Probing the Anisotropy and Non-Gaussianity in the Redshift Space through the Conditional Moments of the First Derivative

Focusing on the redshift space observations with plane-parallel approximation and relying on the rotational dependency of the general definition of excursion sets, we introduce the so-called conditional moments of the first derivative (cmd) measures for the smoothed matter density field in three dimensions. We derive the perturbative expansion of cmd for the real space and redshift space where peculiar velocity disturbs the galaxies’ observed locations. Our criteria can successfully recognize the contribution of linear Kaiser and Finger-of-God effects. Our results demonstrate that the cmd measure has significant sensitivity for pristine constraining the redshift space distortion parameter β = f/b and interestingly, the associated normalized quantity in the Gaussian linear Kaiser limit has only β dependency. Implementation of the synthetic anisotropic Gaussian field approves the consistency between the theoretical and numerical results. Including the first-order contribution of non-Gaussianity perturbatively in the cmd criterion implies that the N-body simulations for the Quijote suite in the redshift space have been mildly skewed with a higher value for the threshold greater than zero. The non-Gaussianity for the perpendicular direction to the line of sight in the redshift space for smoothing scales R ≳ 20 Mpc h −1 is almost the same as in the real space. In contrast, the non-Gaussianity along the line-of-sight direction in the redshift space is magnified. The Fisher forecasts indicate a significant enhancement in constraining the cosmological parameters Ω m , σ 8, and n s when using cmd + cr jointly.


Introduction
In the high-precision cosmology era, drastic attention should be paid to the various robust measures construction for extracting information from random cosmological fields as accurate as possible, particularly from large-scale structures of the matter distribution in the Universe (Peebles 2020;Bernardeau et al. 2002).On the other hand, discrepancies between what we observe through various surveys and theoretical counterparts essentially persuade researchers to include the stochastic notion (Kaiser 1984; Bardeen et al. 1986;Bernardeau et al. 2002;Matsubara 2003;Codis et al. 2013;Matsubara 2020).It is supposed that on the sufficiently large scales, the distribution of galaxies in the real space is homogeneous and isotropic, while, such an assumption is no longer satisfied in the redshift space when the position of structures is plotted as a function of redshift rather than their distances.Dealing with imposed anisotropy requires designing proper methods which are sensitive to both the existence m_jalalikanafi@sbu.ac.ir m.s.movahed@ipm.ir of preferred direction and non-Gaussianity generated form different mechanisms.
The observed redshifts of galaxies which are mainly originated by the Hubble flow are also disturbed by their peculiar velocity along the line of sight.In the vicinity of peculiar velocity which is almost produced by inhomogeneity known as overdensities and underdensities in the local Universe, a difference between galaxies's actual locations and their observed locations as determined by their redshifts exists.This phenomenon is known as the redshift-space distortion (RSD).The Finger-of-God (FoG) effect (Jackson 1972;Peebles 2020) and the linear Kaiser effect (Kaiser 1987) are the different parts of RSD dominated in the small enough and large scales, respectively.The elongation of clusters along the line of sight caused by the random motion of galaxies within the virialized clusters on small scales is so-called FoG, while the linear Kaiser effect refers to the suppression in the clustering of galaxies on large scales due to the coherent motion into the overdense regions of density field leading to squash the shape of clusters in redshift space along the line of sight direction (Sargent & Turner 1977;Hamilton 1992Hamilton , 1998)).Although RSD makes the interpretation of observational data more challenging, it provides an opportunity to extract statistical information to constrain associated cosmological parameters (Hamilton 1992(Hamilton , 1998;;Bernardeau et al. 2002;Weinberg et al. 2013).
In recent years, many researches have been focused on the analysis of RSD from different points of view.As illustration: the correlation between the redshift distortions and cosmic mass distribution makes sense to utilize the RSD for assessing the linear growth rate of density fluctuations (Hamaus et al. 2022;Panotopoulos & Rincón 2021), trying to break the degeneracy between various modified gravity and General Relativity in the presence of massive neutrinos in the context of standard model of cosmology (Wright et al. 2019); the joint analysis of the Alcock-Paczynski effect and RSD to probe the cosmic expansion (Song et al. 2015); combining RSD with weak lensing and baryon acoustic oscillations to improve the observational constraints on the cosmological parameters (Eriksen & Gaztañaga 2015); quantifying the RSD spectrum (Bharadwaj et al. 2020;Mazumdar et al. 2020Mazumdar et al. , 2023)); examining the primordial non-Gaussianity via RSD (Tellarini et al. 2016); disentangling redshift-space distortions and non-linear bias (Jennings et al. 2016).
The central assumptions in many cosmological studies are homogeneity, isotropy, and Gaussianity due to the extension of the central limit theorem domain (see also the Kumar Aluri et al. (2023) for a comprehensive explanation of Cosmological Principle).In the real data sets, not only the violation of Gaussianity is expected, but also the anisotropy can emerge due to different reasons ranging from initial conditions, and phase transitions to the non-linearity among the evolution (Planck Collaboration et al. 2014a, 2016b;Renaux-Petel 2015;Planck Collaboration et al. 2016a;Hou et al. 2009;Springel et al. 2006;Bernardeau et al. 2002;Planck Collaboration et al. 2014b;Vafaei Sadr & Movahed 2021).Subsequently, to explore the large scale structures in the redshift space as the counterpart of the real space, many powerful statistical measures have been considered by concentrating on the non-Gaussianity and anisotropy (Matsubara 1996;Codis et al. 2013;Appleby et al. 2018Appleby et al. , 2019Appleby et al. , 2023)).Recently, Minkowski tensors, an extension of scalar Minkowski functionals (McMullen 1997;Alesker 1999;Beisbart et al. 2002;Hug et al. 2007;Santaló 2004;Kapfer et al. 2010;Schröder-Turk et al. 2013), have been employed on 2-and 3-Dimensional cosmological fields in the real and redshift spaces (Ganesan & Chingangbam 2017;Chingangbam et al. 2017;Appleby et al. 2018Appleby et al. , 2019Appleby et al. , 2023;;Goyal & Chingangbam 2021;Appleby et al. 2022).
Motivated to examine the anisotropy, asymmetry and non-Gaussianity induced in many cosmological random fields, simultaneously, we pursue the mainstream of theoretical measures construction to explore the anisotropy and non-Gaussianity and to quantify the statistical features of a generic field such as density field with z−anisotropic behavior in plane-parallel approximation.When the anisotropy and non-Gaussianity are interested, we advocate the utilizing of measures specially designed to declare the anisotropy rather than using those measures such as Minkowski Functional and contouring analysis which are not in principle directional tools, however, they can recognize the anisotropy and non-Gaussianity, because they generally have the imprint of directional averaging and they may give the spurious results.
It is worth noting that, to introduce a feasible measure, we should notice the following general properties which are necessary to achieve proper cosmological inferences: it should be robust as much as possible against numerical uncertainties and finite size sampling effect.Since we are interested in using the new measure to constrain the cosmological parameters, another aspect that should be taken into account is that it is possible to establish analytical or semi-analytical prediction for the introduced measure, however, in the absence of theoretical prediction for the desired measure, there are some approaches to overwhelm this issue such as Gaussian Processes Regression (Wang 2020).
The novelties and advantages of our approach are as follows: (1) We will provide a comprehensive mathematical description of the so-called conditional moments of the first derivative (cmd) of the fields corresponding to the excursion set and calculate the theoretical prediction of this statistic for a 3-Dimensional isotropic and asymmetric Gaussian field as a function of threshold using a probabilistic framework.We will also take into account the first order correction due to the mildly non-Gaussianity in the context of perturbative approach.Our notable measure is able to recognize the preferred and generally anisotropic directions for any generic field for 2-and 3-Dimension in different disciplines as well as non-Gaussianity (Li et al. 2013;Ghasemi Nezhadhaghighi et al. 2017;Klatt et al. 2022;Kapfer et al. 2010;Schröder-Turk et al. 2013).
(2) The anisotropy imprint by the linear Kaiser effect will be examined by our introduced measure as well as crossing statistics as a particular generalization of Minkowski Functionals in the plane-parallel approximation.Also incorporating the Gaussian and Lorentzian phenomenological models of the FoG effect, the correction to the linear Kaiser limit will be carried out.To make our analysis more complete, we will compare the sensitivity of this statistic to the redshift space distortions parameter concerning other famous measures such as crossing statistics (cr) and Minkowski tensors.
3) Using the N-body simulations provided by the Quijote suite, the capability of cmd and cr statistics will be verified and we will elucidate the non-Gaussianity matter density field in redshift space, especially thorough the line of sight by cmd up to the O(σ 2 0 ), perturbatively.4) By performing Fisher forecasts, we evaluate the power of cmd and cr to constrain the relevant cosmological parameters.5) The sensitivity of cmd and cr to the halo bias will be examined by Quijote halo catalogs in redshift space.
The rest of this paper is organized as follows: Section 2 will be assigned to a brief review the notion of RSD and the relationship between the density field in the redshift and real spaces.In Section 3, we will present a mathematical description of our new measure to capture the preferred direction in the context of a probabilistic framework.The perturbative expansion of theoretical prediction for the cmd in the mildly non-Gaussian regime is also given in this section.Section 4 will be devoted to the characterization of RSD including the linear Kaiser and FoG effects using the geometrical measures.The implementation of geometrical measures, cmd and cr on our synthetic data sets and also N-body simulations by the Quijote team will be presented in section 5. We will give the results of Fisher forecasts and also halo bias dependency and sensitivity, in this section.The last section will be focused on the summary and concluding remarks.

