11institutetext: Instituto de Física, Universidade de São Paulo - C.P. 66318,
CEP: 05315-970, São Paulo, Brazil
11email<EMAIL_ADDRESS>22institutetext: Max-Planck-Institut für
Astrophysik, Karl-Schwarzschild Str. 1, 85741 Garching, Germany
33institutetext: Technische Universität München, Physik-Department T70, James-
Franck-Stra$\beta$e 1, 85748 Garching, Germany 44institutetext: Center for
Gravitation and Cosmology, College of Physical Science and Technology,
Yangzhou University, 225009, China 55institutetext: Instituto Nacional de
Pesquisas Espaciais, Divisão de Astrofísica, Av. dos Astronautas, 1758,
12227-010 - São José dos Campos, SP, Brazil 66institutetext: Department of
Physics and Astronomy, University College London, Gower Street, London,WC1E
6BT, UK 77institutetext: Department of Physics and Electronics, Rhodes
University, PO Box 94, Grahamstown, 6140, South Africa 88institutetext:
Unidade Acadêmica de Física, Univ. Federal de Campina Grande, R. Aprígio
Veloso, 58429-900 - Campina Grande, Brazil 99institutetext: Instituto de
Física, Universidade de Brasília, Brasília, DF, Brazil 1010institutetext:
Centro de Gestão e Estudos Estratégicos - CGEE, SCS Quadra 9, Lote C, Torre C
S/N Salas 401 - 405, 70308-200 - Brasília, DF, Brazil 1111institutetext:
School of Aeronautics and Astronautics, Shanghai Jiao Tong University,
Shanghai 200240, China 1212institutetext: School of Physics and Astronomy,
Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang, Shanghai 200240,
China 1313institutetext: Center for Theoretical Physics of the Universe,
Institute for Basic Science (IBS), Daejeon 34126, Korea 1414institutetext:
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, China 1515institutetext: Jodrell Bank Centre for
Astrophysics, The University of Manchester, Oxford Road, Manchester, M13 9PL,
U.K. 1616institutetext: Laboratoire Astroparticule et Cosmologie (APC),
CNRS/IN2P3, Université Paris Diderot, 75205 Paris Cedex 13, France
1717institutetext: IRFU, CEA, Université Paris Saclay, 91191 Gif-sur-Yvette,
France 1818institutetext: Department of Astronomy, School of Physical
Sciences, University of Science and Technology of China, Hefei, Anhui 230026
1919institutetext: Center for Research and Development in Mathematics and
Applications – CIDMA, Campus de Santiago, 3810-183 Aveiro, Portugal
2020institutetext: MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA
02139, USA 2121institutetext: School of Chemistry and Physics, University of
KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South Africa
2222institutetext: NAOC-UKZN Computational Astrophysics Centre (NUCAC),
University of KwaZulu-Natal, Durban, 4000, South Africa 2323institutetext:
Institut d’Astrophysique Spatiale, Orsay (CNRS-INSU), France
2424institutetext: Instituto de Astrofísica de Canarias, 38200, La Laguna,
Tenerife, Canary Islands, Spain 2525institutetext: Departamento de
Astrofísica, Universidad de La Laguna (ULL), 38206, La Laguna, Tenerife, Spain

# The BINGO project I:

Baryon acoustic oscillations from integrated neutral gas observations

Elcio Abdalla<EMAIL_ADDRESS>Elisa G. M. Ferreira 1122 Ricardo G. Landim 33
Andre A. Costa 44 Karin S. F. Fornazier 11 Filipe B. Abdalla 11556677 Luciano
Barosi 88 Francisco A. Brito 88 Amilcar R. Queiroz 88 Thyrso Villela 55991010
Bin Wang 441111 Carlos A. Wuensche 55 Alessandro Marins 11 Camila P. Novaes 55
Vincenzo Liccardo 55 Chenxi Shan 1212 Jiajun Zhang 1313 Zhongli Zhang 1414
Zhenghao Zhu 1212 Ian Browne 1515 Jacques Delabrouille 161617171818 Larissa
Santos 441111 Marcelo V. dos Santos 88 Haiguang Xu 1212 Sonia Anton 1919
Richard Battye 1515 Tianyue Chen 15152020 Clive Dickinson 1515 Yin-Zhe Ma
21212222 Bruno Maffei 2323 Eduardo J. de Mericia 55 Pablo Motta 11 Carlos H.
N. Otobone 11 Michael W. Peel 24242525 Sambit Roychowdhury 1515 Mathieu
Remazeilles 1515 Rafael M. Ribeiro 11 Yu Sang 44 Joao R. L. Santos 88 Juliana
F. R. dos Santos 11 Gustavo B. Silva 11 Frederico Vieira 55 Jordany Vieira 11
Linfeng Xiao 1111 Xue Zhang 44 Yongkai Zhu 1212

###### Abstract

Context. Observations of the redshifted 21-cm line of neutral hydrogen (Hi)
are a new and powerful window of observation that offers us the possibility to
map the spatial distribution of cosmic Hi and learn about cosmology. Baryon
Acoustic Oscillations from Integrated Neutral Gas Observations (BINGO) is a
new unique radio telescope designed to be one of the first to probe baryon
acoustic oscillations (BAO) at radio frequencies.

Aims. BINGO has two science goals: cosmology and astrophysics. Cosmology is
the main science goal and the driver for BINGO’s design and strategy. The key
of BINGO is to detect the low redshift BAO to put strong constraints on the
dark sector models and test the $\Lambda$CDM (cold dark matter) model. Given
the versatility of the BINGO telescope, a secondary goal is astrophysics,
where BINGO can help discover and study fast radio bursts (FRB) and other
transients, as well as study Galactic and extragalactic science. In this
paper, we introduce the latest progress of the BINGO project, its science
goals, describing the scientific potential of the project for each goal and
the new developments obtained by the collaboration.

Methods. BINGO is a single dish transit telescope that will measure the BAO at
low-z by making a 3D map of the Hi distribution through the technique of
intensity mapping over a large area of the sky. In order to achieve the
project’s goals, a science strategy and a specific pipeline for cleaning and
analyzing the produced maps and mock maps was developed by the BINGO team,
which we generally summarize here.

Results. We introduce the BINGO project and its science goals and give a
general summary of recent developments in construction, science potential, and
pipeline development obtained by the BINGO collaboration in the past few
years. We show that BINGO will be able to obtain competitive constraints for
the dark sector. It also has the potential to discover several FRBs in the
southern hemisphere. The capacity of BINGO in obtaining information from 21-cm
is also tested in the pipeline introduced here. Following these developments,
the construction and observational strategies of BINGO have been defined.

Conclusions. There is still no measurement of the BAO in radio, and studying
cosmology in this new window of observations is one of the most promising
advances in the field. The BINGO project is a radio telescope that has the
goal to be one of the first to perform this measurement and it is currently
being built in the northeast of Brazil. This paper is the first of a series of
papers that describe in detail each part of the development of the BINGO
project.

###### Key Words.:

21-cm cosmology – baryon acoustic oscillations – radio astronomy – BINGO
Telescope

##  1 Introduction

This is the first of a series of papers presenting the Baryon Acoustic
Oscillations [BAO] from Integrated Neutral Gas Observations (BINGO) project.
Here the project is presented from a theoretical point of view, showing the
idea behind it as well as the plan of the whole construction of the BINGO
radio telescope, which evolved from ideas previously discussed elsewhere
(Battye et al. 2012; Wuensche 2019; Wuensche et al. 2020; Peel et al. 2019;
Battye et al. 2016). We also outline the structure of the project as a whole.
The companion papers II through VII show the instrument (Wuensche et al.
2021b), the optics design (Abdalla et al. 2021), the mission simulation
(Liccardo et al. 2021), the component separation analysis (Fornazier et al.
2021), the mock (Zhang et al. 2021), and forecasts on several models (Costa et
al. 2021), respectively. The BINGO concept was presented in Battye et al.
(2013) with later updates presented in Dickinson (2014), Bigot-Sazy et al.
(2015), Battye et al. (2016), and Wuensche (2019).

The BINGO project is a single dish (strictly, it revolves around a single dish
telescope, see paper II (Wuensche et al. 2021b) and paper III (Abdalla et al.
2021) for further details) radio telescope aiming at the observation of the
21-cm line, corresponding to a hyperfine interaction of the atomic hydrogen.
It will survey a sky area of 6000 square degrees in a redshift range from
0.127 to 0.449 (corresponding to a frequency span of 980 to 1260 MHz) with an
angular resolution of $\approx 40$ arcmin. The phase 1 of the project will
have 28 feed horns and receivers with dual polarization. It is designed to
have a system temperature of $\approx 70$ K (see, e.g., Battye et al. 2013;
Bigot-Sazy et al. 2015). These numbers will be finalized depending on
financial and technical achievements. We expect to increase the number of feed
horns, up to 56, in phase 2, also updating the receivers aiming at a smaller
system temperature.

Hydrogen is the most common element of the Universe. We assume that, barring
bias, a hydrogen map is representative of the full matter content in the
Universe. BINGO will be able to deliver detailed maps of the matter
distribution in its redshift range. We can thus extract BAO measurements
comparable to those in the optical observations (Eisenstein et al. 2005).
Moreover, in view of the telescope characteristics, it will be able to
describe phenomena at very short time scales, being thus an interesting device
to study pulsars and fast radio bursts (FRBs; Lorimer et al. 2007, 2012).

The standard approach to probe the large scale structure (LSS) is to perform a
large redshift survey. This yields the positions and redshifts of a large
number of galaxies, that can be used to infer their density contrasts and two-
point correlation functions. Essentially, galaxies are being used as tracers
of the underlying total matter distribution. This approach has been
successfully carried out in the optical band (e.g., SDSS, Blanton et al. 2017
or 2dF, Lewis et al. 2002). In the radio band the natural tracer is the 21-cm
line of neutral hydrogen (Hi), but the volume emissivity associated with this
line is low, meaning that detecting individual galaxies at $z\sim 1$ requires
a very substantial collecting area. This was the original motivation for the
Square Kilometre Array (SKA; Blake et al. 2004; Wilkinson et al. 2004; Abdalla
& Rawlings 2005).

However, measuring each galaxy individually is an expensive and inefficient
way of mapping cosmological volume, and just a few percent of the Universe has
been mapped with such a type of technique. It has been proposed (Battye et al.
2004; Chang et al. 2008; Loeb & Wyithe 2008; Sethi 2005; Visbal et al. 2009)
that mapping the Universe by measuring the collective 21-cm emission of the
underlying matter can be significantly more efficient than the previous
technique using individual galaxies. Such a technique is called intensity
mapping (IM; Madau et al. 1997; Bharadwaj & Sethi 2001). Therefore, exciting
progress can be made with much smaller scale instruments. The novel idea is to
exploit the broad beam at low-frequencies to carry out IM (Battye et al. 2004;
Loeb & Wyithe 2008; Masui et al. 2013; Switzer et al. 2013; Peterson et al.
2006) and hence measure the overall integrated Hi brightness temperature of a
large number of galaxies, taken together as a tracer of the LSS.

A number of approaches has been proposed to conduct IM surveys using an
interferometer array rather than in total intensity with single or double dish
telescopes (see, for instance, van Bemmel et al. 2012; Pober et al. 2013;
Baker et al. 2011). This approach can have a number of advantages, but it also
requires complicated, and hence expensive, electronics to make the
correlations. Using a single telescope with a stable receiver system is the
lowest-cost approach to IM measurements of BAO at low redshifts (Battye et al.
2013), in particular when compared with IM experiments that use interferometer
arrays. Interferometer arrays are likely to be the best approach to probe
higher redshifts at $z\sim 1$, where an angular resolution of $\sim
0.1^{\circ}$ is required. BINGO will be the first telescope in the world
operating in its frequency range whose goal is to study BAO with 21-cm IM.

In recent years new observational results have provided insight into various
aspects of general relativity (GR) and cosmology, especially concerning the
astonishing contribution provided by dark matter (DM) and dark energy (DE) to
the total energy budget of the Universe, contributions that form an
overwhelming majority of the mass of the Universe (see, e.g., Abdalla & Marins
2020). Therefore, it is expected that a new physical picture will emerge once
more is found about the structure of the dark sector (Abdalla et al. 2010;
Wang et al. 2007, 2016). There are many observational programs underway aimed
at elucidating the dark sector of the Universe, involving many different
approaches. Among these, the study of BAO is recognized as one of the most
powerful probes of the properties of DE (see e.g., Albrecht et al. 2006) –
others include weak gravitational lensing (Bartelmann & Schneider 2001;
Refregier 2003), cluster counts (He et al. 2010) and supernova’s luminosity
distance inference (Feng et al. 2008; He et al. 2011). To date, BAO have only
been detected by performing large galaxy redshift surveys in the optical
waveband. However, given the profound implications of these measurements, it
is important that they are confirmed in other wavebands, where systematic
effects are different, and measured over a wide range of redshifts. The radio
band provides a unique and complementary observational window for the
understanding of DE via the redshifted 21-cm Hi emission line from distant
galaxies (Abdalla et al. 2010).

Following the first detection of BAO using the 2dF and SDSS surveys (e.g.,
Eisenstein et al. 2005; Percival et al. 2010), BAO have been measured with
several new generation galaxy and quasar redshift surveys. In particular, the
10 recent measurements with the BOSS survey achieved a precision in the BAO
scale of a few percent in the redshift range $z=[0,2]$ (Anderson et al. 2014;
Alam et al. 2017)111For a compilation of all the results from BOSS, see
https://www.sdss.org/science/final-bao-and-rsd-measurements/.. In the future,
several large experiments designed to measure BAO’s in the optical band, such
as DESI (Aghamousa et al. 2016a, b), Euclid (Refregier et al. 2010), J-PAS
(Benitez et al. 2009, 2014; Bonoli et al. 2020) and WFIRST (Levi et al. 2011;
Green et al. 2011) (as well as LSST for imaging; Tyson et al. 2002), aim to
achieve a precision below the percent level at higher redshifts. These
experiments are large undertakings and will have their first light during the
following decade or are already under commission, like the J-PAS telescope.

Radio Hi IM experiments such as BINGO are very promising complementary routes
to study BAO. Indeed, optical experiments are required to gather data with
very high angular resolution (about an arcsecond) to detect individual
galaxies, while the BAO scale is typically of the order of a degree, depending
on the redshift in question. Radio BAO experiments on the other hand have
angular resolutions that are well matched to the BAO scale performing the
above mentioned experiments in a very cost-effective way. Also, an experiment
such as BINGO will pave the way to the extension of the use of Hi IM to high
redshifts and open a new window on cosmological epochs, which cannot be probed
in the optical or near-infrared bands. Moreover, since radio experiments use
Hi as opposed to stellar light as a tracer of the LSS, they are subject to
different astrophysical effects. IM experiments are more direct tracers of DM,
being less sensitive to “peak statistics” bias and thus complementary to
optical surveys, with which they can be combined to reduce systematic errors
in both.

