11institutetext: Center for Theoretical Physics of the Universe, Institute for
Basic Science (IBS), Daejeon 34126, Korea
11email<EMAIL_ADDRESS>22institutetext: Instituto de Física, Universidade
de São Paulo, C.P. 66.318, CEP 05315-970, São Paulo, Brazil 33institutetext:
Instituto Nacional de Pesquisas Espaciais, Av. dos Astronautas 1758, Jardim da
Granja, São José dos Campos, SP, Brazil 44institutetext: Department of Physics
and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
55institutetext: Center for Gravitation and Cosmology, College of Physical
Science and Technology, Yangzhou University, Yangzhou 225009, China
66institutetext: School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China 77institutetext: IFSA Collaborative
Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, China
88institutetext: Unidade Acadêmica de Física, Universidade Federal de Campina
Grande, R. Aprígio Veloso, 58429-900 - Bodocongó, Campina Grande - PB, Brazil
99institutetext: Max-Planck-Institut für Astrophysik, Karl-Schwarzschild Str.
1, 85741 Garching, Germany 1010institutetext: Technische Universität München,
Physik-Department T70, James-Franck-Stra$\beta$e 1, 85748 Garching, Germany
1111institutetext: Instituto de Física, Universidade de Brasília, DF, Brazil
1212institutetext: School of Aeronautics and Astronautics, Shanghai Jiao Tong
University, Shanghai 200240, China 1313institutetext: Departamento de Física,
Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa,
Paraíba, Brazil

# The BINGO Project VI:

Hi Halo Occupation Distribution and Mock Building

Jiajun Zhang 11 Pablo Motta 22 Camila P. Novaes 33 Filipe B. Abdalla 44 Andre
A. Costa 55 Bin Wang 551212 Zhenghao Zhu 66 Chenxi Shan 66 Haiguang Xu 6677
Elcio Abdalla 22 Luciano Barosi 88 Francisco A. Brito 881313 Amilcar Queiroz
88 Thyrso Villela 331111 Carlos A. Wuensche 33 Elisa G. M. Ferreira 2299 Karin
S. F. Fornazier 22 Alessandro Marins 22 Larissa Santos 55 Marcelo Vargas dos
Santos 88 Ricardo G. Landim 1010 Vincenzo Liccardo 33

(Received xxxx, 2020; accepted xxxx, 2020)

###### Abstract

Context. BINGO (Baryon Acoustic Oscillations from Integrated Neutral Gas
Observations.) is a radio telescope designed to survey from 980 MHz to 1260
MHz, observe the neutral Hydrogen (Hi) 21-cm line and detect BAO (Baryon
Acoustic Oscillation) signal with Intensity Mapping technique. Here we present
our method to generate mock maps of the 21-cm Intensity Mapping signal
covering the BINGO frequency range and related test results.

Aims. We would like to employ N-body simulations to generate mock 21-cm
intensity map for BINGO and study what 21-cm intensity mapping observations
can tell us about structure formation,Hi distribution and Hi mass-halo mass
relation.

Methods. We have fitted an Hi mass-halo mass relation from ELUCID semi-
analytical galaxy catalog and applied to Horizen Run 4 halo catalog to
generate the 21-cm mock map, named as HOD. We have also applied the abundance
matching method, matched the Horizen Run 4 galaxy catalog with Hi mass
function measured from ALFALFA, to generate the 21-cm mock map, named as HAM

Results. We studied the angular power spectrum of the mock maps and the
corresponding pixel histogram. The comparison between two different mock map
generation method (HOD and HAM) is presented. We provided the fitting formula
of $\Omega_{\rm H{\sc i}\,}$, Hi bias and the lognormal fitting parameter of
the maps, which can be used to generate similar maps. We discussed the
possibility of measuring $\Omega_{\rm H{\sc i}\,}$ and Hi bias by comparing
the angular power spectrum of the mock maps and the theoretical calculation.
We also discussed the RSD (Redshift Space Distortion) effect, the nonlinear
effect and the bin size effect in the mock map.

Conclusions. By comparing the angular power spectrum measured from two
different kinds of mock maps and the theoretical calculation, we find that the
theoretical calculation can only fit to the mock result at large scale. At
small scales, neither the linear calculation nor the halofit nonlinear
calculation can provide accurate fitting, which reflects our poor
understanding of nonlinear distribution of Hi and its scale-dependent bias. We
have found that the bias is highly sensitive to the method of populating Hi in
halos, which also means we can put constraints on the Hi distribution in halos
by observing 21-cm intensity mapping. We have also illustrated that only with
thin frequency bins (such as 2 MHz), we can discriminate the effect of Finger-
of-God. All of our investigations using mocks provide useful guide for our
expectation of BINGO experiments and other 21-cm intensity mapping
experiments.

###### Key Words.:

cosmology – 21-cm – intensity mapping

##  1 Introduction

The spin-flip of the electrons in neutral hydrogen (Hi) emit or absorb photons
in the wavelength of 21-cm and therefore indicate their location in the
Universe. As a tracer of the underlying matter density field, 21-cm intensity
mapping can reveal the tomographic Baryon Acoustic Oscillation signal in a
cheap and fast fashion (Wyithe et al., 2008). Therefore, several different
experiments have been proposed to measure the 21-cm intensity map, such as
CHIME (Vanderlinde & Chime Collaboration, 2014), Tianlai (Chen, 2012), BINGO
(Wuensche & the BINGO Collaboration, 2019), FAST (Bigot-Sazy et al., 2016;
Smoot & Debono, 2017; Hu et al., 2019), SKA (Santos et al., 2015), etc. By
analyzing the large scale structure of the Universe using 21-cm intensity
mapping, we can have better understanding about dark matter and dark energy
(Kovetz et al., 2017), which are crucial to understand the Universe. More
detailed overview of BINGO project is described in the our companion papers I
and II (Abdalla et al., 2021a; Wuensche et al., 2021).

At low redshift Hi mostly contained inside galaxies and dark matter
halos(Villaescusa-Navarro et al., 2018). Since $z\sim 10$, the UV photons from
the first generation stars started reionizing the Universe . At $z<5$, the
Universe has been mostly ionized (Becker et al., 2001; Fan et al., 2006b, a).
The Hi can be ”protected” in high density regions(Prochaska et al., 2005;
Wolfe et al., 2005; Zafar et al., 2013), due to sufficient column depth for
self-shielding (Pritchard & Loeb, 2012). The state-of-art hydrodynamic
simulation IllustrisTNG (Villaescusa-Navarro et al., 2018) has shown that more
than $95\%$ Hi gas is inside dark matter halo virial radius.

Therefore, it is valid to assume that Hi gas traces the distribution of
galaxies and to use the distribution of galaxies and dark matter halos to
estimate the distribution of Hi gas with high resolution N-body simulations.
Zhang et al. (2020) has used the ELUCID simulation (Wang et al., 2016)
together with a semi-analytical model (Luo et al., 2016) to construct the Hi
mock. They have carefully studied the Redshift Space Distortion (RSD) effect
and shot noise for intensity mapping.

By observing the 21-cm emission line of Hi, we can determine the redshift of
the emission source. Then, we can estimate the distance of the sources by
applying Hubble’s law. However, they are not the real distances of the
sources. The redshift of the source is mainly contributed from the recessional
velocity and the peculiar velocity. The estimation about the position of the
source is distorted by the peculiar velocity. Such distortion is known as RSD
effect. Since the peculiar velocity of the Hi gas is mainly affected by
gravity from the clustering of matter, the RSD effect also contains rich
information about the large scale structure. The RSD effect has been used to
measure the growth factor of the Universe (Icaza-Lizaola et al., 2019), test
General Relativity (Jullo et al., 2019; Anagnostopoulos et al., 2019) and test
different cosmological models(Costa et al., 2017; An et al., 2019; Yang et
al., 2019; Cheng et al., 2019). The forecast of constraints from BINGO is
described in our companion paper VII (Costa et al., 2021).

The RSD effect can be classified into two parts, Kaiser effect (Kaiser, 1987)
and Finger-of-God (FoG) effect (Jackson, 1972). The moving of matter towards
high density regions under gravity leads to the Kaiser effect. The Kaiser
effect is dominant at large scales, which squeezes the distribution of
galaxies in the line-of-sight direction. On the other hand, the random motion
of matter inside high density regions introduces the FoG effect, which is
dominant at small scales. It elongates the distribution of galaxies, making
them look like fingers pointing at the observer. The RSD effect in 21-cm
intensity mapping is well studied in Sarkar & Bharadwaj (2019), and later
improved in Zhang et al. (2020).

There are various factors that contribute to the 21-cm intensity map, and the
leading terms are the distribution of Hi gas in real space and the RSD effect
(Hall et al., 2013). The contributions from gravitational redshift, the
integrated Sachs–Wolfe effect, gravitational lensing etc., can be neglected
under current consideration. However, more detailed study of these effects
might be necessary in the future experiments with higher sensitivity and
resolution. Using the information of positions and velocity of halos and
galaxies from simulations is sufficient to create a mock 21-cm intensity map
for BINGO.

In Sec. 2, we give the overview about the goals and methods of building mock
map. We introduce the Hi halo occupation distribution (HOD) model built from
ELUCID simulation in Sec. 3. To validate the HOD model and the halo mass
resolution, we test our method using ELUCID simulation in Sec. 4. In Sec. 5,
we introduce the method of building a full-sky 21-cm intensity mock map. The
measurements of the mock are presented in Sec. 6. Finally we summarize the key
points we have got in this study and discuss the future works in Sec. 7. This
is the sixth of a series of companion papers presenting the BINGO project. The
companion paper I is the project paper (Abdalla et al., 2021a), paper II
describes the instrument (Wuensche et al., 2021), paper III the optics
(Abdalla et al., 2021b), paper IV simulations (Liccardo et al., 2021), paper V
data analysis, correlations and component separation (Fornazier et al., 2021),
and paper VII forecasts (Costa et al., 2021).

