11institutetext:  Tartu Observatory, University of Tartu, Observatooriumi 1,
61602 Tõravere, Estonia
11email<EMAIL_ADDRESS>22institutetext: Estonian Academy of
Sciences, Kohtu 6, 10130 Tallinn, Estonia

# Using photometric redshift data to improve the detection of galactic
filaments with the Bisous model

M. M. Muru 11 E. Tempel 1122

###### Abstract

Context. Filament finders are limited, among other things, by the abundance of
spectroscopic redshift data. This limits the sky areas and depth where we can
detect the filamentary network.

Aims. As there are proportionally more photometric redshift data than
spectroscopic, we aim to use data with photometric redshifts to improve and
expand the areas where we can detect the large-scale structure of the
Universe. The Bisous model is a filament finder that uses only the galaxy
positions. We present a proof of concept, showing that the Bisous filament
finder can improve the detected filamentary network with photometric redshift
data.

Methods. We created mock data from the MultiDark-Galaxies catalogue. Galaxies
with spectroscopic redshifts were given exact positions from the simulation.
Galaxies with photometric redshifts were given uncertainties along one
coordinate. The errors were generated with different Gaussian distributions
for different samples. We sample the photometric galaxy positions for each
Bisous run based on the uncertainty distribution. In some runs, the sampled
positions are closer to the true positions and produce persistent filaments;
other runs produce noise, which is suppressed in the post-processing.

Results. There are three different types of samples: spectroscopic only,
photometric only, and mixed samples of galaxies with photometric and
spectroscopic redshifts. In photometric-only samples, the larger the
uncertainty for photometric redshifts, the fewer filaments are detected, and
the filaments strongly align along the line of sight. Using mixed samples
improves the number of filaments detected and decreases the alignment bias of
those filaments. The results are compared against the full spectroscopic
sample. The recall for photometric-only samples depends heavily on the size of
uncertainty and dropped close to 20%; for mixed samples, the recall stayed
between 40% and 80%. The false discovery rate stayed below 5% in every sample
tested in this work. Mixed samples showed better results than corresponding
photometric-only or spectroscopic-only samples for every uncertainty size and
number of spectroscopic galaxies in mixed samples.

Conclusions. Mixed samples of galaxies with photometric and spectroscopic
redshifts help us to improve and extend the large-scale structure further than
possible with only spectroscopic samples. Although the uncertainty sizes
tested in this work are smaller than those for the available photometric data,
upcoming surveys, such as J-PAS, will achieve sufficiently small uncertainties
to be useful for large-scale structure detection.

###### Key Words.:

methods: data analysis – methods: statistical – galaxies: statistics – large-
scale structure of the Universe

##  1 Introduction

The galaxy distribution in the observable Universe is not homogeneous but has
a structure that is dictated by matter distribution and gravitational forces.
The large-scale structure of the Universe defines the environment galaxies
reside in and has a wide range of effects on the properties of those galaxies;
for example, the orientation of galaxies in relation to the filaments (Lee &
Pen, 2000; Aragón-Calvo et al., 2007; Tempel & Libeskind, 2013; Ganeshaiah
Veena et al., 2019; Kraljic et al., 2020), the satellite distribution around
larger galaxies (Knebe et al., 2004; Zentner et al., 2005; Tempel et al.,
2015; Wang et al., 2020), the elliptical-to-spiral ratio, and the star
formation rate (Alpaslan et al., 2015; Kuutma et al., 2017).

Usually, the large-scale structure is divided into four types of substructures
(Libeskind et al., 2018). The densest and most compact are galaxy clusters
that host many gravitationally bound galaxies and are called knots in the
large-scale structure context. The clusters are connected by chains of
galaxies called filaments that populate the intricate cosmic web. Between
clusters and filaments are large under-dense volumes named voids encapsulated
in sheets of filaments called walls or sheets.

There are many different approaches to detecting the different large-scale
structure elements. Usually, the methods use either the relative positions of
the galaxies themselves or different scalar and tensor fields derived from
galaxy positions and properties from observational surveys or simulation data.
For example, the NEXUS+ model (Cautun et al., 2013) uses the Hessian of the
shear tensor field. Models that use galaxy positions also employ different
approaches. For example, DisPerSE (Sousbie, 2011) uses mass estimates and
identifies the cosmic web using topological features of the mass distribution,
and the Bisous model (Tempel et al., 2016) uses the distribution of the
galaxies and marked point process with interactions. Libeskind et al. (2018)
gives an overview and a brief comparison of 12 different methods to detect the
large-scale structure elements.

The accuracy of these finders depends on the completeness and accuracy of the
data. The best results are obtained from the simulations where galaxy
positions and properties are accurate and complete phase-space information is
available. When using data from surveys where the data are incomplete and have
uncertainties, and phase-space information is derived from those observations,
the resulting cosmic web maps deteriorate. Some methods are better suited for
observational data, but all methods are limited by the completeness and
accuracy of spectroscopic redshift data. The current largest spectroscopic
survey is the Sloan Digital Sky Survey (SDSS, Eisenstein et al. 2011; Alam et
al. 2015), which covers
$7221\text{\,}\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{{}^{2}}$ of the sky. There
are upcoming large spectroscopic surveys such as the 4-metre Multi-Object
Spectroscopic Telescope (4MOST, de Jong et al. 2019) surveys and the Dark
Energy Spectroscopic Instrument (DESI, Dey et al. 2019) Bright Galaxy Survey
(BGS, Ruiz-Macias et al. 2021). Future surveys will cover larger areas but
will still be limited by depth and completeness.

Data with photometric redshifts are much more abundant than the spectroscopic
counterpart, as redshifts can be measured in bulk. For example, SDSS Data
Release 12 has 100 times more photometric redshifts than spectroscopic ones
(Beck et al., 2016). The upcoming J-PAS (Benitez et al., 2014) will observe
the sky in 54 narrowband and three broadband filters and is designed to
measure the redshifts for a large number of galaxies with a precision of
$\sigma_{z}\lesssim 0.003(1+z)$. This precision is comparable to low-
resolution spectroscopic surveys and enables wider use of photometric redshift
data for applications that require positions of galaxies, such as large-scale
structure detection.