The main goal in the present project is to detect the BAO present in the maps
as well as to map the 3D distribution of Hi , with observations spanning a few
years. Extensive simulations of the expected BAO signal and of how it can be
extracted from the observations are presented in the companion papers IV and V
(Liccardo et al. 2021; Fornazier et al. 2021). At frequencies below a few GHz
the future of radio astronomy will be dominated by the ambitious SKA project
that will constrain cosmological models (Maartens et al. 2015) with
unprecedented accuracy. Its design has been driven by the aim to probe the Hi
21-cm line over a broad range of radio frequencies. The high importance of the
IM technique is highlighted by the decision to change the baseline design of
the first phase of the SKA instrument to enable it to do IM (Square Kilometre
Array Cosmology Science Working Group et al. 2020a). The SKA will use very
similar techniques to ours and having developed bespoke analysis techniques,
the knowledge involved in BINGO will be in a perfect startup to take advantage
of the increase in the signal-to-noise ratio that the SKA will yield (Bull et
al. 2015; Abdalla et al. 2015).

In particular, one of the big challenges for those measurements is that large
Galactic as well as extragalactic foregrounds have to be removed. The Hi
signal amplitude is typically $\sim 100\,\mu$K whereas the foreground
continuum emission from the Galaxy is $\sim 1$ K in the BINGO frequency band.
Fortunately, the integrated 21-cm emission exhibits characteristic variations
as a function of frequency whereas the continuum emission has a very smooth
spectrum; this allows for the two signals to be separated (Chang et al. 2008;
Liu & Tegmark 2011; Chang et al. 2010; Olivari et al. 2016). The BINGO project
is an excellent ground for testing such ideas using IM as its method of
approach (see paper IV and V of this series: Liccardo et al. 2021; Fornazier
et al. 2021).

The telescope will be located in the hills of “Serra da Catarina” in the
municipality of Aguiar, Paraíba, Brazil. The measured radio frequency
interference (RFI) is at the lowest amplitude we could observe among all
measurements performed in South America, thus excellent for the project (see
Section 2 and the respective BINGO paper on the instrument (Wuensche et al.
2021b). The telescope consists of two dishes concentrating the radiation on to
the set of horns; this radiation then follows through a waveguide to the
receiver, which is also described in Wuensche et al. (2021b) .

The main science goal of BINGO is cosmology –– see Section 3, where we
describe how BINGO will measure BAO and redshift space distortion (RSD) to
constrain cosmological parameters in different cosmological models.
Nonetheless, the BINGO telescope, as designed, also allows studying different
astrophysical phenomena, from transient objects, such as FRBs and pulsars to
Galactic and extragalactic science. Thus, such astrophysical issues are
further scientific goals of the BINGO telescope and are discussed in Section
4. In Section 5 we consider the pipeline, which analyses the signals gathered
by the backends, after which we have the material for the science cases,
cosmology and astrophysics. In Section 6 we summarize other IM experiments
already in operation or planned for the near future. Section 7 is reserved for
conclusions.

##  2 Instrument overview

Detecting signals of $\sim 100\mu$K with a noncryogenic receiver of standard
performance requires that every visited sky pixel contains an accumulated
integration time of more than 1 day over the course of the observing campaign.
Gain variations should be very small on time scales of about $\sim$ 20
minutes, to allow for longer integration times. The design of the optical and
electronics systems for BINGO, as well as its observation strategy, try to
address these concerns from start, and the overall project has been designed
to keep the subsystems as simple as possible, using, when available, “off-the-
shelf” components to minimize the final cost and allow for a quicker
construction phase. This section contains a summary of the instrument, with a
more detailed description presented in the instrument companion paper
(Wuensche et al. 2021b).

BINGO is a fixed transit telescope, employing the rotation of the Earth to
observe the sky as it drifts across the instrument’s field of view (FoV), in
the frequency band $980-1260$ MHz. In doing so it meets the criterion of
maximizing the observation time by revisiting the same patch of the sky every
day. The BINGO optical design follows a crossed-Dragone configuration (Dragone
1978), with an instantaneous FoV of $\sim 14.75^{\circ}\times 6.0^{\circ}$
(DEC, RA), covered by 28 horns in a double rectangular, non-redundant array.
The telescope structure is aligned north-south and the primary reflector
points to $\delta=-15^{\circ}$. The final studies produced very clean optics,
with low sidelobes and good cross-polarization rejection. The telescope focal
length is 63 m, with a full beam of $0.67^{\circ}$, achieved with slight
under-illumination of the secondary mirror to avoid stray radiation. Table 2
contains a summary of the BINGO main parameters, being a subset of Table 1 of
Wuensche et al. (2021b).

The engineering project for the focal plane allows the vertical, elevation and
azimuth, and longitudinal displacement of all feed horns aiming toward a
complete and more uniform coverage of the observed sky area. This focal
arrangement and clean beam are essential for the full success of the BINGO
experiment, as they allow the instrument to resolve structures corresponding
to a linear scale of around 150 Mpc in the BINGO redshift range. The chosen
declination strip also allows for overlapping with important optical surveys,
such as DES, Pan-Starss, LSST and 6dF. The details of the optical design are
reported in Abdalla et al. (2021).

BINGO receivers will operate at room temperature in a correlation receiver
(CR) mode to avoid gain variations and $1/f$ noise levels that might hamper
its capability to detect the Hi signal. The CR radiometer chains were
initially measured in Manchester and later at INPE, and we expect to operate
at nominal $T_{\rm{sys}}\lesssim 70$ K. The prototypes for the front end,
including feed horns, polarizers and magic tees were fully manufactured and
successfully tested in Brazil. The design, construction and testing of the
prototype corrugated horn and polarizer are described in Wuensche et al.
(2021b).

The telescope is being built in a very isolated area in Aguiar, Paraíba,
northeastern Brazil (Lat: 7∘ 2’ 29” S Long: 38∘ 16’ 5” W), in an area
surrounded by hills, with very small nearby population and hardly any mobile
signal in the surroundings. A request for a “radio-quiet” zone has been placed
to the local authorities and to ANATEL, the Brazilian regulator agency for
telecommunications. The site selection process and RFI measurements are
described in Peel et al. (2019). Fig. 1 presents an artist’s view of the
telescope. The picture displays a west-east view (left-right) of the site,
with the sheltering hill in the background and the access road ending in a
service deck. The current design contemplates a control building located under
the deck.

Table 1: Summary of the BINGO telescope parameters - Phase 1

Parameter | Value
---|---
Tsys (K) | 70
Lower frequency cutoff (MHz) - $f_{\rm{min}}$ | 980
Higher frequency cutoff (MHz) - $f_{\rm{max}}$ | 1260
Frequency band - B (MHz) | 280
Number of redshift bandsaa$a$Assumed for science simulations, operational value will be downsampled from the digital backend output | 30
Estimated sensitivitybb$b$Computed for 1 pixel, full 280 MHz band, 1 year of observations and a 60% duty cycle ($\mu$K) | 206
Number of circular polarizations | 2
Focal length (m) | 63.2
Number of feed horns (Phase 1) | 28
Primary dish main diameter (m) |  $\sim$ 40.0
Secondary dish main diameter (m) |  $\sim$ 35.6
Focal plane area ($\textrm{m}^{2}$) | $18.6\times 7.8$
Instantaneous focal plane area ($\textrm{deg}^{\circ}$) | $14.75\times 6.0$
Optics FWHM (deg, at 1.1 GHz) | 0.67
Pixel solid angle (sqr deg) - $\Omega_{\rm{pix}}$ | 0.35
Telescope area (m2) - $A_{\rm tot}$ | 1602
Telescope effective area (m2) - $A_{\rm eff}$ | $\sim 1120$
Telescope azimuth orientation (deg) | 0 (North)
Primary mirror declination pointing (deg) | -15
Declination strip - instantaneous (deg) |  14.75∘
Full survey area - 5 yr(square deg) | $\sim 5324$
Mission duration (Phase 1) | 5 years
Duty cycle (estimated, Phase 1) | 60 - 90% of the time
222

Figure 1: BINGO telescope site - artist’s view. In this figure we show the
hill and the use of the natural inclination of the terrain to place mirrors
and horn structure. In this conception the control cabin is located under the
service deck.

BINGO is funded by several Brazilian and Chinese agencies. From the Brazilian
side, the major supporter is FAPESP, with 60% of the funds. China and the UK
have also contributed in terms of parts and components.

As far as the project timeline is concerned, soil moving services, as land
cleaning and topography, have already been carried out. The project received a
green light to go ahead after a major project review in July 2019. We had a
positive report from five panelists, including two foreign specialists. The
engineering project for the telescope structure, as well as optical project,
have been completed, along the main receiver prototype. Some of the
construction aspects were impacted in 2020 due to the Covid-19 pandemic, but
we had a major advance in early 2021 regarding terrain prospection, roadwork,
and the area cleaning.

We have a major part of funding (around 70%) already approved and available
for use from the FAPESP, the São Paulo State Funding Agency. We also received
contributions from our Chinese partners regarding electronic components
(circulators, isolators, connectors, cables).

##  3 Scientific goals I: cosmology

In this section we describe the Hi IM signal for cosmology and how it can be
used as a tool to measure BAO and RSDs. As a new observation, plenty of work
has to be done to understand the foregrounds and systematics involved in the
experiment, leading to the development of simulation and analysis pipelines.
This work is important to guarantee that BINGO measurements can indeed place
competitive constraints on cosmological parameters. With these tools, we show
how the BAO and RSD measured from BINGO can be used to constrain cosmological
models and test extensions of the $\Lambda$CDM model (Costa et al. 2021).

The aim of this section is to present the 21-cm signal and the corresponding
Hi angular power spectrum. Then using the Fisher matrix formalism we obtain
forecasts for several extensions of the $\Lambda$CDM model and alternative
cosmologies, such as dynamical DE, interacting DE and modified gravity. Bounds
on the sum of neutrino masses and on the 21-cm parameters are also presented.
For further details interested readers can find more information in the
companion paper VII (Costa et al. 2021).

###  3.1 The Hi intensity mapping signal and power spectrum

The 21-cm line emission from Hi is a technique already used for many years in
astronomy to measure the rotation curves of galaxies (Huchtmeier 1975; Bosma
1981; Corbelli et al. 1989; Puche et al. 1991; Olling 1996). Since hydrogen is
the most abundant component in our Universe, measuring its 21-cm emission line
can be used as a tracer of the LSS. This is the idea of the 21-cm Hi as a
cosmological probe. However, after reionization there is little hydrogen left
in neutral form. Hi is mostly hosted inside dense clouds in damped Ly-$\alpha$
(DLA) systems, that provides shielding to preserve the Hi neutral, with
optically-thin Ly-$\alpha$ absorbers in low-density regions being responsible
for a small part of the signal (Prochaska et al. 2005; Curran & Webb 2006).
Inside those hosting clouds, the excited energy level is more populated given
that they have higher temperatures than the energy difference between the
hyperfine levels, although below the DLA transition energy.

The 21-cm emission is a highly forbidden process, with a transition rate of
$2.9\times 10^{-15}\mathrm{s}^{-1}$, however, the amount of hydrogen in
galaxies is very high, allowing for the observation of the emission of the
21-cm line. In view of this faint signal, resolving individual galaxies
requires much longer integration times. Instead, we can perform a low angular
resolution survey that contains a large number of sources and map the
integrated 21-cm emission of those unresolved targets. This is the 21-cm IM
technique, allowing to probe large volumes very efficiently.

Since the wavelength is redshifted by the expansion of the Universe, we have a
precise determination of the redshift of the emitting source. We can thus map
the distribution of the Hi in the Universe at different redshifts, performing
a tomographic analysis that allows studying the evolution of the matter
distribution. This precise and free measurement of the redshift allows for a
precise determination of the RSD signal. These features make 21-cm IM surveys
very efficient for determining RSDs.

The IM technique measures the full intensity field, the brightness
temperature, from the 21-cm emission. In the Rayleigh-Jeans limit and at low
redshifts, the brightness temperature can be calculated as the sum of all the
emitted photons, received by an observer along the line of sight. In a
Friedmann-Lemaître-Robertson-Walker (FLRW) universe in the absence of
perturbations, the brightness temperature is given by (Hall et al. 2013)

$\displaystyle\bar{T}_{b}(z)=\frac{3}{32\pi}\frac{(2\pi\hbar
c)^{3}n_{\rm{HI}}A_{21}}{k_{B}\,E_{21}^{2}(1+z)H(z)}\,,$ (1)

where $E_{21}=5.88\,\mu\mathrm{eV}$ is the energy of a 21-cm photon,
$A_{21}\simeq 2.869\times 10^{-15}\,\mathrm{s}^{-1}$ is the spontaneous
emission coefficient, $n_{\rm{HI}}=\rho_{\rm{HI}}/m_{\rm{HI}}$ is the number
density of Hi atoms of mass $m_{\rm{HI}}$, $\rho_{\rm{HI}}$ the Hi average
energy density. For a homogeneous and isotropic background energy density, we
have $\bar{\rho}_{\rm{HI}}(z)=\Omega_{\rm{HI}}(z)\rho_{\rm{crit}}(1+z)^{3}$,
where $\Omega_{\rm{HI}}=\rho_{\rm{HI}}/\rho_{\rm{crit}}$,
$\rho_{\rm{crit}}=3M^{2}_{\rm{pl}}H_{0}^{2}$ is the critical energy density
and $M_{\rm{pl}}$ is the reduced Planck mass. The brightness temperature is
thus given by

$\bar{T}_{b}(z)=44\,\mu\mathrm{K}\,\left(\frac{\Omega_{\rm{HI}}(z)h}{2.45\times
10^{-4}}\right)\frac{(1+z)^{2}}{E(z)}\,,$ (2)

where $H_{0}=100h\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$ and
$E(z)=H(z)/H_{0}$. This expression describes the evolution of the brightness
temperature with redshift, given a specified cosmology.

The perturbed regime can be obtained considering the metric expanded in the
conformal Newtonian gauge,

$ds^{2}=a^{2}(\eta)\Big{[}(1+2\Psi(\eta,\@vec{x}))d\eta^{2}-(1-2\Phi(\eta,\@vec{x}))d\@vec{x}^{2}\Big{]}\,,$
(3)

where the functions $\Phi$ and $\Psi$ determine the space-time gravitational
potentials. In the presence of perturbations, the brightness temperature
receives corrections coming from
$n_{\rm{HI}}(z,\hat{\mathbf{n}})=\bar{n}_{\rm{HI}}(1+\delta_{n})$, from the
change in the redshift measured by the observer in this perturbed metric and
the perturbed conformal time
$\delta\eta(z,\hat{\mathbf{n}})=\bar{\eta}_{z}+\delta\eta$ around the
unperturbed value $\bar{\eta}_{z}$. To quantify these changes to first order,
we expand the observed brightness temperature,

$T_{b}(z,\hat{\mathbf{n}})=\bar{T}_{b}\left(1+\Delta_{T_{b}}(z,\hat{\mathbf{n}})\right)\,,$
(4)

where the mean brightness temperature $\bar{T}_{b}$ is given in Eq. (2) and
$\Delta_{T_{b}}(z,\hat{\mathbf{n}})$ is the first-order fractional
perturbation to the brightness temperature. Taking into consideration all the
corrections, the first-order fractional perturbation of $T_{b}$ is given by

$\displaystyle\Delta_{T_{b}}(z,\hat{\mathbf{n}})$
$\displaystyle=\delta_{n}-\frac{1}{\mathcal{H}}\,\hat{\mathbf{n}}\cdot\left(\hat{\mathbf{n}}\cdot\nabla\mathbf{v}\right)+\left(\frac{d\ln(a^{3}\bar{n}_{\rm{HI}})}{d\eta}-\frac{\dot{\mathcal{H}}}{\mathcal{H}}-2\mathcal{H}\right)\delta\eta$
$\displaystyle+\frac{1}{\mathcal{H}}\dot{\Phi}+\Psi\,.$ (5)

This expression is fully relativistic and each term in this expression
corresponds to a different physical effect (Hall et al. 2013). The first term
is the perturbation of the Hi density from discrete sources that host the Hi
in Newtonian gauge. The second term is the RSD term that comes from peculiar
velocities of the sources; the third term encodes the effects of the zero-
order brightness temperature calculated at the perturbed time of the observed
redshift; and the last two terms are related to the integrated Sachs-Wolf
(ISW) effect and conversion between increments in redshift to radial distances
in the gas frame, respectively.