Redshift Space Distortions
In this section, for the sake of clarity, we first briefly review the relationship between a typical cosmological stochastic field in the redshifted Universe and corresponding quantity in the real space.Owing to the peculiar velocity field, the observed position of an object in redshift space (s) differs from its real space position, r, and its relation is given by: where v(r) represents peculiar velocity, n is the line of sight direction and H is the Hubble parameter.Equation (1) leads to a distortions in an observed cosmological stochastic field, particularly the observed density field in redshift space.To the linear order, due to the so-called linear Kaiser effect, the distorted density contrast field in redshift space is related to the density contrast field in real space for a given wavenumber, k, by (Kaiser 1987): The (~) symbol is reserved for quantity in the Fourier space throughout this paper.The ⋄ is replaced by (s) and (r) for redshift and real spaces, respectively.Also µ ≡ k. n, f is the linear growth rate of the density contrast and b is the linear bias factor.Equation (2) holds for the matter and biased tracers (e.g.galaxies) density fields, which for the matter case, we have b = 1.Beside to the linear Kaiser effect, there are non-linear effects such as the non-linear Kaiser effect and the FoG effect leading to the distortions of the density field in redshift space with different manners.Therefore, taking into account the nonlinear effects, the Equation (2) can be written in the general form as: (3) in which the operator Õs can be written in the multiplication decomposition of the linear Kaiser part ( Õlin ) and the nonlinear part ( Õnl ) as below: Accordingly, the power spectrum in the redshift space and in the real space have the following relation: Equations ( 3) and ( 5) demonstrate that the Fourier transform of the redshift space density field as well as the power spectrum depends on the direction of wavenumber relative to the line of sight.In other words, the density field in the redshift space is anisotropic and there is an alignment in the line of sight.In this case, we expect that a proper directional statistical measure is capable to distinguish the line of sight direction from the perpendicular directions.It turns out that the mentioned difference should be depended on the amount of anisotropy which is produced by redshift space distortions and even on the sensitivity of the considered directional statistics.For an isotropic density field, there is no difference between various directions.As mentioned in the introduction, any conceivable measure to extract reliable cosmological results should be taken into account such generated anisotropy which is inevitable for the astrophysical context.For this purpose, we will rely on the probabilistic framework in the next section to construct new directional statistical measures and evaluate the corresponding capabilities for desired applications.

Probabilistic Farmework
Suppose that δ (r,s) R denotes the density field contrast in the real and redshift spaces and it is already smoothed by a smoothing window function, W R , in the Fourier space as: We define a so-called set for mentioned smoothed density field in 3-Dimension including the field itself and corresponding first derivative as A (r,s) ≡ δ (r,s) , δ (r,s) and for simplicity, we have omitted the subscript smoothing scale denoted by R and hereafter, the superscript (r, s) of A is dropped.In addition δ (r,s) ,i r,s) and i gets the x, y, z, representing the axises in the Cartesian coordinate.

JPDF of Random Field
The general form of joint probability density function (JPDF) of the set A including 4 elements for the redshift space and real space, separately can be expressed by (Matsubara 2003): where µ1,µ2,...,µn ≡ ⟨A µ1 A µ2 ...A µn ⟩ c represents cumulant and P G (A) is the multivariate Gaussian JPDF of the A and it is given by: where K (2) ≡ ⟨A ⊗ A⟩ c is the 4 × 4 covariance matrix of A known as second cumulant and ⟨⟩ c denotes to connected moment.The matrix form of K (2) can be expressed as: Using the notation i, j ∈ {x, y, z}, the various components in the K (2) becomes: where δ ij is Kronecker delta function.In the above Equation, σ 2 m illustrates the m-order of spectral index and according to the power spectrum of the density field smoothed on the scale R with a given window function, it reads as: and the spectral index for derivative is: Accordingly, we have: and for the isotropic 3-Dimensional field in the real space, we obtain: The observable quantity of any statistical measure, F(A), depending on the A, can be expressed by the following expectation value: where X G ≡ dA P G (A)X.Therefore, in the presence of non-Gaussianity, one can obtain the statistical expectation value of F(A) in terms of Gaussian integrations based on perturbative formalism.