In this paper, we use the Bisous filaments finder, which is developed to
detect filaments from observational data. The Bisous model only needs the
galaxy distribution and uses geometric methods and the marked point process
with interactions to detect the cosmic web (Tempel et al., 2014, 2016). Bisous
has been successfully used in many works, such as Nevalainen et al. (2015),
Kuutma et al. (2017), Ganeshaiah Veena et al. (2019), and Tuominen et al.
(2021). Kruuse et al. (2019) show a significant positive correlation between
the distribution of photometric galaxies and the Bisous filaments, which
suggests that the Bisous model could be able to use photometric data to
improve the detection of filaments.

In this study, we present a proof of concept that data with photometric
redshifts can be used to improve the detection of the filamentary network. For
this, we take a simple approach to use data with significant uncertainties in
position along one axis with the Bisous model. We generate mock data with
photometric and spectroscopic redshifts from a simulation and use samples with
only photometric redshifts, mixed samples of photometric and spectroscopic
redshifts, and, for comparison and benchmarking, also samples with only
spectroscopic redshifts. Using Bisous results from the full spectroscopic
redshift data as a reference, we study the recall and false discovery rate of
the Bisous runs on different samples. Further aspects of interest are whether
or not using data with photometric redshift produces biases in the filaments
and the maximum size of uncertainties that Bisous can handle while still
improving the filamentary network.

The structure of the paper is as follows. In Sect. 2, we describe the
simulation we used to create the mock data and samples in this study. In Sect.
3, we describe the Bisous filament finder and our method to use data with
photometric redshifts. In Sect. 4, we present the results from different
samples. A discussion of the results, problems, possible improvements, and
future applications is presented in Sect. 5 and conclusions are outlined in
Sect. 6.

##  2 Data

###  2.1 Simulation data

The analysis in this paper is based on simulated mock data. For the mock data
set, we used the galaxy catalogue MultiDark-Galaxies which is based on the
MultiDark-Planck 2 (MDPL2111https://www.cosmosim.org/cms/simulations/mdpl2/,
Klypin et al. 2016) simulation with the Sag semi-analytic model for galaxies
described in Knebe et al. (2018). The MDPL2 simulation is based on a dark-
matter-only flat $\Lambda$ cold dark matter ($\Lambda$CDM) model with Planck
cosmological parameters: $\Omega_{\mathrm{m}}=0.307,\
\Omega_{\mathrm{B}}=0.048,\ \Omega_{\Lambda}=0.693,\ \sigma_{8}=0.823,\
n_{s}=0.96$, and $h=0.678$ (Planck Collaboration et al., 2016). The box size
is $1000\text{\,}\mathit{h}^{-1}\,\mathrm{M}\mathrm{p}\mathrm{c}$
($1475.6\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$) with $3840^{3}$ particles
with a mass resolution of
$m_{\mathrm{p}}=$1.51\text{\times}{10}^{9}\text{\,}\mathit{h}^{-1}\,\mathit{M_{\odot}}$$
per dark matter particle.

This work uses a smaller box of the whole simulation with a side of
$250\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ to have a sufficiently large
sample size for statistical analysis but a sufficiently small volume to limit
the calculation time of the Bisous filament finder (see Section 3) applied to
the data. We used a magnitude limit of -20.0 in the SDSS r-band to have galaxy
number density similar to observations (for comparison, see Muru & Tempel
2021). This cut leaves us with $181\,411$ galaxies in a box with a side of
$250\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$, and the galaxy number density is
$0.0116\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}^{-3}$.

###  2.2 Photometric redshift mock data

As the distance measures from spectroscopic surveys are relatively precise,
the spectroscopic redshift mock data are simply data with exact positions from
the simulation, but in order to generate photometric redshift mock data we
have to introduce photometric redshift uncertainties to them. The simulation
data positions form a cube for which we take two axes to represent the sky
plane, and the coordinates represent the sky coordinates and therefore have no
extra uncertainty, and one axis represents the line of sight. We added a
random error to the line of sight coordinate of each galaxy. For simplicity,
all the coordinates are given in megaparsecs
($\mathrm{M}\mathrm{p}\mathrm{c}$), and the errors do not scale with distance.

The random errors for the line-of-sight axis are generated with a Gaussian
distribution ($\mathcal{N}(x,\sigma^{2})$) with different standard deviation
($\sigma$) values for different samples. Within one sample, the standard
deviation value is constant. For this study, we used six different standard
deviation values of
$1\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}2\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}3\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}7\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$.
We also created mixed samples of galaxies with spectroscopic and photometric
distances in different proportions and with different photometric
uncertainties. This is to emulate a realistic situation where one would start
with an observational catalogue of spectroscopic targets and include
photometric targets to improve the detection of the large-scale structure.
Different mixed samples have
$10\text{\,}\%20\text{\,}\%30\text{\,}\%40\text{\,}\%50\text{\,}\%$ of the
brightest galaxies as spectroscopic galaxies; the rest are photometric
galaxies with uncertainties generated with $\sigma=$
$5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$.
This means that a chosen percentage of the brightest galaxies have exact
positions and other galaxies have photometric uncertainties in the line of
sight axis, such as distance.

Figure 1 shows the comparison between a sample of galaxies with no
uncertainties (all spectroscopic redshifts), and two samples of galaxies with
photometric redshifts with uncertainties with distributions
$\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$
and
$\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$.
The leftmost plot shows a visible web-like structure. In the middle plot, the
structure is more diffuse because of the added randomness along the z-axis,
but some of the original structure is still somewhat visible. In the rightmost
plot, the original structure is no longer visible, but rather seems to have
filamentary structures along the z-axis that have been produced by the added
random errors to galaxy positions along the z-axis.