Now, we are ready to define the Hi power spectrum. In many papers in the
literature of 21-cm Hi IM, the 3D $k$-space power spectrum is used (see e.g.,
Bull et al. 2015). However, there are many advantages of using a tomographic
2D angular power spectrum that are specially important for the BINGO science
goals, survey strategy and the way the experiment is designed. Among those
advantages is the fact that this approach is independent of the cosmological
model, since it is not necessary to assume a cosmology to convert into
distance. It also allows the inclusion of lensing effect and wide angle
correlations, and it can be easily used to make cross correlations with other
LSS tracers (for a discussion of the advantages see Shaw & Lewis 2008; Di Dio
et al. 2014; Tansella et al. 2018; Camera et al. 2018). We evaluate the Hi
angular power spectra from the first order brightness temperature
fluctuations. First, we decompose the brightness temperature fluctuation in
spherical harmonics,

$\Delta_{T_{b}}(z,\hat{\mathbf{n}})=\sum_{\ell m}\Delta_{T_{b},\ell
m}(z)Y_{\ell m}(\hat{\mathbf{n}})\,.$ (6)

We then Fourier transform the coefficient $\Delta_{T_{b},\ell m}(z)=4\pi
i^{\ell}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3/2}}\Delta_{T_{b},\ell}(\mathbf{k},z)Y^{*}_{\ell
m}(\hat{\mathbf{k}})$ and re-write the brightness temperature perturbation as
(Hall et al. 2013)

$\displaystyle\Delta_{T_{b},\ell}(\mathbf{k},z)$
$\displaystyle=\delta_{n}j_{\ell}(k\chi)+\frac{kv}{\mathcal{H}}j_{\ell}^{{}^{\prime\prime}}(k\chi)+\left(\frac{1}{\mathcal{H}}\dot{\Phi}+\Psi\right)j_{\ell}(k\chi)$
$\displaystyle-\left(\frac{1}{\mathcal{H}}\frac{d\ln(a^{3}\bar{n}_{\rm{HI}})}{d\eta}-\frac{\dot{\mathcal{H}}}{\mathcal{H}^{2}}-2\right)$
$\displaystyle\times\left[\Psi
j_{\ell}(k\chi)+vj_{\ell}^{{}^{\prime}}(k\chi)+\int_{0}^{\chi}(\dot{\Psi}+\dot{\Phi})j_{\ell}(k\chi^{{}^{\prime}})d\chi^{{}^{\prime}}\right]\,.$
(7)

Here, $\chi$ is the conformal distance to redshift $z$, “ ′ ” denotes the
derivative with respect to $k\chi$ and $j_{\ell}$ is the spherical Bessel
function, ignoring the monopole and dipole terms.

We thus define the tomographic angular power spectrum,

$C_{\ell}(z_{i},z_{j})=4\pi\int d\ln
k\mathcal{P}_{\mathcal{R}}(k)\Delta^{W}_{T_{b},\ell}(k,z)\Delta^{W^{{}^{\prime}}}_{T_{b},\ell}(k,z)\,,$
(8)

where in
$\Delta^{W}_{T_{b},\ell}(\mathbf{k},z)=\int_{0}^{\infty}dz\bar{T}_{b}(z)W(z)\Delta_{T_{b},\ell}(\mathbf{k},z)$
we integrate the temperature fluctuation over the redshift bin with the
normalized window function $W(z)$;
$\Delta^{W}_{T_{b},\ell}(k,z)=\Delta^{W}_{T_{b},\ell}(\mathbf{k},z)/\mathcal{R}(\mathbf{k})$
is the transfer function divided by the primordial curvature perturbation
$\mathcal{R}(\mathbf{k})$, with dimensionless power spectrum given by
$\mathcal{P}_{\mathcal{R}}(k)$.

###  3.2 Baryon acoustic oscillations and redshift space distortion

The BAO are signatures in the matter distribution from the recombination epoch
and have become a prime cosmological probe currently in use. They are based on
small oscillations in the photon baryon cosmic fluid that are imprinted on
matter perturbations after matter and radiation decouple at $z\approx 1090$
and show up in the distribution of galaxies at all redshifts. The scale of
these oscillations ($\approx 150\,\textrm{Mpc}$) is fixed in comoving
coordinates and can be used as a standard ruler to measure the geometry of the
Universe, thus probing the effects of DE (Percival et al. 2010; Beutler et al.
2011; Anderson et al. 2014; Delubac et al. 2015). In Fourier space the BAO
feature appears as wiggles in the power spectrum. Fig. 2 shows the BAO wiggles
in the Hi angular power spectra for the two redshift boundaries of BINGO.

Figure 2: Hi angular power spectra with a bandwidth of 9.33 MHz at redshifts
$z=0.127$ (red) and $z=0.449$ (blue) showing the BAO feature.

.

The RSD is one of the most powerful probes of the rate of growth of structure
(Costa et al. 2017). We have to consider $f\sigma_{8}$, where
$f=-d\ln\delta(z)/d\ln(1+z)$ is the growth rate and $\sigma_{8}$ measures the
amplitude of the (linear) power spectrum on the scale of
$8h^{-1}\,\mathrm{Mpc}$. RSD are effects that appear when we probe observables
in 3D space, where the radial (comoving) distance is determined by the
observed redshift. It receives contributions from peculiar velocities and
makes the redshift-space clustering anisotropic. 21-cm IM experiments are
specially good to determine the RSD signal given its precise method for
determining the redshift. Including RSD information leads to the breaking of
degeneracies between parameters and the improvement in the constraint of many
cosmological parameters, as we show below.

###  3.3 Constraining cosmology

BINGO is a telescope that will measure BAO and RSD using Hi as a tracer of the
matter density in our Universe. This will provide precise measurements of the
expansion history and growth of structures, which can be used to study the
evolution of our Universe, properties of DE, extensions to the standard
$\Lambda$CDM model and Hi physics.

Alternative cosmological models are usually based on either introducing an
exotic cosmological component or by modifying GR (Copeland et al. 2006; Wang
et al. 2016). Each of those classes encompasses many different models with
different properties and parameters. To distinguish between those models, new
and precise measurements of the expansion history and structure growth are
necessary, which is the aim of BINGO.

BINGO will be particularly powerful to improve the constraints on current
cosmological parameters in combination with other experiments, such as optical
surveys and CMB measurements. This will allow to put tighter constraints on
extensions of the $\Lambda$CDM model, which are competitive with current and
future LSS experiments. We have forecast BINGO capacity in several different
scenarios in the companion paper VII (Costa et al. 2021). The forecast was
done using the Fisher matrix formalism considering the instrumental setup as
described in Table 1 but for an observational time of 1 year (and sky coverage
with Galactic mask of 2900 deg) (Costa et al. 2021), and fiducial cosmological
parameters listed in Table 2. In this section, we highlight the main results.

Table 2: Baseline values assumed for the cosmological parameters used in the Fisher forecast analysis, from Switzer et al. (2013); Aghanim et al. (2020). The last two parameters describe the Hi physics. Parameter | Baseline
---|---
$\Omega_{b}h^{2}$ | 0.022383
$\Omega_{c}h^{2}$ | 0.12011
$h$ | 0.6732
$n_{s}$ | 0.96605
$A_{s}$ | $2.1\times 10^{-9}$
$w_{0}$ | -1
$w_{a}$ | 0
$b_{\rm{HI}}$ | 1
$\Omega_{\rm{HI}}$ | $6.2\times 10^{-4}$

####  3.3.1 $\Lambda$CDM model

The $\Lambda$CDM model is the basis for the current description of the
evolution and composition of our Universe. This phenomenological model is our
concordance model in cosmology, exhibiting exquisite agreement with current
precise large scale observations. We assume here that the Universe is
spatially flat and seeded by adiabatic nearly scale invariant
perturbations.333Spatial flatness can be relaxed. This model is composed of
baryons, radiation, and two unknown components that are responsible for the
majority of the energy budget of the Universe today, DM and DE. In this model,
DM is the component responsible for the clustering of perturbations and the
formation of structures, and it is described in the hydrodynamical limit by a
cold, presureless fluid that is at most weakly interacting with baryons, the
CDM. DE is the component responsible for the acceleration of the Universe and
it is compatible, in the $\Lambda$CDM model, with a cosmological constant.

Such a simple and successful model is parametrized by only six
parameters,444Sometimes, instead of $h$, the acoustic angular scale
$\theta_{*}$ is used.

$\left\\{\Omega_{b}h^{2},\,\Omega_{c}h^{2},\,\tau,\,A_{s},\,n_{s},\,h\right\\}\,,$
(9)

where $\Omega_{i}=\rho_{i}/\rho_{\rm{crit}}$ are the baryonic ($i=b$) and CDM
($i=c$) density parameters today, $\tau$ is the reionization optical depth,
$A_{s}$ is the amplitude of the primordial curvature perturbation, $n_{s}$ is
the spectral index of scalar perturbations and $h$ is the Hubble parameter,
defined through the Hubble constant $H_{0}$ by $H_{0}=100h$ km s-1 Mpc-1.
These parameters can be constrained at the level of percent to subpercent by
current cosmological observations (Aghanim et al. 2020). For 21-cm probes, we
have to consider two additional parameters in the analysis describing the Hi
signal: the Hi density parameter and the bias, $\Omega_{\rm{HI}}$ and
$b_{\rm{HI}}$, respectively.

Although BINGO alone does not provide competitive constraints on the
$\Lambda$CDM parameters as compared to Planck (Aghanim et al. 2020), its
combination with Planck measurements can improve the cosmological fits,
especially in the Hubble constant and matter density (by $\sim 25\%$). This
breaking of the degeneracy present in the Planck data is not new and has been
discussed extensively in the data including in Aghanim et al. (2020) and the
main reason for it is the presence of a distance indicator such as the BAOs in
the nearby Universe. Besides, BINGO can help solving the tension in $H_{0}$
between Planck and low redshift measurements (Riess et al. 2019).

####  3.3.2 Dynamical dark energy

In the $\Lambda$CDM scenario, the current acceleration of the Universe is
explained by the introduction of a cosmological constant, also interpreted as
a component with equation of state equal to $w=p/\rho=-1$. Although simple and
well motivated observationally, this description is challenging from a
theoretical point of view.

Moreover, recent data from low redshift measurements present tensions when
compared with Planck results under the $\Lambda$CDM scenario. This motivates
the search for alternative models to explain the current acceleration
(Copeland et al. 2006; Wang et al. 2016). Those models consider that DE can be
a dynamical component. The simplest phenomenological modification from the
cosmological constant is a dynamical model where the equation of state is a
constant, $w=w_{0}$, which is now a free parameter, called the $w$CDM model.
Our forecast shows that BINGO alone could already constrain this parameter
with a better precision than Planck ($\sigma w_{0}^{\rm{BINGO}}=17\%$ against
$\sigma w_{0}^{\rm{Planck}}=26\%$). The combination of BINGO and Planck has
shown to be very powerful to constrain $w_{0}$, with $\sigma
w_{0}^{\rm{comb.}}=3.3\%$, an improvement of more than $87\%$ in the
constraint of the DE equation of state.

Figure 3: Contour plot showing the $68\%$ and $95\%$ confidence level
constraints on the Hubble constant and DE equation of state parameters using
BINGO, Planck and BINGO + Planck, for the dynamical DE model ($w$CDM).

In general, we can consider that the equation of state varies with time (or
with the scale factor). Although there are more sophisticated models leading
to that behavior, we can probe the evolution of $w(z)$ in a phenomenological
way by parametrizing its evolution, as (Chevallier & Polarski 2001; Linder
2003)

$w(z)=w_{0}+w_{a}\frac{z}{1+z}\,,$ (10)

known as the CPL parametrization. This model is also called the
$w_{0}w_{a}$CDM model and has two extra free parameters, $w_{0}$ and $w_{a}$,
changing the normalized Hubble parameter (see companion paper VII, Costa et
al. 2021). Even tough we introduce two new parameters, the model is well
behaved for small redshifts and bounded for high redshifts, which allows a
manageable exploration of its parameter space. One of its advantages is that
it can be used in many scalar field models of DE and, in this sense, it is a
quite general description of a dynamical DE component. Answering the question
whether DE is evolving is one of the biggest questions in cosmology, thus
testing these models with BINGO is an important goal.

In Figures 3 and 4 we show the 68% and 95% confidence level marginalized
constraints for $H_{0}$ and $w_{0}$ (for the wCDM model) and for the equation
of state parameters $w_{0}$ and $w_{a}$ (for the CPL parametrization),
respectively. The combination BINGO + Planck improves considerably the
constraints on those parameters. In the case of the constraints on the DE
equation of state, we compare BINGO’s performance with the one from SKA band 1
and band 2, which correspond to different frequency (redshift) bands of SKA
(Bacon et al. 2020). The constraints on the DE equation of state are the ones
that present the most significant improvement for SKA in comparison to BINGO.
The band 2 of SKA, which corresponds to lower redshifts than band 1
($z=0-0.5$), presents a much stronger constraint than BINGO. The other
cosmological parameters show only a small improvement when SKA is considered,
as it can be seen in the in the companion paper VII (Costa et al. 2021),
together with further details. These results show the scientific potential of
the BINGO project.

Figure 4: Comparison between BINGO, SKA Band 1 and SKA Band 2 for the $68\%$
and $95\%$ confidence level constraints on the DE equation of state parameters
for the CPL parametrization in combination with Planck.

####  3.3.3 Modified gravity

The two new components, DM and DE, have motivated extensions of the standard
theory, such as Brans-Dicke, Hornsdeski and $f(R)$ gravity. They generally
predict a different perturbation dynamics. Scalar perturbations on the FLRW
background are described in the Newtonian gauge by Eq. (3) with the Poisson
equation

$k^{2}\Psi=-4\pi Ga^{2}\mu(a,k)\rho\Delta\,.$ (11)

The parameters $\mu(a,k)$ and $\gamma(a,k)=\frac{\Psi}{\Phi}$ parametrize
deviations from GR, and are $\gamma=\mu=1$ in GR.