The cmd statistical measures
For a 3-Dimensional density field with total volume V sampled on lattice M, we define excursion set Q ϑ as a set of all field points which satisfy condition δ (r,s) . The boundary of mentioned excursion set, denoted by ∂Q ϑ , characterizes the isodensity contours of the density field at threshold ϑ.
As mentioned in the introduction, scalar MFs have been used to characterize the morphology of density contrast field.Focusing on the anisotropy imposed by various phenomena, proper measures which are designed for anisotropy detection are recommended to use.As an illustration, the redshift space distortion affects the isodensity contours of cosmological density fields with different manner in the along and perpendicular to the line of sights, consequently modification of MFs such as so-called Minkowski Valuations (MVs) (see Appendix A for more details) can be proper measures to examine such effect.The rank-2 MVs have been used to asses the anisotropy properties of redshift space and also distortions parameter (Matsubara & Yokoyama 1996;Codis et al. 2013;Appleby et al. 2018Appleby et al. , 2019Appleby et al. , 2023)).The Genus and contour crossing in various dimensions has been examined in redshift space (Matsubara 1996;Codis et al. 2013).Interestingly, those statistics revealing the one-and two-Dimensional slices depend on anisotropy due to peculiar velocities in redshift space.Consequently, we are persuaded that other criteria similar to the well-known measures introduced for the characterization of morphology may have the potential for anisotropy evaluation in the cosmological stochastic field.After introducing the so-called level crossing as a powerful tool for quantifying a typical stochastic time series by S. O. Rice (Rice 1944(Rice , 1945)), the generalized form of that means including the Up-, down-and the conditional crossing statistics have been utilized as complementary methods for diverse applications (Bardeen et al. 1986;Bond & Efstathiou 1987;Ryden 1988;Ryden et al. 1989;Matsubara 1996;Brill 2000;Matsubara 2003;Movahed & Khosravi 2011;Ghasemi Nezhadhaghighi et al. 2017;Klatt et al. 2022).Particularly, the contour crossing statistic corresponds to the mean number of intersections between the isodensity contours of the density field at threshold ϑ , ∂Q v , and a straight line in a specific direction (Ryden 1988).The crossing statistics is given by a specific choose , (see Equation ( 55) in Appendix A) leading to: Using Equations ( 15) and ( 16), the crossing statistic (cr) for a Gaussian 3-Dimensional field can be expressed as (Ryden 1988;Matsubara 1996Matsubara , 2003;;Codis et al. 2013): where i represents the direction of a straight line.
To establish a new tool to quantify the directional dependency of a typical anisotropic field in the context of generalization of the MFs, some options exist incorporating relaxing the "Hadwiger's theorem", inspired by the crossing statistic a straightforward selection which is proper for cosmologi- According to the definition of characteristic function as Z A (Λ) = dAP(A)exp(iΛ.A), we can also generate various orders of cumulants in addition to the moments and the same analysis can be done in a straightforward manner.This modification in the weight of the first partial derivative enables us to capture the footprint of anisotropic e.g.due to RSD.Selecting the regions satisfying the condition given by δ D δ (r,s) − ϑσ (r,s) 0 and by fixing a direction, i, the nth moment of the first derivative of the fields in such direction for the captured regions to be computed.From the theoretical aspect to define a new criterion, as we will show further, an analytical form exits for the cmd measure to make a well-defined relation to desired cosmological parameters.The mathematical description of the cmd criterion for 3−Dimensional density field can be clarified as follows: we utilize the surface to volume integral transformation (Schmalzing & Buchert 1997).About selecting a typical integrand among various options as mentioned in the appendix, we must point out that since our starting point is motivated from the application point of view, we adopt the following properties to propose the functional form of the integrand in Equation ( 18): directional dependency which is encoded in the first derivative of the underlying field and also inspired by the definition of crossing statistics (Ryden 1988); intuitively, our suggestion belongs to the moment and cumulant definition of density field which is more reasonable compared to other complicated functions; taking into account other typical functions, namely r,s) , ...) is in principle allowed but it turns out that the higher derivative the higher computational time consuming and even opens new room for the higher value of numerical uncertainty.Generally, the shear tensor (δ (r,s) ,ij ) and its combination with the first derivative of the field which is represented by a generic definition of spectral parameters γ n ≡ σ 2 n σn−1σn+1 and characteristic radius of local extrema (R * ∼ σ 1 /σ 2 ) are relevant when we are dealing with the local extrema (Bardeen et al. 1986;Vafaei Sadr & Movahed 2021).As long as our purpose is focusing on directional dependency, we do not need to examine the extrema condition expressed by the second derivatives, consequently, a reasonable choice is adopting the first derivative of density field.Using the probabilistic framework presented in the subsection 3.1, the expected values of N (r,s) cmd for a 3-Dimensional Gaussian density field is obtained as follows: where Γ(:) is Gamma function.Equation ( 19), implies that only the even value of n is survived in the Gaussian regime and all odd values of first derivative moments are identically zero.To mitigating the numerical error, the lowest power adopted for the RSD analysis in the context of cmd measure, would be n = 2, throughout this paper.

Perturbative Formalism
In the previous subsection, we introduced our new measure, and in principle according to Equation ( 15), we can derive the perturbative form of the N (r,s) cmd for 3-Dimensional density field in the mildly non-Gaussian regime.To this end, we expand the Equation ( 15) for a typical observable quantity up to O(σ 2 0 ) as: Subsequently, the weakly non-Gaussian form of N (r,s) cr and N (r,s) cmd , up to the O(σ 2 0 ), becomes: where H n (ϑ) represent the probabilists' Hermite polynomials and we have used following definitions: Having Equations ( 21) and ( 22), we can predict the ⟨N cr ⟩ and ⟨N cmd ⟩ for a given field considering the corresponding spectral indices, respectively.We should notice that for implementation on the normalized density contrast field which is usually adopted by following transformation: the nth conditional moment of first derivative for δ ′ with respect to that of for δ, becomes: while the N cr and MVs are invariant against mentioned transformation.

