Probing the anisotropy and non-Gaussianity in redshift space through the derivative of excursion set moments

M. H. Jalali Kanafi
Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran

S. M. S. Movahed
Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran

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 $3$-Dimension. We derive the perturbative expansion of $cmd$ for 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 $\beta=f/b$ and interestingly, the associated normalized quantity in the Gaussian linear Kaiser limit has only $\beta$-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 redshift space has 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 $R\gtrsim 20$ Mpc/h is almost the same as the real space, while the non-Gaussianity for along to the line of sight direction in redshift space is magnified.

§ 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 <cit.>. On the other hand, discrepancies between what we observe through various surveys and theoretical counterparts essentially persuade researchers  to include the stochastic notion <cit.>.  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 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 distortions (RSD). The Finger of God (FoG) effect <cit.> and the linear Kaiser effect <cit.> 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 <cit.>. Although RSD makes the interpretation of observational data more challenging, it provides an opportunity to extract statistical information to constrain associated cosmological parameters <cit.>.

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 <cit.>,  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 <cit.>; the joint analysis of the Alcock-Paczynski effect and RSD to probe the cosmic expansion <cit.>; combining RSD  with weak lensing and baryon acoustic oscillations to improve the observational constraints on the cosmological parameters <cit.>; quantifying the RSD spectrum <cit.>; examining the primordial non-Gaussianity via RSD <cit.>.

As of the importance role of the large scale structures and corresponding observational catalogs, there are many attempts incorporating the geometrical and topological virtues of diverse relevant fields such as the genus statistic <cit.>, contour statistics including 1-, 2- and 3-Dimensional features  <cit.>, Minkowski functionals consisting of  $D+1$ scalar quantities which describe the morphology of isodensity contours of $D$-dimensional field <cit.> (see also  <cit.>).

The central assumptions in many  cosmological studies are homogeneity, isotropy, and Gaussianity due to the extension of the central limit theorem domain. 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 <cit.>.
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 <cit.>.  Recently, Minkowski tensors, an extension of scalar Minkowski functionals  <cit.>, have been employed on 2- and 3-Dimensional cosmological fields in the real and redshift spaces <cit.>.

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.

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 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 <cit.>.

(2) The anisotropy imprint by the linear Kaiser effect will be examined by our introduced measure 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 and Minkowski tensors.

3) Using the N-body simulations provided by the Quijote suite, the capability of $cmd$ 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 $\mathcal{O}(\sigma^2_0)$, perturbatively.

The rest of this paper is organized as follows: Section <ref> 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 <ref>, 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 <ref>  will be devoted to the characterization of RSD including the linear Kaiser and FoG effects using the geometrical measures. The implementation of geometrical measures on our synthetic data sets and also N-body simulations by the Quijote team will be presented in section <ref>. 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 ($\boldsymbol{s}$) differs from its real space position, $\boldsymbol{r}$, and its relation is given by:

\begin{eqnarray}
\boldsymbol{s} = \boldsymbol{r} + \frac{\boldsymbol{v(\boldsymbol{r})}.\hat{\boldsymbol{n}} }{H} \hat{\boldsymbol{n}},
\label{eq:redshift position}
\end{eqnarray}

where $\boldsymbol{v(\boldsymbol{r})}$ represents peculiar velocity, $\hat{\boldsymbol{n}}$ is the line of sight direction and $H$ is the Hubble parameter. Equation   (<ref>) 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, $\boldsymbol{k}$, by <cit.>:

\begin{eqnarray}
\tilde{\delta}^{(s)}(\boldsymbol{k}) = (1+\beta\mu^2)\;\tilde{\delta}^{(r)}(\boldsymbol{k})\;, \hspace{0.5cm} \beta \equiv f/b
\label{eq:linear kaiser}
\end{eqnarray}

The (~) symbol is reserved for quantity in the Fourier space throughout this paper. The $\diamond$ is replaced by ($s$) and ($r$) for redshift and real spaces, respectively. Also $\mu\equiv\hat{\boldsymbol{k}}.\hat{\boldsymbol{n}}$, $f$ is the linear
growth rate of the density contrast and $b$ is the linear bias factor. Equation   (<ref>) 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   (<ref>) can be written in the general form as:

\begin{eqnarray}
\tilde{\delta}^{(s)}(\boldsymbol{k}) = \tilde{O}_{s}(\mu,k\mu)\tilde{\delta}^{(r)}(\boldsymbol{k}).
\label{eq:anisotropic delta}
\end{eqnarray}

in which the operator $\tilde{O}_{s}$ can be written in the multiplication decomposition of the linear Kaiser part ($\tilde{O}_{\rm{lin}}$) and the non-linear part ($\tilde{O}_{\rm{nl}}$) as below:

\begin{eqnarray}
\tilde{O}_{s}(\mu,k\mu)& =& \tilde{O}_{\rm {lin}}(\mu,k\mu)\times\tilde{O}_{\rm {nl}}(\mu,k\mu)\nonumber\\
&=&(1+\beta\mu^2)\tilde{O}_{\rm {nl}}(\mu,k\mu).
\label{eq:operator}
\end{eqnarray}

Accordingly, the power spectrum in the redshift space and in the real space have the following relation:

\begin{eqnarray}
P^{(s)}(\boldsymbol{k}) = (1+\beta\mu^2)^2\big|\tilde{O}_{\rm{nl}}(\mu,k\mu)\big|^2 P^{(r)}(\boldsymbol{k}),
\label{eq:power spectrum}
\end{eqnarray}

Equations   (<ref>) and   (<ref>) 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 $\delta_{R}^{(r,s)}$ 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:

\begin{eqnarray}
\tilde{\delta}^{(r,s)}_R (\boldsymbol{k})= \tilde{W}(kR)\tilde{\delta}^{(r,s)}(\boldsymbol{k}),
\label{eq:field}
\end{eqnarray}

We define a so-called set for mentioned smoothed density field in 3-Dimension including the field itself and corresponding first derivative as  $\mathcal{A}^{(r,s)}\equiv \left\{\delta^{(r,s)},\delta_{,x}^{(r,s)},\delta_{,y}^{(r,s)},\delta_{,z}^{(r,s)}\right\}$ and for simplicity, we have omitted the subscript smoothing scale denoted by $R$ and hereafter, the superscript $(r,s)$ of $\mathcal{A}$ is dropped. In addition $\delta_{,i}^{(r,s)}\equiv \partial_i\delta^{(r,s)}$ 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 $\mathcal{A}$ including $4$ elements for the redshift space and real space, separately can be expressed by <cit.>:

\begin{eqnarray}
\begin{split}
\mathcal{P}(\mathcal{A}) & =  \exp\bigg\lbrack \sum_{j=3}^{\infty}\frac{(-1)^j}{j!}\bigg (
\sum_{\mu_1=1}^{N=4}...\sum_{\mu_j=1}^{N=4}\mathcal{K}^{(j)}_{\mu_1,\mu_2,...,\mu_j}
\\
& \times \frac{\partial^j}{\partial \mathcal{A}_{\mu_1} ... \partial \mathcal{A}_{\mu_j}} \bigg )\bigg\rbrack \mathcal{P}_G(\mathcal{A})
\end{split}
\end{eqnarray}

where $\mathcal{K}^{(n)}_{\mu_1,\mu_2,...,\mu_n}\equiv\left\langle \mathcal{A}_{\mu_1}\mathcal{A}_{\mu_2}...\mathcal{A}_{\mu_n} \right\rangle_c$ represents cumulant and $\mathcal{P}_G(\mathcal{A})$ is the multivariate Gaussian JPDF of the $\mathcal{A}$ and it is given by:

\begin{eqnarray}
\mathcal{P}_G(\mathcal{A}) = \frac{\exp \left(-\frac12
\mathcal{A}^T{\cdot} \left[\mathcal{K}^{(2)}\right]^{-1}{\cdot}\mathcal{A} \right)}{2\pi\; \sqrt{\det \mathcal{K}^{(2)}}}
\end{eqnarray}

where $\mathcal{K}^{(2)}\equiv \left\langle \mathcal{A} \otimes \mathcal{A} \right\rangle_c$ is the $4\times4$ covariance
matrix of  $\mathcal{A}$ known as second cumulant and $\langle\rangle_c$ denotes to connected moment.  The matrix form of  $\mathcal{K}^{(2)}$ can be expressed as:

\begin{eqnarray}
\begin{split}
&\mathcal{K}^{(2)} =&  &\left(\begin{matrix}
\vspace{0.2cm}
\left\langle\delta^{(r,s)}\delta^{(r,s)}\right\rangle_c & \left\langle\delta^{(r,s)}\delta_{,x}^{(r,s)}\right\rangle_c & \left\langle\delta^{(r,s)}\delta_{,y}^{(r,s)}\right\rangle_c & \left\langle\delta^{(r,s)}\delta_{,z}^{(r,s)}\right\rangle_c \\
\vspace{0.2cm}
\left\langle\delta_{,x}^{(r,s)}\delta^{(r,s)}\right\rangle_c & \left\langle\delta_{,x}^{(r,s)}\delta_{,x}^{(r,s)}\right\rangle_c & \left\langle\delta_{,x}^{(r,s)}\delta_{,y}^{(r,s)}\right\rangle_c & \left\langle\delta_{,x}^{(r,s)}\delta_{,z}^{(r,s)}\right\rangle_c  \\
\vspace{0.2cm}
\left\langle\delta_{,y}^{(r,s)}\delta^{(r,s)}\right\rangle_c & \left\langle\delta_{,y}^{(r,s)}\delta_{,x}^{(r,s)}\right\rangle_c & \left\langle\delta_{,y}^{(r,s)}\delta_{,y}^{(r,s)}\right\rangle_c & \left\langle\delta_{,y}^{(r,s)}\delta_{,z}^{(r,s)}\right\rangle_c  \\
\vspace{0.2cm}
\left\langle\delta_{,z}^{(r,s)}\delta^{(r,s)}\right\rangle_c & \left\langle\delta_{,z}^{(r,s)}\delta_{,x}^{(r,s)}\right\rangle_c & \left\langle\delta_{,z}^{(r,s)}\delta_{,y}^{(r,s)}\right\rangle_c & \left\langle\delta_{,z}^{(r,s)}\delta_{,z}^{(r,s)}\right\rangle_c
\end{matrix}\right)&
\end{split}
\end{eqnarray}

Using the notation $ i,j \in \{x,y,z\} $, the various components in the $\mathcal{K}^{(2)}$ becomes:

\begin{eqnarray}
\left\langle\delta^{(r,s)}\delta^{(r,s)}\right\rangle_c &=& \left(\sigma_{0}^{(r,s)}\right)^2\nonumber\\    \left\langle\delta^{(r,s)}\delta_{,i}^{(r,s)}\right\rangle_c &=& 0\nonumber\\
\left\langle\delta_{,i}^{(r,s)}\delta_{,j}^{(r,s)}\right\rangle_c &=&  \left(\sigma_{1i}^{(r,s)}\right)^2\delta_{ij}
\label{eq:matrix components}
\end{eqnarray}

where $\delta_{ij}$ is Kronecker delta function. In the above Equation, $\sigma_m^2$ 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:

\begin{eqnarray}\label{eq:spectral0}
&&\left(\sigma^{(r,s)}_{m}(R)\right)^2 = \int \frac{d^3\boldsymbol{k}}{(2\pi)^3} |\boldsymbol{k}|^{2m} P^{(r,s)}(\boldsymbol{k})\;\tilde{W}^2(kR)
\end{eqnarray}

and the spectral index for derivative is:

\begin{eqnarray}\label{eq:spectral00}
&&\left(\sigma^{(r,s)}_{1i}(R)\right)^2 \equiv \bigg \langle \left(\delta^{(r,s)}_{,i} \right)^2 \bigg \rangle_c\nonumber\\
&&\quad\quad\;\;\;\;\;\;\quad\quad = \int \frac{d^3\boldsymbol{k}}{(2\pi)^3} k_i^{2} P^{(r,s)}(\boldsymbol{k})\;\tilde{W}^2(kR)
\end{eqnarray}

