Wide Angle Effects in Future Galaxy Surveys

TL;DR

This paper develops and applies a full wide-angle formalism for redshift-space galaxy clustering, clarifying how geometric wide-angle effects and velocity contributions relate to the relativistic description of observed fluctuations. It shows that, for SDSS, Euclid, and the BigBOSS, properly averaging the Kaiser-model predictions over the survey volume yields negligible deviations from the distant-observer approximation in the correlation function, with similar conclusions extended to power-spectrum analyses. However, standard FKP-based anisotropic power-spectrum estimators can incur noticeable systematics due to nonuniform pair distributions, highlighting the need for pair-dependent LOS treatments or alternative methods. The work also illuminates the connections and distinctions between the Kaiser framework and relativistic corrections, including corrections arising from luminosity-distance fluctuations and past-light-cone effects, guiding practical analysis choices for upcoming surveys.

Abstract

Current and future galaxy surveys cover a large fraction of the entire sky with a significant redshift range, and the recent theoretical development shows that general relativistic effects are present in galaxy clustering on very large scales. This trend has renewed interest in the wide angle effect in galaxy clustering measurements, in which the distant-observer approximation is often adopted. Using the full wide-angle formula for computing the redshift-space correlation function, we show that compared to the sample variance, the deviation in the redshift-space correlation function from the simple Kaiser formula with the distant-observer approximation is negligible in galaxy surveys such as the SDSS, Euclid and the BigBOSS, if the theoretical prediction from the Kaiser formula is properly averaged over the survey volume. We also find corrections to the wide-angle formula and clarify the confusion in literature between the wide angle effect and the velocity contribution in galaxy clustering. However, when the FKP method is applied, substantial deviations can be present in the power spectrum analysis in future surveys, due to the non-uniform distribution of galaxy pairs.

Figures (11)

• Figure 1: Triangular configuration of the observer $O$ and the galaxy pairs $G_1$ and $G_2$. The opening angle of the galaxy pair is $\Theta\equiv2\theta=\phi_2-\phi_1$, and the line-of-sight direction of the pair is defined as the direction ${\bf \hat{n}}$ that bisects the pair in angle, forming an angle $\phi=(\phi_1+\phi_2)/2$ with the pair separation. With the distant-observer approximation, three angles become identical $\phi=\phi_1=\phi_2$ ($\theta\rightarrow0$).
• Figure 2: Full redshift-space correlation function in galaxy surveys. The full redshift-space correlation function in equation (\ref{['eq:fullcorr']}) depends on the triangular configuration $(s,\mu,\Theta)$ formed by the galaxy pair and the observer in Fig. \ref{['fig:geometry']}. For illustration, we consider equilateral triangular shapes (i.e., $s_1=s_2$ and $\mu=0$). Upper panels show the ratio of the full correlation function to the correlation function computed by using the distant-observer approximation in equation (\ref{['eq:zcorr']}) at the redshift of the galaxy pair ($z_1=z_2$). The deviation therefrom arises due to the wide angle effect and the velocity contribution. ($a$) The selection function is set $\alpha=0$, and there is no velocity contribution, representing the deviation purely due to the wide angle effect. ($b$) The velocity contribution is considered ($\alpha=2$). ($c$) The distance from the observer to each galaxy of the pair. Horizontal lines show the distance to the redshifts in the label. For reference, the SDSS LRG covers the redshift range $z=0.15\sim0.45$ (Euclid: $z=0.7\sim2.0$ and the BigBOSS: $z=2.2\sim3.5$). Most galaxy pairs measured in these surveys will have small opening angles. ($d$) Approximate velocity contribution $\alpha\mathcal{V}/r$ with $V=10^{-3}$ ($300~{\rm km\, s}^{-1}$) to the full Kaiser formula for the redshift-space fluctuation $\delta_s$ in equation (\ref{['eq:fullkaiser0']}). For illustration, we assumed a uniform galaxy sample with $b=1$ (or dark matter). Uniform galaxy samples with higher bias factor would further reduce the deviation from the distant-observer approximation.
• Figure 3: Redshift distribution of the SDSS galaxy sample and the evolution factor $e$ of its number density. In the upper panel, various curves show the normalised redshift distributions (or the radial selection function) in equation (\ref{['eq:pzz']}) for a galaxy sample with a constant comoving number density ($e=3$: solid) and the SDSS LRG sample (dotted). With the FKP weighting in equation (\ref{['eq:FKP']}), the redshift distribution (dashed) is shifted to lower redshift and is closer to the uniform (solid). The bottom panel shows the evolution factor $e$ of the galaxy number density. By definition, the uniform sample has $e=3$, diluting only due to the volume expansion. The dotted curve represents the SDSS LRG sample, indicating that its number density evolves rapidly in time (the comoving number density of the SDSS LRG sample is shown as dashed, and the corresponding selection function $\alpha$ is shown as dot-dashed). The galaxy samples in future surveys are assumed to have a uniform distribution ($e=3$: solid), but with different redshift ranges.
• Figure 4: Probability distribution of triangle shapes formed by SDSS galaxy pairs and the observer. Pair distributions are affected by the number density evolution and the survey geometry. For simplicity, the survey geometry is defined in terms of redshift range and sky coverage only; no holes or disjoint regions in the survey areas are assumed. Upper panels show the distribution of the cosine angle between the pair separation vector ${\bf s}$ and the line-of-sight direction ${\bf \hat{n}}$ bisecting the pair in angle. Bottom panels show the distribution of the opening angle of galaxy pairs seen by the observer at origin. Various curves represent different separation length $s=|{\bf s}|$ in units of ${h^{-1}{\rm Mpc}}$. Thick solid curves show the distribution of the opening angle, averaged over all galaxies with any pair separations. With the FKP weighting, more weight is given to galaxies at lower redshift.
• Figure 5: Probability distribution of triangle shapes formed by galaxy pairs and the observer in Euclid and the BigBOSS, in the same format as Fig. \ref{['fig:adist']}. It is assumed that the sky coverage of Euclid is a half ($f_{\rm sky}=1/2$) and the redshift range is $z=0.7\sim2.0$. The BigBOSS is assumed to cover 14,000 deg$^2$ ($f_{\rm sky}=0.34$) at $z=2.2\sim3.5$. Given the mean distance to galaxy pairs, typical opening angles in these surveys are a lot smaller than in the SDSS. However, the non-uniform distribution of $\mu$ is still present, as it is inherent to survey geometry.