1. Introduction

A spherical object observed at a distance in its longitudinal ^{Alcock & Paczyński (1979)}. The Alcock-Paczyński parameter, hereafter *AP*, basically the ratio of *z* with a strong dependence on the value of the cosmological parameters that make up the Hubble function, introducing a cosmological distortion to the large scale structure observations. This apparently simple comparison is, however, greatly complicated by several factors. First, real-space measurements are not directly attainable and one has to rely on redshift-space. Then, if the proposed object consists of a cluster of quasars or galaxies, or a statistical ensemble of such, proper motions of its constituents (either derived from gravitational collapse or virialized conditions) distort redshift-space measurements causing a degeneracy problem (e.g., ^{Hamilton 1998}). On cosmological scales, clusters of galaxies or quasars are among the simplest geometric structures that one may conceive. Even if single clusters should have non-spherical or filamentary structures, those should be randomly oriented. As we probe more distant clusters, observations become biased towards brighter and widely separated members, and the numbers become statistically insignificant. A superposition of many such clusters may reduce the problem while retaining spherical symmetry. For more than 40 years the two-point correlation function (2PCF), and its Fourier transform, the power spectrum, have been fundamental tools in these studies (e.g., ^{Peebles 1980}).

Overdense clusters or associations separate from the Hubble flow due to their own gravity, which results in peculiar velocities of its members that distort redshift-space observations. When gravitational fields are small, velocities are well described by the linear theory of gravitational collapse (^{Peebles 1980}). In the study of these clusters, the 2PCF was initially conceived as a single entity ξ that could be evaluated in either real (*r*−) or redshift (*s*−) space (^{Peebles 1980}). ^{Davis & Peebles (1983)} even mentioned that when observing the local universe, if the peculiar velocities were relatively small, *s*−space would directly reproduce *r*−space and one would have ^{Huchra et al. 1983}), as described in ^{Davis & Peebles (1983)} peculiar velocities were significant, and the authors chose to go from real *streaming model*. The convolution integral would at the same time convert *r*−space to *s*−space coordinates. However, the same function ^{Kaiser (1987)}, hereafter K87, showed that gravitationally induced peculiar velocities by gravitational collapse of overdense structures in the linear regime produced a power spectrum *P*
^{
(s)
} for *s*−space diﬀerent from the one *P*
^{
(r)
} for *r*−space, that is, two diﬀerent functions for the power spectrum. Both are, however, functions of the *r*−space Fourier frequency k. Then, while *P*
^{
(r)
}
*(k)* is a spherically symmetric function, *P*
^{
(s)
}
*(k)* shows an elongation along the line of sight (LoS) direction. Later on, ^{Hamilton (1992)} translated these results to configuration space obtaining the 2PCF in its two flavors: *r*
^{−γ} is considered, as had been historically accepted (e.g., ^{Peebles 1980}, who favored γ = 1.8 ). Also, in perfect agreement with K87, ^{Hamilton (1998)} presented in great detail the assumptions that led to his results. He started by defining selection functions *n*
^{
(r)
} (*r*) and *n*
^{
(s)
} (*s*) for *r*−space and *s*−space and by numerical conservation obtained a complicated high order expression (his equation 4.28) for the density contrast *s*−space.