Figure 1: Projection of galaxy distributions of samples $\sigma 0,\ \sigma 5,\
\sigma 10$ in a slice with a thickness of
$10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$. Each dot represents a galaxy. The
photometric uncertainties are parallel to the z-axis, which also defines the
line of sight in this work. Only an area of
$150\text{\times}150\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ is shown for
visual clarity. For information about samples; see Sect. 2.3.

###  2.3 Samples

We use the following notation to name the samples. The fraction of galaxies in
the samples with spectroscopic distance estimates is denoted with sXX, where
XX is a number indicating the percentage from the whole sample. The
spectroscopic galaxies are always the brightest galaxies in the sample. For
example, s40 means the sample contains 40% of the brightest galaxies from the
whole sample, all of these have exact distances, and is missing the other 60%
of the galaxies. The photometric samples are denoted with $\sigma$YY, where YY
is a number indicating the size of the uncertainties for photometric distance
estimates. For example, $\sigma$5 means the sample contains galaxies that have
uncertainties in the distance measures that are generated with Gaussian
distribution with a standard deviation of
$5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$. For mixed samples, $\sigma$10s30
for example means 30% of the brightest galaxies have exact distances (i.e.
spectroscopic distance estimates), and the rest, that is 70% of the galaxies
in the sample, have distances with uncertainties generated with Gaussian
distribution with a standard deviation of
$10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$. Table 1 lists samples used in
this work, the distributions used to generate uncertainties for distances, and
the percentages of galaxies with spectroscopic distances.

For brevity, hereafter the term spectroscopic galaxies/data is used as a
synonym for galaxies/data with spectroscopic redshifts, and photometric
galaxies/data is used as a synonym for galaxies/data with photometric
redshift. In this work, the former means data with no uncertainties, and the
latter means data with uncertainties along one axis.

Table 1: Photometric distance uncertainties and percentage of spectroscopic galaxies in each sample. The distance uncertainties column shows the Gaussian distribution used to generate uncertainties for distances of photometric galaxies. The last column shows the percentage of the brightest galaxies with spectroscopic distances, i.e. exact distances. Samples that do not have galaxies with photometric distances are indicated with an em dash (—) in the second column. Name | Distance uncertainties |  Spectroscopic
distances
---|---|---
$\sigma$0 | — | 100%
$\sigma$1 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($1\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 0%
$\sigma$2 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($2\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 0%
$\sigma$3 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($3\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 0%
$\sigma$5 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 0%
$\sigma$7 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($7\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 0%
$\sigma$10 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 0%
$\sigma$5s50 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 50%
$\sigma$5s40 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 40%
$\sigma$5s30 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 30%
$\sigma$5s20 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 20%
$\sigma$5s10 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 10%
$\sigma$10s50 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 50%
$\sigma$10s40 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 40%
$\sigma$10s30 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 30%
$\sigma$10s20 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 20%
$\sigma$10s10 | $\mathcal{N}($0\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,($10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$)^{2})$ | 10%
s50 | — | 50%
s40 | — | 40%
s30 | — | 30%

##  3 The Bisous filament finder

We used the Bisous filament finder to detect the filaments from the mock data.
This finder is a stochastic tool to identify the spines of the filaments using
the spatial distribution of galaxies or haloes (Tempel et al., 2014, 2016).
The Bisous has already been applied to a variety of data and has been proven
to give similar results to other filament finders (Libeskind et al., 2018). We
give a short overview of the method below.

First, the Bisous randomly populates the volume with points with parameters
(called marked points), where each point represents the centre of a cylinder
and the parameters give the size and orientation of the cylinder. The
cylinder’s width is about $1\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$, which
defines the width of the detected filaments. This width is derived from the
gradient of the galaxy density, where there is a peak at approximately
$0.5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ from the filament’s spine. Each
configuration of cylinders in the volume has a defined energy, which depends
on the position of the cylinders in relation to the underlying data of haloes
and the interconnectedness of the filamentary network made up of the
cylinders. Using the Metropolis-Hastings algorithm and the simulated annealing
procedure, the Bisous model optimises the energy function of the system by
suggesting random moves to add, remove, or change the cylinders.

The data of the cylinder configurations are collected over hundreds of
thousands of cycles, each consisting of tens of thousands of moves, which is
the basis for visit map calculations. In general, one realisation of cylinders
in the volume represents the detected filamentary network. As the model is
stochastic, the configuration of cylinders changes from realisation to
realisation. The combination of many realisations allows us to define the
visit map that describes the detected filamentary network. Each coordinate has
a defined visit map value, ranging from 0 to 1. The visit map contains
information on how often a coordinate in space was ‘visited’ by a cylinder,
which signifies how probable it is that a random realisation has a cylinder at
that position.

To decrease the effects of Poisson noise, the Bisous model is run many times,
usually 50-100. This increases the signal-to-noise ratio as a larger number of
independent realisations are combined to obtain the resulting maps.

Muru & Tempel 2021 show how the galaxy number density affects the detected
filamentary network. These authors show that the Bisous method underestimates
the extent of the filamentary network rather than giving false-positive
results. This means that the filament finder underestimates the filamentary
structures at higher distances where the galaxy number density drops. To
improve the quality of the detected filamentary network, we need to increase
the galaxy number density, for example, with photometric data.

### Using photometric data

Filament finders usually need precise data, either scalar or tensor fields or
galaxy positions, and therefore the less accurate photometric data are
ignored. Here we present a method that benefits from photometric data by
having higher input data density and is able to mitigate the uncertainty from
distance measures. This subsection gives an overview of a simple method of how
the Bisous filament finder can use photometric data.

For each galaxy with a photometric redshift estimate and its probability
distribution, we generate $N_{R}$ new distance estimates drawn from the
photometric redshift probability distribution. For the mock data in this
paper, we used a Gaussian distribution to generate the uncertainties, and so
we also use the same Gaussian distribution to generate different distance
estimates for every galaxy. Every Bisous run uses a different distance
estimate for a galaxy with a photometric distance measure. The number of
Bisous runs should be large in order to minimise the Poisson noise in the
results but also small to minimise the computational resources used for the
model. Usually, there are around $50100$ Bisous runs, for this work, we used
$N_{R}=80$, which has been shown to give good results in previous works using
the Bisous model. For mixed data sets of spectroscopic and photometric
targets, only the photometric ones have different distance estimates, whereas
spectroscopic targets have the same distance value in every run.