##  2 Overview

It is very popular in cosmological surveys to use simulated mocks to mimic
real observations before the observations are taken. There are two major
purposes for building the mock:

  * 1

Provide a mock data challenge, test the entire data analysis pipeline and see
whether we can have the correct constraints of cosmological parameters.

  * 2

Construct the covariance matrix, which is essential in understanding the
source of error and uncertainty range of the parameter constraints.

For 21-cm intensity mapping, the data product of the mock is the brightness
temperature distribution in 2D spherical surface at different redshifts. Such
data product consists of the following four components:

  * 1

signal, the 21-cm emission from extra-galactic sources, which mainly comes
from Hi gas in the galaxies and dark matter halos.

  * 2

foreground, coming from radio wave emission in the Milky Way and the extra-
galactic sources, which mainly consists of synchrotron radiation and free-free
emission.

  * 3

noise, including thermal noise, shot noise and other sources of noise.

  * 4

mask, cover part of the sky for removing too strong foreground, usually around
very bright stars or directly toward the Milky Way, or to account for non
observed regions.

In this paper, we will only discuss about the 21-cm signal. The other three
components are discussed in our companion paper IV and V (Liccardo et al.,
2021; Fornazier et al., 2021). In order to generate the mock 21-cm intensity
map, the key part is determining the distribution of Hi. There are 6 ways to
generate the Hi gas distribution:

  * 1

Hydro: Directly read out the gas particle properties from hydrodynamic
simulations, such as IllustrisTNG (Villaescusa-Navarro et al., 2018).

  * 2

SAM: Use the Hi gas mass information from semi-analytical galaxy catalog based
on N-body simulations, such as ELUCID (Zhang et al., 2020).

  * 3

N-body: Use empirical Hi mass-halo mass relation to populate Hi gas in dark
matter halos, such as HIR4 (Asorey et al., 2020).

  * 4

Fast Halo: Again, relate Hi mass with halo mass using empirical relations, but
the dark matter halo catalog is generated by fast simulation method, such as
COLA-HALO (Koda et al., 2016).

  * 5

Model: Populate Hi mass on lognormal dark matter density distribution (Alonso
et al., 2014; Xavier et al., 2016) or even Gaussian realization(Asorey et al.,
2020).

  * 6

Poisson Halo: Use empirical Hi mass-halo mass relation to populate Hi gas in
dark matter halos, but the sub-resolution dark matter halos are generated by
Poisson sampling on dark matter density fields measured in N-body
simulations(Seehars et al., 2016).

Clearly, Hydro is the most expensive and accurate method. However, Hydro is
too expensive to construct a full sky mock. Even the state-of-art hydrodynamic
simulation cannot fulfill our requirement. Semi-analytical galaxy catalog is
built from merger-tree traced by N-body simulations, which requires subhalo
identification. SAM method is less accurate than hydrodynamic simulation, but
more efficient. Populating Hi mass in dark matter halos identified from N-body
simulation or fast simulations is even more efficient, but less accurate.
Using dark matter halos Poisson sampled from dark matter density field can
largely reduce the requirement of simulation resolution, which is very useful
for large scale application. The lognormal (Gaussian) density generation
method is the most efficient among these six methods, but it is the least
accurate one. We need to find the balance between accuracy and efficiency for
different purpose of mock building. We also need to notice that, in order to
construct a full-sky light cone from $z=0.12$ to $z=0.45$ without using
periodic boundary, the box size of such simulation will be larger than
$2.4\text{Gpc}/h$. Currently, we do not have such large scale hydrodynamic
simulation. Therefore, in this paper, we will introduce our empirical
relations from IllustrisTNG and ELUCID semi-analytical model, and apply that
onto Horizon Run 4 simulation (Kim et al., 2015) light cone to generate the
mock map. This method can also be applied on fast halo generation catalogs for
calculating covariance matrix in the future.

##  3 Hi Halo Occupation Distribution

###  3.1 ELUCID SAM catalog

Figure 1: We show the number of galaxies as a function of their host halo mass
and Hi mass in the ELUCID SAM catalog. The number in logarithmic scale is
represented by different colors. We show the result of central (satellite)
galaxies on the left (middle) panel. The mean value and standard deviation are
shown together with the color map to illustrate the trend more clearly. The
average number of satellite galaxies in each halo as a function of mass is
shown on the right panel.

We use the semi-analytical galaxy catalog from ELUCID N-body simulation (Wang
et al., 2016; Luo et al., 2016). The ELUCID simulation is an N-body simulation
with $3072^{3}$ particles in a periodic cubic box of $500h^{-1}$Mpc on a side.
WMAP5 cosmology (Dunkley et al., 2009) was assumed in the ELUCID simulation.
By following the the merger tree of dark matter halos in the simulations, Luo
et al. (2016) constructed the galaxy catalog using their semi-analytical
model, which considered the physical process like gas cooling, star formation,
supernova feedback, etc. In this work for studying the Hi gas halo occupation
distribution, we use the position, velocity, Hi mass and dark matter halo mass
of the galaxies from the galaxy catalog, the lower limit of the dark matter
halo mass is about $1.85\times 10^{9}M_{\odot}/h$. We assume that all the Hi
mass in the Universe is inside galaxies and their hosting halos, concentrated
in the center of the galaxies. As described in Villaescusa-Navarro et al.
(2018), from IllustrisTNG simulation (Naiman et al., 2018; Nelson et al.,
2018; Pillepich et al., 2018; Springel et al., 2018; Nelson et al., 2019)
results, it was shown that at $z<3.0$, more than $90\%$ of the Hi gas is
inside the galaxies and more than $95\%$ of the Hi mass is inside the halos.
Therefore, this assumption is reasonable.

In this galaxy catalog, we also have the information about whether the galaxy
is the central galaxy of the halo or a satellite galaxy. This allows us to
further separate the galaxies into central galaxies and satellite galaxies for
further discussion. Although IllustrisTNG is the state-of-art hydrodynamic
simulation, its box size is still quite small (at most $205\,\mathrm{Mpc}/h$
for IllustrisTNG-300 simulation and smaller for the other IllustrisTNG
simulations). Therefore, a larger box size simulation like ELUCID with high
particle resolution can provide better statistics about the Hi halo occupation
distribution in very massive clusters. The 21-cm intensity mapping using a
semi-analytical model galaxy catalog from MDPL2 simulation has been considered
(Cunnington et al., 2020). However the MDPL2 simulation can only resolve halos
with mass $>3\times 10^{10}M_{\odot}/h$ (Klypin et al., 2016), one order of
magnitude larger than the ELUCID simulation. Without sufficient resolution, it
will be hard to tell the Hi distribution in low mass halos like
$10^{9}M_{\odot}/h$. The ELUCID SAM (semi-analytical model) catalog is a good
choice for studying the Hi halo occupation distribution in both the high mass
end and low mass end. We will use this catalog to study the Hi distribution in
halos throughout this section.

###  3.2 Central and Satellite Galaxies

In general, gas cannot efficiently cool and form galaxies in both too massive
dark matter halos or too light dark matter halos. However, due to different
formation history, central galaxies and satellite galaxies can host very
different fraction of Hi gas. Padmanabhan & Refregier (2017) provided a halo
model and an Hi mass-halo mass relation, but the subhalos inside main halos
was not taken into account separately. An Hi mass-halo mass relation was
provided by Guo et al. (2017), which was fitted to $\sim 16000$ galaxies in
the range of $0.0025<z<0.05$ from ALFALFA. The importance of satellite
galaxies has been illustrated in an updated halo model by Paul et al. (2018).
By studying the different Hi gas distribution in satellite galaxies and
central galaxies using the SAM galaxy catalog at different redshift, we hope
to provide a different Hi mass-halo mass relation, which is more suitable for
our redshift range. We show the number of galaxies as a function of their Hi
mass and host halo mass in Fig. 1, measured from ELUCID SAM catalog at $z=0$.
We can see that for central galaxies, it is more abundant with less massive
host halo and less Hi mass. It is easy to understand because less massive
halos are more abundant, and Hi mass positively correlates to the halo mass if
the halo mass is not more massive than $\sim 10^{12}M_{\odot}/h$.

Figure 2: The Hi mass function measured from the ELUCID SAM catalog at $z=0$
is shown. The result of central (satellte) galaxy is shown in blue (orange),
and the total result is shown in green. The red curve shows the Schechter
function provided by Martin et al. (2010), which represents the fitting
function of ALFALFA observation. Overall, the Hi mass function between the SAM
and observation is consistent.

Together with the number of galaxies represented by different color, we have
also shown the mean value and the standard deviation in Fig.1. We can see the
mean Hi mass of the central galaxies clearly has a peak around host halo mass
$M_{\rm halo}=10^{11.6}M_{\odot}/h$. Below that, Hi mass positively correlates
to the halo mass and beyond that peak, Hi mass drops with larger halo mass
until $M_{\rm halo}\sim 10^{13}M_{\odot}/h$. The mean Hi mass of satellite
galaxies depends weakly on the halo mass. With larger halo mass, the number of
satellite galaxies rises up quickly, and the standard deviation of their Hi
mass also increase. Therefore, the total Hi mass in a dark matter halo
consists of the contribution of the central galaxies and satellite galaxies
differently for different halo masses.

The Hi mass function measured from the ELUCID SAM catalog at $z=0$ is shown in
Fig.2. We have shown the mass function of satellite galaxies and central
galaxies separately, as well as the total Hi mass function. The Schechter
function given in Martin et al. (2010) is shown in red curve for comparison.
The SAM Hi mass function can well fit the Schechter function at
$M_{HI}>10^{8}h^{-1}M_{\odot}$ but deviate from it at low mass end. Overall,
the SAM catalog still provides a reasonably similar mass function to that
fitted from ALFALFA observation.