Accordingly, we have:

\begin{eqnarray}\label{eq:spectral1}
\begin{aligned}
\left(\sigma_{1}^{(r,s)}\right)^2 & = \bigg\langle\left(\delta^{(r,s)}_{,x}\right)^2+\left(\delta^{(r,s)}_{,y}\right)^2+\left(\delta^{(r,s)}_{,z}\right)^2\bigg\rangle_c \\ &=\left(\sigma_{1x}^{(r,s)}\right)^2 + \left(\sigma_{1y}^{(r,s)}\right)^2 + \left(\sigma_{1z}^{(r,s)}\right)^2,
\end{aligned}
\end{eqnarray}

and for the isotropic 3-Dimensional field in the real space, we obtain:

\begin{eqnarray}
\sigma_{1x}^{(r)} = \sigma_{1y}^{(r)} = \sigma_{1z}^{(r)} = \frac{\sigma_{1}^{(r)}}{\sqrt{\;3}}
\end{eqnarray}

The observable quantity of any statistical measure, $\mathcal{F}(\mathcal{A})$, depending on the  $\mathcal{A}$, can be expressed by the following expectation value:

\begin{eqnarray}
\begin{split}
\langle \mathcal{F}(\mathcal{A})\rangle  &= \int d\mathcal{A}\; \mathcal{P}(\mathcal{A})\;\mathcal{F}(\mathcal{A}) \\
&= \bigg\langle\exp\bigg\lbrack \sum_{j=3}^{\infty}\frac{1}{j!} \bigg (
\sum_{\mu_1}^{4}...\sum_{\mu_j}^{4}\mathcal{K}^{(j)}_{\mu_1,\mu_2,...,\mu_j}
\\
&\times \frac{\partial^j}{\partial \mathcal{A}_{\mu_1} ... \partial \mathcal{\mathcal{A}}_{\mu_j}} \bigg ) \bigg \rbrack \mathcal{F}(\mathcal{A}) \bigg\rangle_G
\end{split}\label{eq:expected value}
\end{eqnarray}

$\big\langle X\big\rangle_G \equiv \int d\mathcal{A}\; \mathcal{P}_G(\mathcal{A})X$. Therefore, in the presence of non-Gaussianity, one can obtain the statistical expectation value of $\mathcal{F}(A)$ in terms of Gaussian integrations based on perturbative formalism.

§.§ The $cmd$ statistical measures

For a 3-Dimensional density field with total volume $V$, we define excursion set $Q_v$ as a set of all field points which satisfy condition $ \delta^{(r,s)} \ge \vartheta\sigma_0^{(r,s)}$. The boundary of this excursion set, denoted by $\partial Q_v$, characterizes the isodensity contours of the density field at threshold $\vartheta$. The redshift space distortion affects the isodensity contours of cosmological density fields, more precisely, the scaler and Tensorial forms of Minkowski Functional have been used to examine the anisotropy properties of redshift space and also distortions parameter <cit.>. The Genus and contour crossing in various dimensions has been examined in redshift space <cit.>. Interestingly, those statistics revealing the one- and two-Dimensional slices depend on anisotropy due to peculiar velocities in redshift space. Consequently, we persuade 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 <cit.>, the generalized form of that means including the Up-, down- and the conditional crossing statistics have been utilized as complementary methods for diverse applications <cit.>. Particularly, the contour crossing statistic corresponds to the mean number of intersections between the isodensity contours of the density field at threshold $\vartheta$ , $\partial Q_v$, and a straight line in a specific direction <cit.>. The crossing  statistic ($cr$) can be expressed in a compact form as <cit.>:

\begin{eqnarray}\label{eq:Ncr1}
N_{cr}^{(r,s)}(\vartheta,i) = \left\langle\delta_D\left(\delta^{(r,s)}-\vartheta\sigma^{(r,s)}_{0}\right)\;\bigg| \delta_{,i}^{(r,s)}\bigg|\;\right\rangle
\end{eqnarray}

where $i$ represents the direction of a straight line. Using Equation (<ref>), for a 3-Dimensional field, we have:

\begin{eqnarray}
\left\langle N_{cr}^{(r,s)}(\vartheta,i)\right\rangle_G = \frac{\sigma_{1i}^{(r,s)}}{\pi\sigma_{0}^{(r,s)}}\;e^{-\vartheta^2/2} ,
\label{eq:Ncr}
\end{eqnarray}

Inspired by the crossing statistic, we introduce so-called the conditional moments of the first derivative ($cmd$). 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 isodensity contours at/above a given threshold, $\vartheta$, and by fixing a direction, $i$, the $n$th moment of the first derivative in such direction for the determined pixels to be computed. The mathematical description of the $cmd$ criterion can be clarified as follows:

\begin{eqnarray}\label{eq:Ncmd}
N_{cmd}^{(r,s)}(\vartheta,i;n) &\equiv&\left \langle \delta_D\left(\delta^{(r,s)}-\vartheta\sigma^{(r,s)}_{0}\right)\left(\delta_{,i}^{(r,s)}\right)^n\right\rangle\nonumber\\
&=& \frac{1}{V} \int_{V}  \delta_D\left(\delta^{(r,s)}-\vartheta\sigma^{(r,s)}_{0}\right)\; \left(\delta_{,i}^{(r,s)}\right)^n dV\nonumber\\
& = &\frac{1}{V} \int_{\partial Q_v}  \frac{\left(\delta_{,i}^{(r,s)}\right)^n}{\left|\boldsymbol{\nabla}\delta^{(r,s)}\right|}\; dA
\end{eqnarray}

we utilize the volume to surface integral transformation
<cit.>. Using
the  probabilistic framework presented in the subsection
<ref>, the expected values of
$N_{cmd}^{(r,s)}$ for a  3-Dimensional
Gaussian density field is obtained as follows:

\begin{eqnarray}
\left\langle N_{cmd}^{(r,s)}(\vartheta,i;n) \right\rangle_G =   2^{\frac{n-3}{2}} [1+ \cos(n\pi)] \Gamma \left(\frac{n+1}{2}\right)
\times\frac{\left(\sigma_{1i}^{(r,s)}\right)^n}{\pi\sigma_{0}^{(r,s)}}\;e^{-\vartheta^2/2}
%	\end{split}
\label{eq:Nccr}
\end{eqnarray}

where $\Gamma(:)$ is Gamma function. Equation  (<ref>), 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   (<ref>), we can derive the perturbative form of the $N_{cmd}^{(r,s)}$ for 3-Dimensional density field in the mildly non-Gaussian regime.  To this end, we expand the Equation   (<ref>)
for a typical observable quantity up to $\mathcal{O}(\sigma_0^2)$ as:

\begin{eqnarray}
\begin{split}
\langle \mathcal{F} \rangle =& \langle \mathcal{F} \rangle_G+ \frac{\sigma_{0}}{3!}\sum_{\mu_1,\mu_2,\mu_3}
\mathcal{K}^{(3)}_{\mu_1,\mu_2,\mu_3}\langle\partial {\mathcal{F}}/\partial {\mathcal{A}}_{\mu_1}\partial {\mathcal{A}}_{\mu_2}\partial {\mathcal{A}}_{\mu_3}\rangle_G\\
&+ \mathcal{O}(\sigma_{0}^2)
\end{split}
\label{eq:expansion}
\end{eqnarray}

Subsequently, the weakly non-Gaussian form of $N_{cmd}^{(r,s)}$, up to the $\mathcal{O}(\sigma_0^2)$, becomes:

\begin{eqnarray}\label{eq:cmdnon}
\left\langle N_{cmd}^{(r,s)}(\vartheta,i)\right\rangle_{NG}  =
\frac{\left(\sigma_{1i}^{(r,s)}\right)^2}{\sqrt{2\pi}\;\sigma_{0}^{(r,s)}}\;e^{-\vartheta^2/2} \;
\bigg\{1\; +\;
\bigg [ \frac{1}{6} \;S_0^{(r,s)} H_3(\vartheta) \; + \;S_{1i}^{(r,s)} \bigg ]\sigma_0^{(r,s)} \;+\; \mathcal{O}\bigg ((\sigma_0^{(r,s)})^2\bigg ) \, \bigg\},
\end{eqnarray}

where $H_n(\vartheta)$ represent the probabilists' Hermite polynomials  and we have used following definitions:

\begin{eqnarray}
S_0^{(r,s)}  \equiv \frac{\left\langle\;\left(\delta^{(r,s)}\right)^3\; \right\rangle_c\;}{\left(\sigma_{0}^{(r,s)}\right)^4},
\end{eqnarray}

\begin{eqnarray}
S_{1i}^{(r,s)}  \equiv \;\frac{\left\langle\;\delta^{(r,s)} \left(\delta_{,i}^{(r,s)}\right)^2\; \right\rangle_c}{\sigma_0^{(r,s)}\;\left(\sigma_{1i}^{(r,s)}\right)^2},
\end{eqnarray}

Having Equation (<ref>), we can predict the $\langle N_{cmd}\rangle $ for a given field considering the corresponding spectral indices.

§ 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 $(\hat{{z}})$ (e.g. $\hat{{x}}$ and $\hat{{y}}$), and we use the notation $\hat{\mathcal{I}}$ to show these directions.

§.§ The $cmd$ and $cr$ measures in the Linear Kaiser limit

In the linear Kaiser limit, Equation   (<ref>) reduces to <cit.>:

\begin{eqnarray}
P^{(s)}(\boldsymbol{k}) = (1+\beta\mu^2)^2P^{(r)}(\boldsymbol{k})
\label{eq:linear power spectrum}
\end{eqnarray}

Using Equations    (<ref>), (<ref>)  and   (<ref>), one can obtain:

\begin{eqnarray}
\left(\sigma_{0}^{(s)}\right)^2 &=& C_0\left(\sigma_{0}^{(r)}\right)^2,\nonumber\\
\left(\sigma_{1}^{(s)}\right)^2 &=& C_0\left(\sigma_{1}^{(r)}\right)^2,
\end{eqnarray}

\begin{eqnarray}
\left(\sigma_{{\mathcal{I}}}^{(s)}\right)^2 &=&  \left(\frac{C_0-C_1}{2}\right)\left(\sigma_{1}^{(r)}\right)^2,\nonumber\\ \left(\sigma_{1{z}}^{(s)}\right)^2 &=& C_1\left(\sigma_{1}^{(r)}\right)^2,
\end{eqnarray}

\begin{eqnarray}
C_n(\beta) \equiv \frac12\int_{-1}^{1}\mu^{2n}\;(1+\beta\mu^2)^2\; d\mu\;,
\label{eq:C_n}
\end{eqnarray}

Consequently, in the  linear Kaiser limit, the $cr$ and $cmd$ statistics for a 3-Dimensional Gaussian field in redshift space for $\hat{z}$ and $\hat{\mathcal{I}}$ directions become:

\begin{eqnarray}
N_{cr}^{(s)}(\vartheta,\hat{\mathcal{I}}) = \frac{\sigma_{1}^{(r)}}{\sqrt{2}\pi\sigma_{0}^{(r)}}\;\sqrt{1-\frac{C_1}{C_0}}\;e^{-\vartheta^2/2},
\end{eqnarray}

\begin{eqnarray}
N_{cr}^{(s)}(\vartheta,\hat{z}) = \frac{\sigma_{1}^{(r)}}{\pi\sigma_{0}^{(r)}}\;\sqrt{\frac{C_1}{C_0}}\;e^{-\vartheta^2/2},
\end{eqnarray}

\begin{eqnarray}
N_{cmd}^{(s)}(\vartheta,\hat{\mathcal{I}}) = \frac{\left(\sigma_{1}^{(r)}\right)^2}{\sqrt{2\pi}\;\sigma_{0}^{(r)}}\;{\frac{C_0 - C_1}{2\;\sqrt{C_0}}}\;e^{-\vartheta^2/2},
\end{eqnarray}

\begin{eqnarray}
N_{cmd}^{(s)}(\vartheta,\hat{z}) = \frac{\left(\sigma_{1}^{(r)}\right)^2}{\sqrt{2\pi}\;\sigma_{0}^{(r)}}\;{\frac{C_1}{\;\sqrt{C_0}}}\;e^{-\vartheta^2/2},
\end{eqnarray}