Since then many authors have tried the Kaiser linear approximation facing this dilemma and have introduced similar approximations. In the description of 2PCF in redshift-space, due to the multipole expansion of the inverse Lagrangian operator derived from the corresponding power spectrum in Fourier space (^{Hamilton 1992}), there appears a dependence with µ, the cosine of the angle between the *r* (real space) vector and the LoS: *µ* from redshift-space coordinates as either ^{Matsubara & Suto 1996}; ^{Nakamura et al. 1998}; ^{López-Corredoira 2014}). Yet in some other cases the approximation ^{Tinker et al. 2006}) calling it the “distant observer" approximation. But, as mentioned above, this is really intended to mean a wide separation approximation and does not apply in the small scale regime. Furthermore, the “distant observer" name is also used for the plane-parallel case (e.g., ^{Percival & White 2009}), adding to the confusion. In some other cases the substitution ^{Hawkins et al. 2003}). Another facet of the same problem has been to expand the redshift-space correlation function as a series of harmonics of that same ^{Guo et al. 2015}; ^{Chuang & Wang 2012}; ^{Marulli et al. 2017}). While this is certainly a valid approach, the conclusions of linear theory, like the existence of only monopole, quadrupole and hexadecapole terms in the Legendre polynomial expansion, are not really applicable to the µ(s) case. All these forms of the approximation are really one and the same, and to avoid further confusion (like the term “distant observer") we decided to call it the µ(s) approximation.

When observational data are used to construct the 2PCF ^{Hamaus et al. 2015}), usually called Finger of God (^{Huchra 1988}), hereafter FoG. Prominent examples of FoG were found in the Coma Cluster by ^{de Lapparent et al. (1986)} and in the Perseus cluster by ^{Wegner et al. (1993)}. The FoG feature is also commonly observed in the 2PCF of statistical aggregates (e.g., ^{Hawkins et al. 2003}), making it a common feature of large scale structures.

Many studies have been conducted to explain this discrepancy. In general, non-linear processes are invoked. Sometimes the non-linearities are assigned to virial relaxation in the inner regions of clusters, while others explore the non-linear terms of the approximation in the derivation of the K87 result. In these categories, we mention a small sample of the representative literature. Kinematic relaxation, like the virialized motion of cluster members in the inner regions (^{Kaiser 1987}; ^{Hamaus et al. 2015}), are explored by introducing a distribution of pair-wise peculiar velocities for cluster components. There are at least two ways of doing so: First, the *streaming model* where a velocity distribution *f*(*V*) is convolved with ^{Davis & Peebles (1983)} but diﬀerentiating ^{Seljak & McDonald 2011}; ^{Okumura et al. 2012a},^{b}) by directly obtaining the power spectra in redshift-space as a function of the *s*−space wave-number. Unfortunately, the expression that results for the power spectra is rather complicated, even when it is conveniently expressed as a series of mass weighted velocity moments. However, it is possible to obtain FoG structures in *r*− to *s*−space takes place (e.g., ^{Scoccimarro 2004}). Paradoxically, it is not that easy to obtain the traditional peanut-shape structure that is generally recognized as the K87 limit in *s* ∼ *r* is once again invoked. Second, in the phenomenological *dispersion model* (c.f., ^{Scoccimarro 2004}; ^{Tinker et al. 2006}) a linear K87 spectrum is multiplied in Fourier space by a velocity distribution. This can be seen as a convolution in configuration space, as in ^{Hawkins et al. (2003)}, but the procedure has the disadvantage that it obtains the same function *f* (*V*). It has to be noted, however, that very good fits to the observed data are obtained by this procedure. The same is true for the fits to numerical simulation results at mid spatial frequencies obtained by similar procedures in e.g., ^{Marulli et al. (2017)}. In the streaming model, the velocity distribution function can also be obtained from the interaction of galaxies with dark mater halos (e.g., ^{Tinker et al. 2006}; ^{Tinker 2007}), via the halo occupation distribution formalism.

Apart from kinematics, non-linear terms also arise in the expansion of the mass conservation or continuity equation in *r*− and *s*−spaces to obtain the power spectrum or the 2PCF (e.g., ^{Matsubara 2008}; ^{Taruya et al. 2010}; ^{Zheng & Song 2016}). Preserving only first order terms yields the K87 result. However, a full treatment of all the terms is possible with the use of perturbative methods. There are diverse techniques: standard, Lagrangian, renormalized, resumed Lagrangian (for a comparison see ^{Percival & White 2009}; ^{Reid & White 2011}). The latter authors however, conclude that the failure of these methods to fit the l=2 and 4 terms in the expansion *h*
^{−1} Mpc, must be due to inaccuracies in the mapping between *r*- and *s*-spaces. So, they favor again the *streaming model*. Clearly, there is still substantial debate on this subject.