The novelty of this method is that the runs that have more accurate distance
estimates for the photometric galaxies produce more persistent filaments.
Galaxies with inaccurate distance estimates generate noise. The Bisous model
suppresses the noise by combining a large number of realisations. The more
inaccurate distance estimates there are, the more noise, which means the
Bisous is able to find fewer filaments. This means that uncertainties still
have to be small to produce good results. The generation of new distance
estimates is done separately for each galaxy. In practice, we can use a
different probability distribution for each galaxy.

##  4 Results

This work uses three types of samples: spectroscopic, photometric, and mixed.
The primary purpose of spectroscopic-only samples is to be a reference value
for the other two types of samples. Photometric-only samples show what can be
done using only photometric redshift surveys, and mixed samples show what we
can do by combining the spectroscopic and photometric redshift surveys, for
example in the areas of spectroscopic surveys where galaxies are sparse or at
higher distances where the detection is less complete.

Figure 2: Fraction of galaxies in filaments for different spectroscopic-only,
photometric-only, and mixed samples. The samples are ordered so that the
y-axis values of photometric-only and mixed samples are in increasing order.
The spectroscopic-only samples are used for reference values to show the
increase in the fraction of galaxies in filaments for mixed samples. The
sample s30 is the smallest spectroscopic sample in this study because smaller
samples had too few galaxies to be able to detect the filamentary network.

As mentioned in Section 1, filaments affect the evolution of galaxies and
knowing whether a galaxy is in a filament or not is useful when studying the
galaxy properties. Therefore, one of the simplest metrics with which to
compare the resulting filamentary network is the fraction of galaxies situated
inside filaments. Figure 2 shows the fraction of galaxies inside filaments for
all the samples used in this work. The sample $\sigma 0$ is the most complete
sample (galaxy positions without uncertainties but with the same magnitude
limit as other samples), and the fraction of galaxies in filaments for that
sample could be considered as a reference value for an ideal case. Looking at
samples with only photometric redshift galaxies, we see the expected trend
that the larger the uncertainties for the distance, the fewer galaxies are in
filaments. This comes from the fact that the larger the uncertainties, the
fewer filaments the Bisous model is able to detect (cf. Fig. 3) as the
structure in the galaxy distribution is less obvious, as seen from Figure 1.
Adding spectroscopic redshift galaxies to create the mixed samples
considerably increases the number of galaxies in filaments. For example, 33%
of the galaxies in $\sigma 5$ are in filaments, but when 20% of the brightest
galaxies have spectroscopic redshifts ($\sigma 5$s20) this fraction rises to
45%, and with 50% galaxies with spectroscopic redshifts ($\sigma 5$s50) up to
59% of galaxies are in filaments. This shows that using spectroscopic galaxies
together with photometric galaxies increasingly improves the detected
filamentary network as the number of spectroscopic galaxies in a sample
increases. On the other hand, when comparing spectroscopic-only samples (e.g.
s50 or s40) to mixed samples (e.g. $\sigma 5$s50 or $\sigma 5$s40) where the
galaxy number density is increased with added photometric galaxies, we can see
that the mixed samples have more galaxies in filaments when compared to
spectroscopic samples. This indicates that adding galaxies with photometric
redshifts to increase the number density of galaxies in the sample helps to
improve the detected filamentary network, as it increases the fraction of
galaxies in filaments and is closer to the reference sample ($\sigma 0$).

This metric can also be used to compare the results with observational data,
but different filament finders and different filament definitions give results
that are not directly comparable. For example, Tempel et al. (2014) found that
when using the Bisous model on SDSS data, the fraction of galaxies in
filaments is about 40%, but they use a stricter definition for whether a
galaxy is considered to be in a filament or not. Also, results based on
observational data are likely missing fainter galaxies that are present in
simulations, which affects the fraction of galaxies in filaments.

Figure 3: Projections of maximum visit map values in slices obtained from the
Bisous model using different samples. Only a smaller
$100\text{\times}100\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ area is shown for
visual clarity. The thickness of the slice is
$10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$. Usually, a visit map limit of
0.05 is used to classify whether or not a coordinate is inside a filament.
Therefore, everything besides the blue area is likely part of the filamentary
network. The vertical axis (z) is parallel to the axis of the photometric
uncertainties, i.e. it emulates the line of sight. The leftmost column shows
samples with only photometric galaxies, the middle column shows mixed samples
with medium uncertainties
($\sigma=$5\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$) for photometric
galaxies, and the rightmost column shows mixed samples with the larger
uncertainties ($\sigma=$10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$).
Different rows in the leftmost column have different photometric
uncertainties, and the middle and the rightmost column have different
fractions of the brightest galaxies as spectroscopic galaxies. See Table 1 and
Sect. 2.3 for the sample naming convention used here.

It is a good idea to look at the spatial distribution of filaments produced by
different samples to assess them visually. Figure 3 shows visit map slices
from 12 different samples. The colour indicates the likelihood of a coordinate
being inside a filament. The plot in the upper left corner is the sample we
use as ground truth, the full spectroscopic sample. The vertical axis is
parallel to the axis of photometric uncertainties and emulates the line of
sight. The photometric-only samples in the left column show that photometric
galaxies make it very difficult to detect filaments perpendicular to the line
of sight. Only stretched-out filaments parallel to the line of sight remain.
In the middle and rightmost columns, mixed samples are used. Including the
spectroscopic galaxies helps detect filaments perpendicular to the line of
sight. But even in the mixed samples, when photometric galaxies dominate, as
in the lower rows, the filaments are preferentially parallel to the line of
sight. This does not mean that filaments are parallel to the line of sight,
but that these are the filaments the Bisous model is able to detect with the
corresponding data.

