corrections on primordial non-Gaussianity analyses, highlighting the importance of advanced observational strategies to access these subtle effects.

Abstract

Kaiser redshift-space distortion formula describes well the clustering of galaxies in redshift surveys on small scales, but there are numerous additional terms that arise on large scales. Some of these terms can be described using Newtonian dynamics and have been discussed in the literature, while the others require proper general relativistic description that was only recently developed. Accounting for these terms in galaxy clustering is the first step toward tests of general relativity on horizon scales. The effects can be classified as two terms that represent the velocity and the gravitational potential contributions. Their amplitude is determined by effects such as the volume and luminosity distance fluctuation effects and the time evolution of galaxy number density and Hubble parameter. We compare the Newtonian approximation often used in the redshift-space distortion literature to the fully general relativistic equation, and show that Newtonian approximation accounts for most of the terms contributing to velocity effect. We perform a Fisher matrix analysis of detectability of these terms and show that in a single tracer survey they are completely undetectable. To detect these terms one must resort to the recently developed methods to reduce sampling variance and shot noise. We show that in an all-sky galaxy redshift survey at low redshift the velocity term can be measured at a few sigma if one can utilize halos of mass M>10^12 Msun (this can increase to 10-sigma or more in some more optimistic scenarios), while the gravitational potential term itself can only be marginally detected. We also demonstrate that the general relativistic effect is not degenerate with the primordial non-Gaussian signature in galaxy bias, and the ability to detect primordial non-Gaussianity is little compromised.

Figures (5)

• Figure 1: Redshift dependence of two dimensionless parameters $\mathcal{R}$ and $\mathcal{P}$ in Eq. (\ref{['eq:PR']}). Non-vanishing values of $\mathcal{R}$ and $\mathcal{P}$ represent the general relativistic effects in galaxy clustering, each of which describes the contributions of the gravitational potential and the velocity to the observed galaxy fluctuation field. Three different curves represent galaxy samples in a volume-limited survey ($p$ is constant) with three different limits $L_t$ in luminosity threshold: a sample with low threshold $L_t\ll L_\star$ ($p=0$; dashed), a sample with no magnification bias $L_t\simeq L_\star$$(p=0.4$; solid), a sample at high luminosity tail $L_t\gg L_\star$ ($p=1.5$; dot-dashed). The evolution bias factor $e=3$ is fixed in all cases, representing homogeneous galaxy samples (constant comoving number density) often constructed in large-scale galaxy surveys.
• Figure 2: Number density and average bias of halos above the minimum mass at different redshift slices. Since the multi-tracer method utilizes all the halos of mass above the minimum mass, a large number of halos are required to achieve sufficiently low minimum mass.
• Figure 3: Predicted measurement significance of general relativistic effects $\mathcal{R}$ (upper) and $\mathcal{P}$ (bottom) in the galaxy power spectrum obtained by using a single tracer. All halos of mass above minimum mass are lumped together to construct a single tracer. Four curves represent different survey redshift ranges with corresponding volume $V=2.5$, 7.9, 59, 410 $({h^{-1}\rm Gpc})^3$. For the volume-limited sample (constant $p$) with constant comoving number density ($e=3$), two galaxy samples are constructed to have $p=0$ (left) and $p=0.4$ (right). No uncertainties in theoretical predictions are assumed. With the traditional power spectrum analysis (single tracer), it is difficult to measure the general relativistic effects at any meaningful significance.
• Figure 4: Predicted measurement significance of general relativistic effects $\mathcal{R}$ (upper) and $\mathcal{P}$ (bottom) in the galaxy power spectrum derived by using the multi-tracer method. All halos of mass above minimum mass are utilized to take advantage of the multi-tracer method SELJA09HASEDE11. Various curves are in the same format as in Figure \ref{['fig:single']}. Compared to Figure \ref{['fig:single']}, the measurement significance is substantially enhanced by using the multi-tracer method.
• Figure 5: Predicted constraints on the primordial non-Gaussianity $f_{\rm NL}$ from galaxy power spectrum measurements. To facilitate the comparison, the constraints on $f_{\rm NL}$ are obtained by using the same survey specifications as in HASEDE11: $V\simeq50~({h^{-1}\rm Gpc})^3$ centered at $z=1$. We assume $e=3$ and $p=0.4$, and various curves show $\sigma_{f_{\rm NL}}$ with different priors on $\boldsymbol{e}$ and $\boldsymbol{p}$ ($\sigma_e=0.1$, $\sigma_p=0.05$COEIET08 for the solid curve).