###  3.3 Hi Mass Halo Mass relation

Figure 3: In the upper(lower) panel, we show the Hi mass-halo mass relation at
$z=0(0.66)$. The total Hi mass is shown in blue curve, the contribution from
central galaxies is shown in orange and the contribution from satellite
galaxies is shown in green. No matter z = 0 or 0.66, the central galaxy always
contributes the most Hi mass in low mass halos and satellite galaxies
contribute the most Hi mass in high mass halos. We have also shown the
observed Hi mass-halo mass relation in red dots at $z=0$ (Guo et al., 2020)
for comparison.

For generating mock 21-cm map, the data product from N-body simulations or
fast simulations is usually dark matter halo catalog. The most popular and
simplest information in the halo catalog are the position, the velocity and
the mass of the halo. Therefore, it is most useful if we can find an Hi mass-
halo mass relation, providing Hi mass as a function of halo mass. We can see
in Fig. 3 that, for low mass halos, the major contribution of Hi mass is from
the central galaxies, and for high mass halos, the major contribution comes
from satellite galaxies. Although in massive halos, there are more Hi mass in
central galaxies than in every individual satellite galaxies on average, the
number of satellite galaxies is so huge that they still contribute the most Hi
mass in massive halos. It is very important to consider the satellite galaxy
contribution of Hi mass in massive halos, which is ignored in HIR4 mock
(Asorey et al., 2020). We have also compared the Hi mass-halo mass relation to
the observational results (Guo et al., 2020) at $z=0$, the results from ELUCID
galaxy catalog fits relatively well to the observation.

Figure 4: We fit the measured Hi mass-halo mass relation from ELUCID SAM
catalog with the analytical formula in solid lines. We have chosen the best
fit curve in the mass range $10^{10}M_{\odot}/h<M_{halo}<10^{15}M_{\odot}/h$.

We first fitted the measured Hi mass-halo mass relation from ELUCID SAM
catalog using a linear function plus a Gaussian function with free parameter
$a$, $b$, $c$, $d$ and $f$, then we get the parameters as functions of
redshift $z$, by assuming that the parameters are linearly related with
redshift in $0<z<0.66$. Then we find the following analytical formula, which
can be easily applied to halo catalogs with the information of halo mass,

$\displaystyle\log_{10}(m_{\rm H{\sc
i}\,})=a\log_{10}(m)+b+ce^{-(\log_{10}(m)-d)^{2}/f^{2}}\,,$ (1) $\displaystyle
a=0.78-0.030z\,,$ $\displaystyle b=-0.23+0.68z\,,$ $\displaystyle
c=0.92-0.32z\,,$ $\displaystyle d=11+0.038z\,,$ $\displaystyle
f=0.79+0.18z\,,$

where $z$ is the redshift, $m_{\rm H{\sc i}\,}$ is the Hi mass in unit of
$M_{\odot}/h$, $m$ is the halo mass in unit of $M_{\odot}/h$. The halo mass
definition is based on FoF (Friend-of-Friend) halo finding algorithm, the
linking length is $0.2$ times the mean particle separation distance. The
fitting result is shown in Fig. 4 Notice that the Hi mass-halo mass relation
is very similar between $z=0$ and $z=0.66$, therefore, the assumption that the
parameters are linearly related with redshift in $0<z<0.66$ is reasonable.

##  4 Tests with ELUCID

###  4.1 The Halo Mass Threshold

Figure 5: On the left, we show the Hi power spectrum and on the right, we show
the ratio of power spectrum between the sample with halo mass threshold and
the full sample. The solid curves show the results in real space and the
dashed curves show the results in redshift space. Different colors denote
different samples with different threshold. We have also included the linear
matter power spectrum for reference in cyan color on the left.

When building the mock Hi distribution, the mass resolution and box size of
the halo catalog we use is usually limited. We need to know what is the effect
of limited mass resolution on the Hi power spectrum. Such effect can be
quantified by the bias of Hi gas in the mass limited sample. The Hi gas traces
the matter distribution in a biased way, we usually use the bias parameter
$b_{\rm H{\sc i}\,}$ to describe the difference between Hi distribution and
the matter distribution. Similarly, the dark matter halos also trace matter
distribution in a biased way, and the halo bias is $b$. It has been addressed
in HIR4 (Asorey et al., 2020) that the bias of Hi power spectrum can be
calculated as the weighted average of halo bias,

$b_{\rm H{\sc i}\,}(z)=\dfrac{\int dm\,n(m,z)m_{\rm H{\sc
i}\,}(m,z)b(m,z)}{\int dm\,n(m,z)m_{\rm H{\sc i}\,}(m,z)}\,,$ (2)

where $b(m,z)$ represents the halo bias, $m_{\rm H{\sc i}\,}$ is the Hi mass
and $m$ is the halo mass. The halo bias can be calculated by empirical formula
calibrated by N-body simulations (Tinker et al., 2010). Notice that the halo
bias used here is scale independent, which means the resulted Hi bias model is
also scale independent. In other words, if the Hi bias is actually scale
dependent, our bias model cannot provide a proper description.

With the high resolution ELUCID simulation, we can shed light on whether a
scale independent bias model is a proper description or not. We selected the
galaxies according to their host halo mass with the mass threshold $m>5\times
10^{9}M_{\odot}/h,1\times 10^{10}M_{\odot}/h,1\times
10^{11}M_{\odot}/h,1\times 10^{12}M_{\odot}/h$. In Fig. 5, we show the Hi
power spectrum of the selected samples and the full sample on the left panel,
while on the right panel we show the ratio of Hi power spectrum of the
selected samples and the full sample. The linear matter power spectrum
calculated using Eisenstein & Hu (1998) is shown for reference in cyan color
on the left panel. Considering the RSD effect, the positions of the galaxies
($\@vec{x}$) are mapped to redshift space ($\@vec{s}$) using the plane-
parallel approximation

$\@vec{s}=\@vec{x}+\dfrac{1+z}{H(z)}\@vec{v}_{\rm los}\,,$ (3)

where $\@vec{v}_{los}$ is the peculiar velocity of the galaxies along the
line-of-sight. The redshift space results are shown in dashed lines in Fig. 5.
Notice that we have deducted the shot noise power for the illustration and
calculation of the power spectrum ratio. Only when the ratio of power spectrum
is close to a constant, we can safely use the scale independent halo bias to
calculate the Hi bias. Considering the scale of 1 degree in the sky, which
corresponds to $k\sim 0.5h\text{Mpc}^{-1}$ at $z\sim 0.12$, the power spectrum
ratio of $m>10^{11}M_{\odot}/h$ from $k\sim 0$ to $k\sim 0.5h\text{Mpc}^{-1}$
is close to a constant, but in the $m>10^{12}M_{\odot}/h$ case, the ratio
cannot be well represented by a constant.

Therefore, we conclude that the halo mass resolution for making a mock 21-cm
intensity map requires at least $\sim 10^{11}M_{\odot}/h$. The finer the mass
resolution, the better.

###  4.2 Repopulate Hi Mass

Figure 6: On the left, we show the Hi power spectrum, the solid lines
represent the power spectrum measured from the SAM galaxy catalog, the dashed
lines represent the power spectrum measured from the HOD model. ”real” means
real space point mass power spectrum, ”red” means redshift space point mass
power spectrum and ”red2” means redshift space inside-galaxy Hi distribution
modelled power spectrum (Zhang et al., 2020). On the right, we show the power
spectrum ratio between the results of HOD model and SAM galaxy catalog.

We use the Hi mass-halo mass relation introduced in Sec. 3.3 to repopulate the
main halos in ELUCID simulation at $z=0$ with Hi gas, in order to test the
performance of our Hi HOD model. We have generated three kinds of Hi maps and
calculated their power spectrum,

  * 1

Use the galaxy (halo) positions in real space, treat the galaxies (halos) as
point mass and calculate the Hi distribution in the SAM (HOD) catalog,

  * 2

Use the galaxy (halo) positions in redshift space, treat the galaxies (halos)
as point mass and calculate the Hi distribution in the SAM (HOD) catalog,

  * 3

Use the galaxy (halo) positions in redshift space, treat the galaxies (halos)
as extended mass due to Hi velocity dispersion inside galaxies (halos) and
calculate the Hi distribution in the SAM (HOD) catalog.

We modelled the Hi velocity dispersion inside galaxies at $z=0$ as,

$\sigma_{v}=\dfrac{31\text{km/s}}{\sqrt{3}}\left(\dfrac{m}{10^{10}M_{\odot}/h}\right)^{0.35},$
(4)

which is also used in Zhang et al. (2020), derived from IllustrisTNG
simulation (Villaescusa-Navarro et al., 2018). Usually, we do not consider the
Hi velocity dispersion inside galaxies (halos) when making the mock 21-cm
intensity map using N-body simulations. Zhang et al. (2020) has pointed out
that with the proper consideration of such Hi velocity dispersion, we can have
a more realistic description of Hi mass distribution in redshift space.
Therefore, we have made these three kinds of Hi maps for testing. The results
are shown in Fig. 6. The power spectrum is shown on the left panel and the
ratio between HOD model and SAM catalog is shown on the right panel. On the
left panel, the dashed lines represent the Hi power spectrum measured from HOD
model and the solid lines show the results of the original SAM catalog. We can
see that overall, the HOD model can recover the Hi power spectrum in the SAM
catalog within $10\%$ ($20\%$) accuracy up to $k=0.5h\text{Mpc}^{-1}$
($k=0.8h\text{Mpc}^{-1}$). Therefore, we have validated the method of the HOD
model to populate dark matter halos with Hi mass.