Figure 4: Distributions of the cosine of the angle between filament spines
(fil) and the line of sight (los). For each sample, there are two plots. The
left one is a bar plot of the quartiles of the distribution, where the black
crossbar indicates the second quartile (the median). The right plot is a
violin plot that shows the density curve of the distribution. The horizontal
grey line indicates the median value for a uniform distribution. The closer
the distribution gets to value 1, the more filaments are parallel to the line
of sight (z-axis in other plots).

Figure 3 shows that photometric galaxies, which have large uncertainties along
the line of sight, suppress the detection of filaments perpendicular to the
line of sight. To study this effect, we describe the distribution of angles
between filament spines and the line of sight. These results are shown in
Figure 4. Again, the $\sigma 0$ sample is the baseline for this work and shows
a uniform distribution of angles. Using photometric-only samples skews the
distribution closer to 1, meaning the filaments are mostly parallel to the
line of sight, as is visible from the visit map projections in Figure 3.
Adding spectroscopic galaxies to the samples significantly reduces the bias of
high cosine values in the distributions. This is also visible in Figure 3,
where more filaments are perpendicular to the z-axis in mixed samples.

Figure 5: Recall and false discovery rates for photometric and mixed samples.
All the samples have the same total number of galaxies. The definitions for
recall and false discovery rate are given in Sect. 4. The false discovery rate
uses the secondary vertical axis on the right side of the plot. Including
spectroscopic galaxies improves recall but also increases false discovery
rates. The false discovery rates are below 5% for every sample.

When using the results obtained with a full spectroscopic sample $\sigma 0$ as
ground truth, we can compare other results to it and construct contingency
tables called confusion matrices. We assign a binary label for each coordinate
depending on the visit map value. If the visit map value is equal to or
greater than 0.05, then that coordinate is classified as inside a filament.
With each coordinate labelled, we can assign four kinds of results: true
positive, true negative, false positive, and false negative. To describe the
goodness of the results for sample $s$, we use two statistics: the recall

$\text{Recall}_{s}=\frac{\mathrm{TP}_{s}}{\mathrm{P}_{\sigma 0}}\ ,$ (1)

where $\mathrm{TP}_{s}$ is the number of true-positive values in the sample
$s$, and $\mathrm{P}_{\sigma 0}$ the number of positive values in the
reference sample $\sigma 0$;
and the false discovery rate,

$\text{False discovery rate}_{s}=\frac{\mathrm{FP}_{s}}{\mathrm{P}_{s}}\ ,$
(2)

where $\mathrm{FP}_{s}$ is the number of false-positive values in the sample
$s$, and $\mathrm{P}_{s}$ the number of all positive values in the sample $s$,
which includes both the true-positive and false-positive values. Recall shows
the fraction of filaments the model is able to find compared to the filaments
present in results obtained with the sample $\sigma 0$, which we want to
maximise. The false discovery rate describes the fraction of false filaments
in the results, which we want to minimise.

Figure 5 shows the recall and the false discovery rates for different samples.
As expected, the recall decreases monotonically when photometric uncertainties
increase. Using mixed samples improves the recall even when using small
fractions of spectroscopic galaxies. For example, this improvement can be seen
when comparing the recalls of $\sigma 5$ (0.45) and $\sigma 5$s10 (0.54) or
$\sigma 10$ (0.27) and $\sigma 10$s10 (0.40), both mixed samples use only 10%
of the spectroscopic galaxies. Using 50% of the spectroscopic galaxies boosts
the recall above 0.73, which means almost three-quarters of the original
filaments are detected. As seen in Figure 5, the false discovery rate is below
0.05 for every sample. This shows that the Bisous model produces only little
noise and false-positive values even with photometric redshift data.

In addition, we ran Bisous on mock data without using the method described in
Sect. 3 and using the samples $\sigma 5$, $\sigma$5s30, $\sigma 10$, and
$\sigma$10s30 as they are. This enables us to compare the Bisous model results
obtained with the method in Sect. 3 with results obtained with photometric
data without doing anything special to the photometric galaxies and ignoring
photometric redshift errors. Table 2 lists the different statistics introduced
in this section calculated for these Bisous runs. These results are calculated
as a reference and motivation for using the method described in Sect. 3. In
comparison to the samples introduced in Sect. 2.3 these results show
significantly worse recall values and fewer galaxies in filaments. In some
cases, the false discovery rate can have better results, but this comes from
the fact that when detecting fewer filaments, there are also fewer false-
positive results and therefore a lower false discovery rate.

Table 2: Comparison of Bisous model results with and without using the method
described in Sect. 3. PB in front of the sample name indicates that the
results are obtained with the plain Bisous model. Recall and false discovery
rate are defined by Equations 1 and 2.

Sample | gal in fila | Recall | FDRb
---|---|---|---
PB $\sigma 5$ | 0.217 | 0.296 | 0.037
$\sigma 5$ | 0.329 | 0.452 | 0.029
PB $\sigma 5$s30 | 0.398 | 0.541 | 0.039
$\sigma 5$s30 | 0.501 | 0.679 | 0.042
PB $\sigma 10$ | 0.127 | 0.172 | 0.039
$\sigma 10$ | 0.200 | 0.276 | 0.025
PB $\sigma 10$s30 | 0.348 | 0.473 | 0.039
$\sigma 10$s30 | 0.441 | 0.598 | 0.042
222a Fraction of galaxies in filaments; b false discovery rate

These results qualitatively confirm the results from Kruuse et al. (2019),
showing that galaxies with photometric redshifts are clustered around the
Bisous filaments. We show that the Bisous model can use photometric redshift
data to detect the filamentary network without producing significant amounts
of false-positive results. However, when the uncertainties in the distance
measure increase, the model is able to recall fewer filaments. For example,
with sample $\sigma 10$, the recall is only 0.27, and mostly filaments
parallel to the line of sight are detected. Including spectroscopic galaxies
in the samples considerably improves the recall and helps to mitigate the
issue with filament alignment in the detected filamentary network. Results in
this work also qualitatively follow the results of Muru & Tempel (2021), where
they show how the Bisous filaments depend on the number density of the
galaxies in the input data. In this work, the mixed samples show a similar
trend, and the photometric galaxies boost the number density of galaxies,
although less than the same number of spectroscopic galaxies would.