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 $i$:

\begin{eqnarray}
\begin{split}
&n_{\diamond}^{(r,s)}(\vartheta,i) \equiv\\
& \frac{N_{\diamond}^{(r,s)}(\vartheta,i)}{\int_{-\infty}^{\infty}d\vartheta\left[N_{\diamond}^{(r,s)}(\vartheta,i)+N_{\diamond}^{(r,s)}(\vartheta,j)+N_{\diamond}^{(r,s)}(\vartheta,k)\right]},
\end{split}\label{eq:normal_cr}
\end{eqnarray}

where $\{i,j,k\}\in[\hat{x},\hat{y}, \hat{z}]$ and $i\ne j\ne k$. The $\diamond$ is replaced by $cr$ and $cmd$. Interestingly,  the isotropic Gaussian limit of  Equation   (<ref>) reduces to:

\begin{eqnarray}
n_{cr}^{(r)}(\vartheta) = n_{cmd}^{(r)}(\vartheta)= \frac{1}{3\sqrt{\;2\pi}}\;e^{-\vartheta^2/2}
\label{eq:isotropic limit}
\end{eqnarray}

Equation   (<ref>)  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   (<ref>)  can be considered as the signature of anisotropy and/or non-Gaussianity. While for the redshift space, the normalized quantities can be derived as:

\begin{eqnarray}
n_{cr}^{(s)}(\vartheta,\hat{\mathcal{I}}) = \frac{1}{\sqrt{2\pi}}\;\frac{1}{2+\sqrt{\frac{2C_1}{C_0-C_1}}}\;e^{-\vartheta^2/2},
\label{eq:n1}
\end{eqnarray}

\begin{eqnarray}
n_{cr}^{(s)}(\vartheta,\hat{z}) = \frac{1}{\sqrt{2\pi}}\;\frac{1}{1+\sqrt{2}\sqrt{\frac{C_0}{C_1}-1}}\;e^{-\vartheta^2/2},
\end{eqnarray}

\begin{eqnarray}\label{eq:n44}
n_{cmd}^{(s)}(\vartheta,\hat{\mathcal{I}}) = \frac{1}{2\sqrt{2\pi}}\;\left(1-\frac{C_1}{C_0}\right)\;e^{-\vartheta^2/2},
\end{eqnarray}

\begin{eqnarray}
n_{cmd}^{(s)}(\vartheta,\hat{z}) = \frac{1}{\sqrt{2\pi}}\;\left(\frac{C_1}{C_0}\right)\;e^{-\vartheta^2/2},
\label{eq:n4}
\end{eqnarray}

They have no explicit dependencies on the spectral indices. Thus, for the Gaussian limit, the normalized quantities only depend on the threshold, $\vartheta$, and redshift space parameter, $\beta$, through the $C_0$ and $C_1$. From Equation   (<ref>), we find:

\begin{eqnarray}
C_0(\beta) = 1+\frac{2\beta}{3}+\frac{\beta^2}{5} \;,\hspace{0.4cm} C_1(\beta) = \frac{1}{3}+\frac{2\beta}{5}+\frac{\beta^2}{7}\;.
\end{eqnarray}

In the limit $\beta \rightarrow 0$, Equations (<ref>)-(<ref>) get the isotropic limit presented in Equation   (<ref>). In such a limit, the normalized $cr$ and $cmd$ measures are similar. Therefore, for $\beta\ne0$, 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. <ref>, we plot the analytical predictions of the normalized $cr$ and $cmd$ quantities as a function of threshold, $\vartheta$, for a typical anisotropic Gaussian matter field in redshift space in the presence of the linear Kaiser effect adopted by  $\beta= 0.48$ as a fiducial value. The black solid line illustrates the normalized  $cr$ and $cmd$ for isotropic limit (Equation   (<ref>)). 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 $\hat{\mathcal{I}}$ directions. Noticing the analytical form of $cmd$ criterion (Equations    (<ref>) and (<ref>)), 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 $n_{cmd}(\hat{\mathcal{I}})$ (blue dotted line) and $n_{cmd}(\hat{z})$ (red loosely dashed line) is higher than the corresponding value in the context of $cr$ measure for as fixed value of $\beta$. The lower panel of Fig.  <ref> indicates the difference between $\Delta n_{\diamond}^{(s)}(\vartheta)\equiv n_{\diamond}^{(s)}(\vartheta,\hat{z})-n_{\diamond}^{(s)}(\vartheta,\hat{\mathcal{I}})$.

The theoretical prediction for the normalized $cr$ and $cmd$ measures as a function of threshold for $\beta_{\rm fiducial}=0.48$ as a fiducial value in the linear Kaiser limit. The  “*” symbol is replaced by $\hat{\mathcal{I}}$ and $\hat{z}$ for perpendicular and along to the line of sight, respectively. The $\diamond$ symbol is reserved for $cr$ and $cmd$ statistics.  The black solid line is for $\beta=0$ which shows the isotropic limit. The lower panel depicts the difference $\Delta n_{\diamond}^{(s)}(\vartheta)\equiv n_{\diamond}^{(s)}(\vartheta,\hat{z})-n_{\diamond}^{(s)}(\vartheta,\hat{\mathcal{I}})$ demonstrating that $\Delta n_{cmd}^{(s)}$ is higher than $\Delta n_{cr}^{(s)}$.

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  <cit.> in the context of Minkowski tensors. We introduce following quantities by means of $cr$ and $cmd$ criteria as:

\begin{eqnarray}
\varTheta_{cr}(\vartheta)&\equiv& \frac{N_{cr}^{(s)}(\vartheta,\hat{\mathcal{I}})}{N_{cr}^{(s)}(\vartheta,\hat{z})},\nonumber\\ \varTheta_{cmd}(\vartheta)&\equiv& \frac{N_{cmd}^{(s)}(\vartheta,\hat{\mathcal{I}})}{N_{cmd}^{(s)}(\vartheta,\hat{z})}.
\end{eqnarray}

It turns out that for the Gaussian and linear Kaiser limit, we have:

\begin{eqnarray}\label{eq:42}
\varTheta_{cr}&=& \sqrt{\frac{C_0(\beta)-C_1(\beta)}{2 C_1(\beta)}},\nonumber\\ \varTheta_{cmd}&=& \frac{C_0(\beta)-C_1(\beta)}{2 C_1(\beta)}.
\end{eqnarray}

Error propagator on the $\beta$ for various criteria discussed in the text. Supposing  one percent relative error produced for $\sigma_{\diamond}$ in observation causes lower relative error on redshift space parameter by $cmd$ measure compared to other criteria examined in this paper.

We use the notation  $\varTheta_{\diamond}$ with $ \diamond \in \{cr,cmd,MT1,MT2\}$. Here the $MT1$ and $MT2$ are associated with the type one and two of rank-2 Minkowski tensors as defined by  <cit.>. Based on the theoretical predictions of $\varTheta_{\diamond}$, we can determine the level of accuracy accordingly, we can constrain the value of parameter $\beta$. This accuracy depends on the statistical uncertainty associated with $\varTheta_{\diamond}$, which can be evaluated using the Fisher forecast approach. We rely on  the posterior probability function, $\mathcal{P}_{\diamond}(\beta|\Theta_{\diamond})$, as:

\begin{eqnarray}
\begin{split}
\mathcal{P}_{\diamond}(\beta|\varTheta_{\diamond})&=\langle\delta_{D}(\beta-\Phi^{-1}_{\diamond}(\varTheta_{\diamond}))\rangle\\
&=\int d\varTheta'_{\diamond}\mathcal{P}(\varTheta'_{\diamond})\delta_D(\varTheta_{\diamond}-\varTheta'_{\diamond})\left|\mathcal{J}\right|_{\varTheta'_{\diamond}=\Phi_{\diamond}(\beta)}
\end{split}
\end{eqnarray}

here $\mathcal{J}$ is the Jacobian computed for $\varTheta'_{\diamond}=\Phi_{\diamond}(\beta)$ and $\Phi_{\diamond}(\beta)$ reads by Equation   (<ref>). Finally, according to a given confidence interval (C.L.), the error-bar on $\beta$ is given by:

\begin{eqnarray}
{\rm C.L.}=\int_{\beta_{\rm fiducial}-\sigma^{(-)}_{\beta}}^{\beta_{\rm fiducial}+\sigma^{(+)}_{\beta}}d\beta\; \mathcal{P}_{\diamond}(\beta|\varTheta_{\diamond})
\end{eqnarray}

or equivalently, based on the error propagation formalism up to the first order, we obtain the relative error on redshift space parameter as:

\begin{eqnarray}
\sigma^2_{\beta} = \left( \frac{\partial \ln\varTheta_{\diamond}}{\partial \ln \beta} \right)^{-2} \sigma^2_{\diamond}
\end{eqnarray}

where $\sigma_{\beta}$ and $\sigma_{\diamond}$ represent the fractional uncertainties on $\beta$ and $\varTheta_{\diamond}$, respectively.
In Fig.   <ref>, we plot $\sigma_{\beta}$ in terms of $\sigma_{\diamond}$ for $\beta_{\rm fiducial} = 0.48$ as the fiducial value in the Gaussian limit. We also consider $\sigma_{\diamond} = 0.01$ 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.

Left panel: $\xi_{\diamond}$ as a function of $R$ for the phenomenological Gaussian model of FoG and for $\sigma_u=4$Mpc/h and $\sigma_u=5$Mpc/h. 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 $R_{\star}$ 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 $\xi{\diamond}$ 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 $\xi_{\diamond}$ 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 <cit.> and Lorentzian <cit.> FoG models which are respectively read off by the following Equations:

\begin{eqnarray}
\tilde{O}^{\rm Gauss}_{\rm FoG}(k\mu,\sigma_u) = e^{-\frac{1}{2}\sigma_u^{2} k^2 \mu^2}
\label{eq:Gauss}
\end{eqnarray}

\begin{eqnarray}
\tilde{O}^{\rm Lorentz}_{\rm FoG}(k\mu,\sigma_u) = \frac{1}{1+\frac{1}{2}\sigma_u^{2} k^2 \mu^2} ,
\label{eq:Lorentz}
\end{eqnarray}

where $\sigma_u$ is the one-dimensional velocity dispersion. More precisely, to manipulate the linear Kaiser and FoG effects together, the non-linear part of the Equation   (<ref>) can be replaced by the Equation   (<ref>) or Equation   (<ref>). It is worth noting that the spectral indices given by Equations   (<ref>) and (<ref>) are modified by correction of power spectrum which is in principle constructed by plugging the Equation   (<ref>) or Equation   (<ref>) in the Equation   (<ref>).

To go further, we define $\xi_{\diamond} \equiv \varTheta^{-1}_{\diamond} - 1$ and  $\diamond \in \{cr,cmd\}$. The Gaussian limits of $\xi_{cr}$ and $\xi_{cmd}$ are obtained as:

\begin{eqnarray}\label{eq:xi1}
\xi_{cr}= \frac{\sigma^{(s)}_{1z}}{\sigma^{(s)}_{1\mathcal{I}}}-1, \hspace{0.7cm} \xi_{cmd}= \left(\frac{\sigma^{(s)}_{1z}}{\sigma^{(s)}_{1\mathcal{I}}}\right)^2-1,
\end{eqnarray}

Therefore, in this case, $\xi_{\diamond}$ statistics depend on the following quantities including parameter $\beta$, FoG model,  one dimensional velocity dispersion, $\sigma_u$, smoothing kernel, $\tilde{W}$, smoothing scale, $R$, and power spectrum in the real space, $P^{(r)}(\boldsymbol{k})$. The $\xi_{\diamond}$ 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.

To compare the FoG contribution with the linear Kaiser effect, we adopt a Gaussian smoothing kernel equates to $\tilde{W}(kR) = \exp(-(kR)^2/2)$. We also use CAMB software <cit.> with the fiducial values $\{\,\Omega_{\Lambda} = 0.69179,\, \Omega_{c}h^2 = 0.11865,\, \Omega_{b}h^2 =0.022307,\, \Omega_{\nu}h^2 = 0.000638,\, h = 0.6778,\,\, n_s = 0.9672\, \}$ which is in agreement with the flat $\Lambda$CDM $Planck$ cosmological parameters to compute the matter power spectrum,  <cit.>.
Adopting the linear bias  $b=1$, we obtain $\beta = 0.526$ for these cosmological parameters.