##  5 Mock Building with HR4

###  5.1 Horizon Run 4 Catalog

Horizon Run 4 (HR4 in short) is a cosmological N-body simulation with cubic
box size of $3150\text{Mpc}/h$ and $6300^{3}$ number of particles (Kim et al.,
2015). HR4 simulation adopted WMAP5 $\Lambda$CDM cosmology, where
$\Omega_{m}=0.26,\Omega_{\Lambda}=0.74,h=0.72,\sigma_{8}=0.79,n_{s}=0.96$
(Dunkley et al., 2009). The halos are identified by Friend-of-Friend (FoF)
algorithm with linking length equal to 0.2 times the particle mean distance in
HR4 simulation, at least 30 particles are required for identifying a halo,
which means the minimum halo mass is $2.7\times 10^{11}M_{\odot}/h$.

We use the halo light cone catalog and the mock galaxy light cone catalog to
generate our mock 21-cm intensity map. Usually, in order to generate a full
sky light cone catalog, we will use the periodic boundary conditions of the
simulations to pile up simulation boxes together to get a deep light cone.
This technique will introduce some duplicate structures. However, the HR4
simulation box size is so large that there is no periodic structures in
$z<0.6$ in the full sky light cone catalog.

The redshift range of BINGO is $0.127<z<0.449$, therefore, there is no
duplicate structures in our mock map. In summary, there are four steps in
making the mock map,

  * 1

Populate Hi gas in the galaxy (halo) catalog and calculate the Hi density
distribution $\rho_{\rm H{\sc i}\,}$

  * 2

Calculate the brightness temperature

$T_{b}=189h\dfrac{H_{0}(1+z)^{2}}{H(z)}\dfrac{\rho_{\rm H{\sc
i}\,}}{\rho_{c}}\text{mK}\,,$ (5)

where $\rho_{c}$ is the critical density of the Universe (Villaescusa-Navarro
et al., 2018),

  * 3

Re-scale the map according to required $\Omega_{\rm H{\sc i}\,}$,

  * 4

Smooth the brightness temperature map with a Gaussian kernel FWHM = 40 arcmin,
which is due to the beam of BINGO.

###  5.2 Abundance Matching

Figure 7: Mollview of brightness temperature map using abundance matching
method to populate Hi gas in galaxies in real space from 990 MHz to 1000 MHz.

Halo Abundance Matching (HAM) and Sub-Halo Abundance Matching (SHAM) have been
widely used to link the dark matter halos to galaxies in observations. Under
the assumption that more luminous galaxies (with more Hi mass) are populated
inside more massive halos (sub-halo), we can populate the halos in the
simulations with galaxies, which is adopted in HR4 galaxy catalog generation
(Hong et al., 2016). In HR4 galaxy catalog, the position, velocity and an
”artificial mass-like quantity”, which is also used to estimate the galaxy
luminosity with abundance matching, are provided. We use the Schechter form Hi
mass function measured from the ALFALFA catalog (Martin et al., 2010),

$\displaystyle\phi(m_{\rm H{\sc i}\,})d\log_{10}\left(m_{\rm H{\sc
i}\,}\right)=\ln(10)\phi_{*}\left(\dfrac{m_{\rm H{\sc
i}\,}}{m^{*}}\right)^{\alpha+1}\exp{\left(-\dfrac{m_{\rm H{\sc
i}\,}}{m_{*}}\right)}d\log_{10}(m_{\rm H{\sc i}\,})\,,$ (6)
$\displaystyle\phi_{*}=0.0048$ $\displaystyle\alpha=-1.33$ $\displaystyle
m_{*}=10^{9.96}\left(\dfrac{0.7}{h}\right)^{2}\,,$

where $m_{\rm H{\sc i}\,}$ is in units of $M_{\odot}$. We also calculated the
abundance of galaxy ”mass” in the redshift range $0.127<z<0.449$ in the HR4
galaxy light cone catalog. By matching the abundance of galaxy ”mass” and the
Hi mass function, we get a Hi mass-galaxy ”mass” relation. Therefore, we can
populate Hi mass into the galaxy catalog. Notice that the measurement of Hi
mass function from ALFALFA is at $z=0$, which is not compatible with the light
cone catalog. It is just an approximated way of populating Hi mass into
galaxies. Then we sliced the catalog into redshift bins of 10 MHz each. We
calculate the density of Hi in each redshift using HEALPix pixelization
(Gorski et al., 2005) with $N_{\rm side}=256$. Finally, using Eq. (5), we
calculate the brightness temperature distribution. This set of mock map is
labelled as HAM mock map in the following sections.

In Fig. 7, we show the mollview of brightness temperature map at 990 MHz -
1000 MHz bin, in units of mK, as an example. The map represents the Hi
distribution in real space.

###  5.3 HOD Populate

Figure 8: Mollview of brightness temperature map using HOD method to populate
Hi gas in halos in real space from 990 MHz to 1000 MHz.

Using abundance matching method has the following shortage,

  * 1

Use the Hi mass function at $z=0$ to match all galaxies at different redshift
is not fair, the evolution with redshift is ignored and the Hi mass is
mismatched,

  * 2

The assumption that galaxy with more ”mass” has more Hi mass may not hold,

  * 3

It is not convenient to calculate the bias from theory to compare with the
mock map,

  * 4

If there is only halo catalog, this abundance matching method do not apply,
while many fast simulations only provide halo catalog.

Therefore, we need to use the HOD approach to obtain the mock map.

Using the FoF halo light cone catalog from HR4 simulation, we can use the Hi
mass-halo mass relation described by Eq. (1) to populate Hi mass into the halo
catalog. Then we follow the same process described in the last subsection,
separate the catalog into 30 bins and pixelize the full sky map using HEALPix
with $N_{\rm side}=256$. We get the brightness temperature map and show it in
Fig. 8. This mollview map is from 990 MHz to 1000 MHz, representing the real
space distribution of Hi gas, the same as Fig. 7. Comparing these two maps, we
can see that different ways of populating Hi gas into galaxies (halos) leads
to different results. Though the map looks very similar at large scale, the
brightness temperature fluctuation in the HAM map is much higher than that in
the HOD map. This illustrates different bias of Hi distribution. With
abundance matching method, Hi gas is more clustered than that in the HOD
method. More detailed discussion will be provided in Sec. 6. With intensity
mapping, we can tell different ways of Hi clustering and occupation in halos.

###  5.4 Redshift Space Distortion

Figure 9: Mollview of brightness temperature map using HOD method to populate
Hi gas in halos in redshift space from 990 MHz to 1000 MHz. We have only
considered the velocity of halos as point source in the upper panel, which
mainly only contains Kaiser effect. In the lower panel, we have considered the
velocity dispersion of Hi gas in sources, which means the FoG effect is taken
into account.

The intensity map we observe is binned according to the frequency, which is
determined by the redshift of the 21-cm emission source. Thus, we are
observing in redshift space rather than real space. We need to consider not
only the redshift of emission sources due to the Hubble flow, but also
consider the redshift contribution from the peculiar velocity of the sources.
The positions of the 21-cm emission sources are different in redshift space
from that in real space. This is known as RSD effect, which consists of Kaiser
effect in large scale (Kaiser, 1987) and Finger-of-God (FoG) effect in small
scale (Jackson, 1972). We considered two possible sources of the RSD effect in
the mock map,

  * 1

The peculiar velocity of galaxies (halos) as a point source, which mainly
contributes to Kaiser effect,

  * 2

The velocity dispersion inside the halos, which mainly contributes to FoG
effect.

In Fig. 9, we show the mollview mock map of the brightness temperature
adopting the point source assumption in the upper panel, while the result of
additionally considering velocity dispersion is shown in the lower panel.
Under the point source assumption it is easy to adopt the RSD effect, since
the redshift of the source can be calculated by,

$z^{\prime}=(1+z)(1+v_{r}/c)-1\,,$ (7)

where $z$ is the redshift of the point source calculated according to its
distance to the observer and hubble flow, which is provided in the light cone
catalog, $v_{r}$ is the peculiar velocity along the line-of-sight direction
and $c$ is the speed of light. Then we can bin the sources according to their
redshift $z^{\prime}$, calculate the Hi density with $N_{\rm side}=256$
HEALPix pixelization and calculate the corresponding brightness temperature.
Comparing to the real space mock map, after considering the RSD effect, some
of the sources are moved away from its real space frequency range, and some of
them are moved into the frequency bin. Thus, we can see slightly different
large scale structures between the real space mock map and the redshift space
mock map.

The mock map considering velocity dispersion of Hi gas in halos is built
following the steps,

  * 1

Calculate the velocity dispersion of Hi gas in each halo according to the halo
mass $m$ and real redshift $z$,

$\sigma_{v}(z)=(4.9z+30)\left(\dfrac{m}{10^{10}M_{\odot}/h}\right)^{(0.0091z+0.36)},$
(8)

which is fitted from IllustrisTNG simulation (Villaescusa-Navarro et al.,
2018) and adopted in Zhang et al. (2020) (notice that this is a fitting
formula using multiple data points at different redshifts, the calculated
result at $z=0$ is not exactly the same as Eq. 4),

  * 2

Calculate the corresponding comoving distance variance
$\delta_{s}=\dfrac{1+z}{H(z)}\dfrac{\sigma_{v}(z)}{\sqrt{3}}$ ,

  * 3

For a given redshift bin from $z_{1}$ to $z_{2}$, where $z_{1}<z_{2}$,
calculate the corresponding comoving distance range from $s_{1}$ to $s_{2}$,
select the halos within the range $s_{1}-3\delta_{s}$ and $s_{2}+3\delta_{s}$
in redshift space,

  * 4

Calculate the Hi mass contribution $w$ from the selected halos in this given
bin, for a given halo at $s_{h}$,

$\displaystyle t_{1}=(s_{h}-s_{1})/\delta_{s}\,,$ (9) $\displaystyle
t_{2}=(s_{h}-s_{2})/\delta_{s}\,,$ $\displaystyle w=\dfrac{m_{\rm H{\sc
i}\,}}{\sqrt{2\pi}}\int_{t_{1}}^{t_{2}}e^{-t^{2}/2}dt\,,$

  * 5

Calculate the Hi density in this bin and the related brightness temperature.