##  5 Discussion

Previous works applied the Bisous model to SDSS, which is a spectroscopic
survey, and compiled a catalogue of filaments (Tempel et al., 2014). This work
extends the applicability of the model and demonstrates the effects of using
data with photometric redshifts. The major benefit of using photometric
redshift data comes from its comparatively high availability, and measurements
are made in bulk, and not for single galaxies as in spectroscopic
measurements. The problem with photometric redshift data is the significantly
larger uncertainties when calculating redshifts, which results in larger
uncertainties in distance measurements. This is problematic for filament
finders.

To simulate the large uncertainties in distance measurements, we used
simulation data to create data with added uncertainties. For simplicity, all
the uncertainties are generated with the same Gaussian distribution for each
galaxy. In reality, the uncertainty depends on many properties, and one of the
most relevant is the magnitude of the galaxy. But the dependence of the
uncertainty on the magnitude is different for different surveys. Also, using a
simulation removes any redshift dependence in the data. In observations, there
are two major redshift-dependent effects. Firstly, the number density of
galaxies decreases with redshift as we are able to detect fewer galaxies on
the fainter end, and this affects the ability of the Bisous model to detect
filaments as shown in Muru & Tempel (2021). Secondly, the precision of the
photometric redshift values for galaxies depends on their actual distance.
These dependencies should be studied in greater depth when concentrating on
specific surveys and is outside the scope of this study.

The method we use to overcome this problem of large uncertainties is
straightforward. Essentially, we are just guessing the true positions. Each
galaxy gets 80 different random positions based on the uncertainties of the
redshift estimate. The theory behind this approach is that while random
inaccurate positions produce noise, the positions close to the true position
of the galaxy produce a strong enough signal to be above the noise level.
Regardless of its simplicity, the method shows considerable improvements over
results when not using this method (see Table 2).

Although this simple method improves the results, the problems introduced by
using the photometric redshift data are still prevalent. Using photometric-
only redshift data ($\sigma$XX samples) results in part of the signal being
lost and an incomplete filamentary network. This is visible from the recall
values when compared against $\sigma 0$ (Fig. 5), the fraction of galaxies in
filaments (Fig. 2), and the projections of visit map values (Fig. 3). Another
problem is that with larger uncertainties for distances, the filaments
perpendicular to the line of sight are almost impossible to detect. This
creates a strong bias for filaments parallel to the line of sight (cf Fig. 4).
It is important to note that the false discovery rate (cf Fig. 5) decreases
when data with larger uncertainties are used. This is because galaxies with
larger uncertainties produce less meaningful signals, and therefore there will
be fewer filaments in the results, which also means fewer false-positive
filaments. Low false discovery rate values are good because they demonstrate
the robustness of the results. The model rather outputs fewer filaments than
false-positive filaments.

All of the aforementioned problems are reduced by using mixed samples of
spectroscopic and photometric redshift data instead only photometric, as shown
in Section 4. Figure 2 also shows that using mixed samples to boost the galaxy
number density is better than only using the spectroscopic redshift galaxies.
This could be useful, for example, in the more distant areas of spectroscopic
surveys, where galaxies with spectroscopic redshifts are too sparse to use for
the detection of the large-scale structure. Using mixed data could help us
extend the area where we can reliably detect the filaments.

Still, this method requires photometric redshift data with relatively small
uncertainties, which are not usually achieved by photometric surveys.
Unfortunately, all current photometric surveys have unusably large
uncertainties for the redshifts, but there will be some new surveys with
suitable accuracy in the near future. One prominent candidate for photometric
redshift data is the upcoming Javalambre Physics of the Accelerating Universe
Astrophysical Survey (J-PAS, Benitez et al. 2014; Bonoli et al. 2020; Laur et
al. 2022). J-PAS is designed to measure the positions and redshifts of 14
million galaxies. And the estimated precision for the photometric redshift for
galaxies in the redshift range $0.1<z<1.2$ is $\sigma_{z}\lesssim 0.003(1+z)$.
For example, when using SDSS, the spectroscopic redshift galaxy number density
is high enough to detect some filaments up to a distance of
$400\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$, which is approximately $z=0.1$
(Tempel et al., 2014; Muru & Tempel, 2021). For this distance, the precision
of the redshifts is $\sigma_{z}\lesssim 0.003\times
1.1\approx$14\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$. This is the same order
of magnitude as the $\sigma 10$ samples used in this work. We expect the
uncertainties to be smaller for brighter galaxies. We aim to apply the Bisous
model to J-PAS data when it is released and compile a catalogue of filaments.
To obtain the mixed data of photometric and spectroscopic redshift galaxies,
we plan to use the Sloan Digital Sky Survey (SDSS) (Alam et al., 2015) and the
Dark Energy Spectroscopic Instrument (DESI) Bright Galaxy Survey (BGS) (Dey et
al., 2019; Ruiz-Macias et al., 2021).

Although this study is based on the Bisous filament finder, it is likely that
the general tendencies when using data with photometric redshifts are similar
with other filament finders. Using photometric data will decrease the
effectiveness of the filament finder, and filaments parallel to the line of
sight are more likely to be detected. It is uncertain whether using mixed data
of photometric and spectroscopic redshifts improves the results compared to
using only spectroscopic data when using other filament finders. Also, the
false discovery rates might have different values for other filament finders.
The advantage of the Bisous model is that it models the underlying filamentary
network, and galaxies are only used to constrain the model properties. Hence,
in the Bisous filament finder, it is straightforward to combine spectroscopic
and photometric samples. While fixing the scale of the filaments in the Bisous
model, we are free from smoothing the galaxy distribution, and the Bisous
model is able to detect filaments with a specified scale regardless of the
galaxy density.