Implementation on the Redshift space
In this section, we consider the linear Kaiser and FoG effects as the sources of anisotropy in the density field and evaluate the imprint of these effects on our introduced measures in the previous section.Throughout this paper, we use the plane-parallel approximation and consider the z-axis of the Cartesian coordinate as the line of sight direction, without losing generality.In this approximation, there is no statistical difference between the directions perpendicular to (ẑ) (e.g. x and ŷ), and we use the notation Î to show these directions.
4.1.The cmd and cr measures in the Linear Kaiser limit In the linear Kaiser limit, Equation (5) reduces to (Kaiser 1987): Using Equations ( 11), ( 12) and ( 27), one can obtain: where Consequently, in the linear Kaiser limit, the cr and cmd statistics for a 3-Dimensional Gaussian field in redshift space for ẑ and Î directions become: Note that the r.h.s of the above Equations have been expressed in terms of the real space spectral indices.For further analysis, we define the following normalized quantity for direction i: where {i, j, k} ∈ [x, ŷ, ẑ] and i ̸ = j ̸ = k.The ⋄ is replaced by cr and cmd.Interestingly, the isotropic Gaussian limit of Equation ( 35) reduces to: Equation ( 36) reveals that in the isotropic Gaussian limit, normalized quantities are independent from the spectral indices and therefore the properties of the power spectrum.For a given field, any departure from Equation (36) can be considered as the signature of anisotropy and/or non-Gaussianity.While for the redshift space, the normalized quantities can be derived as: They have no explicit dependencies on the spectral indices.Thus, for the Gaussian limit, the normalized quantities only depend on the threshold, ϑ, and redshift space parameter, β, through the C 0 and C 1 .From Equation (30), we find: In the limit β → 0, Equations ( 37)-(40) get the isotropic limit presented in Equation ( 36).In such a limit, the normalized cr and cmd measures are similar.Therefore, for β ̸ = 0, the among of deviation from the isotropic limit can be considered as a signature for determining the sensitivity of cr and cmd statistics to RSD.In Fig. 1, we plot the analytical predictions of the normalized cr and cmd quantities as a function of threshold, ϑ, for a typical anisotropic Gaussian matter field in redshift space in the presence of the linear Kaiser effect adopted by β = 0.48 as a fiducial value.The black solid line illustrates the normalized cr and cmd for isotropic limit (Equation ( 36)).The green dashed line corresponds to n cr for a line of sight direction, while the purple dashed-dotted line is perpendicular to the line of sight direction.The linear Kaiser effect squeezes the isodensity contours along the line of sight.As a result, according to the definition of cr statistics, we expect that the value of n cr is higher for the line of sight direction compared to Î directions.Noticing the analytical form of cmd criterion (Equations ( 39) and ( 40)), the mentioned imprint of linear Kaiser effect would be magnified leading to make a robust measure compared to the common crossing statistics.Subsequently, the difference between n cmd ( Î) (blue dotted line) and n cmd (ẑ) (red loosely dashed line) is higher than the corresponding value in the context of cr measure for as fixed value of β.The lower panel of Fig. 1 indicates the difference between ∆n (s) ⋄ (ϑ, Î).To make more complete our discussion regarding the capability of cr and cmd measures to put the observational constraint on the RSD parameter, we follow the approach carried on by Appleby et al. (2019) in the context of Minkowski tensors.We introduce following quantities by means of cr and cmd criteria as: Figure 1.The theoretical prediction for the normalized cr and cmd measures as a function of threshold for β fiducial = 0.48 as a fiducial value in the linear Kaiser limit.The "*" symbol is replaced by Î and ẑ for perpendicular and along to the line of sight, respectively.The ⋄ symbol is reserved for cr and cmd statistics.The black solid line is for β = 0 which shows the isotropic limit.The lower panel depicts the difference ∆n It turns out that for the Gaussian and linear Kaiser limit, we have: We use the notation Θ ⋄ with ⋄ ∈ {cr, cmd, M T 1, M T 2}.Here the M T 1 and M T 2 are associated with the type one and two of rank-2 Minkowski tensors as defined by Appleby et al. (2019).Based on the theoretical predictions of Θ ⋄ , we can determine the level of accuracy accordingly, we can constrain the value of parameter β.This accuracy depends on the statistical uncertainty associated with Θ ⋄ , which can be evaluated using the Fisher forecast approach.We rely on the posterior probability function, P ⋄ (β|Θ ⋄ ), as: here J is the Jacobian computed for Θ ′ ⋄ = Φ ⋄ (β) and Φ ⋄ (β) reads by Equation (43).Finally, according to a given confidence interval (C.L.), the error-bar on β is given by: or equivalently, based on the error propagation formalism up to the first order, we obtain the relative error on redshift space parameter as: where σ β and σ ⋄ represent the fractional uncertainties on β and Θ ⋄ , respectively.In the upper panel of Fig. 2, we plot σ β in terms of σ ⋄ for β fiducial = 0.48 as the fiducial value in the Gaussian limit.We also consider σ ⋄ = 0.01 as the comparison base value which is shown by the black vertical solid line in this figure.As mentioned in previous research, incorporating the one percent relative error in the statistical measures as already has been achieved by the current galaxy catalogs, yielding almost higher accuracy in the context of cmd criterion compared to all other statistics including cr and rank-2 Minkowski tensors.Since the functional form of G to establish cmd statistics compared to the common generalization of the MFs, particularly the rank-2 MTs causes to manipulate the presence of the field first derivative for different directions, namely, (σ 1x , σ 1y , σ 1z ), in the denominator of MT1 resulting in almost increasing the sensitivity of cmd.The βdependency of sensitivity with respect to Θ ⋄ (ratio quantity) for different measures is shown in the lower panel of Fig. 2. The relative difference of σ β for M T 1 respect to the cmd measure demonstrates that utilizing cmd yields almost 20% improvement which is almost accepted to achieve high precision evaluation.There is a trade off between the imprint of FoG and linear Kaiser effects for different smoothing scales due to their contradiction behaviors at small and large scales, respectively.At the so-called R⋆ whose value depends on cosmological parameters, the directional dependency of cr and cmd is negligible.The lower part of the left panel illustrates the difference of ξ⋄ for both velocity dispersions.Right panel: the comparison between two phenomenological models for FoG, namely the Lorentzian and the Gaussian models, in the context of cr and cmd statistical measures.The corresponding lower panel depicts the difference between ξ⋄ for the Gaussian and the Lorentzian cases.