To forecast constraints on modified gravity models, in the companion paper VII
(Costa et al. 2021) a specific form for the parametrization above is
considered, which is related to $f(R)$ theories of gravity. The
$B_{0}$-parametrization of $f(R)$ gravity provides a good approximation on
quasi-static scales (Hu & Sawicki 2007; Giannantonio et al. 2010; Hojjati et
al. 2012) and is expressed as

$\displaystyle\mu(a,k)$ $\displaystyle=(1-1.4\times 10^{-8}|\lambda/{\rm
Mpc}|^{2}a^{3})^{-1}\frac{1+4\lambda^{2}k^{2}a^{4}/3}{1+\lambda^{2}k^{2}a^{4}}\,,$
(12) $\displaystyle\gamma(a,k)$
$\displaystyle=\frac{1+2\lambda^{2}k^{2}a^{4}/3}{1+4\lambda^{2}k^{2}a^{4}/3}\,,$
(13)

where $B_{0}\equiv 2H_{0}^{2}\lambda^{2}$. Considering the $\Lambda$CDM
parameters plus $B_{0}$, we obtained the constraints

$\displaystyle\sigma_{B_{0}}$ $\displaystyle=3.1\times
10^{-5}\quad\textrm{(BINGO)}$ (14) $\displaystyle\sigma_{B_{0}}$
$\displaystyle=5.3\times 10^{-2}\quad\textrm{({Planck})}$ (15)
$\displaystyle\sigma_{B_{0}}$ $\displaystyle=1.1\times
10^{-5}\quad\textrm{(BINGO + {Planck})}\,.$ (16)

BINGO alone can already put a better constraint on $B_{0}$ as compared to
Planck and the combination of the two surveys can improve it by $\sim 64\%$.

####  3.3.4 Interacting dark energy

It has also been proposed that DM and DE interact (Amendola 2000; Zimdahl &
Pavon 2001; Chimento et al. 2003). Such interacting models are challenging
because of the unknown nature of both components: it becomes hard to describe
the origin of such an interaction from first principles. Despite that, many
different models have been studied in the literature (for a review see Wang et
al. 2016).

These models are generally described by the divergence of the energy-momentum
tensor of the dark sector components,
$\nabla_{\mu}T_{(i)}^{\mu\nu}=Q_{(i)}^{\nu}$. The quadri-vector
$Q_{(i)}^{\nu}$ provides the energy-momentum transfer between DM and DE, such
that the total conservation law is still obeyed, $\sum_{i}Q_{(i)}^{\nu}=0$.

In order to forecast the power of BINGO in constraining those kinds of models,
we consider a phenomenological interaction described by the DM-DE energy
transfer as

$Q=3H(\xi_{dm}\rho_{dm}+\xi_{de}\rho_{de})\,,$ (17)

where $\xi_{dm}$ and $\xi_{de}$ are the coupling constants. Here we present
the general results for these models, while we refer readers to the more
detailed study of the BINGO forecasts in our companion paper VII (Costa et al.
2021).

We can consider two distinct situations: an interaction proportional to the DM
density or an interaction proportional to the DE density. Because of regions
of instabilities, the last one should be analyzed in two different regions,
$w>-1$ and $w<-1$ (He et al. 2009, 2011). BINGO will provide complementary
constraints on the interacting DE parameters and the equation of state. This
will help break the degeneracy between them and provide additional information
to the CMB data. Considering the BINGO fiducial setup in combination with
Planck data, we obtain

  * •

$Q=3H\xi_{dm}\rho_{dm}$:

$\displaystyle\sigma_{w}$ $\displaystyle=0.078\,,$ (18)
$\displaystyle\sigma_{\xi_{dm}}$ $\displaystyle=0.0007\,.$ (19)

  * •

$Q=3H\xi_{de}\rho_{de}\quad\hbox{and}\qquad w>-1$:

$\displaystyle\sigma_{w}$ $\displaystyle=0.047\,,$ (20)
$\displaystyle\sigma_{\xi_{de}}$ $\displaystyle=0.018\,.$ (21)

  * •

$Q=3H\xi_{de}\rho_{de}\quad\hbox{and}\qquad w<-1$:

$\displaystyle\sigma_{w}$ $\displaystyle=0.056\,,$ (22)
$\displaystyle\sigma_{\xi_{de}}$ $\displaystyle=0.015\,.$ (23)

The combination with BINGO improves the constraints from Planck by at most
$\sim 81\%$ for the DE EoS ($Q\propto\rho_{dm}$) and $\sim 77\%$ for the
interacting parameter ($Q\propto\rho_{de}$, $w>-1$). Recently, Bachega et al.
(2020) have analyzed these models with the latest Planck data and in
combination with current BAO and SNIa data. Our results from BINGO + Planck
put competitive constraints to those obtained combining Planck with five
optical BAO data from 6dFGS ($z=0.106$), SDSS-MGS ($z=0.15$), and BOSS data
release 12 ($z=0.38,0.51,0.61$), or combining Planck with the Pantheon SNIa
data.

####  3.3.5 Massive neutrinos

We know today that neutrinos leave detectable imprints on the LSS of our
Universe. This knowledge comes from results on neutrino oscillations
experiments that show that neutrinos are massive, as a consequence of their
oscillation between flavors (Fukuda et al. 1998; Ahmad et al. 2002).

Given their small mass and large thermal velocities, neutrinos can free stream
away from dense regions before recombination and wash away structures on small
scales, which can be measured as a suppression in the matter power spectrum.
Although this effect is small, it can already be measured by current
experiments and nowadays, with the precision that cosmological experiments
have, it is necessary to take into account the effect of massive neutrinos.
The upper limit on the total neutrino mass from combining Planck temperature
and polarization with BAO data is $\sum
m_{\nu}<0.13\,\mathrm{eV}\,(95\%\,\text{CL})$ (Aghanim et al. 2020). Future
experiments have the goal to improve those bounds.

The presence of massive neutrinos affects cosmology, which is sensitive to its
mass and number of massive species. This change in the components of our
cosmology modifies the angular diameter distance to last scattering. This can
be probed using the angular scale of the CMB first peak. However, this
measurement is degenerate with the density of DE and $H_{0}$ (Olivari et al.
2018). Using LSS data, as the BAO peaks of the Hi power spectrum, can help
reducing this degeneracy and improve the limits on the total mass of
neutrinos.

Including massive neutrino seems to be a necessary extension to $\Lambda$CDM.
In this way we include the $\sum m_{\nu}$ as an extra free parameter and
forecast the improvement BINGO will bring to the bounds of such a parameter.
We estimate that BINGO in combination with Planck data will be able to further
constraint the total neutrino mass as

$\sigma_{\sum m_{\nu}}<0.14\,\textrm{eV}\quad(95\%\,\textrm{CL})\,.$ (24)

This upper bound is slightly higher than the one obtained by the Planck
Collaboration (Aghanim et al. 2020). Therefore, BINGO will provide data
through a different tracer and can impose competitive constraints on the sum
of the neutrino masses.

####  3.3.6 Redshift space distortions

In this work, we have not considered disentangling the contribution of RSD
from the Hi power spectrum and extracted the growth rate yet, nor its
subsequent performance to constrain the cosmological parameters. However, in
the companion paper VII (Costa et al. 2021), we have analyzed the impact RSD
has on the total angular power spectra and how this affects our final
constraints. First, including RSD information leads to the breaking of
degeneracies between the primordial scalar amplitude $A_{s}$ and the Hi bias.
Second, Fig. 5 shows how the constraints on several cosmological parameters
improve adding information from RSD as a function of the number of bins.

Figure 5:  Percentage difference between using or not information from RSD
($\Delta\sigma_{\theta}/\sigma_{\theta}=\sigma_{\theta\,\text{without}}/\sigma_{\theta\,\text{with}}-1$)
in the cosmological parameter constraints from BINGO. $A_{s}$ and
$b_{\textrm{H{\sc i}\,}}$ were not included because they go from complete
ignorance to constraints of about percent level.

####  3.3.7 HI history

The BINGO telescope will not only constrain the cosmological parameters as
described in the previous sections, but it will also help shed light on the Hi
evolution and distribution. Using the 21-cm line of neutral hydrogen as a
tracer of the underlying matter distribution requires the knowledge of the Hi
mean density and its bias. The simplest model assumes that they are constant,
parametrized by $\Omega_{\rm{HI}}$ and $b_{\rm{HI}}$. Therefore, together with
the cosmological parameters, BINGO will constrain these two additional Hi
parameters. However, $\Omega_{\rm{HI}}$ and $b_{\rm{HI}}$ are strongly
correlated. As we discuss above, we also have a complete degeneracy between
$A_{s}$, and $\Omega_{\rm{HI}}$ and $b_{\rm{HI}}$. For this reason, in order
to constrain these quantities independently we need external information. We
show the constraints in $\Omega_{\rm{HI}}$ and $b_{\rm{HI}}$ in Fig. 6, where
we used BINGO in combination with Planck. RSDs information can also help break
this degeneracy, not only between $A_{s}$ and $\Omega_{\rm{HI}}$, but also
between $\Omega_{\rm{HI}}$ and $b_{\rm{HI}}$. In general, those parameters
evolve with time and their measurement describes the Hi history.

Figure 6: Contour plot showing the $68\%$ and $95\%$ confidence level
constraints on the Hi density parameter and bias using BINGO + Planck.

In the companion paper VII, the ability of BINGO to constrain the Hi density
and bias is considered under several circumstances. In Fig. 6 the marginalized
constraints on these two parameters is presented assuming they are constant
over the BINGO redshift range.

####  3.3.8 Systematic effects

One of the main challenges of 21-cm experiments and of the BINGO project are
astrophysical contamination and systematic effects. Systematic effects,
present in the observed Hi signal might affect the constraints on cosmological
parameters that the BINGO telescope can obtain. The main systematics affecting
BINGO and any single-dish 21-cm experiment are $1/f$ noise, standing waves,
atmospheric effects and RFI. Residuals from foreground removal will also be
present and can strongly impact the measured Hi signal. These systematic and
foreground effects are explained in Section 5 in detail, together with the
work made in the BINGO pipeline to mitigate them and their impact in
recovering the Hi signal (see also Olivari et al. 2018).

##  4 Scientific goals II: astrophysics

Although the BINGO telescope is mainly designed to characterize BAO, it is an
experiment that is versatile enough to allow for further scientific goals. The
exceptionally long integration times and spectral stability will enable it to
make important contributions in other fields, such as transient objects
science, with the newly discovered and exciting FRBs and pulsars, as well as
Galactic science.

###  4.1 Transient objects science

The dynamic radio sky includes astrophysical phenomena that emit pulses in the
radio frequency with a large variety of temporal and spectral features. The
period range is from microseconds to a few minutes. Sometimes they can repeat,
sometimes they are single events. The spectral index may show wide variations.
Also diverse are the source objects, from neutron stars to magnetars
(basically highly magnetized and rotating neutron stars) and flare stars. A
better understanding of these phenomena will be profitable to many
astrophysical and cosmological research fields, since they are related to non
conventional emission mechanisms. They could be used as probes of the
intergalactic medium (IGM), indicating the presence of matter and magnetic
fields along the line of sight. The new member of this family, the most
recently discovered, are the so called FRBs. We present here the capabilities
of the BINGO telescope in studying such transients.

####  4.1.1 Fast radio bursts

Discovered in 2007 (Lorimer et al. 2007), FRBs are bright bursts ($0.05-100$
Jy) of radio photons of unknown origin (see Pen 2018; Platts et al. 2019;
Popov et al. 2018; Petroff et al. 2019 for recent reviews), lasting from some
microseconds to a few seconds ($80\,\mu\text{s}-5\text{ s}$). The radio waves
are dispersed by free electrons along the line-of-sight between the source and
the observer, thus the time delay of photons with different frequencies can be
measured. This time delay is proportional to the dispersion measure $\int
n_{e}\,dl$ (where $n_{e}$ is the number of electrons per volume and $l$ is the
distance along the line-of-sight), and it depends on the observed frequencies
$\nu_{1}$ and $\nu_{2}$ as $\Delta t=e^{2}/(2\pi
m_{e}c)(\nu_{2}^{-2}-\nu_{1}^{-2})\int n_{e}dl$, where $m_{e}$ is the electron
mass. The dispersion measure may have contributions from the vicinity of the
FRBs source, the host galaxy and its halo, the IGM, and the Milky Way and its
halo.

There are so far over 100 detected FRBs in a frequency range between 300 MHz
and 8 GHz (Gajjar et al. 2018; Chawla et al. 2020) with dispersion measure
approximately ranging from $100$ pc cm-3 to $2600$ pc cm-3, where a few of
them showed repeated bursts (Spitler et al. 2016; Andersen et al. 2019; Kumar
et al. 2019) and two appear to be periodic (Amiri et al. 2020; Rajwade et al.
2020). Although their origin is still a mystery, most of them are
extragalactic, with only one having been produced within the Milky Way by a
magnetar (Andersen et al. 2020). Due to their still uncertain origin, several
explanations have been proposed (Platts et al. 2019), including, but not
limited to, mergers and interactions between compact objects (such as neutron
stars, black holes and white dwarfs), active galactic nuclei and supernovae
remnants. Due to the distinct repeating feature of some FRBs, it is possible
that non repeating and repeating FRBs do not share the same common progenitor.

Additionally, FRBs can be very useful to various applications in cosmology and
astrophysics, such as constraining cosmological parameters (Zhou et al. 2014;
Yang & Zhang 2016; Walters et al. 2018; Li et al. 2018; Yu & Wang 2017; Wei et
al. 2018; Zitrin & Eichler 2018), testing the Weak Equivalence Principle (Wei
et al. 2015; Tingay et al. 2015; Shao & Zhang 2017; Yu et al. 2018; Bertolami
& Landim 2018; Yu & Wang 2018; Xing et al. 2019), estimating the photon mass
(Xing et al. 2019; Wu et al. 2016; Wei et al. 2017; Bonetti et al. 2016, 2017;
Wei & Wu 2018), studying the properties of the IGM (Deng & Zhang 2014; Zhang
2014; Akahori et al. 2016; Fujita et al. 2017; Shull & Danforth 2018; Ravi,
Vikram 2019), and compact DM (Muñoz et al. 2016; Wang & Wang 2018), among
others (Linder 2020; Landim 2020).

Several telescopes have been detecting FRBs, such as the Parkes Radio
Telescope (Lorimer et al. 2007), the Australian Square Kilometre Array
Pathfinder (ASKAP; Bannister et al. 2017), the Green Bank Telescope (GBT;
Masui et al. 2015), the UTMOST telescope (Caleb et al. 2017), the Arecibo
Observatory (Spitler et al. 2014), the Canadian Hydrogen Intensity Mapping
Experiment (CHIME; Amiri et al. 2018), the Apertif on the Westerbork Synthesis
Radio Telescope (Connor et al. 2020), among others.555See
http://www.frbcat.org/ for a complete list of the detected FRBs.

BINGO has the potential to detect FRBs. Here we show an estimate of the
detection rate for the BINGO telescope. The flux density depends on the
frequency as $S(\nu)=S_{0}\nu^{\alpha}$, where $\alpha$ is the spectral index
and can be either positive or negative. The minimum detectable flux density
for a flat spectrum ($\alpha=0$) is

$S_{\rm{rms}}=\text{S/N}\frac{K\,T_{\rm{sys}}}{G\sqrt{n_{p}\Delta\nu\,\tau}}\,,$
(25)

where $T_{\rm sys}$ is the system temperature, $G$ is the forward gain (in
units of K/Jy), $K=\sqrt{2}$ for a correlation receiver, $n_{p}$ is the number
of polarizations, $\Delta\nu$ is the bandwidth, $\tau$ is the sampling time
and S/N is the signal-to-noise ratio.