In most of these works the necessity to translate their results to observable 2PCFs, *s*-space power spectrum *P*
^{
(s)
} (**
k
**

_{ r }) (e.g.,

^{Matsubara 2008};

^{Okumura et al. 2012a}), but with

*k*

_{ r }in

*r*-space; or display its moments

^{Taruya et al. 2010};

^{Zheng & Song 2016}); or the power spectra with

*k*

_{ s }in

*s*−space

*P*

^{ (s) }(

*k*_{ s },

*µ*

_{ s }) (e.g.,

^{Okumura et al. 2012b}). Other authors display the correlation function in

*r*-space, either as

^{Matsubara 2008}) or

^{Tinker 2007};

^{Reid & White 2011};

^{Okumura et al. 2012a}), or its moments

^{Taruya et al. 2010}) for l=2. Few works try to display directly the 2PCFs

^{Matsubara & Suto 1996};

^{Nakamura et al. 1998};

^{Tinker et al. 2006};

^{López-Corredoira 2014}), but as already mentioned above, usually perform the

To further complicate matters, redshift-space distortions are often treated separately from the cosmological distortions. Both are not easily discernible because both produce stretching or squashing in the LoS direction (^{Hamilton 1998}; ^{Hamaus et al. 2015}). This degeneracy could in principle be resolved because the cosmological and peculiar velocity signals evolve diﬀerently with redshift, but in practice the uncertain evolution of bias (the dimensionless growth rate for visible matter, see equation 26) complicates the problem (^{Ballinger et al. 1996}). Furthermore, ^{Kaiser (1987)} and ^{Hamilton (1992}, ^{1998)} do not consider cosmological distortions in their analysis of peculiar motions. Since the earlier works, the inclusion of cosmological distortions has been attempted by several authors (e.g., ^{Matsubara & Suto 1996}; ^{Hamaus et al. 2015}).

In this paper, we show that a structure quite similar to FoG can be obtained in *streaming model*, nor the non-linearities studied in perturbation theory, but just by avoiding the *µ*(*s*) approximation, in any of its forms *r*(*s*) with the aid of the projected correlation function of both 2PCFs :

We start with a detailed definition of *r*− and *s*−space, noting that frequently *s*−space is expressed in distance units as is *r*−space. But in doing so, one multiplies by a scale factor that invariably introduces a cosmological parameter in the definition; and as a result the named *s*−space is no longer purely observational. Later on, the factor is solved by introducing a fiducial cosmology and solving for the real values. An example can be seen in the analysis made by ^{Padmanabhan & White (2008)} in Fourier space and ^{Xu et al. (2013)} in configuration space. The latter recognize the need of introducing a two-step transformation, one isotropic dilation and one warping transformation, to transform from real fiducial to real space. However, the real fiducial space is actually redshift-space, and this identification is missing in these works.

Therefore, we argue (c.f., § 2) that it is convenient to define the observable-redshift-space *s*−space): *r* by a unitary Jacobian independent of redshift. So, no additional scaling is needed, and the only remaining diﬀerence will be precisely in shape. That is why *s* are more alike, and thus can both be named redshift-space; *s* is the physical redshift-space. Then, the transformation to real-space necessarily goes through redshift distortions.

Furthermore, when we introduce peculiar non-relativistic velocities in this scheme, we will show that it is possible to keep the same relation between observable and physical redshift-spaces, *s* and ^{Kaiser (1987)} eﬀect is recovered independently of redshift (see § 3). That is, now redshift-space will also show an additional gravitational distortion with respect to real-space.

