# One-point Statistics in various cosmic environments in the presence of
massive neutrinos

Mohadese Khoshtinat1, Hossein Hatamnia1, Shant Baghram1,2

1 Department of Physics, Sharif University of Technology, P. O. Box
11155-9161, Tehran, Iran
2 Research Center for High Energy Physics, Department of Physics, Sharif
University of Technology, P.O.Box 11155-9161, Tehran, Iran
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Studying the structures (halos and galaxies) within the cosmic environments
(void, sheet, filament, and node) where they reside is an ongoing attempt in
cosmological studies. The link between the properties of structures and the
cosmic environments may help to unravel the nature of the dark sector of the
Universe. In this paper, we study the cosmic web environments from the spatial
pattern perspective in the context of $\Lambda$CDM and $\nu\Lambda$CDM as an
example of an extension to the vanilla model. To do this, we use the T-web
classification method and classify the cosmic environments for the catalogues
from the gevolution N-body simulations for $\Lambda$CDM and $\nu\Lambda$CDM
cosmology. Then, we compute the first nearest neighbour cumulative
distribution function, spherical contact cumulative distribution function, and
$J$-function for every cosmic environment. In the context of the standard
model, the results indicate that these functions can differentiate the various
cosmic environments. In association with distinguishing between extensions of
the standard model of cosmologies, these functions within the cosmic
environment seem beneficial as a complementary probe.

###### keywords:

(cosmology:) dark matter, (cosmology:) large-scale structure of Universe

††pubyear: 2024††pagerange: One-point Statistics in various cosmic
environments in the presence of massive neutrinos–A

##  1 Introduction

The $\Lambda$CDM model of cosmology successfully predicted almost all results
obtained from the cosmic microwave background (CMB) and the large-scale
structure (LSS). But as the data from ongoing surveys (Aghanim et al., 2020;
Almeida et al., 2023; Adame et al., 2023; Dournac et al., 2024) become
accurate and rich on the finer scales, the observed quantities deviate from
the predicted values based on the $\Lambda$CDM model. Some of the deviations
become serious and challenge the $\Lambda$CDM model, e.g., the Hubble tension
(Di Valentino et al., 2021), the $\sigma_{8}$ tension (Joudaki et al., 2018)
together with the discrepancy on small scales (Perivolaropoulos & Skara,
2022). These challenges, along with the principal questions about the physical
nature of the dark sector of the Universe, created a rich literature around
the probable extension of the standard model of cosmology, i.g., considering
the massive neutrinos (Aker et al., 2022) as a component of the dark
matter–$\nu\Lambda$CDM, the dynamical dark energy model (Peebles & Ratra,
2003; Tabatabaei et al., 2023; Amendola & Tsujikawa, 2015)–$\omega\Lambda$CDM,
interacting dark matter and energy (Amendola, 2000), and the deviation from
the Gaussian initial condition (Bartolo et al., 2004). Most studies so far
apply these models to the datasets using the traditional tools used by
cosmologists, i.e., the 2-point correlation functions in real space or its
Fourier counterpart, the power spectrum. The 2-point correlation function is
the best choice for investigating the cosmological dataset up to the scales in
which the linear perturbation theory still holds (Bernardeau et al., 2002).
Beyond that, due to highly nonlinear gravitational evolution and heavy mode
coupling, the information leaks to the higher-order correlation function, and
the 2-point correlation function fails to capture all the information hidden
in the dataset. The leaked information at these scales is crucial as the
deviation from the $\Lambda$CDM model occurs mostly at these scales. To
extract this information, one can follow two widely used strategies. One is to
equip the 2-point correlation with extra details accounting indirectly for the
higher-order information. Some examples are the marked power spectrum for
halos or galaxies (Stoyan, 1984; Massara et al., 2021), the redshift space
studies (Kaiser, 1987), and computing the 2-point function in different cosmic
web environments and components—void, sheet, filament, and node—in the
coordinate and the redshift spaces (Bonnaire et al., 2022, 2023). The latter
highlights the potential of environmental studies on constraining the
cosmological parameters and breaking the degeneracies between different
extensions of the $\Lambda$CDM model. Zeldovich (1970), and many others, e.g.,
Doroshkevich & Shandarin (1978), noted that the formation of these structures
is rooted in the collapse of anisotropic fluctuations in a specific
environment via gravitational instabilities in an expanding universe.
The second solution to get the most out of data is to find other interpretable
statistics that, in principle, contain all the information of n-point
correlations but are computed for one point, e.g., White (1979) used the void
probability function to study the clustering of the tracers, Banerjee & Abel
(2020); Banerjee et al. (2022); Yuan et al. (2023) introduced the k-nearest
neighbours cumulative distribution functions (CDF) or peaked-CDF to study the
properties of the non-Gaussian clustering at the small scale. Fard et al.
(2022) and Khoshtinat et al. (2024) used the letter functions to study the
clustering of the dark matter halos in the context of the $\Lambda$CDM and the
$\nu\Lambda$CDM, respectively and Kousha et al. (2024) for studying the
distribution of LSS in Fuzzy dark matter models. the The letter
functions—$G(r)$ or the nearest neighbour CDF, and $F(r)$ or the spherical
contact CDF— are statistics used to investigate the clustering aspect of the
distribution of points in the space.
In this paper, we study the behaviour of the letter functions in various
cosmic web environments identified by the T-web classification method (Hahn et
al., 2007; Forero-Romero et al., 2009) in the context of the $\Lambda$CDM and
then use the $\nu\Lambda$CDM as an example of the extended model of cosmology.
In Section 2, we summarize the theoretical ingredients used in this work.
Section 3 describes the simulations and the algorithm used to identify the
cosmic environments. Section 4 represents the results of the analysis.
Finally, in Section 5, we conclude the discussion and point to the future
direction.