As mentioned in Sect. 1, one common application for filaments is to study the
alignment of galaxies and their host filaments. This means that obtaining the
accurate filament orientation from the data is instrumental. In future
studies, we aim to improve the Bisous model to reduce the alignment bias of
filaments when using data with photometric redshifts.

##  6 Conclusions

Filament finders are limited, among other things, by the abundance of
spectroscopic redshift data. This limits the sky areas and depth where we can
detect the filamentary network. As photometric redshift data can be obtained
on shorter timescales, because you can observe many objects simultaneously,
there are many more photometric redshift data available. We present a method
that enables the Bisous filament finder to use data with considerable
uncertainties in one coordinate; for example photometric redshift data. We use
MultiDark-Galaxies, a dark matter-only simulation with semi-analytical
galaxies, to generate the data for analysis. Spectroscopic redshift data are
simply the exact positions of galaxies from the simulation, and photometric
redshift galaxies have added random error with Gaussian distribution in one
axis, where this latter represents the line of sight. This work uses three
types of samples. Firstly, spectroscopic samples with different magnitude cuts
for reference values for other samples. Secondly, photometric samples using
different standard deviations from $\sigma=$
$1\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$
to generate the errors with different sizes for distances. Thirdly, mixed
samples, where in different samples $10\text{\,}\%50\text{\,}\%$ of the
brightest galaxies have spectroscopic redshifts, that is, they have exact
distance measurements, and other galaxies have distances with uncertainties.
An overview of the samples used in this work is given in Sect. 2.3.

The Bisous model uses a marked point process to fit cylinder-like objects to
the underlying galaxy distribution and optimises the distribution of objects
based on the galaxy distribution and the interconnectedness of the cylinder
network. To use the photometric redshift data with uncertainties along one
axis, we modified the coordinates along that axis. Knowing the distribution of
the uncertainties for the distance of photometric redshift galaxies, we use
the same distribution to add a random value to the distance of a galaxy. For
each Bisous run, we generated a new galaxy distribution, where each
photometric redshift galaxy has a different random value added to its distance
based on the uncertainty distribution. Each Bisous model uses 80 runs. The
theory underpinning this approach is that those runs, where some galaxies have
random distance values that are closer to their true distances, produce strong
signals, and others with scrambled galaxy distributions just produce noise,
which will be removed in the post-processing.

Using photometric-only samples shows that when uncertainties are very small,
Gaussian distribution with $\sigma=$
$1\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ or
$2\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$, the Bisous model can find most of
the same filaments as in the full spectroscopic sample $\sigma 0$.
Unfortunately, these uncertainties are unachievable for modern or even future
planned photometric surveys. With larger uncertainties in the photometric-only
samples, the ability to recall the filaments in the reference sample drops
below 50%, and the filaments align with the line of sight. Using mixed samples
of photometric and spectroscopic data helps to reduce the mentioned problems.
For example, a comparison between three samples: a spectroscopic-only sample
s30, which uses only 30% of the brightest galaxies, a photometric-only sample
$\sigma$10, which uses data with errors generated with
$\sigma=$10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$, and a mixed sample
$\sigma$10s30, which uses the same standard deviation
($\sigma=$10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$) for errors and the same
amount of spectroscopic galaxies (30% of the whole sample). Using the
spectroscopic data, which contain only 30% of the brightest galaxies, results
in 36% of galaxies being inside filaments. Using only photometric data, which
contain all the galaxies, but have uncertainties in one coordinate, we find
that 20% of galaxies are inside filaments. And finally, using the mixed data,
which contain more data than the spectroscopic sample and, in contrast to the
photometric sample, also incorporate 30% of the spectroscopic data, the Bisous
model finds that 40% of the galaxies are inside filaments. The reference value
for these galaxies and the volume is from the full spectroscopic sample, which
gives a value of 71% of galaxies in filaments. Adding the spectroscopic
galaxies from the sample s30 to the photometric sample $\sigma$10 increases
the recall of filaments from 27% to 60%. This shows that using mixed data is
beneficial when spectroscopic data are too sparse and photometric data have
excessively large uncertainties to be used without spectroscopic data.

J-PAS is an upcoming photometric survey that is designed to produce data with
sufficiently small uncertainties to be applicable to a method such as the one
in this article. The expected precision of the redshifts is
$\sigma_{z}\lesssim 0.003(1+z)$ (Benitez et al., 2014). For a distance of
about $z=0.1$, this is $\sigma_{z}\lesssim 0.003\times
1.1\approx$14\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$, which is close to the
values used in this work. The next step is to apply the Bisous model to J-PAS
data once available.

###### Acknowledgements.

We thank the referee for their comments and suggested improvements. Part of
this work was supported by institutional research funding PRG1006 of the
Estonian Ministry of Education and Research. We acknowledge the support by the
Centre of Excellence ”Dark Side of the Universe” (TK133). Part of this work
was carried out in the High-Performance Computing Center of the University of
Tartu (University of Tartu, 2018). The CosmoSim database used in this paper is
a service by the Leibniz-Institute for Astrophysics Potsdam (AIP). The
MultiDark database was developed in cooperation with the Spanish MultiDark
Consolider Project CSD2009-00064. The authors gratefully acknowledge the Gauss
Centre for Supercomputing e.V. (www.gauss-centre.eu) and the Partnership for
Advanced Supercomputing in Europe (PRACE, www.prace-ri.eu) for funding the
MultiDark simulation project by providing computing time on the GCS
Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de). The
data exploration was done using TOPCAT (Taylor, 2005), and analysis and
plotting were done using Julia Language (Bezanson et al., 2017) and the
following packages: DrWatson.jl (Datseris et al., 2020), Pluto.jl (van der
Plas et al., 2022), Makie.jl (Danisch & Krumbiegel, 2021), DataFrames.jl
(White et al., 2020), Distributions.jl (Besançon et al., 2021),
ColorSchemes.jl, which uses Scientific colour maps (Crameri, 2021).

