One important consequence of the wide-angle effect is that when naively applying the multipole expansion in a certain LOS definition, it produces new contributions not only at even multipoles but also at odd multipoles. For cosmological data analyses including the wide-angle effect, see e.g. Indeed, the impact of wide-angle effect on RSD have been long studied in both analytical and numerical approaches, and there is thus a large number of literature on this topic, including early works (Fisher, Scharf & Lahav 1994 Heavens & Taylor 1995 Hamilton & Culhane 1996 Zaroubi & Hoffman 1996 Szalay, Matsubara & Landy 1998 Matsubara 2000).

Since the statistical homogeneity is not fully ensured in the presence of RSD, and the techniques developed so far in both measurement and theoretical predictions heavily rely on statistical homogeneity, the impact of its violation, often quoted as wide-angle effect, can introduce systematics in constraining cosmology with RSD measurement, potentially leading to a biased cosmological result. However, one must be careful in characterizing the galaxy clustering. With the wealth of large data set from future galaxy surveys, the statistical precision will be substantially improved, and it will help to further tighten the cosmological constraints (see Weinberg et al. 2014 Carlson, Reid & White 2013 Alam et al. Linder 2008 Percival & White 2009 Song & Percival 2009 Taruya, Nishimichi & Saito 2010 Vlah et al. This explains why there have been so far numerous works in both theory and observation to model, predict, and measure the anisotropies of galaxy clustering, leading to fruitful cosmological constraints (e.g. In this respect, the measurement of clustering anisotropies caused by RSD offers an exciting opportunity to probe gravity on cosmological scales. On large scales, such anisotropies are described by linear theory only with few low multipoles, which tell us that the strength of anisotropies is directly related to the growth of cosmic structure induced by gravity (Kaiser 1987 Hamilton 1992). Indeed, taking the distant-observer or plane-parallel limit, the statistical homogeneity is approximately restored, and the apparent anisotropies induced by RSD are characterized well by the multipole expansion with respect to the LOS direction of the distant observer. RSD generally complicates the data analysis and cosmological interpretation of the observed galaxy clustering, but one advantage may be that RSD provides an additional information on the velocity field at large scales. In particular, the galaxy distribution observed via spectroscopic survey appears distorted along the observer’s line of sight (LOS) due to the contribution of peculiar velocities to the measured redshift of a galaxy, referred to as the redshift-space distortions (RSD).

However, the observation can break homogeneity and isotropy. While the spatial inhomogeneity of matter and galaxy distribution is in nature random and stochastic, it is supposed to be statistically homogeneous and isotropic. The large-scale structure of the Universe, as partly seen by galaxy distributions, has evolved dominantly under the influence of gravity and cosmic expansion. Methods: analytical, galaxies: distances and redshifts, cosmology: theory, large-scale structure of Universe 1 INTRODUCTION

Redshift space distortions in illustris