In the left panel of Fig.   <ref>, we illustrate the $\xi_{\diamond}$ as a function of the smoothing scale when the phenomenological Gaussian  model is taken into account for the FoG effect. Scaling dependency of $\xi_{\diamond}$ is clearly due to the FoG and interestingly we obtain that at a smoothing scale denoted by $R_{\star}$, the $\xi_{\diamond}$ 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 ($R_{\star}$), the linear Kaiser effect and the FoG effect cancel each other out, and $\xi_{cr}$ and $\xi_{cmd}$ reach zero. In scales smaller than $R_{\star}$, the FoG effect is dominant, and both $\xi_{cr}$ and $\xi_{cmd}$ have negative values, but for scales larger than $R_{\star}$, the linear Kaiser effect becomes significant and both $\xi_{cr}$ and $\xi_{cmd}$ 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 $\Delta \xi_{\diamond}\equiv\xi_{\diamond}(\sigma_u=5{\rm Mpc/h})-\xi_{\diamond}(\sigma_u=4{\rm Mpc/h})$ and our results confirm that $\xi_{cmd}$ is higher than $\xi_{cr}$ for the two fixed values of $\sigma_u$.

In the left panel of Fig. <ref>, the $\xi_{\diamond}$ 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  $\Delta \xi_{\diamond}\equiv\xi_{\diamond}(Lorentzian\; FoG)-\xi_{\diamond}(Gaussian\;FoG)$ and the bottom part of left panel shows that $\xi_{cmd}$ for small smoothing scale has higher dependency on the model of FoG than the $\xi_{cr}$.

§ 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 approaches are considered: at first, according to the computed matter power spectrum consistent with flat $\Lambda$CDM model, we simulate Gaussian random field. Secondly, we will rely on the N-body simulations known as Quijote simulations <cit.>.

Upper panel: The crossing statistics as a function of $\vartheta$ 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.

§.§ 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, $\delta^{(r)}$, sampled on a cubical lattice with the total volume size $V = (1{\rm Gpc/h})^3$ which consist of $N_{pix} = 512^3 $ pixels. Applying the Fourier transform on the simulated isotropic density field, we construct an anisotropic field according to the following transformation:

\begin{eqnarray}
\tilde{\delta}^{(s)}(\boldsymbol{k}) = \left(1+\beta\frac{k_z^2}{k^2}\right)\tilde{\delta}^{(r)}(\boldsymbol{k}),
\end{eqnarray}

and therefore, we obtain the redshift space density field in Fourier space. Then we smooth $\tilde{\delta}^{(s)}$ by a Gaussian kernel with scale $R = 20$ (Mpc/h).  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, $\vartheta\in[-4.0,4.0]$,
and then we do the ensemble average. Increasing the number of realizations had no significant effect on our ensemble average.
Fig.   <ref> presents the $cr$ and $cmd$ as a function of $\vartheta$ 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 $1\sigma$ 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 error-bar compared to $cr$.

In the rest of this subsection, motivated by introducing a proper measure to put observational constraint on $\beta$, we define the following weighted summation to marginalize the effect of threshold bins:

\begin{eqnarray}\label{eq:xibar}
\bar{\xi}_{\diamond} = \frac{\sum_{\vartheta_i=\vartheta_{\min}}^{\vartheta_{\max}}\omega_{\diamond}(\vartheta_i) \xi_{\diamond}(\vartheta_i)}{\sum_{\vartheta_i=\vartheta_{\min}}^{\vartheta_{\max}}\omega_{\diamond}(\vartheta_i)},
\end{eqnarray}

where the weight is defined by means of statistical error as $\omega_{\diamond}(\vartheta_i)\equiv \sigma^{-1}_{\xi_{\diamond}}(\vartheta_i)$.

In Fig.   <ref>, we plot the $\bar{\xi}_{\diamond}$ statistics as a function of $\beta$ extracted from theoretical (solid line) and computational (symbols) approaches.  The higher slope for $cmd$ measure concerning the $cr$ statistics versus $\beta$ reveals more robustness in discriminating different values of $\beta$.

The $\bar{\xi}_{\diamond}$ as a function of $\beta$. The filled  circle symbols correspond to the numerical analysis of $cr$, while the filled triangles represent the numerical results for the $cmd$. The solid lines indicate the corresponding theoretical predictions. Here we took $R=20$Mpc/h.

The $N^{(r,s)}_{cmd}{\rm[Mpc/h]}^{-2}$ versus threshold for the Quijote simulations. Upper left panel: The expectation value of conditional moment of the first derivative in real and redshift spaces for both $\hat{\mathcal{I}}\in[\hat{x},\hat{y}]$ and $\hat{z}$ directions adopting $R=40$Mpc/h. Upper right panel: The $N^{(r)}_{cmd}(\hat{z})$ 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 (<ref>)). The filled black circle symbols correspond to the numerical analysis including their $1\sigma$ level of confidence. The lower panels are the same as upper right panel just for redshift space in $\hat{z}$ (left) and $\hat{x}$ (right) directions.

§.§ 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 <cit.>.
Each our ensemble extracted form the Quijote simulations has following properties: $N_{\rm particles}=512^3$, box size of $V=(1{\rm Gpc/h})^3$, the fiducial cosmological parameter are based on flat $\Lambda$CDM including $\Omega_m=0.3175$, $\Omega_b=0.049$, $h=0.6711$, $n_s=0.9624$ and $\sigma_8=0.834$ (see <cit.> for more details). To construct the proper density field in applying our numerical pipeline, we use  <cit.> and at redshift $z=0$, exploiting the routine cloud in cell (CIC) for mass assignment, the density field contrast, $\delta(\boldsymbol{r})$, would be retrieved. Finally, convolving the $\delta(\boldsymbol{r})$ with a Gaussian window function characterized by a smoothing scale, $R$, the matter density contrast is constructed in the real space, $\delta_R^{(r)}(\boldsymbol{r})$. To create the corresponding field in redshift space in plane-parallel approximation, $\delta_R^{(s)}(\boldsymbol{r})$, for each value of  $\boldsymbol{r}$, we use   which in principle considers the Equation   (<ref>).

Fig. <ref> depicts the $N^{(r,s)}_{cmd}$ versus threshold for the Quijote simulations. Upper left panel corresponds to $N^{(r,s)}_{cmd}$ in real and redshift spaces for $\hat{\mathcal{I}}\in[\hat{x},\hat{y}]$ and $\hat{z}$ directions taking $R=40$Mpc/h. As we expect, there are no significant deviations between various directions in real space, while in the redshift space, the $N^{(s)}_{cmd}(\vartheta,\hat{x})= N^{(s)}_{cmd}(\vartheta,\hat{y})\ne N^{(s)}_{cmd}(\vartheta,\hat{z})$. The upper right panel indicates the $N^{(r)}_{cmd}(\vartheta,\hat {z})$ for the Gaussian prediction (green dashed-dot line) while the red dashed line indicates the theoretical non-Gaussian prediction for $cmd$ (Equation (<ref>)). The filled black circle symbols correspond to the numerical analysis including their $1\sigma$ level of confidence. The numerical result is mildly skewed and it is tilted to the higher thresholds yielding the non-Gaussian behavior. The lower left panel illustrates the $N^{(s)}_{cmd}(\vartheta,\hat{z})$ for Gaussian model (green dashed-dot line), non-Gaussian model up to the $\mathcal{O}(\sigma_0^2)$ (red dashed line) and filled circle symbols correspond to the numerical results for $R=40$Mpc/h. The lower right panel shows the results for $N^{(s)}_{cmd}(\vartheta,\hat{x})$ which is perpendicular to the line of sight direction.

The difference between the numerical computation of $N^{(r,s)}_{cmd}$ and the corresponding theoretical Gaussian prediction are depicted in Fig. <ref>. 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 upper left panel, 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 $N^{(s)}_{cmd}(\vartheta,\hat{z})$, $N^{(s)}_{cmd}(\vartheta,\hat{x})$ and $N^{(r)}_{cmd}(\vartheta,\hat{z})$ computed numerically for N-body simulations and associated Gaussian models, respectively, in the upper right panel. For this part, we consider the smoothed scale equates to $R=40$ Mpc/h. In the lower panels, we display the same quantities as expressed for the upper panel but for $R=20$ Mpc/h (lower left panel) and $R=30$ Mpc/h (lower right panel). 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 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 extracted from N-body simulations is justified due to the reason that for the lower value of the smoothing scale, the $\sigma_0$ 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 (<ref>) to achieve more accurate formula for $\left\langle N_{cmd}^{(r,s)}(\vartheta,i)\right\rangle_{NG}$ (Equation (<ref>)).

The $N^{(r,s)}_{cmd}-\langle N^{(r,s)}_{cmd}\rangle_G$ with dimension ${\rm[Mpc/h]}^{-2}$ versus threshold for Quijote simulations. Upper left panel shows the difference between $N^{(r)}_{cmd}(\vartheta,\hat{z})$ and the value corresponds to the theoretical Gaussian prediction for $R=40$ Mpc/h. The red dashed solid line is for the theoretical prediction for deviation from Gaussian field up to the second order expansion, while the filled black circle symbols are for numerical deviation from  Gaussian theory. The same quantities just for redshift space are plotted in the upper right panel. Notice that, for the redshift space, the results for the $\hat{x}$ direction has been also given. The lower panels are the same as upper panels just we took $R=20$ Mpc/h (lower left) and $R=30$ Mpc/h (lower right).

§ SUMMARY AND CONCLUSIONS

The redshift space distortions caused by the linear and non-linear 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, 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 ($\vartheta$) 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 $\mathcal{O}(\sigma^2_{0})$ 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 ($\beta$). The normalized quantity $n_{\diamond}$ depending on direction, threshold, and the $\beta$ parameter has been introduced and our results demonstrated that the $n_{cmd}^{(s)}$ is more sensitive than the $n_{cr}^{(s)}$ to the anisotropy as depicted in the lower panel of Fig.   <ref>, particularly for intermediate threshold. According to the error propagation approach and by considering the relative error on the $\varTheta_{\diamond}$ equates to one percent indicating that the $cmd$ enables to put tight constraint compare to other statistics in Gaussian (Fig.   <ref>) and non-Gaussian regimes.

To make our evaluation more complete, we defined $\xi_{\diamond}$ (Equation   (<ref>)) 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. <ref>). This quantity for a high enough value of $R$ also asymptotically goes to the fixed value implying the $\beta$ 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. <ref>). To adopt the observational constraining robustly, we also defined a weighted parameter, $\bar{\xi}_{\diamond}$ (Equation (<ref>)), and for various values of $\beta$ in simulation, we revealed that $\bar{\xi}_{cmd}$ has higher $\beta$-dependency (Fig. <ref>).

Although, we have implemented our methodology on the N-body simulations publicly available by the Quijote suite. The $N^{(r,s)}_{cmd}$ as a function of $\vartheta$ 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 $\vartheta\gtrsim 0$ ($\vartheta\lesssim 0$) (Fig. <ref>). To make more sense regarding the non-Gaussianity in the N-body 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. <ref>). 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 $cmd$ measure.

To go further, we suggest the 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 plane-parallel 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 $cmd$ measure opens new room to evaluate bias factor. 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 an 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   <cit.>.

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.