Finger of God impact on the cr and cmd measures
Thus far, we have applied cr and cmd statistics to the redshift space density field in the presence of the linear Kaiser effect.In this subsection, we take into account the FoG phenomenon in addition to the linear Kaiser effect as the anisotropy sources of the density field in redshift space and try to characterize their impacts on our statistical measures.
To elaborate on the FoG effect describing the elongation of the clusters along the line of sight on small scales, there are several phenomenological models in the literature.Here we consider Gaussian (Peacock & Dodds 1994) and Lorentzian (Percival et al. 2004;Ballinger et al. 1996) FoG models which are respectively read off by the following Equations: and where σ u is the one-dimensional velocity dispersion.More precisely, to manipulate the linear Kaiser and FoG effects together, the non-linear part of the Equation ( 4) can be replaced by the Equation (47) or Equation (48).It is worth noting that the spectral indices given by Equations ( 11) and ( 12) are modified by correction of power spectrum which is in principle constructed by plugging the Equation (47) or Equation (48) in the Equation (5).
To go further, we define ξ ⋄ ≡ Θ −1 ⋄ − 1 and ⋄ ∈ {cr, cmd}.The Gaussian limits of ξ cr and ξ cmd are obtained as: Therefore, in this case, ξ ⋄ statistics depend on the following quantities including parameter β, FoG model, one dimensional velocity dispersion, σ u , smoothing kernel, W , smoothing scale, R, and power spectrum in the real space, P (r) (k).The ξ ⋄ can be numerically computed for a desired cosmological field and supposing the Gaussian model, this can be considered as a new model-dependent observational measure to constrain the associated cosmological parameters.
In the left panel of Fig. 3, we illustrate the ξ ⋄ as a function of the smoothing scale when the phenomenological Gaussian model is taken into account for the FoG effect.Scaling dependency of ξ ⋄ is clearly due to the FoG and interestingly we obtain that at a smoothing scale denoted by R ⋆ , the ξ ⋄ pierces the zero threshold.This means that at such a scale, the directional dependency of cr and cmd measures are diminished due to the competition between the linear Kaiser and FoG effects which behave on the contrary ways.In other words, the FoG and the linear Kaiser effects lead to the stretching and hardening of the iso-density contours along the line of sight, respectively.On a specific scale (R ⋆ ), the linear Kaiser effect and the FoG effect cancel each other out, and ξ cr and ξ cmd reach zero.In scales smaller than R ⋆ , the FoG effect is dominant, and both ξ cr and ξ cmd have negative values, but for scales larger than R ⋆ , the linear Kaiser effect becomes significant and both ξ cr and ξ cmd take positive values.In addition, by increasing the velocity dispersion, the dominant range of FoG grows, which is in agreement with the analytical modeling of FoG.In the lower part of left panel, we plot the ∆ξ ⋄ ≡ ξ ⋄ (σ u = 5 Mpch −1 ) − ξ ⋄ (σ u = 4 Mpch −1 ) and our results confirm that ξ cmd is higher than ξ cr for the two fixed values of σ u .
In the right panel of Fig. 3, the ξ ⋄ for the Gaussian and Lorentzian models of FoG are compared.The higher value of smoothing scale leads to diminishing the contribution of higher k resulting in the two mentioned models converge to each other.We must point out that, the sensitivity of cr measure is almost less than cmd criterion to non-linearity in redshift space.Therefore, the capability of cmd in distinguishing different FoG models is higher than cr statistics.To make more sense, we also compute ∆ξ ⋄ ≡ ξ ⋄ (Lorentzian F oG) − ξ ⋄ (Gaussian F oG) and the bottom part of left panel shows that ξ cmd for small smoothing scale has higher dependency on the model of FoG than the ξ cr .
As indicated by the right panel of Fig. 3, for small smoothing scale the ξ cmd has more R-dependency leading to have more dependency to the scale dependent bias, while the ξ cr has almost smaller amplitude and it shows weak Rdependency causing to have less dependency to scale dependent bias.This means that for the cosmological inferences from the linear regime, utilizing the cr statistics reveals robust pipeline.Meanwhile, to put almost stringent constraints on the cosmological parameters and to examine the peculiar velocity field (Jiang et al. 2022), the cmd measure may give promising results.

Application on Mock data
In this section, we are going to numerically extract the cr and cmd statistical measures for simulated anisotropic density field and compare our results with the theoretical predictions obtained in previous section.Two following are considered: at first, according to the computed matter power spectrum consistent with flat ΛCDM model, we simulate Gaussian random field.Secondly, we will rely on the Nbody simulations known as Quijote simulations (Villaescusa-Navarro et al. 2020).

Gaussian synthetic field
We consider the linear Kaiser effect as a source of anisotropy and therefore generate the anisotropic Gaussian  field.To this end, using the linear power spectrum of matter determined by CAMB, we generate an isotropic Gaussian density field, δ (r) , sampled on a cubical lattice with the total volume size V = (1Gpc h −1 ) 3 which consist of N pix = 512 3 pixels.Applying the Fourier transform on the simulated isotropic density field, we construct an anisotropic r)  cr (ϑ, ŷ) N (r)  cr (ϑ, ẑ) N (s)  cr (ϑ, x) N (s)  cr (ϑ, ŷ)  22)).The filled black circle symbols correspond to the numerical analysis including their 1σ level of confidence.The panels (a.3) and (a.4) are the same as the panel (a.2) just for redshift space in ẑ and x directions, respectively.The lower panels are the same as the upper panels just for N (r,s) cr .field according to the following transformation: and therefore, we obtain the redshift space density field in Fourier space.Then we smooth δ(s) by a Gaussian kernel with scale R = 20 Mpc h −1 .We generate N sim = 100 realizations of Gaussian isotropic and anisotropic fields, and then apply the mentioned numerical methods to the simulated density fields to obtain the cr and cmd measures as a function of threshold.For each realization, we extract these statistics in threshold range, ϑ ∈ [−4.0, 4.0], and then we do the ensemble average.Increasing the number of realizations had no significant effect on our ensemble average.Fig. 4 presents the cr and cmd as a function of ϑ for perpendicular and along to line of sight in redshift space and for real space.The upper panel corresponds to cr, while the lower panel shows the cmd.The solid lines are associated with theoretical predictions and the symbols are for corresponding numerical results.The error bar represents the 1σ level of confidence demonstrating the good consistency between the numerical and theoretical results.However, due to the presence of the first derivative of the smoothed field in cmd measure, we expect to obtain an almost higher value of errorbar compared to cr.
In the rest of this subsection, motivated by introducing a proper measure to put observational constraint on β, we define the following weighted summation to marginalize the effect of threshold bins: where the weight is defined by means of statistical error as ω ⋄ (ϑ i ) ≡ σ −1 ξ⋄ (ϑ i ).In Fig. 5, we plot the ξ⋄ statistics as a function of β extracted from theoretical (solid line) and computational (symbols) approaches.The higher slope for cmd measure concerning the cr statistics versus β reveals more robustness in discriminating different values of β.