To solve for the relation between real-space and redshift-space, we will rely on projected correlations. Projections of the 2PCF in the plane of the sky have been widely used to avoid the complications of dealing with unknown components in redshift-space (e.g., ^{Davis & Peebles 1983}). This has the advantage that in the case of a symmetric 2PCF in real-space, the 3-D structure can be inferred from the projection. We will show in § 4 that since the projections of the 2PCF in real-space and in redshift-space are bound to give the same profile, a relationship can be obtained for the real-space coordinate

2. Redshift-space

Consider the Friedmann-Lemaître-Robertson-Walker metric (e.g., ^{Harrison 1993}) written in units of distance and time as follows:

with *S*
_{
k
} = (sin, *Identity*, sinh) for *k* = (1, 0,−1). Then the co-moving present-time length of an object *dr*
^{0} that is observed longitudinally is related to a variation in the observed redshift *dz* by

where H(*z*) is the Hubble function and the ^{0} superindex is used to define the present time *t*
_{0}. Similarly, an object with a transversal co-moving dimension *d*θ given by the angular co-moving distance (e.g., ^{Hogg 1999}) as

where *a*
_{0} is the present day scaling parameter of the metric.

Observationally one measures redshift differences *dz* and subtended angles *d*θ. We then define the observable redshift-space adimensional quantities

and

The physical redshift-space sizes

and

where

and

with

and

It is clear then that the ^{Alcock & Paczyński (1979)} function *AP*(*z*), that tests redshift distortions of a particular cosmology, can be written as

Furthermore, from the transformation of physical redshift-space with coordinates

In order for this transformation to preserve scale we need a unitary Jacobian. This condition can be achieved simply by the following condition:

as can be seen from equations (10) to (12). Here the dependence on the cosmology is made explicit through the Hubble function. Note that the resulting scale factor *z*. In fact we have (see also ^{Xu et al. 2013}):

and

Peculiar velocities modify the observed redshift, and therefore alter the relation between real-space and redshift-space giving rise to kinematic distortions. Suppose the near-end of an object is at rest at redshift *z*, while the far-end is moving with peculiar non-relativistic velocity ^{Matsubara & Suto 1996}; ^{Hamaus et al. 2015}):

where

where *ds*
_{v} (in physical redshift-space) is given by

and

Through the similarity of equations (19) and (20) with equations (6) and (4), we note that the concepts of observable redshift-space and physical redshiftspace can be extended to include peculiar motions as well.

3. Two point correlation function

Let *r* be real-space Euclidean co-moving coordinates in the close vicinity of a point at redshift *z*, defined as *dr* in the previous section. Then for azimuthal symmetry around the line of sight (aligned to the third axis) we have

where we have used the unitary condition on equation (13) to eliminate the *r* to *s* space, the density change can be related to the change in volume *V*, and the Jacobian by the equation

This can also be expressed in terms of the contrast density ratios in *s* and *r* spaces defined such that

where ^{Hamilton 1998}), which makes it rarely a first choice. However, the procedure given below allows us precisely to solve for the function **
r
** (

*s*).

In linear theory, ^{Peebles (1980)} shows by equations 14.2 and 14.8 that an overdensity of mass ^{Thornton & Marion 2004}) times a constant which is time (or redshift) dependent. That is

where

Here *D*(*z*) is the growth factor, the temporal component of density. Note that in ^{Peebles (1980)} coordinates are given in the expanding background model *x*, which relate to present time real-space coordinates by *z*) factor in equation (24). The m subscript to *b*(*z*) and to define the dimensionless growth rate for visible matter

Then from equations (21) to (24) we get:

where ^{Matsubara & Suto 1996}).

The square modulus of the Fourier transform of equation (27) gives an expression for the power spectrum, or the Fourier transform of the autocorrelation function (2PCF) *z*

where **
k
**

_{ r }in real-space; it arises by the Fourier transform property of changing differentials into products. Note that wave number vectors in real-space also differ from their counterparts in redshift-space by the unknown velocity field in equation (17).