By doing so, we have assumed that the mass distribution of Hi in one halo in
redshift space follows Gaussian distribution. This process can provide a rough
estimation of FoG effect in the mock map.

###  5.5 Rescale the Map

The average brightness temperature $\bar{T}_{b}$ can be calculated from the
mock maps at different redshifts, it also represents the Hi abundance
$\Omega_{\rm H{\sc i}\,}$ since the mean temperature is proportional to it.
The relation between $\bar{T}_{b}$ and $\Omega_{\rm H{\sc i}\,}$ is

$\bar{T}_{b}=189h\dfrac{H_{0}(1+z)^{2}}{H(z)}\Omega_{\rm H{\sc
i}\,}\text{mK}\,.$ (10)

In the mock map, since we have neglected the contribution from halos whose
mass is smaller than the resolution, we may have missed some of the Hi gas in
small halos. On the other hand, we may have also overestimated the Hi mass in
halos comparing to the real observation by various reasons, such as
overestimating the Hi mass-halo mass relation. Therefore we need to rescale
the maps to meet the requirement of $\Omega_{\rm H{\sc i}\,}$. We can rescale
the maps according to the required $\Omega_{\rm H{\sc i}\,}$ value, by
applying $T^{\prime}_{b}=T_{b}\dfrac{\Omega^{\prime}_{\rm H{\sc
i}\,}}{\Omega_{\rm H{\sc i}\,}}$.

##  6 Results of the Mock

###  6.1 Angular Power Spectra

We would like to address three questions in this subsection,

  * 1

What is the effect of different ways of linking Hi gas with galaxies (halos)?

  * 2

What is the effect of RSD in angular power spectrum?

  * 3

Can we identify the above difference with BINGO telescope?

In Fig. 13, we have shown the zoom-in view of a $10\times 10$ degree2
brightness temperature map of the full sky mock map from 990 MHz to 1000 MHz.
We have compared the map generated using HOD method (labeled as HOD),
abundance matching method (labeled as HAM) and re-scale the HAM map to the
same $\Omega_{\rm H{\sc i}\,}$ as the HOD map (labeled as Re-HAM). We have
also shown the map smoothed by a Gaussian kernel with FWHM = 40 arcmin.The
same color map has been given in each row for a fair comparison. The $T_{b}$
contrast in HAM map is clearly much higher than that in HOD map, which can be
more easily identified in the Re-HAM map. In the HAM map, there are more Hi
gas inside clusters than in the HOD map. Since the bias of more massive halos
is higher than less massive halos, we will expect that with the same
$\Omega_{\rm H{\sc i}\,}$, the angular power spectrum of the mock map
generated by abundance matching method will be higher than the map generated
using HOD.

Figure 10: The comparison of angular power spectrum among HOD mock, HAM mock
and Re-HAM mock in real space. With the same $\Omega_{\rm H{\sc i}\,}$, Re-HAM
mock has higher angular power spectrum than the HOD mock, which means it has a
higher bias. Though very close at large scale, HAM mock has higher angular
power spectrum than the HOD mock at small scale due to the existence of
substructures in HAM mock.

In Fig. 10, our expectation is verified. Re-HAM and HOD share the same
$\Omega_{\rm H{\sc i}\,}$, however the angular power spectrum of the Re-HAM
mock map is much higher than the HOD mock map. The HAM mock shows very close
angular power spectrum to the HOD mock by coincidence. We can also notice that
at large $l$ ($l>200$), the real space angular power spectrum of the HAM mock
is slighter larger than the HOD mock. This phenomena can also be noticed in
Fig. 6. We know that in the HOD mock, the Hi gas are assumed to locate at the
center of the halo, without any substructures. While in the mocks that contain
subhalos, the subhalo distribution inside main halos leads to additional
clustering at small scale, which is known as one-halo term. This can be seen
when comparing the HR4 HAM mock and HR4 HOD mock in Fig. 10. It can also be
seen when comparing ELUCID SAM mock and ELUCID HOD mock in Fig. 6. Therefore,
it is expected that at small scale, the real space power spectrum of HOD mock
has smaller amplitude than that of the HAM mock or SAM mock in real space.
However, when considering RSD effect, since the subhalos in halos are also
moving randomly in the frame of the center of mass of the main halo, there
will be FoG effect that can suppress the power spectrum at small scale. Thus,
in the ”red” mock, the higher power from one-halo term and the lower power
from FoG counter each other, the power spectrum between HOD mock and SAM from
ELUCID simulation are getting closer. Furthermore, in the ”red2” mock, the FoG
effect is taken into account in the HOD mock, the difference between HOD and
SAM are revealed again.

Figure 11: The comparison of angular power spectrum between the HOD mock in
real space and redshift space. The power in redshift space is higher than that
in real space at large scale due to Kaiser effect. No matter we treat the Hi
gas in halos as point sources (”red”) or extended sources due to internal
velocity dispersion (”red2”) in redshift space, the difference between them is
not significant.

In Fig. 11, we have shown the comparison between the HOD real space mock
(labeled as ”real”), the HOD redshift space point source mock (labeled as
”red”) and HOD redshift FoG effect included mock (labeled as ”red2”). The
angular power spectrum difference between the ”real” mock and ”red” mock is
clear at large scale due to Kaiser effect, but the difference between ”red”
mock and ”red2” mock due to FoG effect at small scale is not significant. This
is because of the bin width of the mock being 10 MHz, that corresponds to more
than $20\text{ Mpc}/h$, while the typical velocity dispersion is no larger
than $1000\text{ km/s}$, which roughly corresponds to $10\text{ Mpc}/h$ in
redshift space. Therefore, the power spectrum suppression introduced by the
velocity dispersion is averaged out by the large bin width.

Figure 12: We show the angular power spectrum of the HOD mock, HAM mock and
Re-HAM mock in redshift space. Re-HAM mock is clearly higher due to a higher
bias. The difference between HAM mock and HOD mock due to one-halo term and
the difference between ”red” mock and ”red2” mock due to FoG effect at small
scale are not significant because of smoothing introduced by Gaussian kernel
FWHM = 40 arcmin.

In Fig. 12, we have compared the angular power spectrum of the mocks smoothed
by Gaussian kernel FWHM = 40 arcmin, which is the beam size of BINGO. After
smoothing, the difference at small scale is easily covered. The difference of
bias can still be clearly identified by comparing HOD mock and Re-HAM mock,
which share the same $\Omega_{\rm H{\sc i}\,}$ value. Therefore, we can answer
the questions raised in the beginning of this subsection,

  * 1

Different ways of linking Hi gas with galaxies (halos) lead to different bias
and one-halo term, the bias difference can be seen in the total power of
angular power spectrum and the difference of one-halo term can be seen in the
small scale,

  * 2

Kaiser effect leads to higher angular power spectrum at large scale. FoG
effect leads to suppression of power spectrum at small scale, but cannot be
discriminated due to 10 MHz bin size,

  * 3

Gaussian kernel smoothing eliminates the difference at small scale, but the
bias difference shown in all scales and the Kaiser effect shown in large
scales can still be easily identified.

Figure 13: Zoom-in view of the central $10\times 10$ degree2 brightness
temperature distribution of the mock map from 990 MHz to 1000 MHz.

###  6.2 Pixel Histogram

Figure 14: The pixel histogram of the brightness temperature distribution of
the mock map from 990 MHz to 1000 MHz in redshift space. In the upper panel,
we show the histogram comparison of the HOD map and the HAM map from 0 mK to
1.5 mK. In the lower panel, we zoom in to the range from 0 mK to 0.4 mK. The
lognormal fitted curves are also shown. The HAM map has clearly more extremely
high brightness temperature pixels, as expected by looking at Fig. 13.

We have shown that the brightness temperature fluctuation in the HAM mock is
larger than in the HOD mock in Fig. 13. More quantitatively, we calculated the
histogram of the pixels in the unsmoothed mock map. In the upper panel of Fig.
14, we showed the histogram of the pixels, notice that we have excluded all
the pixels which are zero. As expected, there are more extremely high
brightness temperature pixels in the HAM mock than in the HOD mock.

On the other hand, the lognormal distribution was widely used to simulate dark
matter tracer fields (Xavier et al., 2016), which is the 21-cm intensity map
in our case. We calculated the histogram of the pixels from 0 mK to 0.4 mK and
fitted with the lognormal distribution using the stats routine in scipy
(Virtanen et al., 2019). The result is shown in the lower panel of Fig. 14. We
found that the lognormal distribution can reasonably represent the measured
results from our mock map. This means that using lognormal distribution to
simulate the brightness temperature map is reasonable, but with clear
limitations as well.

The difference between the pixel histogram of the HOD mock and the HAM mock
shows that the way how the Hi gas is distributed inside halos can largely
affect the intensity map, not only in the angular power spectrum, but also in
other statistical measurements.

###  6.3 Lognormal Fitting

Figure 15: In the upper left panel, we have shown the histogram of the pixels
of the brightness temperature distribution of the redshift space mock map from
990 MHz to 1000 MHz smoothed by a 40 arcmin Gaussian kernel. After smoothing,
the histogram can be very well fitted by the lognormal distribution. The
fitted parameters, $s$, $loc$, and $scale$, are shown as a function of the
redshift (frequency) in the other three panels. For each of them we have also
fitted a parabola function, which turns out to be a good fit. The lognormal
fitted parameters are shown in red color for the HOD mock and blue color for
the HAM mock, crosses for the real space mock and dots for the redshift space
mock. The parabola fitting curves are shown in red color for the HOD mock and
blue color for the HAM mock, solid lines for the real space mock and dashed
lines for the redshift space mock.