N-body simulations
To examine the non-Gaussian impact on the conditional moments of derivative, we use the three-dimensional large scale structure made by publicly available N-body simulations from the Quijote complex (Villaescusa-Navarro et al. 2020).Each our ensemble extracted form the Quijote simulations has following properties: N particles = 512 3 , box size of V = (1Gpc/h) 3 , the fiducial cosmological parameter are based on flat ΛCDM including Ω m = 0.3175, Ω b = 0.049, h = 0.6711, n s = 0.9624 and σ 8 = 0.834 (see (Villaescusa-Navarro et al. 2020) for more details).To construct the proper density field in applying our numerical pipeline, we use Pylians (Villaescusa-Navarro 2018) and at redshift z = 0, exploiting the routine cloud in cell (CIC) for mass assignment, the density field contrast, δ(r), would be retrieved.Finally, convolving the δ(r) with a Gaussian window function characterized by a smoothing scale, R, the matter density contrast is constructed in the real space, δ (r) R (r).To create the corresponding field in redshift space in plane-parallel approximation, δ (s) R (r), for each value of r, we use Pylians which in principle considers the Equation (1).We also use The difference between the numerical computation of N (r,s) (cmd,cr) and the corresponding theoretical Gaussian prediction are depicted in Fig. 7.The dashed lines are for the deviation of perturbative non-Gaussian theory concerning Gaussian prediction, while the symbols are the same quantities computed from simulations, numerically.In the left panel (c.1), we depict the difference between the numerical results and theoretical Gaussian predictions in addition to the variation of the theoretical non-Gaussian model concerning the Gaussian form for real space.The green filled circle symbols, blue triangle symbols, and red rectangle symbols indicate the difference between N (s) cmd (ϑ, ẑ), N (s) cmd (ϑ, x) and N (r) cmd (ϑ, ẑ) computed numerically for N-body simulations and associated Gaussian models, respectively, in the panel (c.2).For this part, we consider the smoothed scale equates to R = 40 Mpc h −1 .We display the same quantities as expressed for the panel (c.2) but for R = 20 Mpc h −1 in the panel (c.3) and R = 30 Mpc h −1 in the panel (c.4).Our results confirm that the deviation from Gaussianity perpendicular to the line of sight directions in redshift space is almost the same as the real space.It has been shown that keeping the directional dependency in computing power spectrum and also in derived quantities causes to mitigate the degeneracy between RSD and non-linearity consequences (Jennings et al. 2016).We also advocate that the separation of perpendicular to the line of sight analysis from the ẑ direction in the redshift space can reduce the RSD impact on the cosmological inferences such as non-Gaussianity.It is worth noting that, to compute the corresponding theoretical results, we adopt the spectral indices numerically from simulations.The inconsistency between theory and numerical results ex-tracted from N-body simulations is justified due to the reason that for the lower value of the smoothing scale, the σ 0 gets higher value for the lower value of R, consequently, to obtain more precise consistency, we have to take into account the higher terms in perturbative formula according to Equation (20) to achieve more accurate formula for N (r,s) (Equation ( 22)).The behavior of N (r,s) cr − ⟨N (r,s) cr ⟩ G for different situations are illustrated in the lower part of Fig. 7. Our results demonstrate that to have more consistent results from numerical analysis and theoretical prediction for cr measure, we need to take into account higher order terms beyond 1st-order.In addition, for some lower thresholds, the results for the hatx in the redshift space deviates from that of in real space in the context of N cr confirming that to mitigate the RSD non-Gaussianity imprint, the ϑ ≳ 0 should be taken while this limitation almost does not exist for the N cmd statistics.

Fisher Forecasts
In order to present a quantitative description regarding the capability of various criteria explained before, particularly cmd and cr to put constraint on relevant cosmological parameters, we compute the Fisher matrix in this subsection (see e.g.(Bassett et al. 2011;Wolz et al. 2012) for the reviews on Fisher forecast and its applications in cosmology).Using the likelihood L, the Fisher matrix can be defined as: where we consider {α} = {σ 8 , Ω m , n s } as the set of model parameters.Accurate constraining of cosmological parameters using information available at small scales requires modeling the nonlinear effects of matter clustering, galaxy bias, and redshift space distortions (e.g.FoG effect), which are theoretically challenging and here we confine ourselves to use most famous cosmological parameters among the full set of them.A way to overcome these challenges is to use the simulation based inference approach (Papamakarios & Murray 2016;Alsing et al. 2019;Cranmer et al. 2020;Hahn et al. 2022).Assuming that L is a multivariate normal distribution, the Fisher matrix element reads as: where {D} = {N cmd , N cr , Θ cmd , Θ cr } represents the data vector consisting of observables and C indicates the covariance matrix.It is worth mentioning that, we use the transformation as δ → δ ′ = δ/σ 0 , therefore all statistics in the data vector are fully numerically computed for δ ′ .As an example to compute the cmd statistics, from the density field numerically, we utilize the discrete form of the volume integrals presented in Equation ( 18), which is given by: where δ ′ (p, q, u) and δ ′ ,i (p, q, u) represent the value and first derivative of the field and in a pixel identified by (p, q, u) indexes in the Cartesian coordinates (x, y, z), respectively.The ∆ represents the pixel size.We also use the discrete Dirac delta function (Schmalzing & Buchert 1997).
To estimate the partial derivatives in Equation ( 53) for Quijote fiducial values of cosmological parameters, we consider 500 corresponding realizations.To extract the covariance matrix, we also utilize 5000 realizations of the fiducial simulations.To examine the influence of cmd and cr statistics on parameter constraining, we also takes R = 40 Mpc h −1 .Fig. 8 indicates the Fisher forecasts for some relevant parameters.The constraints in the Ω m − σ 8 plane for ratio component (Θ) and joint analysis of cmd + cr statistics are depicted in the upper left panel.Taking into account the Θ cmd instead of Θ cr results in almost ∼ 10% and ∼ 20% improvements on constraining the Ω m and σ 8 , respectively.The joint analysis of Θ cmd + Θ cr also enhances the constraint on the σ 8 about 35%, while for Ω m we have 45% compared to Θ cr measure.The joint analysis of different components of cmd measures increase the capability to constraint in the Ω m −σ 8 plane.Incorporating the R = 40 Mpc h −1 and computing the Fisher matrix elements, reveal that constraint interval on the σ 8 becomes large as we expect, while the impact of Ω m due to Kaiser effect remains almost unchanged.It is worth noting that the Θ ⋄ can reduce the degeneracy in the Ω m − n s plane respect to the N ⋄ .A final remark is that, since we have used the unit variance density field, therefore the coefficient of N cmd is independent of σ 8 similar to N cr , consequently, the constraint on σ 8 by N cmd for large scale is relatively weak (lower left panel of Fig. 8).