Fourier transforming back into coordinate space gives the ^{Hamilton (1992)} result:

Note that this equation is written in a way that all terms in the right hand side are real-space coordinates **
r
** dependent, as is the case for the derivatives and inverse Laplacian. Recalling that the solution of the Laplace equation in spherical coordinates consists of spherical harmonics in the angular coordinates and a power series in the radial part, one can write for the case of azimuthal symmetry

where *P*
_{
l
} (*μ*(**
r
** )) are the Legendre polynomials,

explicitly defined for real-space coordinates, and the harmonics are given by the coefficients

Substituting equation (29) in (32) for the case of spherical symmetry in real-space ^{Hamilton (1992)}, see also ^{Hawkins et al. (2003)}. That result consists of only three terms: monopole, quadrupole and hexadecapole (*l* = 0, 2, 4), all the others evaluate to zero. It is important to note that this is not true when the expansion of equation (30) is done in *μ*(*s*) as assumed by several authors (e.g., ^{Guo et al. 2015}; ^{Chuang & Wang 2012}; ^{Marulli et al. 2017}).

When the 2PCF can be approximated by a power-law,

where ^{Hamilton 1992}; ^{Matsubara & Suto 1996}; ^{Hawkins et al. 2003}). One of these is the following:

This function takes values greater than 1 for the equatorial region

However, we will show below that the stretching of redshift scale along the LoS will counteract this apparent squashing producing a structure similar to a FoG. In order to stay within the linear regime, we ensure not to reach the turnaround velocity by keeping

We now remark that **
s
** and

**vectors refer to the same point. However, we remark that**

*r*The result in our equation (33) has been derived for **
s
** and

**are not just independent names for position, and there exists a relation**

*r**s*(

**) between them that is not linear. Specifically, the parallel component is**

*r***. In the case of small disturbances we expect small velocities (below turnover) that result in a bi-univocal map**

*r**s*(

**) and its inverse. So, if we want to display the resulting**

*r***space, we must proceed first to evaluate**

*s**r*=

*r*(

**) and then**

*s**s*-space.

On the other hand, if the *µ*(**
s
** ) approximation is used one obtains structures that are squashed in the LoS direction, and with a characteristic peanut-shaped geometry close to the polar axis (see for example

^{Hawkins et al. 2003}). One concludes that this geometry fails to reproduce the structure known as “Finger of God" (FoG). The consequence is that other processes are called upon to account for it, such as random motions arising in the virialized inner regions of clusters. We show below that avoiding this approximation allows us to obtain a geometrical structure quite similar to the FoG feature.

4. Projected correlation function

In order to avoid the complications that redshift-space distortions introduced in the correlation function, such as those produced by gravitationally induced motions or virialized conditions, the projected correlation function ^{Binney & Tremaine 1987}) numerically. See also ^{Pisani et al. (2014)} for other possibilities. In the case where ^{Krumpe et al. 2010}).

We will show that the projected correlation function can be used to obtain the *µ*(**
r
** ). We start by noting that the projection on the plane of the sky may be performed either by using the

and

where

The integral limits should go to infinity to get the total projected functions. However, one can project the correlation function up to a particular real space distance ^{Nock et al. 2010}). On the one hand, slices in *r*-space (equation 36) do not depend on the observer’s perspective, while on the other (equation 35) the limit of the integral (boundary condition) becomes a function that is precisely going to be evaluated. Carrying on, due to number conservation, the projections in redshift- and real-space multiplied by the corresponding area elements that complete the volume where the number of pairs are counted, must be equal. This leads to

for all values of

Then, changing variables to

where the dependence

5. Resulting redshift-space and real-space relation

We integrate equation (39) numerically using equation (34), to obtain the

So, it can be noted that for on-axis separations (where **
s
** ) approximation. On the other hand, for

**) approximation) that ultimately converges to the limit**

*s*These geometrical distortions can be better appreciated by their eﬀect on the 2PCF presented in Figure 2. Here we start from a grid in *s*−space, and transform to *r*-space using the integral relation (equation 39) for the parallel component and equation (9) for the perpendicular one. From there, we calculate *µ*(**
r
** ) (equation 31),

^{Alcock & Paczyński 1979}).