For a general spectral index the detectable flux density (25) is the peak flux
density
$S_{\rm{peak}}=\text{S/N}\,T_{\rm{sys}}/(G\sqrt{n_{p}\Delta\nu\,\tau})$, which
is written in terms of the bolometric luminosity $L_{\rm bol}$ as (Lorimer et
al. 2013)

$S_{\rm{peak}}=\frac{1}{4\pi
d_{L}(z)^{2}}\frac{L_{\rm{bol}}}{\nu_{\rm{high}}^{1+\alpha}-\nu_{\rm{low}}^{1+\alpha}}\left(\frac{\nu_{2}^{1+\alpha}-\nu_{1}^{1+\alpha}}{\nu_{2}-\nu_{1}}\right)\,,$
(26)

where $d_{L}(z)$ is the luminosity distance, $\nu_{1}$ and $\nu_{2}$ are the
observed frequencies, and $\nu_{\rm{high}}$ and $\nu_{\rm{low}}$ are the
highest and lowest frequencies, respectively, in which the source emits, as
seen by the observer.

For a general spectral index the received power delivered by an antenna is

$\displaystyle P_{\rm{rec}}$
$\displaystyle=\frac{1}{2}A_{e}\int_{\nu_{1}}^{\nu_{2}}S_{0}\nu^{\alpha}\,d\nu\,,$
$\displaystyle=\frac{1}{2}A_{e}S_{0}\frac{\nu_{2}^{1+\alpha}-\nu_{1}^{1+\alpha}}{1+\alpha}\,,$
(27)

while the Johnson–Nyquist noise power remains the same as in the case of flat
spectrum

$P_{\rm{noise}}=k_{B}T_{A}(\nu_{2}-\nu_{1})\,,$ (28)

where $T_{A}$ is the antenna temperature. Using the radiometer equation

$\text{S/N}=\frac{T_{A}}{T_{sys}}\sqrt{\Delta\nu\,\tau}\,,$ (29)

the total flux density for $n$ channels is

$S_{\rm{tot}}=\text{S/N}_{\rm
tot}\frac{K\,T_{\rm{sys}}}{G\sqrt{n_{p}\Delta\nu\,\tau}}\,,$ (30)

where the total S/N is given by

$\text{S/N}_{\rm{tot}}=\text{S/N}\frac{\nu_{2}-\nu_{1}}{\nu_{2}^{1+\alpha}-\nu_{1}^{1+\alpha}}\frac{(1+\alpha)}{\sqrt{n}}\left(\sum_{p=1}^{n}\nu_{p}^{2\alpha}\right)^{1/2}\,.$
(31)

The factor $\sqrt{n}$ comes from the division of bandwidth by the number of
channels $n$ and

$\nu_{p}\equiv\nu_{1}+\left(p-\frac{1}{2}\right)\frac{\Delta\nu}{n}\,.$ (32)

The behavior of $\text{S/N}_{\rm{tot}}/\text{S/N}$, from Eq. (31), is shown in
Fig. 7 for the BINGO telescope. Since the peak flux density is also given by
Eq. (25) the relation between $S_{\rm{peak}}$ and $S_{\rm{tot}}$ is

$S_{\rm{tot}}=\frac{\text{S/N}_{\rm{tot}}}{\text{S/N}}S_{\rm{peak}}\,.$ (33)

Figure 7: Deviation of Eq. (31) from S/N (for a flat spectrum) as a function
of the spectral index $\alpha$, for different number of channels $n$. It can
be seen that the total S/N is only $\sim$ 1.5 times larger for very large
spectral index $|\alpha|$.

Having determined the flux density for a given survey through Eq. (25), the
detection rate of FRBs can be calculated using the same approach as Luo et al.
(2018, 2020), where the Schechter function (Schechter 1976) was assumed as the
distribution of the FRBs luminosity and the intrinsic FRB width distribution
is given by a log-normal function (Connor 2019)

$\displaystyle\lambda(\Omega,S_{\rm{peak}})=$
$\displaystyle\int_{0}^{\Omega}d\Omega\int_{0}^{z_{max}}dz\frac{c}{1+z}\frac{r(z)^{2}}{E(z)}\int_{w_{i}^{\prime}}^{\infty}\frac{e^{-\frac{(\log
w_{i}-\mu_{w})^{2}}{2\sigma_{w}^{2}}}}{\sqrt{2\pi\sigma_{w}^{2}}}\,d\log
w_{i}$ $\displaystyle\int_{L_{\rm
min}}^{\infty}\phi^{*}\left(\frac{L}{L^{*}}\right)^{\gamma}e^{-\frac{L}{L^{*}}}\frac{dL}{L^{*}}\,,$
(34)

where $r(z)$ is the comoving distance, $w_{i}=w_{0}/(1+z)$, $w_{0}$ is the
observed pulse width, and the upper cut-off luminosity $L_{*}$ and the
normalization $\phi_{*}$ are parameters constrained in Luo et al. (2020). The
integral of the Schechter function gives the incomplete gamma function
$\phi^{*}\Gamma(1+\gamma,L_{\rm{min}}/L^{*})$, where a (Gaussian) beam
response has already been integrated out (Luo et al. 2020). The minimum
luminosity is given by the $L_{\rm{min}}=\max(L_{0},L_{\rm{bol}})$, where the
bolometric luminosity is given by Eq. (26) and we take the cut-off luminosity
to be $L_{0}\leq 9.1\times 10^{41}\text{erg s}^{-1}$ (Luo et al. 2020).

Using the constrained parameters found in Luo et al. (2020) ($L^{*}=2.9\times
10^{44}\text{erg s}^{-1}$, $\phi^{*}=339\text{ Gpc}^{-3}\text{yr}^{-1}$,
$\gamma=-1.79$, $\mu_{w}=0.13$ and $\sigma_{w}=0.33$), $z_{\rm{max}}=6$,
$w_{0}=1$ ms and S/N $=5$ we show the detection rate for a flat spectrum for
the BINGO telescope in Fig. 8, along with the corresponding rates for other
surveys. Those rates are also given in Table 3 with different S/Ns.

In Fig. 9 we estimate the detection rate for BINGO, for a range of spectral
indices between $-5\leq\alpha\leq+15$ for three values of S/N. The highest and
lowest frequencies ($\nu_{\rm high}$ and $\nu_{\rm low}$) are chosen as
$\nu_{\rm high}=1400$ MHz and $\nu_{\rm low}=400$ MHz to give illustrative
results of the impact of the spectral index on the detection rate. The case
$\alpha=0$ recovers the results presented in Fig. 8 (Luo et al. 2020).
According to the Schechter luminosity function, populations with larger
luminosities are more suppressed than the ones with smaller luminosities, thus
the detection rate (as seen in Fig. 9) is smaller for more negative spectral
indices, since they lead to larger luminosities. In order to investigate the
impact of $\nu_{\rm{high}}$ and $\nu_{\rm{low}}$ on the detection rate, we
calculated the rate $\lambda_{\alpha}/\lambda_{0}$ for two other frequency
ranges, such that their differences are still 1 GHz: $\nu_{\rm{high}}=1260$
MHz and $\nu_{\rm{low}}=260$ MHz, and $\nu_{\rm{high}}=1980$ MHz and
$\nu_{\rm{low}}=980$ MHz. For the lower frequency band ($\nu_{\rm{high}}=1260$
MHz and $\nu_{\rm{low}}=260$ MHz) the peak flux density (26) is higher for
positive spectrum indices than its corresponding value for the flat spectrum
($S_{\rm{peak},\alpha>0}>S_{\rm{peak},\alpha=0}$), while for the higher
frequency band ($\nu_{\rm{high}}=1980$ MHz and $\nu_{\rm{\rm{low}}}=980$ MHz)
the peak flux density is higher for negative spectrum indices
($S_{\rm{peak},\alpha<0}>S_{\rm{peak},\alpha=0}$). Increasing the peak flux
density increases the detection rate as well because given a S/N threshold a
signal that would be smaller than this threshold now can be larger. For FRBs
with spectral indices between $-5$ and $5$, for example, the difference in the
detection rate from the lower band choice ($\nu_{\rm{high}}=1260$ MHz and
$\nu_{\rm{low}}=260$ MHz) to the higher band choice ($\nu_{\rm{low}}=260$ MHz,
and $\nu_{\rm{high}}=1980$ MHz) is only $\sim 2.5$.

Figure 8: FRB detection rate per day (4.1.1) as a function of the FoV and the minimum detectable flux density. Following Luo et al. (2020) we show the BINGO detection rate along with other surveys: Arecibo (Spitler et al. 2014), ASKAP (Shannon et al. 2018), CHIME (Amiri et al. 2018), FAST (Jiang et al. 2020), GBT (Masui et al. 2015), MeerKAT (Bailes et al. 2020), Parkes (Keith et al. 2010), UTMOST (Caleb et al. 2016) and Tianlai (Li et al. 2020). The BINGO detection rate is shown by the dot within its logo. The minimum flux density (corresponding to S/N$=1$) is 0.41 Jy and the FoV is 10.5 $\text{deg}^{2}$.  Table 3: Mean time elapsed between two detections (in hours), for the surveys presented in Fig. 8.  Survey | S/N $\geq$ 3 | S/N $\geq$ 5 | S/N $\geq$ 10
---|---|---|---
CHIME | 1.5 | 2.4 | 4.6
MeerKAT | 22.6 | 34.4 | 61.6
UTMOST | 42.5 | 68.7 | 135.3
ASKAP | 50 | 90.7 | 210.2
GBT | 72.2 | 112.2 | 207.5
BINGO | 112.4 | 189.8 | 398.5
FAST | 133.5 | 201.2 | 352.5
Parkes | 223.3 | 351.4 | 662.5
Tianlai | 245.1 | 463.9 | 1139
Arecibo | 799.1 | 1212.3 | 2150.6
Figure 9: Detection rate for the BINGO telescope for different spectral
indices and S/Ns, normalized to the corresponding values for $\alpha=0$.

In order to achieve the capability of resolving FRB spectral signatures, BINGO
will need a complementary setup in the digital backend output, with a sampling
time of $\thicksim 0.05-0.1$ ms (much shorter than what is needed for
intensity mapping) and 1024 – 2048 output channels. This fine resolution is
needed for RFI flagging and data reduction. After the data cleaning and FRB
searching process, data channels can be binned in the nominal value of 30,
enough for BAO data analysis. The localization of FRBs on the sky will be
possible using outrigger telescopes and it will be explored in future work.

####  4.1.2 Pulsars

Pulsars are rapidly spinning highly magnetized neutron stars that emit beams
of electromagnetic waves. They take their name from their periodic emission
caused by the rotation frequency of the radiation beam in our line of sight,
like a lighthouse. In most cases, the pulse profile is extremely stable, both
in form and in the arrival phase at the rotational frequency. This stability
allows for the precision timing of pulsars, with remarkable applications
(Beskin et al. 2015).

A fixed position in the sky will take $\sim$ 28 minutes to pass over the focal
plane of BINGO. However, due to the sparse configuration of the horns (see
Figures 4 and 5 of Wuensche et al. 2021b), considering the areas with
responses higher than $50\%$ of the peak gain, a fixed position will pass over
for $\sim$ 40 arc-minutes accumulated, which corresponds to $\sim$ 3 minutes
and is taken as the integration time for one-day observation in the following
estimations.

Assuming that BINGO will have buffers large enough to store the pulsar data
within 3 minutes, the sensitivity of pulsar detection ($S_{\rm{pulsar}}$) can
be estimated via

$S_{\rm
pulsar}=\rm{S/N}\frac{\sqrt{2}\,T_{sys}}{G\sqrt{n_{p}\Delta\nu\tau}}\,,$ (35)

where $T_{\rm sys}=70$ K is the system temperature, $G=0.32$ K/Jy is the gain,
$n_{p}=2$ is the polarization number, $\Delta\nu=280$ MHz is the bandwidth,
and $\tau=3$ minutes is the integration time. This gives a sensitivity of 2.92
and 4.87 mJy with S/N equals 3 and 5, respectively. In the BINGO FoV (strip
between DEC$=-10\deg$ to DEC$=-25\deg$), there are 372 known pulsars in the
Australia Telescope National Facility (ATNF) pulsar
catalog666https://www.atnf.csiro.au/research/pulsar/psrcat/ (Manchester et al.
2005). We estimated their fluxes at 1.1 GHz (the mid-frequency of BINGO) using
a power-law fitting to their existing flux data in other frequencies, or a
spectral index of $-1.6$ is assumed. The 38 brightest known pulsars can be
detected over S/N$=3$ with one-day observation, and 4 of them are millisecond
pulsars.

Taking the advantage of BINGO’s daily monitoring of the same sky stripe for
five years, we could at least focus on the following sciences with these
$\sim$ 40 brightest pulsars in the FoV.

Refractive scintillation (RISS): RISS of pulsars when their emissions pass
through the interstellar medium (ISM) with large-scale inhomogeneities
(Kumamoto et al. 2021). The pulsar RISS phenomenon normally has a timescale of
one or few weeks, which perfectly fits into long-term daily observations. The
understanding of the RISS is still relatively poor although the underlying
physics of RISS is clear, which is mainly because of the uncertainty about the
ISM inhomogeneities in our Galaxy. The observational result from BINGO will
help to improve the understanding the ISM distribution in our Galaxy.

Monitor the change in pulsar periods: The pulsed emission of pulsar may appear
to be completely ceased over the time scale of the pulsar period to several
years, called pulse nulling (e.g., Young et al. 2015). The physics behind
pulse nulling has not been fully understood yet. Besides, some pulsars have
quasi-periodic switching, which might be caused by magneto-spherical
switching. Pulsar periods can also change due to the orbital motion on the
timescale of several years (e.g., Keane 2013).

Pulsar glitch: The glitch of a pulsar refers to a sudden increase in its
rotating frequency, and the exact reason that causes a glitch is still unknown
(Haskell & Melatos 2015). Up to now, only few glitches are observed in Crab
and Vela pulsars. BINGO has the penitential to discover more example of pulsar
glitches (also for fainter pulsars in the FoV) and to study the underlying
physical nature.

Rotating radio transients (RRATs): RRATs are a type of pulsar discovered
through single pulse searches, instead of the usual Fourier domain searches.
Their detected radio pulses are more sporadically than the normal class. The
reason for this anomalous feature is not yet known but could be due to
fallback of material from a supernova debris disk, presence of an asteroid
belt around the pulsar or just a normal pulsar showing extreme nulling
(McLaughlin et al. 2006; Burke-Spolaor 2013; Keane & McLaughlin 2011; Shapiro-
Albert et al. 2018).

The pulsar science could be extended if a hydrogen maser is to be installed in
the telescope. We will be able to combine observations of more than one day to
achieve much higher sensitivities. For 10- and 100-day integrations, the
$S_{\rm pulsar}$ goes down to 0.92 mJy and 0.29 mJy with S/N$=3$, which
corresponds to the detection of 103 and 207 known pulsars in the FoV,
respectively. We could at least study their pulse shape and polarization,
although flux estimation would be difficult. With the hydrogen maser, precise
pulsar timing could be done with the brightest pulsars as well. The
millisecond pulsars among them would contribute to the pulsar timing array for
the detection of the gravitational waves at nHz, which might be generated from
super-massive black-hole binaries at a cosmological distance (Wang & Mohanty
2017), and measure the gravitational constant as well as examine the
gravitational theory (e.g., Hobbs & Dai 2017).