The lognormal distribution can provide a reasonable fitting to the pixel
histogram of the mock map, both in redshift space and in real space.
Furthermore, after we considered the 40 arcmin Gaussian kernel smoothing for
the mock map, we found that the lognormal function can fit the pixel histogram
very well. We illustrated one slice as an example in Fig. 15. We fitted the
histogram by a lognormal distribution using the stats routine in scipy
(Virtanen et al., 2019). For every redshift bin, we fitted the histogram by
lognormal distribution and showed the parameters in Fig. 15. The lognormal
probability density function is

$\displaystyle
f(y,s)=\dfrac{1}{sy\sqrt{2\pi}}\exp\left(-\dfrac{-\log^{2}(y)}{2s^{2}}\right)/\text{scale}\,,$
(11) $\displaystyle y=(x-\text{loc})/\text{scale}\,,$

where the $x$ variable is given by $T_{b}$ in our case.

We have used the parabola function, $f(z)=a+bz+cz^{2}$, to fit these three
parameters of the lognormal distribution as a function of redshift $z$. The
fitted results are given in Tab. 1.

Mock | parameter | $a$ | $b$ | $c$
---|---|---|---|---
HOD Real | s | 0.99 | -2.8 | 2.9
HOD Red | s | 1.1 | -2.9 | 3.0
HAM Real | s | 1.6 | -4.4 | 4.6
HAM Red | s | 1.7 | -4.7 | 4.9
HOD Real | loc | -0.0016 | -0.047 | -0.025
HOD Red | loc | -0.00037 | -0.050 | 0.0046
HAM Real | loc | 0.00095 | -0.021 | 0.0030
HAM Red | loc | 0.0015 | -0.024 | 0.0080
HOD Real | scale | 0.021 | 0.18 | -0.046
HOD Red | scale | 0.017 | 0.19 | -0.081
HAM Real | scale | 0.0058 | 0.16 | -0.12
HAM Red | scale | 0.0042 | 0.16 | -0.13
Table 1: The parabola fitting parameters of lognormal parameters as a function
of redshift $z$

Given these parameters, one can generate a similar mock map as ours using
lognormal distribution (Xavier et al., 2016). It will be helpful for further
study and comparison.

###  6.4 Effect of Bin Size

Figure 16: The comparison of HOD and HAM mocks in real space is shown with 10
(5, 2) MHz in red (blue, green) color. In the upper panel, the HOD results are
shown in thick solid lines and the HAM results are shown in faint solid lines.
Notice that for better illustration, we have artificially amplified the curves
by 0.5 (1, 2) times for 10 (5, 2) MHz results. In the lower panel, we show the
ratio of angular power spectrum between HOD mocks and HAM mocks. Clearly, HAM
mocks have higher angular power spectrum at small scale and it depends weakly
on the bin width.

As discussed in Sec. 6.1, different ways of populating Hi gas in galaxies
(halos) will lead to different bias and one-halo term, and the bin width of 10
MHz will largely wipe out the effect of FoG. One step further, it is
interesting to ask the following question, if we use different bin width, for
example 2 MHz or 5 MHz, will the conclusion be different? Can we see more
significant effect of FoG or different small scale power introduced by one-
halo term with smaller bin width? To answer this question, we have generated
the mock map with HOD and HAM method in 2 MHz bin width and 5 MHz bin width,
following the same steps as we used for generating the 10 MHz bin width mock
map.

In Fig. 16, we show the comparison of angular power spectrum among the mocks
with 10 MHz, 5 MHz and 2 MHz. For a more fair comparison, we take the
frequency range from 990 MHz to 1000 MHz, which consists of 1 bin in the 10
MHz mock, 2 bins in the 5 MHz mock and 5 bins in the 2 MHz mock. We take the
average of the angular power spectrum measured from these 1 (2, 5) bin(s) of
10 (5, 2) MHz mocks. We have also amplified the results to keep the curves of
different bin width separated for better illustration. We can clearly see the
higher angular power spectrum of HAM mocks compared to the HOD mocks at small
scale ($l>100$). The angular power spectrum of HOD mocks comparing to HAM
mocks are lower by more than $10\%$ at $l>300$. It is also obvious that the
difference hardly depends on the bin width. It is well expected, since no
matter if the difference of small scale power is due to scale dependent bias
or one-halo term, they are all isotropic. The width of frequency bin is
expected to play little role in the power spectrum.

Figure 17: The comparison of ”red” (point source assumption) HOD mock and
”red2” (velocity dispersion induced FoG effect modelled) HOD mock is shown
with 10 (5, 2) MHz in red (blue,green) color. The ”red” result are shown in
thick solid lines and the ”red2” results are shown in faint solid lines.
Notice that for better illustration, we have artificially amplified the curves
by 0.5 (1, 2) times for 10 (5, 2) MHz results. The FoG suppression at small
scale is more clear with smaller bin width.

In contrast, the FoG effect is anisotropic. The FoG effect is pointing toward
the line-of-sight direction. We expect to see stronger suppression due to FoG
effect at small scale with smaller bin width, since there is less smoothing.
The result is shown in Fig. 17, which is exactly what we have expected. We
have taken point source assumption in the ”red” mock, and take the velocity
dispersion into account in the ”red2” mock. Therefore, the FoG effect is
modelled in the ”red2” sample. We can see that the FoG effect is very clear in
the 2 MHz mock, weaker in the 5 MHz mock and almost indistinguishable in the
10 MHz mock. We can conclude that smaller frequency bin width is very
effective in identifying the FoG effect in 21-cm intensity mapping
observations. However, we should also notice that with smaller bin width, the
noise level is also higher. Therefore, it is an open question whether we can
identify the FoG effect in the real observations by BINGO.

###  6.5 Comparison with UCLCl

Figure 18: We show the angular power spectrum of HOD mocks (solid faint
curves) comparing to the calculation of UCLCl code with linear calculation
(dashed lines) and halofit nonlinear correction (dotted lines) in redshift
space. Red (orange, green, blue) lines show the result from 960 (1050, 1150,
1160) MHz to 970 (1060, 1160, 1260) MHz, middled at 965 (1055, 1155, 1255)
MHz, which are amplified by 0.25 (0.5, 1, 2) times for better illustration.
HOD mock measurements are higher than the linear calculation, but lower than
the halofit calculation at small scale.

If we take the mock maps and the measured angular power spectrum as real
observation, we need a fast model prediction to compare with the map and see
whether we can fit it to the observation. It is essential for parameter
fitting and cosmological parameter constraints. UCLCl (Unified Cosmological
Library for $C_{\ell}$’s) is a package that calculates the angular power
spectra (McLeod et al., 2017; Loureiro et al., 2019) from the power spectrum
of density fluctuations through

$C_{l}^{ij}=4\pi\int\dfrac{dk}{k}W_{l}^{i}(k)W_{l}^{j}(k)\Delta^{2}(k)\,,$
(12)

where $W_{l}(k)$ is the window function that absorbs all the processes
involved in the evolution, convolution and the projection effects, and
$\Delta^{2}(k)$ is the dimensionless matter power spectrum. By using the
linear perturbation calculation of the matter power spectrum, we can get the
linear angular power spectrum. By using the halofit nonlinear correction
(Takahashi et al., 2012), we can take the one-halo term nonlinear effect into
account. We have calculated the angular power spectrum using the UCLCl package
with the cosmological parameters consistent with the HOD mock.

In Fig. 18, we have compared the results of the UCLCl and our HOD mock
measurements in redshift space. The point mass assumption was taken in this
HOD mock for comparison. The HOD mock results are higher than the linearly
calculated ones, but lower than the halofit corrected results at small scale.
At large scale, the UCLCl results are consistent with the HOD mock except for
the obvious fluctuations due to cosmic variance. At lower redshift (higher
frequency), we can see larger small scale nonlinear correction than at higher
redshift (lower frequency). There are two reasons for that, one is that at
higher redshift, the one-halo term plays a smaller role than at lower
redshift, the other one is that at lower redshift, the shell of Hi
distribution that we can measure is closer to us, which means that at lower
redshifts we can see more details from small scales with the same angular
resolution when compared to higher redshift shells.

If we take the HOD mock as a benchmark, we can see that the linear case have
underestimated the angular power spectrum at small scales and the halofit
correction has overestimated the angular power spectrum. At higher redshift,
the UCLCl calculation is closer to the measurements of the HOD mock. At
$l<80$, the UCLCl linearly calculated results, which adopted the linear bias
assumption, can fit to the HOD mock from 1250 MHz to 1260 MHz. From 960 MHz to
970 MHz, the UCLCl linear calculation can fit the HOD mock result up to
$l<200$. This comparison clearly tells us the shortage of our modelling for
the 21-cm intensity mapping angular power spectrum using UCLCl package.
Without further improvement in the modelling at small scale and better
understanding about the Hi distribution in halos, it is hard to use the full
information of angular power spectrum at $l>80$ at the lowest redshift and
$l>200$ at the highest redshift for BINGO.

###  6.6 Hi Bias and Abundance

Figure 19: We show the $\Omega_{\rm H{\sc i}\,}$ as a function of frequency
(redshift) in our mock map, the dotted lines show the result of the HOD mock
and the dashed curves show the result of HAM mock. ”real” means the real space
mock, ”red” means the redshift space mock under point source assumption. The
black solid lines show the parabola fitting results to the real space mock.
Figure 20: We show the $b_{\rm H{\sc i}\,}$ as a function of frequency
(redshift) in our mock map, the crosses show the result of the HOD mock and
the dots show the result of the HAM mock. Both the real space and redshift
space mock bias are measured, and they are almost identical. The solid black
lines show the parabola fitting results.