##  2 Theoretical Background

The following section provides essential theoretical aspects required for this
work. The first subsection discusses the cosmic web, the second addresses the
impact of the neutrino mass on cosmology, and the final one covers one-point
statistics.

Figure 1: A $5\ h^{-1}\rm{Mpc}$ thick slice of one sub-box of the $\Lambda$CDM
simulation at $z=0$. The topmost panel shows all the halos in the slice. The
other four illustrate the halos in various environments: voids, sheets,
filaments, and nodes, respectively. The T-web formalism is used to identify
the environments.

###  2.1 Cosmic Web Environment

It was evident from early galaxy surveys to recent ones (Huchra et al., 1983;
de Lapparent et al., 1986; Stoughton et al., 2002; Abazajian et al., 2009) and
also from the N-body simulations (Springel, 2005) that the distribution of
matter in the Universe displays a filamentary pattern, known as the cosmic
web. The theoretical description—Zeldovich approximation—of the formation of
such a filamentary configuration goes back to the works of Zeldovich (1970);
Doroshkevich & Shandarin (1978); Arnold et al. (1982). Zeldovich approximation
predicts the formation of the cosmic web components due to the local level of
tidal anisotropy for the linear growth of perturbation, i.e., in the
Lagrangian description based on the series expansion of the equation of
motion, test particles along their trajectories experience zero-order
acceleration and first-order deformation determined by the tidal field, which
ultimately leads to the formation of the voids, sheets, filaments, and nodes.
This physical picture sets a mathematical guideline for classifying the
cosmological environment based on the eigenvalues of a deformation tensor
(Libeskind et al., 2018; Ayçoberry et al., 2023) and a venue to count the
number of cosmic structures (Sheth & van de Weygaert, 2004; Colberg et al.,
2005; Fard et al., 2019). Among various definitions for a deformation tensor,
we use the T-web classification scheme extended by Forero-Romero et al.
(2009). In this approach, they defined the tidal shear tensor $T$ as follows:

$\\\ T_{ij}=\frac{\partial^{2}\Phi}{\partial x_{i}\partial x_{j}},$ (1)

where $\Phi$ is the gravitational potential, $i$ and $j$ stands for the
spatial coordinates. The environments can then be classified using the
following criteria for the eigenvalues of the $T$ tensor:

  * •

Node: Three eigenvalues are above the threshold.