###  4.2 Galactic science

Four satellite missions were designed to map CMB fluctuations: Relikt (1983)
(Strukov & Skulachev 1984; Strukov et al. 1993), COBE (1989 – 1993) (Bennett
et al. 1992b), WMAP (2001 – 2010) (Gold et al. 2011) and Planck (2009 – 2013)
(Planck Collaboration et al. 2020). Three of them (COBE, WMAP, Planck)
produced, as part of their ancillary science data, sets of maps with Galactic
synchrotron, bremsstrahlung, and dust emission (unpolarized and some
polarized). The lower frequencies where these satellites operated ($22$, $30$,
$37$, $40$, $44$, $53$, $70$ GHz) cover the frequency band of minimum Galactic
contribution to CMB measurements.

However, around $1\mathrm{GHz}$, where BINGO will operate, there will be a
clear predominance of synchrotron emission dominating the Galactic foreground
contribution (Berkhuijsen 1972; Reich & Reich 1986; Reich et al. 2001),
seconded by free-free and then, depending on the chosen model, by Anomalous
Microwave emission (AME) (Dickinson et al. 2018). Although BINGO will not map
a vast area of the sky, we expect to collect data that can contribute to
cross-check observations of ongoing sky mapping experiments.

For instance, at 1,465 MHz, a frequency close to the BINGO band, the total sky
brightness has been mapped in two adjacent 60-degree declination bands with
the Galactic emission mapping (GEM) experiment, which employs a portable 5.5-m
parabolic reflector to survey the radio continuum of the sky in decimeter and
centimeter wavelengths (Tello et al. 2000). The GEM experiment also mapped the
sky at 2.3 GHz (Tello et al. 2013). BINGO will add well-calibrated data to the
Galactic microwave spectrum, contributing to a better determination of the
spectral index of the synchrotron emission at the low-frequency microwave
region.

###  4.3 Extragalactic science – associated Hi absorption

Associated Hi absorption is detected against bright radio continuum emission
sources with a compact core, like active galactic nuclei (AGNs) and nuclear
starbursts, due to absorption by Hi clouds within the continuum sources
themselves. They are some of the best objects to study the interplay of AGN
and stellar feedback on the interstellar medium of galaxies (Morganti &
Oosterloo 2018). There are less than 100 known detections of associated Hi
absorption even with targeted searches, especially at $z>0.25$ (e.g., Roberts
1970; Dickey 1986; van Gorkom et al. 1989; Carilli et al. 1998; Gallimore et
al. 1999; Moore et al. 1999; Pihlström et al. 2003; Vermeulen et al. 2003;
Morganti et al. 2005; Gupta et al. 2006; Chandola et al. 2011; Darling et al.
2011; Allison et al. 2012; Chandola et al. 2013; Allison et al. 2014; Geréb et
al. 2015; Maccagni et al. 2017; Aditya & Kanekar 2018a, b; Aditya 2019).

In a Hi IM survey such as BINGO, the bright radio continuum emission sources
will typically by flagged out at the data reduction stage. But given the high
sensitivity of such surveys, many of the radio continuum sources observed by
them will have detectable associated Hi absorption in them. Roychowdhury et
al. (2019) specifically calculated the expected number of associated Hi
absorptions that will be detected with BINGO (see their Appendix B for
details). For the full survey, they expect the total number of Hi absorption
detections will be almost twice the number known today. Given the low spatial
resolution of BINGO, each such detection should ideally be followed-up by
interferometric observations. Thus BINGO can provide targets to follow-up for
the existing SKA precursors like ASKAP and MeerKAT, which are also observing
the southern sky.

##  5 Simulations and data analysis pipeline

The BINGO simulation pipeline is an ongoing effort, being developed alongside
the instrument construction, covering input modules for sky emission models
and experiment characteristics, a mission simulation module, production of
time ordered datasets (TOD), calibration and cleaning in the time domain, map-
making, generation of multifrequency maps, component separation for foreground
removal, reconstructed Hi sky maps and power spectrum estimation. The
information flow in the BINGO mission simulation and data analysis pipeline is
shown in Fig. 10, and this section describes the pipeline modules according to
it.

Figure 10: Flowchart of the BINGO simulation pipeline.

###  5.1 Sky module

The observed brightness temperature of the sky,
$T_{\rm{sky}}(\nu,\mathbf{x})$, at frequency $\nu$ and direction $\mathbf{x}$,
is composed by several different sources (Wilson et al. 2013; Condon & Ransom
2016)

$\displaystyle T_{\mathrm{\rm{sky}}}(\nu,\mathbf{x})$
$\displaystyle=T_{\mathrm{\rm{gal}}}(\nu,\mathbf{x})+T_{\mathrm{eg}}(\nu,\mathbf{x})$
$\displaystyle+T_{\mathrm{CMB}}(\nu,\mathbf{x})+T_{\mathrm{atm}}(\nu,\mathbf{x})+T_{\mathrm{cosmo}}(\nu,\mathbf{x}),$
(36)

where $T_{\mathrm{gal}}$ is the diffuse Galactic radiation, $T_{\mathrm{eg}}$
is the emission from extragalactic sources, $T_{\mathrm{CMB}}(\nu,\mathbf{x})$
is the CMB temperature, $T_{\mathrm{atm}}(\nu,\mathbf{x})$ is the atmospheric
emission, and $T_{\mathrm{cosmo}}$ is the cosmological Hi emission. We briefly
describe their contributions to the sky module below.

####  5.1.1 Hi emission

The synthetic Hi maps used in the BINGO simulation pipeline are produced in
two ways. One of them is to use the FLASK (Full-sky Log-normal Astro-fields
Simulation Kit) package777 Available at http://www.astro.iag.usp.br/~flask/.
(Xavier et al. 2016). FLASK uses as input the angular auto- and cross-power
spectra, $C^{ij}(\ell)$, calculated for each of the $i$ and $j$ fields and
redshifts, to produce two-dimensional tomographic realizations (spherical
shells around the observer) of an arbitrary number of random astrophysical
fields, reproducing the desired cross-correlations between them, derived
either from a multivariate Gaussian distribution or a multivariate log-normal
distribution.

An option to using FLASK as the Hi input data is to simulate 21cm IM mock
maps, based upon N-body simulations and assuming all the Hi gas is contained
within galaxies or dark matter halos. We used the N-body simulation Horizon
Run 4 (HR4; Kim et al. 2015), which consists of $6300^{3}$ particles in a
cubic box of volume $3150^{3}\,\mathrm{Mpc}^{3}/h^{3}$ to produce a full sky
mock map using the halo and galaxy light cone catalog where the halo mass
resolution is $2.7\times 10^{11}M_{\odot}/h$. We use the Hi HOD model and
abundance matching to generate the full sky 21cm IM mock map. The details of
the mock catalog production are discussed in companion paper VI (Zhang et al.
2021).

The project plans to combine the fast log-normal realization with FLASK and
the more optimal N-body simulation mock for pipeline testing.

####  5.1.2 Galactic foregrounds

It is well known that the faint Hi signal expected to be detected in IM
experiments will be dominated by different astrophysical foregrounds, broadly
classified in Galactic diffuse emission and the extragalactic, point source
emission. Foregrounds removal and component separation methods have been
explored by the Planck Collaboration (Adam et al. 2016a) and here we use some
of these techniques.

The Galactic foreground is mainly constituted by four components. The main
contribution comes from synchrotron emission from relativistic cosmic ray
electrons interacting with the Galactic magnetic field, producing a power law
that dominates the Galactic foreground at the BINGO frequencies. Results from
the WMAP team present strong evidence of a flatter spectral index in the
plane, steepening with Galactic latitude (Bennett & Larson 2013). Typically,
it is expected that the synchrotron spectral index varies from
$\beta_{s}\approx 2.6$ in most of the galactic plane to $\beta\approx 3.1$ in
the halo around 1 GHz (Bennett et al. 2003).

The second strongest contribution comes from free-free emission caused by
electrons scattering off ions in the interstellar medium. The $H\alpha$
emission line is usually used to trace the free-free emission as they both
depend on the emission measure $\text{EM}=\int n_{e}^{2}dl$ (Dickinson et al.
2003). The free-free spectral index is a slowly varying function of frequency
with a slight dependence on the local value of $T_{e}$ (Bennett et al. 1992a).
At $1\,\mathrm{GHz}$ and considering $T_{e}=7000$ K as a typical value for the
electron temperature, the spectrum has a temperature spectral index of
$\beta_{\rm{ff}}=2.1$. Both synchrotron ($C_{\ell}^{Synch}$) and free-free
emission ($C_{\ell}^{Free-Free}$) can be seen to dominate the signal power
spectrum, together with the contribution from strong radio sources, in Fig.
11.

Anomalous microwave emission (AME) produced by the the electric dipoles in the
small spinning dust grains (Leitch et al. 1997; Draine & Lazarian 1999) is one
of the dominant foregrounds in the frequency range of $10-60\,\mathrm{GHz}$
(Dickinson et al. 2018; Adam et al. 2016b). AME is not an important
contaminant to the 21-cm in BINGO frequency range, being subdominant with
power spectrum well-bellow the one from signal. The same applies for the
thermal dust emission from dust grains heated by interstellar radiation field
(Abergel, A. et al. 2011). Between 980 and 1260 MHz, thermal dust and AME
($C_{\ell}^{AME}$) contributions are well below the Hi signal
($C_{\ell}^{21_{40}}$), where the Hi power spectrum was obtained from maps
convolved with a 40 arcmin beam. Their contribution in the BINGO central
frequency is also shown in Fig. 11.

####  5.1.3 Extragalactic point sources and CMB

Extragalactic emission in the BINGO band mainly comes from discrete radio
point sources. They comprise all radio emissions from astrophysical objects,
such as the inhomogeneous mix of radio galaxies, quasars, star-forming
galaxies, etc. Strong sources whose emission can be well characterized by a
power-law in the radio band can be identified and masked out during the
analysis, essentially eliminating their contribution to the incoming signal.

There is also a diffuse contribution from weaker, unresolved, radio sources
that is indistinguishable from the Galactic, spatially continuous, foreground.
It contaminates the incoming signal to an intensity level slightly lower than
the Galactic free-free emission for lower ($\ell\lesssim 100$) multipoles and
overcoming the Galactic synchrotron emission for higher ($\ell\gtrsim
300-400$) multipoles. It is expected to be significant at high Galactic
latitudes, but can be simulated as another diffuse background and combined
with the other continuous foreground emissions. The contribution of the faint
radio point sources in the power spectrum ($C_{\ell}^{FRPS}$) can be seen in
Fig. 11.

The CMB, with a mean temperature today of $T_{\rm{CMB}}\simeq 2.72548$ K
(Fixsen 2009), contributes to the Hi signal with its temperature fluctuations
$\Delta T/T\sim 10^{-5}$, being a relevant foreground at $\approx
1\,\mathrm{GHz}$. Even though it is not the main contaminant for the 21-cm
signal, its temperature power spectrum ($C_{\ell}^{\mathrm{CMB}}$) is still
higher than the 21-cm power spectrum ($C_{\ell}^{21_{40}}$) for the multipole
range up to $\ell\simeq 300$ for the BINGO central frequency band as shown in
Fig. 11.

A detailed description of the sky simulations, including the foreground
models, can be found in the BINGO companion papers IV and V (Liccardo et al.
2021; Fornazier et al. 2021).

Figure 11: Angular power spectra (up to $\ell=10^{3})$ of the foregrounds
described above, compared with the signal for the central BINGO frequency band
($1125\,\mathrm{MHz}$) ($C_{\ell}^{21_{40}}$), considering a sky coverage of
about 3500 square degrees ($f_{\rm{sky}}\approx 8.7\%$).

###  5.2 Mission simulation

This module takes as input the simulated sky maps (HEALPix maps) described in
section 5.1 and reproduces the observation processes to create TODs. The
mission simulation includes various instrument information, such as receiver
temperature, focal length, horn coordinates in the focal plane, number of
receivers, beam profile, number of frequency channels, thermal and $1/f$
noise, noise power amplitude, correlations among different channels etc. More
details on the parameters used in the simulations can be found in the
companion paper II (Wuensche et al. 2021b). Different horn configurations,
mission duration and different foreground combinations have been tested and
are discussed in the companion paper IV (Liccardo et al. 2021).

####  5.2.1 Beam profiles

A useful simulated sky map is produced by convolving the frequency-dependent
beam response pattern with the sky signal creating a TOD as similar to the
real data as possible. The BINGO observing strategy of letting the sky drift
across the beam makes the beam summations much simpler, effectively it occurs
in only one direction. Frequency response is currently modeled with a Gaussian
beam, with plans to use the beam profile measured for the prototype horn
(Wuensche et al. 2020) in future works. At this point we have not implemented
polarization effects yet (leakage etc.).

####  5.2.2 Instrumental noise

Thermal noise in the TOD can be accurately modeled in the simulations from a
Gaussian white noise distribution. The stochastic variations in the amplifier
voltage are characterized by the receiver system temperature $T_{\rm{sys}}$
using the radiometer equation (e.g., Wilson et al. 2013). BINGO receivers are
also subject to gain variations, which, incorporated to the radiometer
equation, is presented in the form (Burke et al. 2019)

${\sigma_{N}}=K{T_{\rm{sys}}}\sqrt{\frac{1}{{\tau\Delta\nu}}+{{\left({\frac{{\Delta
G}}{G}}\right)}^{2}}}\,,$ (37)

where $\Delta G$ describes the stochastic variations in the mean gain.
$\sigma_{N}$ is the minimum noise variation that can be detected in the TOD,
$\tau$ is the receiver sampling rate and $\Delta\nu$ is the receiver
bandwidth. For the BINGO correlation receiver, $K=\sqrt{2}$.

The $1/f$ noise is a different phenomenon from thermal noise, being a form of
correlated noise present in radio receiver systems, caused by small gain
fluctuations (see, e.g., Maino et al. 1999; Seiffert et al. 2002; Meinhold et
al. 2009). It appears in the data as large scale spatial fluctuations that are
not trivial to separate from the true underlying sky signal. The removal of
1/f noise has also become an area of research that has resulted in several
advanced map-making methods, discussed in section 5.3. We model the $1/f$
contribution as (Harper et al. 2018)

$P\left(\nu\right)={\sigma^{2}}\left[{1+{{\left({\frac{\nu_{\rm{k}}}{{{\nu}}}}\right)}^{\alpha}}}\right]\,,$
(38)

where $P$($\nu$) is the spectral power of each temporal mode, $\sigma$ is the
radiometer sensitivity as described by Eq. (37), $\nu$ is the sampling
frequency, $\nu_{\rm{k}}$ is the temporal knee frequency, in which the
correlated noise and white noise contribute to the signal with the same level
of power, and $\alpha$ is the spectral index of the correlated noise.
Typically, the spectral index will vary between $1$ (usually referred to as
“pink noise” and $2$ (“brown noise”). Simulations allow the analysis of the
effects of white noise, knee frequency and $1/f$ spectral index variations
during a given mission length.