The overall amplitude of the angular power spectra of 21-cm intensity mapping
is closely related to $\Omega_{\rm H{\sc i}\,}$ and $b_{\rm H{\sc i}\,}$.
Therefore, observations can give constraints on these parameters. The detailed
forecast is provided in our companion paper VII (Costa et al., 2021). Here we
provide the ”answer” of the mock map for the future parameter fitting and
pipeline building. We calculate the $\Omega_{\rm H{\sc i}\,}$ in each mock map
by adding up all the Hi mass in each redshift bin, and then divided by the
volume of each bin. We show $\Omega_{\rm H{\sc i}\,}$ as a function of
frequency (redshift) in Fig. 19. The measurement of HOD mock is shown in
dotted lines and the HAM mock is shown in dashed lines. The population of Hi
mass is different between the real space mock map and the redshift space mock
map in the same frequency bin due to RSD effect. We have illustrated the
result of real space mock and redshift space mock in different colors, but
clearly there is little difference between the real space and redshift space.
The RSD has little impact in the $\Omega_{\rm H{\sc i}\,}$ in each bin. We
have provided a parabola fitting to the real space mocks, which gives

$(\text{HOD})\Omega_{\rm H{\sc i}\,}=2.7\times 10^{-4}+1.0\times
10^{-4}z-8.0\times 10^{-5}z^{2},$ (13)

for the HOD mock and

$(\text{HAM})\Omega_{\rm H{\sc i}\,}=2.5\times 10^{-4}-4.0\times
10^{-5}z-7.0\times 10^{-5}z^{2},$ (14)

for the HAM mock, shown in the black solid lines.

For the HOD mock, we use the publicly available python package COLOSSUS
(Diemer, 2018) to calculate the bias of each halo using the empirical relation
provided by Tinker et al. (2010), and by considering the following relation in
real space

$\dfrac{Cl_{\rm HAM}}{Cl_{\rm HOD}}=\dfrac{\bar{T}_{\rm
HAM}^{2}}{\bar{T}_{HOD}^{2}}\dfrac{b_{\rm HAM}^{2}}{b_{\rm HOD}^{2}}\;,$ (15)

we estimated the bias of the HAM mock. We show the bias $b_{\rm H{\sc i}\,}$
as a function of frequency (redshift) in Fig. 20. The crosses represent the
result of the HOD mock and the dots represent the result of the HAM mock.
Since the real space and redshift space results are almost identical, we just
show the results of real space mock for better illustration. Similarly, we
have also provided the parabola fitting results

$(\text{HOD})b_{\rm H{\sc i}\,}=0.96+0.36z+0.17z^{2},$ (16)

for HOD mock and

$(\text{HAM})b_{\rm H{\sc i}\,}=1.4+0.092z+0.50z^{2},$ (17)

for HAM mock, shown in the black solid lines. The big difference among the
bias of the HOD mock and the HAM mock shows that different ways of Hi gas
occupation can lead to very different linear bias, which is observable in the
angular power spectrum. The 21-cm intensity mapping can also tell us about the
Hi gas distribution, related to the galaxy formation and evolution.

There are three advantages of using the fitting formula:

  * •

it is useful for reproducing the map with other methods, such as using
Gaussian realization or FLASK, which can easily generate large amount of
realizations for constructing covariance matrix;

  * •

it can largely reduce the number of free parameters about $\Omega_{\rm H{\sc
i}\,}$ and $b_{\rm H{\sc i}\,}$ with fitting formula, when using maps at many
different redshifts;

  * •

this fitting formula is independent of the number of bins, it is useful when
we need to change the number of bins.

##  7 Summary and Discussion

In this paper, we have introduced the method of building mock 21-cm intensity
maps using N-body simulation data. In summary, we have achieved the following
points,

  * 1

We have used the ELUCID N-body simulation and its semi-analytical galaxy
catalog to study the Hi halo occupation distribution. We have concluded that
for halos less massive than $10^{12}M_{\odot}/h$, the major contribution of Hi
gas is from the central galaxies. While for halos more massive than
$10^{13}M_{\odot}/h$, the major contribution of Hi gas is from the satellite
galaxies. Neglecting the Hi contribution in satellite galaxies is not a good
approximation.

  * 2

We have provided a good fitting formula of Hi mass-halo mass relation, make it
easier to populate Hi gas into dark matter halos. The fitting function is
given by Eq. (1).

  * 3

The HOD method was verified using the ELUCID simulation by comparing to the
SAM galaxy catalog, within $10\%$ accuracy, the power spectrum of the HOD
repopulated Hi distribution is correct up to $k=0.4h\text{Mpc}^{-1}$.

  * 4

The mass resolution of the N-body simulation can lead to scale dependent bias.
With the halo mass limit down to around $10^{11}M_{\odot}/h$, the linear bias
assumption, which assume a constant bias number, is still valid. The linear
bias assumption is not valid with $10^{12}M_{\odot}/h$ halo mass limit. This
sets a resolution requirement for N-body simulation to be used to build Hi
mock map.

  * 5

Using HR4 simulation, we have built the full-sky 21-cm intensity mock maps
covering from 960 MHz to 1260 MHz, covering the range of BINGO (from 980 MHz
to 1260 Mhz). Both the HOD method and abundance matching (HAM) method are
applied to get two kinds of mocks.

  * 6

The redshift space distortion (RSD) was taken into consideration, we
introduced the method to consider both the point source redshift space
distortion and the method to consider the velocity dispersion to account for
finger-of-god (FoG) effect.

  * 7

The lognormal distribution can provide a reasonable fitting to the pixel
histogram of the mock map, if the map is smoothed by 40 arcmin Gaussian
kernel, the fitting will be very good.

  * 8

We have provided the lognormal fitting parameters for our mock maps. A further
parabola fitting to the lognormal parameters as functions of redshift was
done. The fitting results are given. It is easy to realize a similar
brightness temperature distribution with lognormal distribution using our
parameters.

  * 9

We have discussed about the effect of Gaussian beam smoothing, RSD and the
difference of HOD mock and HAM mock by comparing the angular power spectrum.
Different bias and Kaiser effect can be discriminated. The nonlinear effect at
small scale and FoG effect will not be important if the 40 arcmin Gaussian
smoothing is taken into account.

  * 10

The effect of different frequency bin sizes was discussed. FoG effect can be
more easily identified with smaller bin size. Different bin size has little
impact in telling the difference between HOD mock and HAM mock.

  * 11

The theoretical calculation was done using UCLCl package and compared to our
mock map measurements. We found that the UCLCl results can fit to the mock at
large scale, but failed at small scale. The lower the redshift, the worse.
They fit well at $l<80$ for the highest frequency bin and at $l<200$ for the
lowest frequency bin of BINGO. The linear theory calculation has
underestimated the angular power spectra at small scale and the halofit
nonlinear correction has overestimated it. We need a better modelling to make
full use of the whole range of angular power spectra.

  * 12

We have provided the measurement of $\Omega_{\rm H{\sc i}\,}$ and $b_{\rm
H{\sc i}\,}$ as a function of frequency (redshift) in the HOD mock and HAM
mock. The parabola function can provide very good fitting to the measurements.
The fitted functions in each case are listed in Sec. 6.6.

Beside constraining cosmological parameters, 21-cm intensity mapping is
expected to constrain the astrophysical processes related to Hi gas. There is
no conclusion about the best method, HAM and HOD, two different methods
applied in this paper for populating Hi gas in haloes, can be regarded as
valid tests. Since we can distinguish HAM from HOD using 21-cm intensity
mapping by constraining the bias parameter, it proves the ability of
discriminating the Hi population in haloes. The mock we have generated here
will serve as the source signal to test the BINGO data analysis pipeline. This
will include parameter fitting, BAO signal fitting and many other methods
developed for intensity mapping.

The methodology we have introduced here will be useful in building mocks based
on fast simulations such as COLA-HALO (Koda et al., 2016). After generating
hundreds of mocks, we can measure the covariance matrix, which is essential
for estimating the cosmological parameters. On the other hand, this
methodology can also be applied for other 21-cm intensity mapping experiments
such as SKA. Next step, we will combine the foreground, noise and mask of
BINGO to generate a more realistic mock map for testing our data analysis
pipeline. In the future, we will also include more effects that might change
the map, such as point radio sources, lensing effects, gravitational redshift
etc. By making more and more realistic mock maps, we can improve the
understanding about intensity mapping and provide better data challenge for
the pipeline.

###### Acknowledgements.

This work was supported by the Ministry of Science and Technology of China
(grant nos. 2018YFA0404601) and National Natural Science Foundation of China
(grant nos. 11973033, 11835009). The BINGO project is supported by FAPESP
grant 2014/07885-0; the support from CNPq is also gratefully acknowledged
(E.A.). C.A.W. acknowledges a CNPq grant 2014/313.597. T.V. acknowledges CNPq
Grant 308876/2014-8. J.Z. was supported by IBS under the project code,
IBS-R018-D1. C.P.N. would like to thank São Paulo Research Foundation
(FAPESP), grant 2019/06040-0, for financial support. A.A.C. acknowledges
financial support from the China Postdoctoral Science Foundation, grant number
2020M671611. F.B.A. acknowledges the UKRI-FAPESP grant 2019/05687-0, and
FAPESP and USP for Visiting Professor Fellowships where this work has been
developed. R.G.L. thanks CAPES (process 88881.162206/2017-01) and the
Alexander von Humboldt Foundation for the financial support. A.R.Q., F.A.B.,
L.B., and M.V.S. acknowledge PRONEX/CNPq/FAPESQ-PB (Grant no. 165/2018). H.X.
and Z.Z. are supported by the Ministry of Science and Technology of China
(grant No. 2018YFA0404601 and 2020SKA0110200), and the National Science
Foundation of China (grant Nos. 11621303, 11835009 and 11973033). K.S.F.F.
thanks São Paulo Research Foundation (FAPESP) for financial support through
grant 2017/21570-0. V.L. acknowledges the postdoctoral FAPESP grant
2018/02026-0. L.S. is supported by the National Key R&D Program of China
(2020YFC2201600).