Sensitivity to Halo bias
The visible matters of the Universe forming inside the gravitationally bound dark matter halo is considered as a representative to trace the dark matter distribution on the cosmological scales.This mechanism inevitably prevents the galaxies perfectly trace the underlying mass distribution.Subsequently, to achieve the proper cosmological inferences by the galaxies observation, the halo bias indicating the relationship between dark matter halos and dark matter distribution on large scale, should be clarified as much as possible (Desjacques et al. 2018;Lucie-Smith et al. 2023).In our approach, the ratio quantity (Θ (cmd,cr) ) depends on β according to Equations (43) for the Gaussian and linear Kaiser limit.The β depends on bias, also, as we defined before, the weighted summation on thresholds ξ(cmd,cr) (see Equation ( 51)) ξ(cmd,cr) is related to ξ (cmd,cr) and equivalently to Θ (cmd,cr) .Subsequently, we expect that by computing the ξ(cmd,cr) from the available observational catalogs or from mock data to manage various parameters, in redshift space for different mass cuts is able to reflect the halo bias.To show how our defined parameter in the context of cmd and cr statistics can clarify the halo bias dependency and sensitivity of cmd and cr measures, we calculate ξ(cmd,cr) for the mock halo catalogs of the Quijote fiducial simulations to quantify the sensitivity of N cmd and N cr  statistics to the halo (or galaxy) bias.we take four mass cuts, [13.1, 13.4, 13.7, 14.0].For each mass cut, we construct a density contrast field by using halos that have a mass greater than the selected mass cut sampled on a regular lattice with N pix = 256 3 .Then, we smooth the obtained density contrast field using a Gaussian kernel with smoothing scale R = 40 Mpc h −1 .Finally, we calculate the ξcmd and ξcr corresponding to each M cut .Fig. 9 illustrates the ξ⋄ as a function of log 10 M cut / M ⊙ h −1 .The mass cut dependency of cmd and cr according to the Fig. 9 with monotonic behavior means that for a given mass cut value, one can find a unique value for ξ, consequently the ξ⋄ can be considered as a new indicator for examining the halo (or galaxy) bias dependency.The 1σ confidence interval for symbols in Fig. 9 decreases by increasing mass cut value.Such behavior can be justified that for the lower value of mass cut, the diversity of halo bias becomes significant leading to obtain the dispersion on averaged halo bias indicated by higher statistical uncertainty.On the contrary, for the higher value of mass cut with statistically enough population, the similarity in the set of derived halo bias increase leading to achieve lower value in the computed statistical error.The results illustrated in Fig. 9 have been derived for fiducial simulation and since the β depends on the other cosmological parameters in addition to the bias factor, therefore, one cannot yield constraint on the halo (or galaxy) bias.Practically, to mitigate this discrepancy, incorporating the quantity possessing a lower footprint of the bias should be considered (Appleby et al. 2020(Appleby et al. , 2021)).In other word, to put constraint on the bias factor which plays as nuisance parameter, we need to know the values of other cosmological parameters such as power spectrum.The cmd is also more sensitive to halo bias compared to cr, consequently aiming for constraining on the cosmological parameters in the presence of bias factor considered as a nuisance parameter, the cr statistics is recommended to implement.

Summary and Conclusions
The redshift space distortions caused by the linear and nonlinear effects lead to anisotropy in the density field in the redshift space.To clarify the mention anisotropy as well as non-Gaussianity, we have developed a geometrical measure which is quite sensitive to the anisotropic distribution of density fields.
In this work, inspired by the contour crossing (cr) statistic and generalization of MFs, we have introduced the so-called conditional moments of derivative (cmd) criteria, which can capture the preferred direction and also are sensitive to the induced anisotropy together with the non-Gaussianity in the underlying cosmological stochastic field.Using a probabilistic framework, we have analytically calculated the theoretical expectation value of cmd measure as a function of threshold (ϑ) for isotropic and anisotropic Gaussian density field in terms of associated spectral indices.Also, for the weakly non-Gaussian field, we have perturbatively extended our analysis up to the O(σ 2 0 ) contribution due to the general non-Gaussianity in real and redshift spaces.In addition, to perform a comparison between cmd and cr statistics, we have carried out similar computations for the cr as well.
Taking into account the Gaussianity and incorporating the linear Kaiser effect as the source of anisotropy in the redshift space density field, we have compared the sensitivity of cr and cmd statistics to the redshift space parameter (β).The normalized quantity n ⋄ depending on direction, threshold, and the β parameter has been introduced and our results demonstrated that the n (s) cmd is more sensitive than the n (s) cr to the anisotropy as depicted in the lower panel of Fig. 1, particularly for intermediate threshold.According to the error propagation approach and by considering the relative error on the Θ ⋄ equates to one percent indicating that the cmd enables to put tight constraint compare to other statistics in Gaussian (Fig. 2) and non-Gaussian regimes.
To make our evaluation more complete, we defined ξ ⋄ (Equation ( 49)) and its smoothing scale dependency for various values of relevant parameters for the treatment of the influence of FoG and the comparison with the linear Kaiser effect.This quantity is carefully recognized in the range scale where the contribution of FoG or linear Kaiser effect becomes dominant (Fig. 3).This quantity for a high enough value of R also asymptotically goes to the fixed value implying the β value.
Implementation of the synthetic data numerically has also supported the good consistency between numerical results and theoretical prediction of cr and cmd measures for the Gaussian field (Fig. 4).To adopt the observational constraining robustly, we also defined a weighted parameter, ξ⋄ (Equation ( 51)), and for various values of β in simulation, we revealed that ξcmd has higher β-dependency (Fig. 5).
Although, we have implemented our methodology on the N-body simulations publicly available by the Quijote suite.The N (r,s) (cmd,cr) as a function of ϑ implied that there is a deviation from Gaussian theory along the line of sight for both real and redshift spaces and the numerical results are higher(less) than the Gaussian prediction for ϑ ≳ 0 (ϑ ≲ 0) (Fig. 6).To make more sense regarding the non-Gaussianity in the Nbody simulations provided by the Quijote, we obtained that the amount of non-Gaussianity in the context of N cmd for perpendicular to the line of sight directions in redshift space is almost the same as for along of line of sight in real space (Fig. 7).This means that to mitigate the non-Gaussianity produced by RSD and to examine non-Gaussianity due to other mechanisms such as primordial ones, we should consider the analysis on the plane perpendicular to the line of sight in redshift space.The peculiar velocity magnifies the non-Gaussianity along the line of sight in redshift space which is well recognized by cmd and cr measures.
To quantify the constraining power of cmd and cr measured, we have done Fisher forecasts.Numerically determining the associated matrix elements clarified the influences of our statistical measures individually accompanying the joint analysis on the relevant cosmological parameters (Fig. ( 8)).We achieved that constraints on σ 8 and Ω m according to joint analysis of Θ cmd +Θ cr are better by 35% and 45% relative to considering Θ cr , respectively.Taking into account the ratio quantity of our measures leads to reduce the degeneracy in the Ω m − n s plane compared to that of given by N (cmd,cr) .
We have also attempted to address the sensitivity of the cmd and cr measures to the halo bias in redshift space.Using the Quijote halo catalogs, we have calculated the ξ(cmd,cr) for the density field constructed from halos whose masses are greater than a specific mass cut (Fig. 9).Our results confirmed that ξ can be a promising measure to evaluate halo bias.While the cr measure with lower dependency on mass cut revealed robust quantity when the bias factor is considered as a nuisance parameter.
To go further, we suggest to do following tasks as the complementary subjects in the banner of excursion sets and RDS and will be left for the future study: however the planeparallel approximation provides a reliable approach in accounting for the peculiar velocity along the line of sight in the observed distribution of galaxies, but to obtain more accurate evaluation, spherical redshift distortions could be interesting to pursue.We also focused on the matter density field and it is useful to consider the galaxies catalogs and other real data sets instead, consequently utilizing the cmd and cr measures open new room to evaluate bias factor.Utilizing the cmd and cr for cosmological parameters constraining approach need to do more complementary analysis like assessing the parameters associated with nonlinear phenomena such as σ u , and a recommended method is to use the simulation based inference approach (Papamakarios & Murray 2016;Alsing et al. 2019;Cranmer et al. 2020;Hahn et al. 2022).In addition, various model of primordial non-Gaussianity can be evaluated.
Generally, the existence of a preferred direction in cosmology and for various scales has remained under debate topic, to this end, our measures can provide a pristine framework.Thanks to scaling window analysis and modifying the cmd criteria, hopefully, makes it more capable by scanning over the underlying field to capture the scale and location dependency of directional behavior (Li et al. 2013;Ghasemi Nezhadhaghighi et al. 2017;Klatt et al. 2022;Kapfer et al. 2010;Schröder-Turk et al. 2013).
The authors are very grateful to Ali Haghighatgoo for his extremely useful comments on different parts of this paper.Also thanks to Ravi K. Sheth for his constructive discussions.SMSM appreciates the hospitality of the HECAP section of ICTP where a part of this research was completed.We also thank the Quijote team for sharing its simulated data sets and providing extensive instruction on how to utilize the data.Finally, we appreciate the anonymous referee who helped us to focus on the most relevant topics which led to improved our paper.