  * •

Filament: Two eigenvalues are above the threshold.

  * •

Sheet: One eigenvalue is above the threshold.

  * •

Void: No eigenvalue is above the threshold.

The threshold is a free parameter of the process, obtained by complementary
observations such as the number density of each element in the cosmic web or
their volume fraction. We will come back to it in Section 3.2 .
In the context of the standard model, analysis of the properties of the halos
from the numerical simulation (Hahn et al., 2007; Wechsler et al., 2006)
suggests that these are not only dependent on the mass of the halo (as
considered in the semi-analytical models like the excursion set theory (Bond
et al., 1991; Zentner, 2007)) but on the environments where the halo is formed
and evolve during the cosmic time (Sheth & Tormen, 2004; Parkavousi et al.,
2023). At each epoch, the heaviest halos residue in the nodes having the
broadest range of halo mass decades contrasted to the other environments, with
the voids having the shortest mass interval. Additionally, Hahn et al. (2007)
observed that the typical mass-scale collapsing at each epoch, $M_{col}(z)$,
acts as a natural criterion for the halo mass. Halos that are affected the
most by the characteristics of their surrounding environments are those with
masses less than $M_{col}(z)$. For example, halos with a mass less than
$M_{col}(z)$ tend to align more with the preferred direction of their hosting
filament or sheet, spin-wise or shape-wise. Another notable characteristic is
that the voids have the youngest halos with the mass $\lesssim M_{col}(z)$ as
opposed to the other environments (Hahn et al., 2007). With these observations
in mind, in this work, we study the environment dependence of one-point
statistics.

###  2.2 Massive Neutrino

The neutrino oscillation phenomenon is an experimental fact that the total
mass of the three neutrino species can not be considered zero. The latest
release from the KATRIN (Aker et al., 2022) experiment suggests an upper bound
for the total neutrino mass of about $\Sigma\ m_{\nu}\lesssim 0.8$ eV ($90\%$
C. L.). Regarding the neutrinos, their mass puts them in the category of the
light relic in cosmological studies. Light relic species leave the primordial
thermal equilibrium as relativistic particles. As the Universe expands, these
particles lose their energy and eventually become non-relativistic when their
mass surpasses their kinetic energy. This procedure induces a length
scale—$\lambda_{fs}$, below which these particles stream freely because of
their thermal velocities. For a single neutrino species $m_{\nu}$ at any given
redshift $z$, this length scale is defined accordingly:

$\\\ k_{fs}(z)\approx 0.063\,h\ {\rm Mpc}^{-1}\,\frac{m_{\nu}}{0.1{\rm eV}}\
\frac{1}{(1+z)^{2}}\ \frac{H(z)}{H_{0}}\ ,$ (2)

where $H(z)$ and $H_{0}$ are Hubble parameter and constant, respectively
(Agarwal & Feldman, 2011). The free-streaming of neutrinos below
$\lambda_{fs}\sim k_{fs}^{-1}$ smooths the fluctuation of the matter field,
which translates to power suppression contrasted to the massless scenario.
Beyond the free-streaming length scale, neutrinos behave like cold dark matter
particles. At redshift zero, the neutrino energy density is equal to:

$\\\ \omega_{\nu}=\frac{\sum m_{\nu}}{94.13\ \rm eV}\ ,$ (3)