## References

  * Abdalla et al. (2021a) Abdalla, E., Ferreira, E. G., Landim, R. G., et al. 2021a, The BINGO project I: Baryon Acoustic Oscillations from Integrated Neutral Gas Observations
  * Abdalla et al. (2021b) Abdalla, F. B., Marins, A., et al. 2021b, The BINGO project III: optical design and optimisation of the focal plane
  * Alonso et al. (2014) Alonso, D., Ferreira, P. G., & Santos, M. G. 2014, Monthly Notices of the Royal Astronomical Society, 444, 3183
  * An et al. (2019) An, R., Costa, A. A., Xiao, L., Zhang, J., & Wang, B. 2019, MNRAS, 489, 297
  * Anagnostopoulos et al. (2019) Anagnostopoulos, F. K., Basilakos, S., & Saridakis, E. N. 2019, Phys. Rev. D, 100, 083517
  * Asorey et al. (2020) Asorey, J., Parkinson, D., Shi, F., et al. 2020, Monthly Notices of the Royal Astronomical Society, 495, 1788
  * Becker et al. (2001) Becker, R. H., Fan, X., White, R. L., et al. 2001, AJ, 122, 2850
  * Bigot-Sazy et al. (2016) Bigot-Sazy, M. A., Ma, Y. Z., Battye, R. A., et al. 2016, Astronomical Society of the Pacific Conference Series, Vol. 502, HI Intensity Mapping with FAST, ed. L. Qain & D. Li, 41
  * Chen (2012) Chen, X. 2012, in International Journal of Modern Physics Conference Series, Vol. 12, International Journal of Modern Physics Conference Series, 256–263
  * Cheng et al. (2019) Cheng, G., Ma, Y., Wu, F., Zhang, J., & Chen, X. 2019, arXiv e-prints, arXiv:1911.04520
  * Costa et al. (2021) Costa, A. A., Landim, R. G., et al. 2021, The BINGO Project VII: Cosmological Forecasts from 21-cm Intensity Mapping
  * Costa et al. (2017) Costa, A. A., Xu, X.-D., Wang, B., & Abdalla, E. 2017, J. Cosmology Astropart. Phys., 2017, 028
  * Cunnington et al. (2020) Cunnington, S., Pourtsidou, A., Soares, P. S., Blake, C., & Bacon, D. 2020, Monthly Notices of the Royal Astronomical Society
  * Diemer (2018) Diemer, B. 2018, ApJS, 239, 35
  * Dunkley et al. (2009) Dunkley, J., Komatsu, E., Nolta, M. R., et al. 2009, ApJS, 180, 306
  * Eisenstein & Hu (1998) Eisenstein, D. J. & Hu, W. 1998, ApJ, 496, 605
  * Fan et al. (2006a) Fan, X., Carilli, C. L., & Keating, B. 2006a, ARA&A, 44, 415
  * Fan et al. (2006b) Fan, X., Strauss, M. A., Becker, R. H., et al. 2006b, AJ, 132, 117
  * Fornazier et al. (2021) Fornazier, K. S. F., Abdalla, F. B., et al. 2021, The BINGO Project V: Further steps in Component Separation and Bispectrum Analysis
  * Gorski et al. (2005) Gorski, K. M., Hivon, E., Banday, A. J., et al. 2005, The Astrophysical Journal, 622, 759
  * Guo et al. (2020) Guo, H., Jones, M. G., Haynes, M. P., & Fu, J. 2020, ApJ, 894, 92
  * Guo et al. (2017) Guo, H., Li, C., Zheng, Z., et al. 2017, ApJ, 846, 61
  * Hall et al. (2013) Hall, A., Bonvin, C., & Challinor, A. 2013, Phys. Rev. D, 87, 064026
  * Hong et al. (2016) Hong, S. E., Park, C., & Kim, J. 2016, ApJ, 823, 103
  * Hu et al. (2019) Hu, W., Wang, X., Wu, F., et al. 2019, arXiv e-prints, arXiv:1909.10946
  * Icaza-Lizaola et al. (2019) Icaza-Lizaola, M., Vargas-Magaña, M., Fromenteau, S., et al. 2019, arXiv e-prints, arXiv:1909.07742
  * Jackson (1972) Jackson, J. C. 1972, MNRAS, 156, 1P
  * Jullo et al. (2019) Jullo, E., de la Torre, S., Cousinou, M. C., et al. 2019, A&A, 627, A137
  * Kaiser (1987) Kaiser, N. 1987, MNRAS, 227, 1
  * Kim et al. (2015) Kim, J., Park, C., L’Huillier, B., & Hong, S. E. 2015, JKAS, 48, 213
  * Klypin et al. (2016) Klypin, A., Yepes, G., Gottlöber, S., Prada, F., & Heß, S. 2016, MNRAS, 457, 4340
  * Koda et al. (2016) Koda, J., Blake, C., Beutler, F., Kazin, E., & Marin, F. 2016, Monthly Notices of the Royal Astronomical Society, 459, 2118
  * Kovetz et al. (2017) Kovetz, E. D., Viero, M. P., Lidz, A., et al. 2017, arXiv e-prints [arXiv:1709.09066]
  * Liccardo et al. (2021) Liccardo, V., Mericia, E. J., Wuensche, C. A., et al. 2021, The BINGO project IV: Simulations for Mission Performance Assessment
  * Loureiro et al. (2019) Loureiro, A., Moraes, B., Abdalla, F. B., et al. 2019, MNRAS, 485, 326
  * Luo et al. (2016) Luo, Y., Kang, X., Kauffmann, G., & Fu, J. 2016, MNRAS, 458, 366
  * Martin et al. (2010) Martin, A. M., Papastergis, E., Giovanelli, R., et al. 2010, The Astrophysical Journal, 723, 1359
  * McLeod et al. (2017) McLeod, M., Balan, S. T., & Abdalla, F. B. 2017, MNRAS, 466, 3558
  * Naiman et al. (2018) Naiman, J. P., Pillepich, A., Springel, V., et al. 2018, MNRAS, 477, 1206
  * Nelson et al. (2018) Nelson, D., Pillepich, A., Springel, V., et al. 2018, MNRAS, 475, 624
  * Nelson et al. (2019) Nelson, D., Springel, V., Pillepich, A., et al. 2019, Computational Astrophysics and Cosmology, 6, 2
  * Padmanabhan & Refregier (2017) Padmanabhan, H. & Refregier, A. 2017, MNRAS, 464, 4008
  * Paul et al. (2018) Paul, N., Choudhury, T. R., & Paranjape, A. 2018, MNRAS, 479, 1627
  * Pillepich et al. (2018) Pillepich, A., Nelson, D., Hernquist, L., et al. 2018, MNRAS, 475, 648
  * Pritchard & Loeb (2012) Pritchard, J. R. & Loeb, A. 2012, Reports on Progress in Physics, 75, 086901
  * Prochaska et al. (2005) Prochaska, J. X., Herbert-Fort, S., & Wolfe, A. M. 2005, ApJ, 635, 123
  * Santos et al. (2015) Santos, M., Bull, P., Alonso, D., et al. 2015, Advancing Astrophysics with the Square Kilometre Array (AASKA14), 19
  * Sarkar & Bharadwaj (2019) Sarkar, D. & Bharadwaj, S. 2019, MNRAS, 487, 5666
  * Seehars et al. (2016) Seehars, S., Paranjape, A., Witzemann, A., et al. 2016, J. Cosmology Astropart. Phys., 2016, 001
  * Smoot & Debono (2017) Smoot, G. F. & Debono, I. 2017, A&A, 597, A136
  * Springel et al. (2018) Springel, V., Pakmor, R., Pillepich, A., et al. 2018, MNRAS, 475, 676
  * Takahashi et al. (2012) Takahashi, R., Sato, M., Nishimichi, T., Taruya, A., & Oguri, M. 2012, ApJ, 761, 152
  * Tinker et al. (2010) Tinker, J. L., Robertson, B. E., Kravtsov, A. V., et al. 2010, ApJ, 724, 878
  * Vanderlinde & Chime Collaboration (2014) Vanderlinde, K. & Chime Collaboration. 2014, in Exascale Radio Astronomy, Vol. 2
  * Villaescusa-Navarro et al. (2018) Villaescusa-Navarro, F., Genel, S., Castorina, E., et al. 2018, ApJ, 866, 135
  * Virtanen et al. (2019) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2019, arXiv e-prints [arXiv:1907.10121]
  * Wang et al. (2016) Wang, H., Mo, H. J., Yang, X., et al. 2016, ApJ, 831, 164
  * Wolfe et al. (2005) Wolfe, A. M., Gawiser, E., & Prochaska, J. X. 2005, ARA&A, 43, 861
  * Wuensche & the BINGO Collaboration (2019) Wuensche, C. A. & the BINGO Collaboration. 2019, in Journal of Physics Conference Series, Vol. 1269, Journal of Physics Conference Series, 012002
  * Wuensche et al. (2021) Wuensche, C. A., Villela, T., Abdalla, E., Abdalla, F. B., et al. 2021, The BINGO project II: instrument description
  * Wyithe et al. (2008) Wyithe, J. S. B., Loeb, A., & Geil, P. M. 2008, MNRAS, 383, 1195
  * Xavier et al. (2016) Xavier, H. S., Abdalla, F. B., & Joachimi, B. 2016, 459, 3693
  * Yang et al. (2019) Yang, W., Shahalam, M., Pal, B., Pan, S., & Wang, A. 2019, Phys. Rev. D, 100, 023522
  * Zafar et al. (2013) Zafar, T., Péroux, C., Popping, A., et al. 2013, A&A, 556, A141
  * Zhang et al. (2020) Zhang, J., Costa, A. A., Wang, B., et al. 2020, The Astrophysical Journal, 895, 34