#####  5.2.2.1 Atmosphere

Radio waves at $\sim 1\,$GHz cross the atmosphere essentially without being
absorbed. However, atmospheric emission, dominated by Oxygen with a small
fraction of precipitable water vapor (PWV), contributes with
$1.81\textrm{K}\leq T_{atm}\lesssim 1.82\textrm{K}$ to the final system
temperature (Bigot-Sazy et al. 2015). Atmospheric fluctuations around
$\bar{T}_{atm}$ can mimic a correlated noise, due to varying opacity from
water, and can be treated as a residual $1/f$ in the later phases of the data
analysis. These fluctuations depend on on-site atmospheric conditions,
observing frequency band and also on the sky scan strategy (Church 1995; Lay &
Halverson 2000, e.g.), and are caused by the anisotropic distribution of water
vapor molecules in the upper atmosphere. Bigot-Sazy et al. (2015) estimate the
atmospheric fluctuations as seen by a telescope such as BINGO should be
$\Delta T_{\rm atm}=0.01$ mK, below the instrumental noise in Table 2 of the
companion paper II Wuensche et al. (2021b). The atmosphere is not expected to
have any intrinsic contribution to the polarization observations at BINGO
frequencies (e.g., Hanany & Rosenkranz 2003), but it is still important as a
contribution to $I$ into $Q/U$ leakage. The simulations generated for the
companion paper V (Liccardo et al. 2021) do not take the atmospheric
contribution into account.

####  5.2.3 Radio frequency interference

RFI is the usual term to nominate the emission of radio waves by man-made
sources, such as personal computers, cell phones, airplanes, power lines, and
orbiting satellites, such as those used for radio-navigation transmissions
(Harper & Dickinson 2018).

Removal of the transient RFI spikes requires the definition of a threshold for
identifying spikes. The pixel that contains the brightest observed Galactic
emission within the frequency band and the survey area is a convenient choice
for this threshold. Higher intensity spikes in the TOD are then flagged by
iteration, to check if they are astrophysical transient events. In negative
cases, the data chunks are excised out from the next data analysis steps. Our
simulations currently include the presence of ground transients appearing as
an intense spike in the TOD data and contributions from geostationary
satellites, for which orbit and transmitting power are well known.

###  5.3 Map-making

The map making step of the BINGO pipeline deals with the conversion from TOD
to a coordinate-based, pixel-stacked data. Mathematically, the map-making
equation, in its simplest linear form, is given by

$m\approx\hat{s}=Wd\,,$ (39)

where m is the map, $\hat{s}$ is our estimate of the signal, d is the data
(the TOD) and W is a matrix that when applied to the signal performs the
required steps. The BINGO pipeline currently has three different map-making
routines available: naive filtering and binning, destripping, and maximum
likelihood.

One of the implementations of the BINGO pipeline currently uses the simple
averaging and binning routines, in a process that is usually referred to as
“naive map-making”. This map-making technique performs no additional data-
processing in the timelines. The data are simply re-mapped onto the image
plane, by projecting the full power seen by a detector onto the nearest sky
map pixel. The output is an image map, an error map (as the standard deviation
of data samples divided by the square root of the number of data samples) and
a coverage map, effectively a hit map for samples/pixel.

The naive technique is the fastest of the three map-making schemes. It is
relatively easy to implement but might yield crude maps due to the simplified
noise estimation. When applying the naive technique we assume that the filters
in use in the pipeline manage to remove enough of the noise correlations, such
that we can estimate the remaining data to be signal and white noise only. For
the moment, we assume that all detectors read the same noise level. In this
case the data can simply be written as

$d=Ps+{n_{w}}\,,$ (40)

where P is a $(n_{t},\,n_{\mathrm{pix}})$-matrix taking the signal from the
time-domain into the pixel-domain, $s$ is the signal of interest, $n_{w}$ is
the uncorrelated white noise. Eq. (40), when solved for the estimated signal
$\hat{s}$, can be written as:

$\hat{s}={\left({{P^{T}}P}\right)^{-1}}{P^{T}}d\,.$ (41)

The result is a simple average of all observations per pixel. This equation
can be modified to account for individual weighting when calculating the bin-
average by adding a diagonal matrix.

###  5.4 Component separation

The component separation module of the pipeline is a very important part of
the data analysis, where the desired Hi signal must be separated from all
foregrounds and analyzed according to the BINGO noise model. The foreground
power spectra presented in Section 5.1.2 can be approximated by power-laws,
enabling their separation from the Hi signal. There are several component
separation methods to extract the target cosmological signal from the
foregrounds that can be divided into parametric and “blind” techniques. The
parametric methods assume each component is characterized by a specific
dependent emission law known a priori, whereas the “blind” techniques, as the
name suggests, do not assume any specific parametric model for the
foregrounds.

The pipeline work currently assumes a non parametric approach, using the
Generalized Needlet Internal Linear Combination (GNILC) methodology as the
standard component separation technique (Remazeilles et al. 2011; Olivari et
al. 2016) to recover the 21-cm signal. GNILC takes into consideration both the
frequency and spatial information of the observed (simulated) data in order to
separate the components. A set of needlet windows is defined in harmonic space
to isolate different ranges of angular scales in the input map. Each frequency
map can be split in several needlet maps, allowing for a more localized
analysis of the signal. Figure 12 shows maps covering the BINGO patch of the
sky with three different signals: input Hi plus noise, GNILC reconstructed Hi
plus noise, and residuals. The maps are produced at $\approx$ 1.1 GHz. It is
clear how GNILC is able to reconstruct the main features of the input signal
with a r.m.s residual equal to 0.06 mK For the scenarios considered we found
values off by $\leq$ 10%. More details regarding this analysis can be be found
in the companion paper V (Fornazier et al. 2021).

We intend to use other foreground cleaning methods, along with GNILC, in our
simulations in the future, as well as developing specific methods optimized
for Hi IM experiments. The GNILC algorithm can be improved to obtain a more
efficient component separation. The BINGO collaboration is working on the
optimization of the code and on a more precise estimation of the errors
arising during the Hi recovery process. A detailed description of the current
status of GNILC performance and the results of the sky simulations, including
the foreground models, can be found in the companion papers IV and V (Liccardo
et al. 2021; Fornazier et al. 2021).

Figure 12: Comparison between BINGO Hi plus noise maps ($N_{\rm{side}}=128$)
with a Galactic mask applied at $\approx$ 1.1 GHz (first strip), the GNILC
reconstructed map (second strip) and the residuals map (last strip). See the
companion paper IV for more details (Liccardo et al. 2021). Temperatures are
given in K.

###  5.5 Cosmology modules

This final part of the pipeline is about obtaining the science products from
the 21-cm Hi IM data. These products are science-dependent. For cosmology, for
instance, we would estimate the covariance, the Hi power spectra, perform a
Bayesian analysis to estimate the cosmological parameters of interest, and
estimate the bispectrum.

####  5.5.1 Covariance estimation

Constructing a covariance matrix is a key part for estimating observable
uncertainties and further constraining the parameter space of the cosmological
parameters. In our case, the observable is the angular power spectra of the Hi
IM map. For constructing the covariance matrix of the angular power spectrum,
a number of realizations are needed. However, a large scale N-body simulation
is computationally very expensive. The Gaussian random realization using the
calculated angular power spectra is applied in Asorey et al. (2020).

We use two methods to generate hundreds of realizations of the tomographic Hi
IM maps: the log-normal density field realization using the FLASK code (Xavier
et al. 2016) as well as fast light cone halo catalog generation using the
L-PICOLA code (Howlett et al. 2015). We can generate full-sky IM maps, despite
BINGO actually observing only about 15% ($6000$ deg2) of the sky. Therefore,
in every realization, we also mask the full-sky in different random directions
to achieve about eight times more maps from each realization. This reduces
significantly the computational cost for constructing the covariance matrix.

####  5.5.2 Bispectrum estimation

There is additional information in the 21-cm maps than the one that can be
obtained by the angular power spectrum. Non-gaussianities (NG), which are
deviations in our maps from Gaussian statistics, will be present in our data
at all redsfhits, but most notably in the low redshift 21-cm signal. They
might come from early-times evolution of the Universe, what we call primordial
NG; and/or will get imprinted onto the 21-cm maps by gravity itself; and/or
will get imprinted on the 21-cm maps when these maps are cleaned, by residuals
from the galactic foreground distribution. We use higher order statistics to
characterize these NG.

We focus here on the three-point correlation function, or its Fourier
transform, the bispectrum. The three-point correlation function in flat sky
approximation is given by (Liguori et al. 2010; Komatsu 2001; Smith &
Zaldarriaga 2011)

$\langle\Delta_{T_{b},\ell_{1}}\Delta_{T_{b},\ell_{2}}\Delta_{T_{b},\ell_{3}}\rangle\equiv
B_{l_{1}l_{2}l_{3}}^{m_{1}m_{2}m_{3}}=\langle
B_{l_{1}l_{2}l_{3}}\rangle\Bigg{(}\begin{matrix}l_{1}&l_{2}&l_{3}\\\
m_{1}&m_{2}&m_{3}\end{matrix}\Bigg{)}\,,$ (42)

where $B_{l_{1}l_{2}l_{3}}^{m_{1}m_{2}m_{3}}$ is the angular bispectrum
($B_{\ell}$ from now on), and $B_{l_{1}l_{2}l_{3}}$ is the averaged angular
bispectrum. The matrix denotes the Wigner-$3j$ symbol, which is invariant
under permutations. It describes three angular momenta that form a triangle
$\mathbf{L}_{1}+\mathbf{L}_{2}+\mathbf{L}_{3}=0$, where $m_{1}+m_{2}+m_{3}=0$.
The bispectrum can be written as

$B_{\ell}=G_{l_{1}l_{2}l_{3}}^{m_{1}m_{2}m_{3}}\,b_{l_{1}l_{2}l_{3}}\,,$ (43)

where $b_{l_{1}l_{2}l_{3}}$ is the reduced power spectrum and
$G_{l_{1}l_{2}l_{3}}^{m_{1}m_{2}m_{3}}$ is the Gaunt integral (for more
details see Komatsu 2001; Fornazier et al. 2021), non zero only if
$l_{1}+l_{2}+l_{3}=\mathrm{even}$, where all the information about the
Wigner-$3j$ symbol is present. The bispectrum can have different shapes. Here
we are going to focus on two: the equilateral shape, given by
$l_{1}=l_{2}=l_{3}$ and the isosceles shape, where $l_{2}=l_{3}\neq l_{1}$.

While the primordial NG is a signal we would like to extract from the data,
the residual NG in the maps coming from foreground subtraction and radio
frequency interference mitigation is a signal that is on the way of extracting
information about the cosmology from the cleaned 21-cm signal. For this reason
it is important to be able to detect the presence of these NG features in the
maps and characterize them to understand their origin and characteristic.

The goal of this module is to be able to identify cosmological NG information
in the reconstructed maps, after the component separation module (GNILC)
removed the foregrounds and recovered the cosmological 21-cm signal. The
module looks more specifically for NG with the equilateral and isosceles
shapes, and can evaluate the mean bispectrum $B_{\ell}$.

In order to verify how well our pipeline is recovering the cosmological Hi
data, we perform a few tests to verify if the cosmological NG 21-cm
information (previously included in the simulation) is recovered in the
reconstructed maps from GNILC, and, mainly, if the bispectrum module can
identify this information in those maps, by comparing those with graphical
results of foregrounds. To perform such tests, we compare the maps generated
by FLASK. It contains an NG information from the input log-normal
distribution, with the same maps after they are reconstructed by GNILC,
following the procedure explained in Section 5.4.

To compare these different maps and test the reconstruction procedure we use
the shape of the NG, and mean value of the $B_{\ell}$ obtained from each map.
Using the shape, the tests showed that we cannot recover the full 21-cm
information at some redshift bins, with the recovery becoming better and
yielding acceptable results for higher redshift bins. The isosceles shape
shows slightly more satisfactory results, as it can be seen in Fig. 13. The
foreground graphs are shown to be very different from the FLASK and recovered
maps, since it contains mixed NG and Gaussian information from different types
of contaminants. The reconstructed map has the 21-cm recovered information,
while the so called 21-cm prior has the pure cosmological signal, with NG
information. It is expected that these two maps have the same order of
magnitude and same component. However, this is not the case in our analysis,
and it is still not clear if this discrepancy comes from the reconstruction
procedure or from the bispectrum module.

Comparing the average amplitude of the bispectrum confirms these results,
where there is a consistent recovery, but not the entire signal. A more
detailed analysis can be found in the companion paper V (Fornazier et al.
2021).

Figure 13: Contour charts for bispectrum with equilateral and isosceles shape
for central channel ($\approx$ 1125 MHz), white noise case. The first column
corresponds to equilateral case and the second to isosceles one. The first row
corresponds to foreground information, the second row corresponds to the 21-cm
map from the FLASK simulation and the third row from the reconstructed map
(GNILC).

####  5.5.3 Minkowski functionals estimation

A different and complementary tool to the previous optimal NG estimator based
on the bispectrum is the Minkowski functionals (MF; Minkowski 1903; Novikov et
al. 1999; Komatsu et al. 2003). Sensitive to weak and arbitrary NG signals,
the MF not only provide information about the spatial correlation of a random
field, but can also provide morphological information and map the shape of
structures.

For the 2D Hi brightness temperature field, defined on the sphere ${\cal
S}^{2}$, $T=T(\theta,\phi)$, with mean $\langle T\rangle$ and variance
$\sigma_{0}^{2}$, three MF are enough to provide a test of non-Gaussian
features by assessing properties of connected regions in a map. The global MF
are the total area
$V_{0}(\nu_{t})=(1/4\pi)\int_{\Sigma}d\Omega=\sum_{i}a_{i}$, perimeter
$V_{1}(\nu_{t})=(1/8\pi)\int_{\partial\Sigma}dl=\sum_{i}l_{i}$ and genus
(number of holes)
$V_{2}(\nu_{t})=(1/8\pi^{2})\int_{\partial\Sigma}\kappa\leavevmode\nobreak\
dl=\sum_{i}(1-n_{i})=N_{\mathrm{over}}-N_{\mathrm{under}}$ calculated over all
the connected region $R_{i}$ in an excursion888An excursion set is defined as
the region of a sky patch where the Hi field exceed a given threshold
$\nu_{t}$, $\delta T(\theta,\phi)/\sigma_{0}\equiv\nu>\nu_{t}$, with $\delta
T\equiv T(\theta,\phi)-\langle T\rangle$. $\Sigma\subset{\cal S}^{2}$ (Novikov
et al. 1999; Komatsu et al. 2003; Novaes et al. 2018; Ducout et al. 2013),
where $d\Omega$ and $dl$ are the elements of solid angle and line,
respectively, $\kappa$ is the geodesic curvature, and $a_{i}$, $l_{i}$ and
$n_{i}$ are the area, perimeter and genus of one connected region $R_{i}$.
This method can be efficiently applied to masked skies or in small regions of
the sphere, being specially useful in the case of the Hi measurements from
BINGO, with a partial sky coverage. In fact, such versatility of the MF
motivated their usage as one of the modules for Hi IM data analysis.

The MF module is under development to work using two main approaches: (1) to
access the cosmological information provided by the Hi signal in constraining
cosmological parameters; and (2) to evaluate the recovered Hi signal by using
different component separation methods, exploring the efficiency and
robustness of each method and comparing their results, as a complement to the
bispectrum module.