Figure 2 (a) shows the case that corresponds to the parameters used for Figure 1:

In the other three Figures, 2(b), 2(c) and 2(d), we explore the eﬀect of cosmological and gravitational alterations. Figure 2 (b) shows that the eﬀect of increasing AP is a geometrical distortion that concentrates the structure towards the polar axis direction for AP = 2 that corresponds to ΛCDM cosmology at *z* = 2.6. In Figure 2 (c) we explore the eﬀect of changing the dimensionless growth-rate for visible mater β. This gravitational eﬀect is to enhance the FoG structure as its value increases (recall that its limit value is 2/3). On the other hand, if

Although it has not been the purpose of this paper, we may consider diﬀerent values of the power-law index

As we have indicated, a rounder 2PCF at mid spatial scales is favored by some works that use the _{⊥} > s_{||}. An increase in the AP parameter may also contribute to alleviate the situation.

Another possibility, that was not intended to be covered here, is the case of a more realistic 2PCF ^{Slosar et al. 2013}) or those obtained by the CAMB code (^{Seljak & Zaldarriaga 1996}). In order to apply the results of this paper to such cases, one could try breaking the inferred

instead of equation (39). We would also have to find a way to estimate the multipole moments

We conclude that a whole range of possibilities in shape and strength of the FoG structure and the squashing of the equatorial zone can be obtained by tuning the parameters ^{Satpathy et al. 2017}).

6. Conclusions

We emphasize the importance of distinguishing three spaces in cluster and large scale structure studies: the observable redshift-space **
s
** , and the real-space

**. The transformation between**

*r**s*is an isotropic dilation that introduces a scale factor dependent on the cosmology.

On the other hand, the transformation between **
s
** and

**occurs through a unitary Jacobian independent of redshift, and only distorts the space by factors related to the Alcock-Paczyński AP function (c.f., equations 15 and 16).**

*r*Furthermore, when we introduce non-relativistic peculiar velocities in this scheme, we demonstrate that the same relation between observable and physical redshift-spaces ^{Kaiser (1987)} eﬀect independent of redshift in Fourier space, and ^{Hamilton (1992)} results in configuration space.

We remark that a dependence with µ in real-space **
s
** for

**in the equations. To avoid further confusion we have called this the**

*r*Since r_{||} is usually unknown, we propose a method to derive it from s_{||} using number conservation in the projected correlation function in both real- and redshift-spaces. This leads to a closed form equation (39) for the case where the real 2PCF can be approximated by a power-law. From this, we solve for µ(**
r
** ) in real-space, and show that a diﬀerent view of the redshift-space 2PCF emerges. The main result is that the redshift-space 2PCF presents a distortion in the LoS direction which looks quite similar to the ubiquitous FoG. This is due to a strong anisotropy that arises purely from linear theory and produces a stretching of the scale as one moves into the on-axis LoS direction. Moving away from the LoS the structures appear somewhat more squashed than those obtained by the

*µ*(

*s*) approximation for equivalent values of β. The implications of this remains an open question.

The development presented here produces structures that qualitatively reproduce the observed features of the 2PCF of galaxies and quasars large scale structure. A squashing distortion in the equatorial region is attributed to a mixture of cosmological and gravitational eﬀects. The FoG feature that is usually attributed to other causes is instead ascribed to the same gravitational eﬀects derived from linear theory.

We conclude that a whole range of possibilities in shape and strength of the FoG structure, and the squashing of the equatorial zone, can be obtained by tuning the parameters γ, β, and AP . This provides a path towards solving the usual degeneracy problem between cosmological and gravitational distortions. In a future paper (Salas & Cruz-González in preparation) we will apply these results to the galaxies and quasar data obtained by current large scale surveys.

I.C.G. acknowledges support from DGAPA-UNAM (Mexico) Grant IN113417.