§ BAYESIAN CREDIBLE REGION

In the Bayesian context, given the posterior probability density,
$p(\theta|{\bf D})$, of a parameter, $\theta$, given the dataset,
${\bf D}$, the highest posterior density region (or credible
region), $R$, is defined by

C = ∫R dθ p(θ|D)

where C is the probability contained in the credible region.  The
region $R$ is selected such that the posterior probability of any
point inside $R$ is larger than that of any point outside.  We like
Eq. (<ref>).


and $\Phi_{k}^{f}$ $(k=1,2,\cdots,N)$ solves the following hierarchy of
equations

$\displaystyle\frac{i}{n}\partial_{Y}^{2}\Phi_{k}^{f}-c\Phi_{k}^{f}=-U_{s}\Phi_{k-1}^{f}+2\partial_{Y}^{-1}\left(\partial_{Y}U_{s}\Phi_{k-1}^{f}\right),~{}\Phi_{k}^{f}(Y;c)|_{Y=0}=0.$
(5.7)

From (5.3) and (5.4), we know that

$(1-\tau_{4}^{-1}\alpha^{\nu_{0}})\alpha^{-2(1+\nu_{0})}\leq|nc|\leq(1+\tau_{4}^{-1}\alpha^{\nu_{0}})\alpha^{-2(1+\nu_{0})},$

and

$-\frac{\pi}{2}<\arg(-inc)<-\frac{\pi}{2}+\tau_{4}^{-1}\alpha^{\nu_{0}}.$

Then $\varpi=(-inc)^{\frac{1}{2}}$ is well-defined, analytic in $c$ and
satisfies

$\displaystyle|\varpi|=\alpha^{-1-\nu_{0}}(1+O(1)\alpha^{\nu_{0}}),~{}-\frac{\pi}{4}<\arg\varpi<-\frac{\pi}{4}+\frac{\tau_{4}^{-1}}{2}\alpha^{\nu_{0}},~{}\frac{\sqrt{2}}{3}\alpha^{-1-\nu_{0}}\leq\text{Re}\varpi\leq\frac{2\sqrt{2}}{3}\alpha^{-1-\nu_{0}}.$
(5.8)

Thus we can solve the leading order profile

$\displaystyle\Phi_{0}^{f}(Y;c)=e^{-\varpi Y}$

from (5.6) and solve $\Phi_{k}^{f}$ from (5.7) inductively by

$\Phi_{k}^{f}(Y;c)=in\int_{0}^{Y}e^{-\varpi(Y-Y^{\prime})}{\rm
d}Y^{\prime}\int_{Y^{\prime}}^{\infty}e^{\varpi(Y^{\prime}-Y^{\prime\prime})}\left(-U_{s}\Phi_{k-1}^{f}+2\partial_{Y}^{-1}(\partial_{Y}U_{s}\Phi_{k-1}^{f})\right){\rm
d}Y^{\prime\prime},~{}k=1,2,\cdots,N.$ (5.9)

The approximate fast mode of magnetic field is defined accordingly as

$\displaystyle\Psi_{\text{app}}^{f}(Y;c)=\sum_{k=0}^{N}\Psi_{k}^{f}(Y;c),~{}\text{with
}\Psi_{k}^{f}(Y;c)=-\partial_{Y}^{-1}\Phi_{k}^{f}(Y;c).$ (5.10)

It is straightforward to check that the fast mode
$(\Phi_{\text{app}}^{f},\Psi_{\text{app}}^{f})$ satisfies

$\left\\{\begin{aligned}
&\frac{i}{n}\partial_{Y}^{4}\Phi_{\text{app}}^{f}+(U_{s}-c)\partial_{Y}^{2}\Phi_{\text{app}}^{f}-\partial_{Y}^{2}U_{s}\Phi_{\text{app}}^{f}=\partial_{Y}^{2}\left(U_{s}\Phi_{N}^{f}-2\partial_{Y}^{-1}(\partial_{Y}U_{s}\Phi_{N}^{f})\right),~{}Y>0,\\\
&-\partial_{Y}^{2}\Psi_{\text{app}}^{f}-\partial_{Y}\Phi_{\text{app}}^{f}=0,~{}\Phi_{\text{app}}^{f}(Y;c)|_{Y=0}=1.\end{aligned}\right.$

In summary, we define the approximate growing mode as

$\displaystyle\Phi_{\text{app}}(Y;c)$
$\displaystyle=\Phi_{\text{app}}^{s}(Y;c)-\Phi_{\text{app}}^{s}(0;c)\Phi_{\text{app}}^{f}(Y;c),$
(5.11) $\displaystyle\Psi_{\text{app}}(Y;c)$
$\displaystyle=\Psi_{\text{app}}^{s}(Y;c)-\Phi_{\text{app}}^{s}(0;c)\Psi_{\text{app}}^{f}(Y;c),$

where $\Psi_{\text{app}}^{s}$ is the solution to (LABEL:3.1) with
$\varphi_{b}=\Phi_{\text{app}}^{s}(0;c)\Psi_{\text{app}}^{f}(0;c)$ and
$f=i\alpha H_{s}\Phi_{\text{app}}^{s}+\partial_{Y}\Phi_{\text{app}}^{s}.$

In the following lemma, we will give some pointwise estimates on
$\Phi_{k}^{f}$ and $\Psi_{k}^{f}.$

######  Lemma 5.1.

The following bounds hold

$\displaystyle|\partial_{Y}^{j}\Phi_{k}^{f}(Y;c)|$
$\displaystyle\lesssim\alpha^{k\nu_{0}-j(1+\nu_{0})}e^{-\frac{1}{4}\alpha^{-(1+\nu_{0})}Y},$
(5.12) $\displaystyle|\partial_{Y}^{j}\Psi_{k}^{f}(Y;c)|$
$\displaystyle\lesssim\alpha^{k\nu_{0}-(j-1)(1+\nu_{0})}e^{-\frac{1}{4}\alpha^{-(1+\nu_{0})}Y},$
(5.13)

where $Y\geq 0$, $k=0,1,2,\cdots,N$ and $j=0,1,2$.

###### Proof.

Define

$\mathcal{G}_{k}\triangleq\sup_{Y\geq
0}\left(1+\alpha^{-(1+\nu_{0})}Y\right)e^{\frac{1}{4}\alpha^{-(1+\nu_{0})}Y}|\Phi_{k}^{f}(Y)|.$

Recall that $\Phi_{k}^{f}$ satisfies (5.9). Then

$\displaystyle\left|U_{s}\Phi_{k-1}^{f}(Y)\right|+2\left|\partial_{Y}^{-1}(\partial_{Y}U_{s}\Phi_{k-1}^{f})(Y)\right|$
$\displaystyle\lesssim\left\|\frac{U_{s}}{Y}\right\|_{L^{\infty}}|Y\Phi_{k-1}^{f}(Y)|+\int_{Y}^{\infty}|\Phi_{k-1}^{f}|(X){\rm
d}X$
$\displaystyle\lesssim\alpha^{1+\nu_{0}}e^{-\frac{1}{4}\alpha^{-(1+\nu_{0})}Y}\mathcal{G}_{k-1}.$

By plugging it into (5.9), we obtain that

$\displaystyle|\Phi_{k}^{f}(Y)|$ $\displaystyle\lesssim
n\alpha^{1+\nu_{0}}\mathcal{G}_{k-1}\int_{0}^{Y}e^{-\text{Re}\varpi(Y-Y^{\prime})}{\rm
d}Y^{\prime}\int_{Y^{\prime}}^{\infty}e^{\text{Re}\varpi(Y^{\prime}-Y^{\prime\prime})}e^{-\frac{1}{4}\alpha^{-(1+\nu_{0})}Y^{\prime\prime}}{\rm
d}Y^{\prime\prime}.$ (5.14)
$\displaystyle\lesssim\frac{n\alpha^{1+\nu_{0}}}{(\text{Re}\varpi)^{2}}\mathcal{G}_{k-1}e^{-\frac{1}{4}\alpha^{-(1+\nu_{0})}Y}\lesssim\alpha^{\nu_{0}}\mathcal{G}_{k-1}e^{-\frac{1}{4}\alpha^{-(1+\nu_{0})}Y}.$

Here we have used (5.3) and (5.8). Applying an inductive argument to (5.14),
we obtain for $k=1,2,\cdots,N$ that

$\sup_{Y\geq
0}e^{\frac{1}{4}\alpha^{-1-\nu_{0}}Y}|\Phi_{k}^{f}(Y)|\lesssim\alpha^{k\nu_{0}}\sup_{Y\geq
0}\left(1+\left(\alpha^{-(1+\nu_{0})}Y\right)^{k}\right)e^{\frac{1}{4}\alpha^{-(1+\nu_{0})}Y}|\Phi_{0}^{f}(Y)|\lesssim\alpha^{k\nu_{0}}.$

This gives (5.12) when $j=0$. All other inequalities in (5.12) and (5.13) can
be obtained similarly. We omit the detail for brevity. ∎

We now define $\tilde{\Gamma}_{0}(c)=\partial_{Y}\Phi_{\text{app}}(0;c)$. With
the above pointwise estimates, we can show the existence of $c$ for
$\tilde{\Gamma}_{0}(c)=0$ in the following lemma.

######  Lemma 5.2.

Let $\beta\in(1/12,1/8)$ and $r_{3}\in(0,\frac{\sqrt{2}}{2})$. There exists
$\varepsilon_{5}\in(0,1)$, such that for any
$\varepsilon\in(0,\varepsilon_{5})$ and $\alpha=\varepsilon^{\beta}$, the
function $\tilde{\Gamma}_{0}(c)$ has a unique zero point inside $D_{**}$.
Moreover, on the circle $\partial D_{**}$, it holds that

$\displaystyle|\tilde{\Gamma}_{0}(c)|\geq\frac{r_{3}}{2}.$ (5.15)

######  Remark 5.3.

By (5.4), for $\alpha=\varepsilon^{\beta}$ and $c\in D_{**}$, it holds that

$\alpha\text{Im}c\geq\tau_{4}\alpha^{\frac{1}{4\beta}}=\tau_{4}\varepsilon^{\frac{1}{4}}.$

###### Proof.

The proof is similar to the one for Proposition 3.2. In fact, note that

$\displaystyle\tilde{\Gamma}_{0}(c)$
$\displaystyle=\partial_{Y}\Phi_{\text{app}}^{s}(0;c)-\Phi_{\text{app}}^{s}(0;c)\partial_{Y}\Phi_{\text{app}}^{f}(0;c)$
(5.16)
$\displaystyle=\partial_{Y}\Phi_{\text{app}}^{s}(0;c)-\Phi_{\text{app}}^{s}(0;c)\left(-\varpi+\sum_{k=1}^{N}\partial_{Y}\Phi_{k}^{f}(0;c)\right)$
$\displaystyle=\partial_{Y}\Phi_{\text{app}}^{s}(0;c)+\alpha^{-(1+\nu_{0})}e^{-\frac{1}{4}\pi
i}\Phi_{\text{app}}^{s}(0;c)+\Phi_{\text{app}}^{s}(0;c)\left(\varpi-\alpha^{-(1+\nu_{0})}e^{-\frac{1}{4}\pi
i}\right)$
$\displaystyle\qquad-\Phi_{\text{app}}^{s}(0;c)\left(\sum_{k=1}^{N}\partial_{Y}\Phi_{k}^{f}(0;c)\right).$

In view of asymptotic behavior (3.9), (3.10) and the pointwise bounds (5.12),
we define

$\tilde{\Gamma}_{\text{ref}}(c)=1+\alpha^{-(1+\nu_{0})}e^{-\frac{1}{4}\pi
i}(-c+\alpha).$

One can see that $\tilde{\Gamma}_{\text{ref}}$ has $c_{**}$ given in (5.1) as
its unique zero point in the whole complex plane. On the circle $\partial
D_{**}$, it holds that

$\displaystyle|\tilde{\Gamma}_{\text{ref}}(c)|=r_{3}>0.$

To further estimate the difference
$|\tilde{\Gamma}_{0}(c)-\tilde{\Gamma}_{\text{ref}}(c)|$, we observe from
(3.9), (5.3) and (5.4) that

$\left|\Phi_{\text{app}}^{s}(0;c)\right|\lesssim|c-\alpha|+\frac{1}{n}+\alpha|\hat{c}\log\text{Im}\hat{c}|\lesssim\alpha^{1+\nu_{0}}(1+\alpha^{1-\nu_{0}}|\log\alpha|)\lesssim\alpha^{1+\nu_{0}},$
(5.17)

where we have used $\nu_{0}\in(0,1)$ in the last inequality. Thus by using
(5.8), (5.12) and (5.17) we get