A. Generalization of Minkowski Functionals
The so-called Minkowski Functionals (MFs) possess scalar property.To characterize the morphology of a typical d-dimensional field, there are (d + 1) MFs which are unique and complete in the sense of Hadwigers's theorem and satisfying the motion invariance (e.g.rotations and translations), additivity and conditional continuity.It is well-known that the additivity and motion-invariance properties of the MFs lead to prevent the MFs from discriminating different anisotropic patterns in a field (Beisbart et al. 2002).Depending on starting point objective which is in principle devoted to the mathematical side as well as from the applications, substantial progress can be considered to generalize scalar MFs.Relaxing the above conditions allows to consider the following generalization of the MFs in d−dimension: where s ν is a functional form of curvatures and ν = 0, ..., (d − 1).The G is a general functional form of (s ν ; r, δ, ∇δ, ...).A reasonable extension of scalar MFs on Euclidean space has been done by introducing a specific functional form for G which is known as the "Minkowski valuations" (MVs) (McMullen 1997;Alesker 1999;Hug et al. 2007).In this regard, we have: here ⊗ reveals the tensor product.Accordingly, the vectorial form is derived for (p = 1, q = 0), while the W (p,q) ν for (p = 0, q = 1) by definition is vanished.Also for rank-2 tensor form, the condition p + q = 2 should be satisfied in Equation ( 56).

Figure 2 .
Figure2.Upper panel: Error propagator on the β for various criteria discussed in the text.Supposing one percent relative error produced for σ⋄ in observation causes lower relative error on redshift space parameter by cmd measure compared to other criteria examined in this paper.Lower panel: The β-dependancy of relative error (σ β ) for various statistics.We considered σ⋄ = 0.01.

ξFigure 3 .
Figure3.Left panel: ξ⋄ as a function of R for the phenomenological Gaussian model of FoG and for σu = 4 Mpc h −1 and σu = 5 Mpc h −1 .There is a trade off between the imprint of FoG and linear Kaiser effects for different smoothing scales due to their contradiction behaviors at small and large scales, respectively.At the so-called R⋆ whose value depends on cosmological parameters, the directional dependency of cr and cmd is negligible.The lower part of the left panel illustrates the difference of ξ⋄ for both velocity dispersions.Right panel: the comparison between two phenomenological models for FoG, namely the Lorentzian and the Gaussian models, in the context of cr and cmd statistical measures.The corresponding lower panel depicts the difference between ξ⋄ for the Gaussian and the Lorentzian cases.

Figure 4 .Figure 5 .
Figure 4. Upper panel: The crossing statistics as a function of ϑ for theoretical predictions (solid lines) and corresponding numerical reconstructions (symbols).Lower panel: cmd measure versus threshold.The solid lines correspond to theoretical predictions, while the symbols indicate the results given by numerical simulations.Here we took R = 20 Mpc h −1 .

Figure 6 .
Figure 6.The N (r,s) (cmd) [Mpc h −1 −2 and N (r,s) (cr) [Mpc h −1 ] −1/2 versus threshold for the Quijote simulations.Panel (a.1):The expectation value of conditional moment of the first derivative in real and redshift spaces for both Î ∈ [x, ŷ] and ẑ directions adopting R = 40 Mpc h −1 .Panel (a.2):The N (r) cmd (ẑ) for the Gaussian prediction considering corresponding spectral indices (green dashed-dot line) while the red dashed line indicates the theoretical non-Gaussian prediction for cmd (Equation (22)).The filled black circle symbols correspond to the numerical analysis including their 1σ level of confidence.The panels (a.3) and (a.4) are the same as the panel (a.2) just for redshift space in ẑ and x directions, respectively.The lower panels are the same as the upper panels just for N

Figure 8 .Figure 9
Figure 8. Fisher forecasts: Upper left panel indicates the constraints in the Ωm − σ 8 plane for ratio component, Θ, and joint analysis of cmd + cr.Upper right panel shows the constraints on Ωm and σ 8 by using N cmd and Ncr criteria.Lower panels are for R = 40 Mpc h −1 , when the non-linear effects are suppressed.In this case the constraint interval on the σ 8 increases as we expect, while the impact of Ωm due to Kaiser effect remains almost unchanged.Other point is that the Θ⋄ can reduce the degeneracy in the Ωm − ns plane respect to N⋄. Al contours have been determined for 68% confidence interval.