with the neutrino mass fraction of $f_{\nu}={\omega_{\nu}}/{\omega_{m}}$,
where $\omega_{m}=\Omega_{m}h^{2}$ is the dark matter density normalized to
the critical density.
In addition to the suppression of the relative power spectrum ($\nu\Lambda$CDM
to the standard model), predicted by both linear and non-linear perturbation
theory, data from the N-body simulations with massive neutrinos show a
turnover of the relative power spectrum around $k\sim 1\ h\ {\rm Mpc}^{-1}$,
also known as spoon-like as opposed to predicted slide-like behaviour.
Hannestad et al. (2020) claims from the halo model (Navarro et al., 1996;
Cooray & Sheth, 2002) perspective suppression of the relative power spectrum
is due to the suppression of the two-halo clustering caused by neutrinos.
Whereas the rise in the relative power spectrum at $k\sim 1\ h\ {\rm
Mpc}^{-1}$ is controlled by the one-halo term, where the virialization process
eradicates the initial condition of perturbations deep inside a halo.
Regarding the spatial distribution of halos, Khoshtinat et al. (2024) observe
for a randomly placed point in a universe with massive neutrinos, the
neighbouring is empty than the $\Lambda$CDM universe, and the degree of
emptiness highly depends on the total neutrino mass. They also show that the
immediate neighbour of halos in the presence of massive neutrinos happens at
further distances as opposed to the nearest neighbour for the $\Lambda$CDM
universe. Concisely, the presence of massive neutrinos reduces the overall
clustering in the LSS. These observations motivate us to study the
aforementioned effects in different cosmic environments.

###  2.3 One-point Statistics

The spatial distribution of the dark matter halos, formed by gravity, carries
loads of information about the underlying dark matter field (Ravi K, 1996;
Bardeen et al., 1986). In the absence of gravity and other kind of
interactions, the pattern exhibited by the halos would be an example of the
Poisson point process111 For instance, a set of $N$ points in a cubic box of
side $L$ generated from a uniform distribution is an example of a Poisson
process with the intensity of $\bar{n}=NL^{-3}$., i.e., complete spatial
randomness. The main characteristic of a Poisson point process is the absence
of clustering, which is not the case for the LSS in the Universe. In the
context of the spatial point analysis (Hand, 2008), the letter functions—
$G(r)$, $F(r)$, and $J(r)$—are tools used to examine the clustering and its
extent for a spatial distribution of points. The $G$ and $F$ are of the nature
of cumulative distribution functions and are defined as follows:

  * •

$G(r)$: The CDF of the first nearest neighbour distance between the halos.

  * •

$F(r)$: The CDF of the distances from randomly generated points to the first
neighbouring halos.

$G(r)$, for a single halo, means the probability of having its first nearest
neighbour on the surface of a sphere with the radius $r$. Whereas for all
halos in the catalogue, it indicates the proportion of the halos having their
first nearest neighbour on a sphere of radius $r_{i}\leq r$ around them where
$r_{i}$ stands for the sphere centered at the location of $i$-th halos. $F(r)$
states the same but for the randomly generated points in the corresponding
volume. $J-$function is defined:

$\\\ J(r)\equiv\frac{1-G(r)}{1-F(r)}.$ (4)

The value of the $J$-function for complete spatial randomness is one, while in
the presence of clustering, it takes values less than unity.

Aimed at studying the large-scale structure, one should address the
information these statistics hold and their connection to n-point correlation
functions. In the seminal work of White (1979), the conditional correlation
functions $\Xi_{i}$ defined as,

$\displaystyle\Xi_{i}(\textbf{r}_{1},...,\textbf{r}_{i};V)=$
$\displaystyle\sum_{j=0}^{\infty}\dfrac{(-n)^{j}}{j!}$ (5)
$\displaystyle\int...\int\xi_{i+j}(\textbf{r}_{1},...,\textbf{r}_{i+j})\,dV_{i+1}...dV_{i+j},$

establish a way to link these statistics and the n-point correlations. In
equation 5, $\xi_{i}$ are the n-point correlation functions where by
definition $\xi_{0}=0$ and $\xi_{1}=1$. And the integrals are over an empty
volume of interest $V$, except for the points $r_{i}$ and $\xi_{i}$.
The void probability, $P_{0}(V)$–the probability of finding an empty sphere of
radius $r$, is defined as $P_{0}(V)=\exp\left[\Xi_{0}(V(r))\right]$. By
definition, the complement of $P_{0}(V)$ is the $F(r)$.
The probability for a halo to have its nearest neighbour on the surface of a
volume $V(r)$ is the complement of the absence of any halo in this volume,
$G(r)=1-\Xi_{1}(\textbf{r}_{o};V(r))\exp\left[\Xi_{0}(V(r))\right]$.