$\left|\Phi_{\text{app}}^{s}(0;c)\left(\varpi-\alpha^{-(1+\nu_{0})}e^{-\frac{1}{4}\pi
i}\right)\right|+\left|\Phi_{\text{app}}^{s}(0;c)\left(\sum_{k=1}^{N}\partial_{Y}\Phi_{k}^{f}(0;c)\right)\right|\lesssim\alpha^{\nu_{0}}.$

By plugging this bound into (5.16), we then obtain that

$\displaystyle|\tilde{\Gamma}_{0}(c)-\tilde{\Gamma}_{\text{ref}}(c)|$
$\displaystyle\lesssim\alpha|\log\text{Im}\hat{c}|+\alpha^{-(1+\nu_{0})}\left(\frac{1}{n}+\alpha|\hat{c}\log\text{Im}\hat{c}|\right)+\alpha^{\nu_{0}}$
$\displaystyle\lesssim\alpha^{\nu_{0}}+\alpha^{1-\nu_{0}}|\log\alpha|,$

where we have used $\nu_{0}\in(0,1)$ again. Taking $\varepsilon_{5}\in(0,1)$
suitably small, such that for any $\varepsilon\in(0,\varepsilon_{5})$,

$\alpha^{\nu_{0}}+\alpha^{1-\nu_{0}}|\log\alpha|<\frac{r_{3}}{2},$

we obtain that

$|\tilde{\Gamma}_{0}(c)-\tilde{\Gamma}_{\text{ref}}(c)|<\frac{1}{2}|\tilde{\Gamma}_{\text{ref}}(c)|,~{}\text{for
any }c\in\partial D_{**}.$

We then complete the proof by Rouché’s theorem. ∎

By (5.5), (5.10) and (5.11), it is straightforward to check that

$\left\\{\begin{aligned}
&\text{OS}_{c}(\Phi_{\text{app}},\Psi_{\text{app}})=\left(E_{\beta}^{s}(Y;c)+E^{f}_{\beta}(Y;c),~{}F_{\beta}^{f}(Y;c)\right),~{}Y>0,\\\
&\Phi_{\text{app}}(Y;c)|_{Y=0}=\Psi_{\text{app}}(Y;c)|_{Y=0}=0.\end{aligned}\right.$
(5.18)

Here $E_{\beta}^{s}(Y;c)=\partial_{Y}E_{\beta,1}^{s}(Y;c)+i\alpha
E_{\beta,2}^{s}(Y;c)+E_{\beta,3}^{s}(Y;c)$ is the error term generated by the
slow mode $\Phi_{\text{app}}^{s}$ and it is exactly the same as (3.22), that
is, $E_{\beta,j}^{s}(Y;c)=E_{j}^{s}(Y;c),~{}j=1,2,3.$ The error term
$E_{\beta}^{f}(Y;c)$ generated by the fast mode is slightly different from
(3.23). In fact,

$E_{\beta}^{f}(Y;c)=\partial_{Y}E_{\beta,1}^{f}(Y;c)+i\alpha
E_{\beta,2}^{f}(Y;c)+E_{\beta,3}^{f}(Y;c),$

where

$\displaystyle E^{f}_{\beta,1}(Y;{c})\triangleq$
$\displaystyle-\Phi_{\text{app}}^{s}(0;{c})\bigg{(}-\frac{i}{n}(2\alpha^{2}+1)\partial_{Y}\Phi_{\text{app}}^{f}+U_{s}\partial_{Y}\Phi_{N}^{f}-\partial_{Y}U_{s}\Phi_{N}^{f}$
$\displaystyle\qquad-\sqrt{\varepsilon}H_{s}\partial_{Y}\Psi_{\text{app}}^{f}-\frac{\alpha}{n}(U_{s}-{c})\Psi_{\text{app}}^{f}+\frac{\alpha}{n}H_{s}\Phi_{\text{app}}^{f}\bigg{)},$
$\displaystyle E^{f}_{\beta,2}(Y;{c})\triangleq$
$\displaystyle-\Phi_{\text{app}}^{s}(0;{c})\left(\frac{\alpha}{n}(\alpha^{2}-1)\Phi_{\text{app}}^{f}+i\alpha(U_{s}-\hat{c})\Phi_{\text{app}}^{f}-i\alpha\sqrt{\varepsilon}H_{s}\Psi_{\text{app}}^{f}\right),$
$\displaystyle E^{f}_{\beta,3}(Y;{c})\triangleq$
$\displaystyle-\Phi_{\text{app}}^{s}(0;{c})\left(\sqrt{\varepsilon}\partial_{Y}H_{s}\partial_{Y}\Psi_{\text{app}}^{f}+\sqrt{\varepsilon}\partial_{Y}^{2}H_{s}\Psi_{\text{app}}^{f}\right).$

And the error term $F_{\beta}^{f}(Y;c)$ in the magnetic equation remains the
same as (3.24), that is, $F_{\beta}^{f}(Y;c)=F^{f}(Y;c).$ For these error
terms, we have the following bound estimates.

######  Lemma 5.4.

Let $\beta\in(1/12,1/8)$. There exists
$\varepsilon_{6}\in(0,\varepsilon_{5})$, such that if
$\alpha=\varepsilon^{\beta}$ and $c\in D_{**}$ where $D_{**}$ is given in
(5.2), then we have

$\displaystyle\|E_{\beta,1}^{s}(\cdot~{};c)\|_{L^{2}}+\|E_{\beta,2}^{s}(\cdot~{};c)\|_{L^{2}}\lesssim\alpha^{2+\frac{1}{2}(1+\nu_{0})},$
(5.19)
$\displaystyle\||U_{s}^{\prime\prime}|^{-\frac{1}{2}}E_{\beta,3}^{s}(\cdot~{};c)\|_{L^{2}}\lesssim\alpha^{\frac{3}{2}},$
(5.20)
$\displaystyle\|E_{\beta,1}^{f}(\cdot~{};c)\|_{L^{2}}+\|E_{\beta,2}^{f}(\cdot~{};c)\|_{L^{2}}+\||U_{s}^{\prime\prime}|^{-\frac{1}{2}}E_{\beta,3}^{f}(\cdot~{};c)\|_{L^{2}}\lesssim\alpha^{\frac{5}{2}(1+\nu_{0})},$
(5.21)
$\displaystyle\|F_{\beta}^{f}(\cdot~{};c)\|_{L^{2}}\lesssim\alpha^{1+\frac{3}{2}(1+\nu_{0})}.$
(5.22)

###### Proof.

We first estimate the slow mode $\Psi_{\text{app}}^{s}$ of magnetic field,
which solves (LABEL:3.1) with
$\varphi_{b}=\Phi_{\text{app}}^{s}(0;c)\Psi_{\text{app}}^{f}(0;c)$ and
$f=i\alpha H_{s}\Phi_{\text{app}}^{s}+\partial_{Y}\Phi_{\text{app}}^{s}.$
Taking $\varepsilon_{6}\in(0,\varepsilon_{5})$ suitably small, we have

$D_{**}\subsetneqq\\{\text{Im}c>0,~{}|\hat{c}|\leq\gamma_{2}\\}\cap\\{|c|<\gamma_{0}\\}~{}\text{and
}\alpha\in(0,\alpha_{0}),$

where $\gamma_{0}$ and $\alpha_{0}$ are the numbers given in Proposition 2.1
and $\gamma_{2}$ is the number given in Lemma 3.1. Using (5.13) with $j=0$ and
(5.17), we have

$|\varphi_{b}|=\left|\Phi_{\text{app}}^{s}(0;c)\Psi_{\text{app}}^{f}(0;c)\right|\lesssim\alpha^{2(1+\nu_{0})}.$

Then as the proof of Lemma 3.8, we obtain

$\displaystyle\alpha\|\Psi_{\text{app}}^{s}(\cdot~{};{c})\|_{L^{\infty}_{\alpha}}+\alpha^{\frac{1}{2}}\|\partial_{Y}\Psi_{\text{app}}^{s}(\cdot~{};{c})\|_{L^{\infty}_{\alpha}}+\|\partial_{Y}^{2}\Psi_{\text{app}}^{s}(\cdot~{};{c})\|_{L^{\infty}_{\alpha}}\lesssim
1,$ (5.23)

and

$\displaystyle\alpha\|\Psi_{\text{app}}^{s}(\cdot~{};{c})\|_{L^{2}}+\alpha^{\frac{1}{2}}\|\partial_{Y}\Psi_{\text{app}}^{s}(\cdot~{};{c})\|_{L^{2}}+\|\partial_{Y}^{2}\Psi_{\text{app}}^{s}(\cdot~{};{c})\|_{L^{2}}\lesssim\alpha^{-\frac{1}{2}}.$
(5.24)

Now we turn to estimate error terms $E_{\beta}^{s},~{}E_{\beta}^{f}$ and
$F_{\beta}^{f}.$ As for (3.38), we use the bound estimates (3.37),
(LABEL:5.7), (5.23) and (5.24) to obtain

$\displaystyle\|E_{1}^{s}(\cdot~{};{c})\|_{L^{2}}\lesssim$
$\displaystyle\frac{1}{n}\left(\|\partial_{Y}^{3}\psi_{\alpha,1}\|_{L^{2}}+\alpha\|\partial_{Y}^{3}\Phi_{1}^{s}\|_{L^{2}}+\alpha^{2}\|\partial_{Y}\psi_{\alpha,1}\|_{L^{2}}+\alpha^{3}\|\partial_{Y}\Phi_{1}^{s}\|_{L^{2}}\right)$
$\displaystyle+\varepsilon^{\frac{1}{2}}\|\partial_{Y}\Psi_{\text{app}}^{s}\|_{L^{2}}+\frac{\alpha}{n}\left(\|\Psi_{\text{app}}^{s}\|_{L^{2}}+\|\psi_{\alpha,1}\|_{L^{2}}+\alpha\|\Phi_{1}^{s}\|_{L^{2}}\right)$
$\displaystyle\lesssim$
$\displaystyle\frac{1}{n}\left(1+\frac{\alpha}{|\text{Im}\hat{c}|^{\frac{3}{2}}}+\alpha^{2}+\alpha^{3}\right)+\frac{\varepsilon^{\frac{1}{2}}}{\alpha}+\frac{\alpha}{n}\left(\frac{1}{\alpha^{\frac{3}{2}}}+\frac{1}{\alpha^{\frac{1}{2}}}+\alpha^{\frac{1}{2}}\right)$
$\displaystyle\lesssim$
$\displaystyle\alpha^{3+2\nu_{0}}\left(1+\alpha^{1-\frac{3}{2}(1+\nu_{0})}\right)+\alpha^{\frac{5}{2}+2\nu_{0}}\lesssim\alpha^{2+\frac{1}{2}(1+\nu_{0})}.$

Here we have used (5.3) and (5.4) in the last line. Similar as Proposition
3.3, one can get

$\displaystyle\|E_{2}^{s}(\cdot~{};{c})\|_{L^{2}}\lesssim\frac{\alpha^{\frac{5}{2}}}{n}+\frac{\varepsilon^{\frac{1}{2}}}{\alpha^{\frac{1}{2}}}+\frac{\alpha^{\frac{1}{2}}}{n}\lesssim\alpha^{\frac{7}{2}+2\nu_{0}},$

and

$\||U_{s}^{\prime\prime}|^{-\frac{1}{2}}E_{3}^{s}(\cdot~{};{c})\|_{L^{2}}\lesssim\alpha^{\frac{3}{2}}+\frac{\varepsilon^{\frac{1}{2}}}{\alpha}+\frac{\varepsilon^{\frac{1}{2}}}{\alpha^{\frac{3}{2}}}\lesssim\alpha^{\frac{3}{2}}.$

This completes the proof of (5.19) and (5.20). As for the fast mode, by
applying pointwise estimates (5.12), (5.13) and (5.17) to
$E_{\beta,1}^{f}(Y;c)$, we obtain that