Using the algorithm provided by Gay et al. (2012), and Ducout et al. (2013) to
calculate the MF, the first approach has been tested on simulated data
produced by the FLASK code. In summary, the methodology of this approach
concerns the calculation of the MF from a large set of synthetic Hi maps,
produced assuming different cosmologies (varying a set of cosmological
parameters) and redshift bins that will be observed by BINGO, and use a
machine learning technique to map the MF features in terms of the cosmological
parameters. As an illustrative example, Fig. 14 shows the evolution of the Hi
signal with redshift, as measured by BINGO, for a given fixed cosmology. After
widely testing it in synthetic data, evaluating the impacts of residual
contamination and instrumental effects, we will be able to apply the
methodology to the BINGO data.

Figure 14: Three MF: area (left), perimeter (middle), and genus (right),
averaged over $1000$ Hi IM simulations at each redshift (frequency) bin to be
measured by BINGO. Here we consider 30 redshift bins, equally spaced in
frequency, from $0.127$ ($980$ MHz) to $0.449$ ($1260$ MHz), indicated by the
color scale. The simulation are generated assuming the $\Lambda$CDM
cosmological parameters measured by Aghanim et al. (2020).

##  6 Scientific synergies with other instruments

###  6.1 BINGO in the wider context

There has been a recent burgeoning of interest in new radio astronomy
instruments such as ALMA, LOFAR, and of course the SKA and its precursors
MeerKAT (Booth et al. 2009) and ASKAP (Johnston et al. 2008; DeBoer et al.
2009). All these activities have been driven by the realization that radio
astronomical observations have a unique contribution to make to cosmology,
astrophysics and even fundamental physics.

For comparison, in Fig. 15 we show some of the IM experiments already in
operation or planned for the near future, with covered area in the sky as a
function of redshift. Albeit operating in the same redshift range of Tianlai,
FAST and SKA telescopes, BINGO can still be a strong competitor, producing
good quality data very early.

Figure 15:  Top: Comparative plot of angular resolution versus redshift
coverage for different IM experiments. The red rectangle with the arrow shows
the approximate spot for BINGO (adapted from Kovetz et al. 2017). Bottom:
Figure from Wuensche et al. (2021a), showing a comparative plot of the sky and
redshift coverage for different IM experiments. From those, CHIME, FAST and
Tianlai are already in operation.

###  6.2 Square kilometer array

The SKA project is an idea under discussion since the turn of the second
millennium and is well known as the largest and most important observational
project in cosmology and astrophysics up to now and in the decades to come.

SKA is a project aiming to look at the cosmos from 10 million years of age,
when the first generation of stars lighted, up to the DE-dominated Universe,
basically today. A telescope looking at these matters should have a total size
of one square kilometer, hence its name and structure. Clearly, SKA has a
tremendous scope: astrobiology, tests of gravity, exotic transient phenomena
(pulsars, FRBs, etc.) planet observations, magnetic fields, interplanetary
medium, DE and DM, among further more specific astrophysical and cosmological
aims.

A new science book has been produced by the SKA consortium (Square Kilometre
Array Cosmology Science Working Group et al. 2020a). The previously proposed
SKA surveys, which usually were thought of as the main cosmology surveys to be
carried out with the SKA, were the SKA continuum survey and the SKA galaxy
redshift survey (Abdalla et al. 2015; Santos et al. 2015; Square Kilometre
Array Cosmology Science Working Group et al. 2020b). In the recent science
prioritization, three summary chapters were produced to outline the main SKA
cosmology goals, two of them being those for the above mentioned surveys. An
additional chapter has been produced by the SKA team to cover the science that
can be done with IM (Bourke 2014). This shows the growing realization that IM
is one of the most powerful cosmological tools available. The main summary
chapter for cosmology with the SKA argues that without using techniques to
obtain intensity maps, such as the ones produced by BINGO, the project will be
severely hampered in its capability of producing cosmological transformational
science in its first phase. BINGO will be an excellent pathfinder to
investigate issues that will likely be faced with the SKA.

The main observations common to SKA and BINGO, actually the ”raison d’être” of
the BINGO project is the (indirect) observation of the dark sector, by means
of the BAO, which can be observed by both projects, leading to a unique
fingerprint on cosmological observations and in turn being used to constrain
cosmological parameters.

Good observations by BINGO can give hints to what to look for in the case of
SKA, which will observe again the BINGO volume and go much beyond into the
cosmic dawn. Further aims of SKA include search of protoplanetary disks and
exoplanets as well as magnetic fields and other galaxy structures. In Galactic
astrophysics, details of the synchroton emission and Faraday rotation will be
clearly observed.

###  6.3 MeerKAT

The MeerKAT is a project designed as a SKA pathfinder. Located in the Karoo
region, South Africa, it consists in an array of 64 antennas and is very
sensitive, with a resolution of circa 1 arc second at 1420 MHz, seeking
regions at $z\sim 4$ or even higher. A low frequency band, from 850 to 2500
MHz (which includes the BINGO band) will look at most epochs of modern
Universe, and will be able to deeply map the Hi distribution. In particular,
it covers the epoch of reionization, thus informing about DM evolution in the
recent Universe. Galactic studies, magnetic fields and the tiny variations in
matter distribution are also within the reach of low frequency band of
MeerKAT. MeerKAT also operates in a high frequency band (8 to 14.5 GHz)
searching for transients and other astrophysical objects.

###  6.4 CHIME

Another experiment aiming to measure BAO is the Canadian Hydrogen Intensity
Mapping Experiment - CHIME, a radio interferometer located at the Dominion
Radio Astrophysical Observatory (DRAO, CHIME; Bandura et al. 2014). CHIME
operates in the frequency range of 400 MHz to 800 MHz, with a synthesized beam
resolution between 20 and 40 arcminutes, in order to produce 21-cm sky maps
for about 50% of the sky, across the redshift range $z=0.8-2.5$ (Shaw et al.
2014).

CHIME is complementary to BINGO in the frequency and correspondingly redshift
range ($0.8\leq z\leq 2.5$) as well as in the form, consisting of 4 adjacent
cylindrical reflectors of $20$ m diameter $\times 100$ meters length, oriented
north-south. Similarly to BINGO, CHIME has no moving parts.

The main cosmological goal for CHIME is also measuring BAO using Hi IM.
However, CHIME is a very efficient telecope to detect FRBs and, at the time of
this writing, has already found dozens of them, with a predicted discovery
rate of $2-50$/day. Pulsars are also a strong target for CHIME, making it a
partner for BINGO (Amiri et al. 2018, 2019).

In CHIME a separate pipeline for FRBs has been constructed, with dedicated
electronics, for identifying FRBs as well as storing the data. Smaller
dispersion measures are easier to handle, and CHIME has a maximum capability
of handling large dispersion measures with high resolution.

###  6.5 Five-hundred meter aperture spherical radio telescope (FAST)

The Five-hundred-meter Aperture Spherical radio Telescope (FAST) is currently
the largest single dish radio telescope in the world. It is located in the
Guizhou province, China (Nan et al. 2011). FAST is a spherical reflector with
a diameter of 500 m, fit in the Dawodang depression. Because of its size, the
FAST dish itself does not move. However, the telescope is able to observe
sources within 40 degrees of the zenith because the shape of the dish’s
surface made up of 4,450 aluminum panels that are tilted and turned by about
2,000 mechanical winches and the feed cabin, where the radio waves are
focused, is suspended from six cables and moves around the surface of the
dish. Inside the cabin, multibeam and multiband receivers are installed,
covering a large frequency range of 70 MHz – 3 GHz (Nan et al. 2011; Li 2016).
Besides being a galaxy survey, FAST is also an IM survey (Smoot & Debono 2017;
Bigot-Sazy et al. 2016; Hu et al. 2020). As it was pointed out in these
references IM cannot be done in this entire frequency range. In Smoot & Debono
(2017) for a 500 m aperture, they could do IM between redshifts $0.5-2.5$,
while in Bigot-Sazy et al. (2016) for the illuminated aperture os 300 m it is
suggested a frequency range for IM of $350-1350$ MHz. In Hu et al. (2020) the
performance of IM with FAST is studied in details for a redshift range of
$0.05<z<1.25$. From this study, in this other mode of operation, FAST IM can
provide good cosmological measurements, with forecasts showing that FAST can
reach the constraining level of dark energy task force (DETF) stage IV
experiments.

FAST is a Chinese contribution to the international efforts to build the SKA.
It is the most sensitive single dish radio telescope ever built. FAST is
expected to map the Hi gas in the Milky Way Galaxy at a very high resolution
and to increase the number of known pulsars from almost 2,000 to about 6,000.

The FAST project took 5.5 years from the commencement of work in March of 2011
and was completed in September 2016. FAST has been in final tests for three
years, and it has already spotted 102 brand new pulsars (half of the pulsars
in the Galaxy are in the reach of FAST). It also detected a few FRBs. In
September 2019, the telescope passed a series of performance assessments and
was given the green light open to astronomers. The project was funded by the
Chinese government, and FAST is operated by the National Astronomical
Observatories of the Chinese Academy of Sciences.

We can see that the redshift range of FAST coincides with the one from BINGO,
specially the frequency where it is possible to do IM, as shown in Fig. 15
where we used the IM redshift range from Hu et al. (2020). This makes FAST not
only a good competitor of BINGO, but also complementary, since BINGO will be
built in the southern hemisphere while FAST is in the northern hemisphere.
Having different experiments measuring in the same frequency is also important
to confirm the science explored by both projects.

###  6.6 The Tianlai project: a 21cm radio telescope array

The Tianlai project (“Sounds from the Heaven”) is a similar project to survey
the northern sky, making IM observations of the redshifted 21-cm line from the
Hi below redshift 3, to measure the BAO signal of LSS and measure the DE
equation of state so as to constrain DE models (Chen 2012). It is also a
pathfinder for testing the basic principles and key technologies of 21-cm IM.
The project is an international effort with participants from China, France,
the USA and Canada and is being built in a very quiet radio site in Balikun,
Xinjiang, China.

The Tianlai pathfinder includes a dish array with 16 dishes of $6$m diameter,
each equipped with a dual linear-polarization feed, compactly arranged in two
concentric rings. These dishes are equipped with electronically controlled
motor drives in the altitude-azimuth mount, which allow people to steer the
dishes to point to almost any desirable directions above the horizon. The
dishes operate alongside a cylinder array with three $15$-m diameter
$\times\,40$-m length, north-south oriented, cylinders. The cylinders are
equipped with 96 dual linear-polarization feeds that can be moved freely along
the guide rail, which allows observers to change the separation between the
feeds flexibly. Tianlai array receivers are designed to have broad band
response (400 – 1500 MHz).

The construction of the two arrays was completed in 2015, and the first trial
observations were done in September 2016. An initial report of observations
was done by Wu et al. (2021).

##  7 Conclusions

This paper is the first of a series of papers describing the developments and
capabilities of the BINGO telescope. In this paper, we have described the
BINGO project and its science goals: cosmology, which is the main driver of
the project, and astrophysics. We summarized the most recent plans of
construction and key observations strategy, developments of the pipeline, and
cosmological and astrophysical forecasts for the instrument.

BINGO will use the IM technique to map the redshifted Hi between $z=0.13$ and
$z=0.45$, and measure the BAO, providing independent cosmological data. It
will put complementary constraints on plenty of cosmological models and
alternative cosmologies, such as dynamical DE, interacting DE and modified
gravity, whose combination with Planck  results will lead to a gain in
precision of the cosmological parameters. Additionally, it will be possible to
constrain the 21-cm parameters (Hi bias and density parameter) and the sum of
neutrino masses.

Besides its main goal, BINGO has the possibility to detect FRBs at a rate of
one each $4-5$ days for a signal-to-noise ratio greater than 3 and further
investigation is going to be presented in future work (dos Santos et al.
2021). Additional goals can include pulsar science and new data in an
unexplored frequency range, that can be used in combination with ongoing sky
mapping and radio experiments in other frequencies.

We also present a general description of the BINGO pipeline and its different
modules. It covers input modules for sky emission models and experiment
characteristics, a mission simulation module, production of TOD, calibration
and cleaning in the time domain, map-making, generation of multifrequency
maps, component separation for foreground removal, reconstructed Hi sky maps
and cosmology models, comprising covariance matrix, bispectrum and Minkowski
functional estimations. Additional information can be found in the companion
papers IV, V and VI (Liccardo et al. 2021; Fornazier et al. 2021; Zhang et al.
2021).

The BINGO project aims to produce a map of the nearby Universe in a redshift
range that, though observing the later universe, covers a large amount of the
Universe volume (about 22 Mpc3), thus being able to analyze most of the
important structures of the Universe at a low cost. Indeed, it covers the turn
from the epoch when matter dominated over DE ($z\approx 0.45,$, when the
Universe was about 9 Gyr) to the accelerated era ($z\approx 0.13$, when the
Universe was 12.05 Gyr), thus including the turning point, $z\approx 0.32$,
and providing an excellent study of the properties of DE.

The BINGO results for cosmology, prime attention given to the BAO structure at
radio frequencies, provide constraints to the cosmological parameters, as well
as presumably hinting at alternative cosmologies and further constraints
thereof. Moreover, a number of astrophysical questions can be addressed, such
as transient objects’ science, for example, FRBs and pulsars, Galaxy science,
and the nearby Universe. Despite being a so-called ”off-the-shelf” telescope,
there were (and are) a number of technical/technological challenges BINGO is
facing to meet the conceptual design, such as the construction of an efficient
and low-noise antenna and front end, the fabrication of 4.3 m long corrugated
horns and 40 m-class dishes, at affordable costs and with the required
performance standards.

Finally, the project involves cultural and educational aspects as well as
outreach activities that are developed simultaneously to its technical and
scientific goals.

###### Acknowledgements.

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. A.A.C. acknowledges
financial support from the China Postdoctoral Science Foundation, grant number
2020M671611. Y.Z.M. acknowledges the support from NRF-120385, NRF-120378 and
NRF-109577. B. W. and A.A.C. were also supported by the key project of NNSFC
under grant 11835009. R.G.L. thanks Rui Luo and Kejia Lee for comments and
clarifications on the FRB detection rate, CAPES (process 88881.162206/2017-01)
and the Alexander von Humboldt Foundation for the financial support. 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. J.Z. was
supported by IBS under the project code, IBS-R018-D1. C.P.N. and K.S.F.F.
would like to thank FAPESP for grants 2019/06040-0 and 2017/21570-0. F.V. and
E.J.M. acknowledge the support by CAPES. A.R.Q., F.A.B., L.B., J.R.L.S., 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). S.A. acknowledges funding
for a technical internship associated to FAPESP project 2014/07885-0 M.P.
acknowledges funding from a FAPESP Young Investigator fellowship, grant
2015/19936-1. V.L. acknowledges the postdoctoral FAPESP grant 2018/02026-0.
L.S. is supported by the National Key R&D Program of China (2020YFC2201600).
M.R. acknowledges funding from the European Research Council Grant CMBSPEC
(No. 725456). Some of the results in this paper have been derived using the
HEALPix package (Gorski et al. 2005). This research made use of
Astropy,999http://www.astropy.org. a community-developed core Python package
for Astronomy (Astropy Collaboration et al. 2018), NumPy (Harris et al. 2020)
and Healpy (Zonca et al. 2019).

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