These relations and equation 5 show that, by definition, these statistics
depend on the correlations of all orders, which makes them a suitable probe to
study the LSS, especially at the non-linear scales.

##  3 Simulations and Cosmic Web classification

In the first subsection, we introduce the simulation suit used in this work.
In the subsequent subsection, we review the details of the cosmic web
classification procedure.

$\omega_{c}$ | $\omega_{b}$ | $h$ | $n_{s}$ | $T_{\rm CMB}$ | $A_{s}$
---|---|---|---|---|---
$0.12038$ | $0.022032$ | $0.67556$ | $0.9619$ | $2.7255$ | $2.215\times 10^{-9}$
Table 1: The cosmology parameters of the standard model where
$\omega_{c}=\Omega_{c}h^{2}$, $\omega_{b}=\Omega_{b}h^{2}$ are baryonic and
dark matter densities normalizes to the critical density. $h=H_{0}/100$,
$n_{s}$, $T_{\rm CMB}$, $A_{s}$ are respectively the reduced Hubble parameter,
spectral index, the CMB temperature, and the amplitude amplitude of scalar
perturbations.

###  3.1 Simulations

We use existing halo catalogues for the $\nu\Lambda$CDM models with a suitable
$\Lambda$CDM reference catalogue to study the environmental effects using the
letter functions. Each catalogue is prepared from a snapshot of the gevolution
222https://github.com/gevolution-code N-body simulation (Adamek et al., 2016a,
b) by the ROCKSTAR halo finder (Behroozi et al., 2013). The simulations follow
$4096^{3}$ particles and grids in a cubic comoving box $L=2048$ $h^{-1}{\rm
Mpc}$ initialized at redshift $z=100$. The total neutrino mass in the
$\nu\Lambda$CDM catalogues are $\sum m_{\nu}=0.06,\ 0.2,\ 0.3$ eV. The cold
dark matter density parameter for each of these catalogues is consistent with
the $\omega_{c}\equiv\Omega_{c}h^{2}=0.12038-\omega_{\nu}$, so the total
matter density ($\omega_{\nu}+\omega_{b}+\omega_{c}$) is untouched. Other
cosmological parameters are summarized in Table 1. From each snapshot, we
construct 64 sub-boxes of side $512h^{-1}{\rm Mpc}$ and find the cosmic
environments. Figure 1 shows the results for a $5h^{-1}{\rm Mpc}$ thick slice
of a sub-box from $\Lambda$CDM simulation box.

###  3.2 Cosmic Web Classification

In this paper, we use the T-web classification method introduced and developed
by Hahn et al. (2007); Forero-Romero et al. (2009). In Figure 2, we summarize
the workflow of the classification process. This process starts by
constructing the density field $\rho(\textbf{x})$ on a simple cubic lattice,
with a lattice parameter $l_{p}$, from the coordinates and mass of halos using
the cloud-in-cell (CiC) scheme. Then, we compute the density contrast field at
each lattice point by $\delta(\textbf{x})=\rho(\textbf{x})/\bar{\rho}-1$.
After transforming the density contrast field to the Fourier space, we smooth
the field by a Gaussian window function with a smoothing scale of $l_{s}$.
Later, we compute the normalized gravitational potential using the normalized
Poisson equation, $\nabla^{2}\Phi=4\pi G\bar{\rho}\ \delta$. After the inverse
Fourier transforms, we calculate the T-web tensor, equation 1, at each grid
and find the eigenvalues of it at each lattice point. Then, we determine the
environment of each lattice point by comparing the eigenvalues with the
$\lambda_{th}$ with the criteria mentioned in Section 2.1.
Three parameters—$l_{p}$, $l_{s}$, and $\lambda_{th}$—introduced in this
process. The typical interspacing between halos and the computation cost
limits the choice for $l_{p}$. Based on these limits, we construct a lattice
with $300^{3}$ points in each of the 64 sub-boxes for each simulation
snapshot. This choice sets the $l_{p}$ around $1.6\ h^{-1}{\rm Mpc}$. This
value is in the same order as the peak of the first nearest neighbour
probability distribution function.