$\displaystyle\|E_{\beta,1}^{f}(\cdot~{};c)\|_{L^{2}}\lesssim$
$\displaystyle\alpha^{1+\nu_{0}}\bigg{(}\frac{1}{n}\|\partial_{Y}\Phi_{\text{app}}^{f}\|_{L^{2}}+\left\|\frac{U_{s}}{Y}\right\|_{L^{\infty}}\|Y\partial_{Y}\Phi_{N}^{f}\|_{L^{2}}+\|\Phi_{N}^{f}\|_{L^{2}}$
(5.25)
$\displaystyle+\sqrt{\varepsilon}\|\partial_{Y}\Psi_{\text{app}}^{f}\|_{L^{2}}+\frac{\alpha}{n}\left\|\frac{U_{s}}{Y}\right\|_{L^{\infty}}\|Y\Psi_{\text{app}}^{f}\|_{L^{2}}+\frac{\alpha|c|}{n}\|\Psi_{\text{app}}^{f}\|_{L^{2}}+\frac{\alpha}{n}\|\Phi_{\text{app}}^{f}\|_{L^{2}}\bigg{)}$
$\displaystyle\lesssim$
$\displaystyle\alpha^{1+\nu_{0}}\left(\frac{1}{n\alpha^{\frac{1+\nu_{0}}{2}}}+\alpha^{N\nu_{0}+\frac{1}{2}(1+\nu_{0})}+\sqrt{\varepsilon}\alpha^{\frac{1}{2}(1+\nu_{0})}+\frac{\alpha^{1+\frac{1}{2}(1+\nu_{0})}}{n}\right)$
$\displaystyle\lesssim$
$\displaystyle\alpha^{\frac{5}{2}(1+\nu_{0})}\left(\alpha+\alpha^{N\nu_{0}-(1+\nu_{0})}\right),$

where we have used (5.3) in the last inequality. Similarly, we obtain that

$\displaystyle\|E_{\beta,2}^{f}(\cdot~{};c)\|_{L^{2}}$
$\displaystyle\lesssim\alpha^{1+\nu_{0}}\left(\frac{\alpha}{n}\|\Phi_{\text{app}}^{f}\|_{L^{2}}+\alpha\left\|\frac{U_{s}}{Y}\right\|_{L^{\infty}}\|Y\Phi_{\text{app}}^{f}\|_{L^{2}}+\alpha|\hat{c}|\|\Phi_{\text{app}}^{f}\|_{L^{2}}+\alpha\sqrt{\varepsilon}\|\Psi_{\text{app}}^{f}\|_{L^{2}}\right)$
(5.26)
$\displaystyle\lesssim\alpha^{1+\nu_{0}}\left(\frac{\alpha^{1+\frac{1}{2}(1+\nu_{0})}}{n}+\alpha^{2+\frac{1}{2}(1+\nu_{0})}+\sqrt{\varepsilon}\alpha^{1+\frac{1}{2}(1+\nu_{0})}\right)\lesssim\alpha^{\frac{5}{2}(1+\nu_{0})}\alpha^{1-\nu_{0}}.$

By using (1.26), we deduce that

$\displaystyle\||U_{s}^{\prime\prime}|^{-\frac{1}{2}}E_{\beta,3}^{f}(\cdot~{};{c})\|_{L^{2}}$
$\displaystyle\lesssim\alpha^{1+\nu_{0}}\bigg{(}\varepsilon^{\frac{1}{2}}\left\|\frac{H_{s}^{\prime}}{|U_{s}^{\prime\prime}|^{\frac{1}{2}}}\right\|_{L^{\infty}}\|\partial_{Y}\Psi_{\text{app}}^{f}\|_{L^{2}}+\varepsilon^{\frac{1}{2}}\left\|\frac{H_{s}^{\prime\prime}}{|U_{s}^{\prime\prime}|^{\frac{1}{2}}}\right\|_{L^{\infty}}\|\Psi_{\text{app}}^{f}\|_{L^{2}}\bigg{)}$
(5.27) $\displaystyle\lesssim\alpha^{2+\frac{7}{2}(1+\nu_{0})}.$

By taking $N$ suitably large so that $N\nu_{0}>1+\nu_{0}$ and by noting that
$\nu_{0}\in(0,1)$, we obtain (5.21) from (5.25), (5.26) and (5.27). In
summary, we have

$\displaystyle\|F_{\beta}^{f}(\cdot~{};c)\|_{L^{2}}$
$\displaystyle\lesssim\alpha^{1+\nu_{0}}\left(\alpha^{2}\|\Psi_{\text{app}}^{f}\|_{L^{2}}+\alpha\left\|\frac{U_{s}}{Y}\right\|_{L^{\infty}}\|Y\Psi_{\text{app}}^{f}\|_{L^{2}}+\alpha|c|\|\Psi_{\text{app}}^{f}\|_{L^{2}}+\alpha\|\Phi_{\text{app}}^{f}\|_{L^{2}}\right)$
$\displaystyle\lesssim\alpha^{1+\frac{3}{2}(1+\nu_{0})},$

which is (5.22). We then complete the proof of Lemma 5.4. ∎

Proof of Theorem 1.1 for $\beta\in(3/28,1/8)$. We look for the solution
$(\Phi,\Psi)(Y;c)$ to the Orr-Sommerfeld equation (1.22) with (1.23) in the
following form

$(\Phi,\Psi)(Y;c)=(\Phi_{\text{app}},\Psi_{\text{app}})(Y;c)-(\Phi_{\beta,1},\Psi_{\beta,1})(Y;c)-(\Phi_{\beta,2},\Psi_{\beta,2})(Y;c).$

Here $(\Phi_{\text{app}},\Psi_{\text{app}})$ is the approximate solution
constructed in (5.11) and it satisfies (5.18),
$(\Phi_{\beta,1},\Psi_{\beta,1})$ satisfies

$\left\\{\begin{aligned}
&\mbox{OS}_{c}(\Phi_{\beta,1},\Psi_{\beta,1})=\left(E_{\beta,3}^{s}+E_{\beta,3}^{f},~{}F_{\beta}^{f}\right)\\\
&\Phi_{\beta,1}(Y;c)|_{Y=0}=(\partial_{Y}^{2}-\alpha^{2})\Phi_{\beta,1}(Y;c)|_{Y=0}=\Psi_{\beta,1}(Y;c)|_{Y=0}=0,\end{aligned}\right.$

and $(\Phi_{\beta,2},\Psi_{\beta,2})$ satisfies

$\left\\{\begin{aligned}
&\mbox{OS}_{c}(\Phi_{\beta,2},\Psi_{\beta,2})=\left(\partial_{Y}E_{\beta,2}^{s}+i\alpha
E_{\beta,2}^{s}+\partial_{Y}E_{\beta,2}^{f}+i\alpha
E_{\beta,2}^{f},~{}0\right)\\\
&\Phi_{\beta,2}(Y;c)|_{Y=0}=(\partial_{Y}^{2}-\alpha^{2})\Phi_{\beta,2}(Y;c)|_{Y=0}=\Psi_{\beta,2}(Y;c)|_{Y=0}=0.\end{aligned}\right.$

For $\alpha=\varepsilon^{\beta}$ with $\beta\in(1/12,1/8)$ and $c\in D_{**}$,
by using (5.3) and (5.4) we can take $\varepsilon_{0}$ suitably small so that
for $\varepsilon\in(0,\varepsilon_{0})$, the following holds:

$\text{Im}\hat{c}\geq\tau_{4}\alpha^{1+\nu_{0}}\geq\tau_{4}n^{-1}\alpha^{-(2+\nu_{0})}\geq
2\tau_{2}^{-1}n^{-1},~{}\text{Im}\hat{c}\geq\tau_{4}n^{-1}\alpha^{-\frac{1}{2}}\alpha^{-(\frac{3}{2}+\nu_{0})}\geq
2\tau_{3}^{-1}n^{-1},$

and

$\frac{1}{\alpha^{\frac{1}{2}}n(\text{Im}\hat{c})^{2}}\lesssim\alpha^{3+2\nu_{0}-\frac{1}{2}-2(1+\nu_{0})}\lesssim\alpha^{\frac{1}{2}}\ll
1.$

Here $\tau_{2}$ and $\tau_{3}$ are the numbers given in Lemma 4.3 and Lemma
4.5 respectively. Therefore, $D_{**}\subsetneqq\Sigma_{d}\cap\Sigma_{s}$, and
in view of (4.41) and Remark 4.2, the solution sequence defined in (4.34)
converges. This gives the existence of $(\Phi_{\beta,1},\Psi_{\beta,1})(Y;c)$
and $(\Phi_{\beta,2},\Psi_{\beta,2})(Y;c)$. Moreover, in view of (4.43),
(4.44) and (4.45), since

$\frac{1}{\alpha^{\frac{1}{2}}n(\text{Im}\hat{c})^{2}}\lesssim\alpha^{\frac{1}{2}},~{}\frac{1}{(\alpha
n\text{Im}\hat{c})^{\frac{1}{2}}}\lesssim\alpha^{\frac{1}{2}(1+\nu_{0})},~{}n^{\frac{1}{2}}(\text{Im}\hat{c})^{\frac{3}{2}}\lesssim\alpha^{\frac{1}{2}\nu_{0}},$

we can show that $(\Phi_{\beta,1},\Psi_{\beta,1})(Y;c)$ and
$(\Phi_{\beta,2},\Psi_{\beta,2})(Y;c)$ satisfy the same bounds as (4.3) and
(4.4) respectively. In particular, it holds that

$\displaystyle\left|\partial_{Y}\Phi_{\beta,1}(0;c)\right|$
$\displaystyle\lesssim\frac{1}{\text{Im}\hat{c}}\left(\||U_{s}^{\prime\prime}|^{-\frac{1}{2}}E_{\beta,3}^{s}\|_{L^{2}}+\||U_{s}^{\prime\prime}|^{-\frac{1}{2}}E_{\beta,3}^{f}\|_{L^{2}}\right)+\|F^{f}_{\beta}\|_{L^{2}},$
(5.28) $\displaystyle\left|\partial_{Y}\Phi_{\beta,2}(0;c)\right|$
$\displaystyle\lesssim\frac{1}{(\text{Im}\hat{c})^{2}}\left(\|(E_{\beta,1}^{s},E_{\beta,2}^{s})\|_{L^{2}}+\|(E_{\beta,1}^{f},E_{\beta,2}^{f})\|_{L^{2}}\right).$

Recall $\tilde{\Gamma}_{0}(c)$ is the mapping given in Lemma 5.2. Now we
define the mapping

$\tilde{\Gamma}(c)\triangleq\partial_{Y}\Phi(0;c):~{}D_{**}\mapsto\mathbb{C}.$

By using (5.19), (5.20), (5.21), (5.22) and (5.28), we obtain

$\displaystyle\left|\tilde{\Gamma}(c)-\tilde{\Gamma}_{0}(c)\right|$
$\displaystyle\leq\left|\partial_{Y}\Phi_{\beta,1}(0;c)\right|+\left|\partial_{Y}\Phi_{\beta,2}(0;c)\right|$
(5.29)
$\displaystyle\lesssim\frac{1}{\text{Im}\hat{c}}\left(\alpha^{\frac{3}{2}}+\alpha^{\frac{5}{2}(1+\nu_{0})}\right)+\alpha^{1+\frac{3}{2}(1+\nu_{0})}+\frac{1}{(\text{Im}\hat{c})^{2}}\left(\alpha^{2+\frac{1}{2}(1+\nu_{0})}+\alpha^{\frac{5}{2}(1+\nu_{0})}\right)$
$\displaystyle\lesssim\alpha^{\frac{1}{2}-\nu_{0}}+\alpha^{\frac{1}{2}(1-3\nu_{0})}.$

For $\beta\in(3/28,1/8)$, we have $\nu_{0}\in(0,1/3)$. Then taking
$\varepsilon_{0}$ smaller if necessary, we deduce that

$\left|\tilde{\Gamma}(c)-\tilde{\Gamma}_{0}(c)\right|\lesssim\alpha^{\frac{1}{2}-\nu_{0}}+\alpha^{\frac{1}{2}(1-3\nu_{0})}\lesssim\frac{r_{3}}{4}.$
(5.30)

The theorem then follows from Lemma 5.2, (5.15) and an application of Rouché’s
Theorem. ∎.

Acknowledgment: The research of C.-J. Liu was supported by the National Key
R&D Program of China (No.2020YFA0712000) and National Natural Science
Foundation of China (11801364). The research of T. Yang was supported by the
General Research Fund of Hong Kong CityU No. 11303521.