## References

  * Alam et al. (2015) Alam, S., Albareti, F. D., Allende Prieto, C., et al. 2015, ApJS, 219, 12
  * Alpaslan et al. (2015) Alpaslan, M., Driver, S., Robotham, A. S. G., et al. 2015, MNRAS, 451, 3249
  * Aragón-Calvo et al. (2007) Aragón-Calvo, M. A., van de Weygaert, R., Jones, B. J. T., & van der Hulst, J. M. 2007, ApJ, 655, L5
  * Beck et al. (2016) Beck, R., Dobos, L., Budavári, T., Szalay, A. S., & Csabai, I. 2016, MNRAS, 460, 1371
  * Benitez et al. (2014) Benitez, N., Dupke, R., Moles, M., et al. 2014, arXiv e-prints, arXiv:1403.5237
  * Besançon et al. (2021) Besançon, M., Papamarkou, T., Anthoff, D., et al. 2021, Journal of Statistical Software, 98, 1
  * Bezanson et al. (2017) Bezanson, J., Edelman, A., Karpinski, S., & Shah, V. B. 2017, SIAM Review, 59, 65
  * Bonoli et al. (2020) Bonoli, S., Marín-Franch, A., Varela, J., et al. 2020, arXiv e-prints, arXiv:2007.01910
  * Cautun et al. (2013) Cautun, M., van de Weygaert, R., & Jones, B. J. T. 2013, MNRAS, 429, 1286
  * Crameri (2021) Crameri, F. 2021, Scientific colour maps, The development of the Scientific colour maps is not funded any longer, but will continue as a pro bono project for the scientific community. - Fabio
  * Danisch & Krumbiegel (2021) Danisch, S. & Krumbiegel, J. 2021, Journal of Open Source Software, 6, 3349
  * Datseris et al. (2020) Datseris, G., Isensee, J., Pech, S., & Gál, T. 2020, Journal of Open Source Software, 5, 2673
  * de Jong et al. (2019) de Jong, R. S., Agertz, O., Berbel, A. A., et al. 2019, The Messenger, 175, 3
  * Dey et al. (2019) Dey, A., Schlegel, D. J., Lang, D., et al. 2019, AJ, 157, 168
  * Eisenstein et al. (2011) Eisenstein, D. J., Weinberg, D. H., Agol, E., et al. 2011, AJ, 142, 72
  * Ganeshaiah Veena et al. (2019) Ganeshaiah Veena, P., Cautun, M., Tempel, E., van de Weygaert, R., & Frenk, C. S. 2019, MNRAS, 487, 1607
  * Klypin et al. (2016) Klypin, A., Yepes, G., Gottlöber, S., Prada, F., & Heß, S. 2016, MNRAS, 457, 4340
  * Knebe et al. (2004) Knebe, A., Gill, S. P. D., Gibson, B. K., et al. 2004, ApJ, 603, 7
  * Knebe et al. (2018) Knebe, A., Stoppacher, D., Prada, F., et al. 2018, MNRAS, 474, 5206
  * Kraljic et al. (2020) Kraljic, K., Davé, R., & Pichon, C. 2020, MNRAS, 493, 362
  * Kruuse et al. (2019) Kruuse, M., Tempel, E., Kipper, R., & Stoica, R. S. 2019, A&A, 625, A130
  * Kuutma et al. (2017) Kuutma, T., Tamm, A., & Tempel, E. 2017, A&A, 600, L6
  * Laur et al. (2022) Laur, J., Tempel, E., Tamm, A., et al. 2022, A&A, 668, A8
  * Lee & Pen (2000) Lee, J. & Pen, U.-L. 2000, ApJ, 532, L5
  * Libeskind et al. (2018) Libeskind, N. I., van de Weygaert, R., Cautun, M., et al. 2018, MNRAS, 473, 1195
  * Muru & Tempel (2021) Muru, M. M. & Tempel, E. 2021, A&A, 649, A108
  * Nevalainen et al. (2015) Nevalainen, J., Tempel, E., Liivamägi, L. J., et al. 2015, A&A, 583, A142
  * Planck Collaboration et al. (2016) Planck Collaboration, Adam, R., Ade, P. A. R., et al. 2016, A&A, 594, A1
  * Ruiz-Macias et al. (2021) Ruiz-Macias, O., Zarrouk, P., Cole, S., et al. 2021, MNRAS, 502, 4328
  * Sousbie (2011) Sousbie, T. 2011, MNRAS, 414, 350
  * Taylor (2005) Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 29
  * Tempel et al. (2015) Tempel, E., Guo, Q., Kipper, R., & Libeskind, N. I. 2015, MNRAS, 450, 2727
  * Tempel & Libeskind (2013) Tempel, E. & Libeskind, N. I. 2013, ApJ, 775, L42
  * Tempel et al. (2016) Tempel, E., Stoica, R. S., Kipper, R., & Saar, E. 2016, Astronomy and Computing, 16, 17
  * Tempel et al. (2014) Tempel, E., Stoica, R. S., Martínez, V. J., et al. 2014, MNRAS, 438, 3465
  * Tuominen et al. (2021) Tuominen, T., Nevalainen, J., Tempel, E., et al. 2021, A&A, 646, A156
  * University of Tartu (2018) University of Tartu. 2018, UT Rocket
  * van der Plas et al. (2022) van der Plas, F., Dral, M., Berg, P., et al. 2022, fonsp/Pluto.jl: v0.19.11
  * Wang et al. (2020) Wang, P., Libeskind, N. I., Tempel, E., et al. 2020, ApJ, 900, 129
  * White et al. (2020) White, J. M., Kamiński, B., powerdistribution, et al. 2020, JuliaData/DataFrames.jl: v0.22.1
  * Zentner et al. (2005) Zentner, A. R., Kravtsov, A. V., Gnedin, O. Y., & Klypin, A. A. 2005, ApJ, 629, 219