Figure 2: The workflow of the cosmic environments classification. The process
starts by constructing the density field $\rho(x)$ on a simple cubic lattice
(SC-lattice) using the coordinates and masses of the halos in the catalogue
and ends by specifying the environment in which a halo resides. The lattice
parameter—$l_{p}$, the smoothing scale—$l_{s}$, and the threshold
value—$\lambda_{th}$ for the eigenvalues of the $T_{ij}$ tensor are three
parameters introduced in this process.  Figure 3: Studying the effect of
change of the smoothing scale–$l_{s}$ and $\lambda_{th}$ on the environments.
The top panel shows the heat map of the percentage of halo residue in the
voids when the two parameters vary. The bottom left plot indicates the volume
fraction of various environments for fixed $l_{s}=2.2\ h^{-1}{\rm Mpc}$ and
different $\lambda_{th}$. The bottom right plot shows the mass fraction of
various environments for fixed $\lambda_{th}=0.4$ and different $l_{s}$. In
the bottom plots, the solid and dash-dotted lines stand for nodes and
filaments while the dotted and dash lines present void and sheet,
respectively.

The choice for ($l_{s}$, $\lambda_{th}$) is a bit challenging. To find the
suitable values for these two parameters, we run the environment
classification process several times by changing them slightly at each run.
Figure 3 summarizes the results. The heat map on the top of Figure 3 indicates
the percentage of the halos belonging to voids concerning all halos. Our
choice of parameters should allow for a percentage that provides statistical
significance for our analysis. We pick (0.4, 2.2) for ($l_{s}$,
$\lambda_{th}$), with these at least five percent of halos belonging to the
voids. To check the physical rationality behind this choice of parameters, we
adjust one parameter while allowing the other parameter to vary within a
specific interval. With $l_{s}=2$, the choice for $\lambda_{th}$ must be in
the region where the hierarchy of the volume fraction of various environments
stays the same, the bottom left plot of Figure 3. For $\lambda_{th}=0.4$, the
value for the $l_{s}$ should be above the points where the mass fraction of
various environments experience a considerable change, the bottom right of
Figure 3.
In Appendix A, we study the impact of varying the $\lambda_{th}$ and $l_{s}$
on the statistics used in this work.

##  4 Results :Letter functions and environment

Figure 4: Letter functions computed in various cosmic environments for
$\Lambda$CDM model, from top to bottom $G$, $F$, and $J$ functions. In all
panels, the solid line shows the behaviour of all the data points, and the
squares, stars, triangles, and circles represent the points belonging to the
voids, sheets, filaments, and nodes, respectively. The error bars are smaller
than the markers of the data points.

In the following section, we study different cosmic environments using the
letter functions—$G$ or the nearest neighbour CDF, the $F$ or the spherical
contact CDF, and $J$-function as defined by equation 4—in the $\Lambda$CDM
model and cosmologies with massive neutrinos. As explained in Section 3, we
divide each simulation snapshot into 64 sub-boxes of side 512 $h^{-1}\rm{Mpc}$
and run the environment classification algorithm on each of these 64 sub-
boxes, and then compute our favored quantities (e.g., $G$ in voids) in each of
these sub-boxes. In the Figures 4 and 5, the data points and the error bars
are the mean and standard deviation of 64 values from each sub-box.

###  4.1 $\Lambda$CDM model

Figure 4 shows the letter functions computed in various environments within
the $\Lambda$CDM model. The solid line represents the behaviour of all the
points regardless of their habitats. Squares and stars stand for the points
residue in the voids and on the sheets; Triangles and circles represent the
points in the filaments and the nodes. The top panel of Figure 4 shows the $G$
function—the CDF of the first nearest neighbour for the halos. It is possible
to make the following observations from this diagram:

  * •

The $G_{all}$ function follows mostly the $G$ for filamentary halos (i.e., the
triangles). This behaviour is rooted in the fact that the filamentary halos
outnumbered all the halos in the other three environments. Ganeshaiah Veena et
al. (2021) among many others, pointed out that most halos in the Universe live
in filaments.