##  6\. Appendices

###  6.1. Properties of Airy function

Recall the classical Airy function, cf. [18],

$\displaystyle Ai(z)=\frac{1}{2\pi
i}\int_{L}\exp\left(zt-\frac{t^{3}}{3}\right){\rm d}t,$ (6.1)

where the contour $L$ is given by

$L=\left\\{re^{-\frac{2\pi
i}{3}},r\in(1,+\infty)\right\\}\cup\left\\{e^{i\theta},\theta\in[-\frac{4\pi}{3},-\frac{2\pi}{3}]\right\\}\cup\left\\{re^{\frac{2\pi
i}{3}},r\in(1,+\infty)\right\\}.$

Its first, second and third order primitive functions are given by

$\displaystyle Ai(k,z)\triangleq\frac{1}{2\pi
i}\int_{L}t^{-k}\exp\left(zt-\frac{t^{3}}{3}\right){\rm d}t,~{}k=1,2,3.$ (6.2)

It is straightforward to check that

$\partial_{z}^{2}Ai(z)-zAi(z)=0,$

and

$\partial_{z}Ai(k,z)=Ai(k-1,z),~{}k=1,2,3,~{}Ai(0,z)=Ai(z).$

The following lemma gives the asymptotic behavior of $Ai(k,z)$ for large
$|z|$, see [5, 10, 18]. The proof of (6.3) for the case $k=0,1,2$ can be found
in [18], while the case $k=3$ can be treated similarly. We omit detail of the
proof for brevity.

######  Lemma 6.1.

There exists a positive constant $M>1$, such that for $|z|\geq M$ and $|\arg
z|\leq\frac{5\pi}{6}$, we have the following asymptotic formula:

$\displaystyle
Ai(k,z)=\frac{(-1)^{k}}{2\sqrt{\pi}}z^{-\frac{1+2k}{4}}e^{-\frac{2}{3}z^{\frac{3}{2}}}(1+O(z^{-\frac{3}{2}})),~{}k=0,1,2,3.$
(6.3)

###  6.2. Some explicit formula

######  Lemma 6.2.

Let $\kappa\geq 1$ be a positive integer. There exists
$\tilde{\gamma}\in(0,1)$, such that for any $c\in$
$\\{\text{Im}c>0,~{}|\hat{c}|\leq\tilde{\gamma}\\}$, it holds that

$\displaystyle\int_{0}^{1}|\log(U_{s}(Y)-\hat{c})|^{\kappa}{\rm d}Y$
$\displaystyle\lesssim 1,~{}\kappa\geq 1,$ (6.4)
$\displaystyle\int_{0}^{1}|U_{s}(Y)-\hat{c}|^{-1}{\rm d}Y$
$\displaystyle\lesssim 1+|\log\text{Im}\hat{c}|,$ (6.5)
$\displaystyle\int_{0}^{1}|U_{s}(Y)-\hat{c}|^{-\kappa}{\rm d}Y$
$\displaystyle\lesssim 1+|\text{Im}\hat{c}|^{-\kappa+1},~{}\kappa>1.$ (6.6)

###### Proof.

We give the proof of (6.4) only because (6.5) and (6.6) can be proved
similarly. Take
$\tilde{\gamma}=\min\\{\frac{1}{2}U_{s}(1),\frac{1}{2}(1-U_{s}(1))\\}$. Then
one has

$|U_{s}(Y)-\hat{c}|\leq U_{s}(1)+|\hat{c}|<1\text{ for }Y\in[0,1]\text{ and
}|\hat{c}|<\tilde{\gamma}.$

Note that

$\int_{0}^{1}|\log(U_{s}(Y)-\hat{c})|^{\kappa}{\rm
d}Y\lesssim\int_{0}^{1}\left(-\log\left(|U_{s}(Y)-\text{Re}\hat{c}|+\text{Im}\hat{c}\right)\right)^{\kappa}{\rm
d}Y+1.$

We now consider two cases.

Case 1: $\text{Re}\hat{c}\in\left(0,\frac{U_{s}(1)}{2}\right)$. For this case,
let $Y_{1}\in[0,1]$ be the point such that $U_{s}(Y_{1})=\text{Re}\hat{c}$.
Then we have

$\displaystyle\int_{0}^{1}\left(-\log\left(|U_{s}(Y)-\text{Re}\hat{c}|+\text{Im}\hat{c}\right)\right)^{\kappa}{\rm
d}Y$
$\displaystyle\qquad=\int_{0}^{Y_{1}}\left(-\log\left(\text{Re}\hat{c}-U_{s}(Y)+\text{Im}\hat{c}\right)\right)^{\kappa}{\rm
d}Y+\int_{Y_{1}}^{1}\left(-\log\left(U_{s}(Y)-\text{Re}\hat{c}+\text{Im}\hat{c}\right)\right)^{\kappa}{\rm
d}Y.$

For the first integral, denote
$Q(Y)\triangleq-\log\left(\text{Re}\hat{c}-U_{s}(Y)+\text{Im}\hat{c}\right)$.
Since $U_{s}^{\prime}(Y)>0$, we have

$\displaystyle\int_{0}^{Y_{1}}Q^{\kappa}(Y){\rm d}Y=$
$\displaystyle-\int^{Y_{1}}_{0}\frac{{\rm d}}{{\rm
d}Y}\left\\{\left(\text{Re}\hat{c}-U_{s}(Y)+\text{Im}\hat{c}\right)Q^{\kappa}(Y)\right\\}\frac{1}{U_{s}^{\prime}(Y)}{\rm
d}Y+\kappa\int^{Y_{1}}_{0}Q^{\kappa-1}(Y){\rm d}Y$ $\displaystyle\lesssim$
$\displaystyle 1+\kappa\int_{0}^{Y_{1}}Q^{\kappa-1}(Y){\rm
d}Y+\int^{Y_{1}}_{0}\left|\left(\text{Re}\hat{c}-U_{s}(Y)+\text{Im}\hat{c}\right)Q^{\kappa}(Y)\right|\left|\frac{U_{s}^{\prime\prime}(Y)}{(U_{s}^{\prime}(Y))^{2}}\right|{\rm
d}Y$ $\displaystyle\lesssim$ $\displaystyle
1+\kappa\int_{0}^{Y_{1}}Q^{\kappa-1}(Y){\rm d}Y,$

where in the last inequality we have used $x|\log
x|^{\kappa}\lesssim_{\kappa}1$ for $x\in[0,1]$. Inductively, we can obtain

$\int_{0}^{Y_{1}}Q^{\kappa}(Y){\rm d}Y\lesssim 1.$

Similarly, we also have

$\int_{Y_{1}}^{1}\left|\log\left(\text{Re}\hat{c}-U_{s}(Y)+\text{Im}\hat{c}\right)\right|^{\kappa}{\rm
d}Y\lesssim 1.$

Case 2: $\text{Re}\hat{c}\in\left(-\frac{U_{s}(1)}{2},0\right]$. In fact, this
case can be treated similarly. We then conclude (6.4) and also the proof the
lemma. ∎

Next we list some explicit formula related to $\Phi_{1}^{s}(Y;{c})$ which is
defined in (3.4).

$\displaystyle\partial_{Y}\Phi_{1}^{s}(Y;{c})=$
$\displaystyle-2(\partial_{Y}-\alpha)\psi_{0,1}(Y)e^{-\alpha
Y}\int_{0}^{Y}U_{s}^{\prime}(X)\psi_{0,2}(X){\rm d}X$ (6.7)
$\displaystyle-2(\partial_{Y}-\alpha)\psi_{0,2}(Y)e^{-\alpha
Y}\int_{Y}^{\infty}U_{s}^{\prime}(X)\psi_{0,1}(X){\rm d}X,$
$\displaystyle\partial_{Y}^{2}\Phi_{1}^{s}(Y;{c})=$
$\displaystyle-2(\partial_{Y}-\alpha)^{2}\psi_{0,1}(Y)e^{-\alpha
Y}\int_{0}^{Y}U_{s}^{\prime}(X)\psi_{0,2}(X){\rm d}X$
$\displaystyle-2(\partial_{Y}-\alpha)^{2}\psi_{0,2}(Y)e^{-\alpha
Y}\int_{Y}^{\infty}U_{s}^{\prime}(X)\psi_{0,1}(X){\rm
d}X+2U_{s}^{\prime}e^{-\alpha Y},$
$\displaystyle\partial_{Y}^{3}\Phi_{1}^{s}(Y;{c})=$
$\displaystyle-2(\partial_{Y}-\alpha)^{3}\psi_{0,1}(Y)e^{-\alpha
Y}\int_{0}^{Y}U_{s}^{\prime}(X)\psi_{0,2}(X){\rm d}X$
$\displaystyle-2(\partial_{Y}-\alpha)^{3}\psi_{0,2}(Y)e^{-\alpha
Y}\int_{Y}^{\infty}U_{s}^{\prime}(X)\psi_{0,1}(X){\rm d}X$
$\displaystyle+2U_{s}^{\prime\prime}(Y)e^{-\alpha Y}-6\alpha
U_{s}^{\prime}(Y)e^{-\alpha Y}.$

Finally we state the following lemma about analyticity.

######  Lemma 6.3.

Let $\Omega\subset\mathbb{C}$ be an open subset and $J,K$ be two Banach
spaces. Denote by $B(J,K)$ the space of bounded linear operators from $J$ to
$K$. Assume that

$L(\cdot~{};c):\Omega\mapsto B(J,K)$

is analytic in $c$ with values in $B(J,K)$ and

$f(c):\Omega\mapsto J$

is analytic in $c$ with values in $J$. Then $L(f;c)$ is analytic in $c$ with
values in $K$.

###### Proof.

Let $c_{0}\in\Omega$ be a fixed number. Since $L(\cdot~{};c)$ is analytic at
$c_{0}$, there exists $r_{1}>0$ and a sequence of bounded linear operator
$\\{L_{n}\\}_{n=0}^{\infty}\subset B(J,K)$, such that

$L(\cdot~{};c)=\sum_{n=0}^{+\infty}(c-c_{0})^{n}L_{n}(\cdot),$

with

$\sum_{n=0}^{+\infty}|c-c_{0}|^{n}\|L_{n}\|_{B(J,K)}<\infty~{}\text{ for
}|c-c_{0}|<r_{1}.$

Also by analyticity of $f$ we can find $r_{2}>0$ and a sequence of elements
$f_{n}\in J$ such that

$f(c)=\sum_{n=0}^{+\infty}(c-c_{0})^{n}f_{n},$

with

$\sum_{n=0}^{+\infty}|c-c_{0}|^{n}\|f_{n}\|_{J}<\infty~{}\text{ for
}|c-c_{0}|<r_{2}.$

By linearity, for $|c-c_{0}|<\min\\{r_{1},r_{2}\\}$ we deduce that

$\displaystyle
L\left(f(c);c\right)=\sum_{n=0}^{+\infty}(c-c_{0})^{n}L_{n}\left(\sum_{m=0}^{+\infty}(c-c_{0})^{m}f_{m}\right)=\sum_{n=0}^{+\infty}(c-c_{0})^{n}\left(\sum_{m=0}^{n}L_{m}f_{n-m}\right).$
(6.8)

For $|c-c_{0}|<\min\\{r_{1},r_{2}\\}$, it holds

$\displaystyle\sum_{n=0}^{+\infty}|c-c_{0}|^{n}\left\|\sum_{m=0}^{n}L_{m}f_{n-m}\right\|_{K}\leq$
$\displaystyle\sum_{n=0}^{+\infty}|c-c_{0}|^{n}\left(\sum_{m=0}^{n}\|L_{m}\|_{B(J,K)}\|f_{n-m}\|_{J}\right)$
$\displaystyle\leq$
$\displaystyle\left(\sum_{n=0}^{+\infty}|c-c_{0}|^{n}\|L_{n}\|_{B(J,K)}\right)\times\left(\sum_{n=0}^{+\infty}|c-c_{0}|^{n}\|f_{n}\|_{J}\right)<\infty,$

which implies that the series on the right hand side of (6.8) absolutely
converges in $K$. Therefore, $L(f;c)$ is analytic and the proof of Lemma 6.3
is completed. ∎

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