  * •

The $G$ functions of less dense environments (i.e., stars for sheets and
squares for voids) are below the $G_{all}$ for all distances, which translates
to less probability of finding the immediate neighbour in these environments.

  * •

The $G$ functions of extreme environments, node (i.e., circles), and void
(i.e., squares) envelope the $G$ of other environments. In addition, at the
scale that almost all node halos find their immediate neighbours, most of the
void halos are still at the center of an empty sphere without any halos on the
boundary.

The middle panel of Figure 4 indicates the $F$ function—the CDF of the
distance to the first nearest halos from randomly generated points. To
calculate this, first, we prepare a randomly generated set of points in the
sub-box, then determine the environments for each random point, and finally
find the distance to the nearest neighbouring halo (regardless of the
environment that the halo belongs to). Observations from the diagram:

  * •

The $F_{all}$ tends toward the $F$ functions of sheets and voids since these
environments are voluminous and contain the most randomly generated points.

  * •

The $F$ functions of the extreme environments envelope the other $F$ function.

In the bottom panel of Figure 4, we plot the $J$-function. As mentioned in
Section 2.3, the $J$-function quantifies the extent of the clustering of a
point pattern when compared with unity. A significant deviation from the unity
indicates a high degree of clustering in the point pattern. In this diagram,
as anticipated, voids are less clustered environments, while nodes are highly
clustered. It is worth noting that the turnover in the $J$-function is because
the denominator of the equation 4, $1-F(r)$, becomes zero.
Figure 4 shows that the letter functions can differentiate between different
environments.

###  4.2 $\nu\Lambda$CDM Versus $\Lambda$CDM

In Figure 5, we plot the difference in letter functions between the
$\Lambda$CDM model and cosmologies with massive neutrinos. In this plot, plus,
cross, and diamond indicate the total neutrino mass of $0.06,0.2,0.3$ eV.
Looking at this diagram, one can note the following:

  * •

$G$ function can distinguish between different total masses in every
environment. The deviation from the $\Lambda$CDM model increases as the total
neutrino mass increases. The degree of deviation from the standard model
decreases as the cosmic environment becomes more dense.

  * •

$F$ function can distinguish between different total neutrino masses in the
voluminous environment. In dense and compact environments (i.e., filaments and
nodes), due to small-scale fluctuation of randomness, the $F(r)$ can not tell
apart the total neutrino mass.

  * •

$J$-function indicates that as the total neutrino mass increases, the extent
of clustering decreases as opposed to the $\Lambda$CDM model.

By investigating the samples using letter functions, one can cross-check the
presence of the massive neutrino mass. The presence of this deviation in all
three is the sanity of the method. If the effect were absent in one of the
functions, this would be an alarm for the model. In the recent work by
Khoshtinat et al. (2024), we showed that the letter functions can distinguish
between different cosmologies. We demonstrate in this paper that the letter
functions computed in distinct cosmic environments can serve as complementary
tools to differentiate between various cosmological models.

Figure 5: The difference between letter functions in $\Lambda$CDM model and
cosmologies with massive neutrinos. The plus, cross, and diamond represent
cosmologies with a total neutrino mass of $\Sigma\ m_{\nu}=0.06,0.2,0.3$ eV,
respectively.

##  5 Conclusions and Future remarks

The distribution of galaxies contains information that can constrain and
distinguish cosmological models. Galaxies reside inside dark matter halos,
which make the cosmic skeleton known as the cosmic web. The cosmic web
consists of different elements: nodes, filaments, sheets, and voids. In this
work, we compute the one-point statistics—$G$, $F$, and $J$ functions—in
different cosmic environments for a suit of the gevolution simulation. We
classify the cosmic environments via the T-web method, described in Section
2.1.

In the first step, we study the standard model and show that the letter
functions can distinguish the different environments. $G$ and $J$ functions
for all halos mimic the results obtained from the halos in filaments. However,
in the case of the $F$ function, the whole sample resembles the sheet. One
step further, we study these functions in cosmologies with massive neutrinos.
We propose that the letter functions computed for different cosmic
environments fit as complementary tools for differentiating the extension of
the standard model. In the case of cosmologies with massive neutrinos, the
voluminous underdense regions–voids and sheets–are more prominent for doing
this task.

In the T-web algorithm, the smoothing scale $l_{s}$ of the density field and
the threshold criteria for eigenvalues of T-web $\lambda_{th}$ are the two
main parameters of the environment-finding algorithm. To find the plausible
values for $l_{s}$ and $\lambda_{th}$, we repeat the algorithm for several
values of $l_{s}$ and $\lambda_{th}$ and based on the heat map for the number
of halos in voids and the behaviour of the volume and mass fractions choose
$2.2h^{-1}\rm{Mpc}$ and $0.4$ for $l_{s}$ and $\lambda_{th}$, respectively.

For future studies, one should incorporate the physics of halo occupation
distribution to assign each halo a galaxy to probe the one-point statistics in
galaxy surveys. It is also critical to study the possible degeneracies in the
one-point statistics raised from other extensions of $\Lambda$CDM, such as
dark energy models and modified cold dark matter models.

## Acknowledgements

We are very grateful to Farbod Hassani, Julian Adamek, and Martin Kunz for
giving us access to the halo catalogues of high-resolution neutrino
simulations presented in (Adamek et al., 2017). SB is partially supported by
the Abdus Salam International Center of Theoretical Physics (ICTP) under the
regular associateship scheme. Moreover, MK and SB are partially supported by
the Sharif University of Technology Office of Vice President for Research
under Grant No. G4010204.

## Data Availability

The halo catalogues we used are based on the gevolution simulation (Adamek et
al., 2017).

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##  Appendix A Effect of classification parameters on the letter function

To check the impact of the classification parameters on the statistics we use
in this work, we run the classification algorithm for six pairs of ($l_{s}$,
$\lambda_{th}$), where $l_{s}=0.4,0.5$ and $\lambda_{th}=2.1,\ 2.3,\ 2.5\
h^{-1}\rm{Mpc}$. All these values meet the criterion discussed in Section 3.2.
Figure 6 shows the $G$ functions in various environments for the $\Lambda$CDM
model. The markers of environments are the same as in Section 4. As evident in
Figure 6, the shape and the hierarchy of $G$ functions do not change. A slight
change in values of the $G$ functions is detected, rooted in the variation of
the number of halos in each environment.
Figure 7 shows the impact of changing the $l_{s}$ and $\lambda_{th}$ on the
difference of the $G$ functions between the standard model and the cosmologies
with massive neutrinos. The order and the shape of the difference in $G$
functions do not change, yet a slight change in the values of the difference
is detected.
To conclude, if the parameters meet the criterion in Section 3.2., the results
we represent in this paper do not change drastically. The letter functions
computed in various environments could serve as a complementary probe for
differing the extension of the standard model of cosmology.

Figure 6: The effect of classification parameters on $G$ function in different
cosmic environments for the $\Lambda$CDM models. The values for the parameters
are $l_{s}=\ 2.1,\ 2.3,\ 2.5\ h^{-1}\rm{Mpc}$ and $\lambda_{th}=\ 0.4,\ 0.5$.
The solid lines stand for the whole halos. The squares, stars, triangles, and
circles depict the $G$ of the voids, sheets, filaments, and nodes,
respectively. The order and the shape of $G$ functions stay the same, but a
slight change in the values at different scales happens. This change is due to
the change in the number density of halos in different environments.  Figure
7: The effect of classification parameters on the difference of $G$ function
between $\Lambda$CDM model and $\nu\Lambda$CDM model with total neutrino mass
of $0.3$ eV in various cosmic environments. The shape again stays the same
with a slight change in the values. This change is due to the change in the
number density of halos in different environments.