11institutetext:  Instituto Nacional de Pesquisas Espaciais, Av. dos
Astronautas 1758, Jardim da Granja, São José dos Campos, SP, Brazil
11email<EMAIL_ADDRESS>22institutetext: Shanghai Astronomical
Observatory, Chinese Academy of Sciences, Shanghai 200030, China
33institutetext: University College London, Gower Street, London, WC1E 6BT, UK
44institutetext: Instituto de Física, Universidade de São Paulo, R. do Matão,
1371 - Butantã, 05508-09 - São Paulo, SP, Brazil 55institutetext: Department
of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140,
South Africa 66institutetext: CNRS-UCB International Research Laboratory,
Centre Pierre Binétruy, IRL2007, CPB-IN2P3, Berkeley, USA 77institutetext:
Laboratoire Astroparticule et Cosmologie (APC), CNRS/IN2P3, Université Paris
Diderot, 75205 Paris Cedex 13, France 88institutetext: IRFU, CEA, Université
Paris-Saclay, 91191 Gif-sur-Yvette, France 99institutetext: Department of
Astronomy, School of Physical Sciences, University of Science and Technology
of China, Hefei, Anhui 230026 1010institutetext: Instituto de Física de
Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, E-39005
Santander, Spain 1111institutetext: Jodrell Bank Centre for Astrophysics,
Department of Physics and Astronomy, The University of Manchester, Oxford
Road, Manchester, M13 9PL, UK 1212institutetext: Center for Gravitation and
Cosmology, College of Physical Science and Technology, Yangzhou University,
225009, China 1313institutetext: School of Aeronautics and Astronautics,
Shanghai Jiao Tong University, Shanghai 200240, China 1414institutetext:
Technische Universität München, Physik-Department T70, James-Franck-Strasse 1,
85748, Garching, Germany 1515institutetext: Unidade Acadêmica de Física,
Universidade Federal de Campina Grande, R. Aprígio Veloso, Bodocongó,
58429-900 - Campina Grande, PB, Brazil 1616institutetext: Centro de Gestão e
Estudos Estratégicos SCS Qd 9, Lote C, Torre C S/N Salas 401 a 405, 70308-200
- Brasília, DF, Brazil 1717institutetext: Instituto de Física, Universidade de
Brasília, Campus Universitário Darcy Ribeiro, 70910-900 - Brasília, DF, Brazil
1818institutetext: College of Science, Nanjing University of Aeronautics and
Astronautics, Nanjing 211106, China 1919institutetext: Kavli IPMU (WPI),
UTIAS, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583,
Japan

# The BINGO project VIII:

On the recoverability of the BAO signal on Hi intensity mapping simulations

Camila P. Novaes<EMAIL_ADDRESS>Jiajun Zhang 22 Eduardo J. de
Mericia 11 Filipe B. Abdalla 11334455 Vincenzo Liccardo 11 Carlos A. Wuensche
11 Jacques Delabrouille 66778899 Mathieu Remazeilles 10101111 Larissa Santos
12121313 Ricardo G. Landim 1414 Elcio Abdalla 44 Luciano Barosi 1515 Amilcar
Queiroz 1515 Thyrso Villela 1116161717 Bin Wang 12121313 Francisco A. Brito
1515 André A. Costa 12121818 Elisa G. M. Ferreira 441919 Alessandro Marins 44
Marcelo V. dos Santos 441515

(Received ; accepted )

###### Abstract

Context.  A new and promising technique for observing the Universe and study
the dark sector is the intensity mapping of the redshifted 21 cm line of
neutral hydrogen (Hi). The Baryon Acoustic Oscillations [BAO] from Integrated
Neutral Gas Observations (BINGO) radio telescope will use the 21 cm line to
map the Universe in the redshift range $0.127\leq z\leq 0.449$, in a
tomographic approach, with the main goal of probing BAO.

Aims. This work presents the forecasts of measuring the transversal BAO signal
during the BINGO Phase 1 operation.

Methods. We use two clustering estimators: the two-point angular correlation
function (ACF), in configuration space, and the angular power spectrum (APS),
in harmonic space, and a template-based method to model the ACF and APS
estimated from simulations of the BINGO region and extract the BAO
information. The tomographic approach allows the combination of redshift bins
to improve the template fitting performance. We compute the ACF and the APS
for each of the 30 redshift bins and measure the BAO signal in 3 consecutive
redshift blocks (lower, intermediate and higher) of 10 channels each.
Robustness tests are used to evaluate several aspects of the BAO fitting
pipeline for both clustering estimators.

Results. We find that each clustering estimator shows different sensitivities
to specific redshift ranges, although both of them perform better at higher
redshifts. In general, the APS estimator provides slightly better estimates,
with smaller uncertainties and larger probability of detection of the BAO
signal, achieving $\gtrsim 90$% at higher redshifts. We investigate the
contribution from instrumental noise and residual foreground signals and find
that the former has the greater impact, getting more significant as the
redshift increases, in particular the APS estimator. Indeed, including noise
in the analysis increases the uncertainty up to a factor of $\sim 2.2$ at
higher redshifts. Foreground residuals, in contrast, do not significantly
affect our final uncertainties.

Conclusions. In summary, our results show that, even including semi-realistic
systematic effects, BINGO has the potential to successfully measure the BAO
scale in radio frequencies.

###### Key Words.:

cosmology – 21 cm intensity mapping – baryon acoustic oscillations

##  1 Introduction

Recent cosmological observations, such as distance measurements with type Ia
supernovae, observations of cosmic microwave background (CMB) temperature and
polarization anisotropies, and the detection of baryon acoustic oscillations
(BAO) in large scale structure spectroscopic surveys, have been key in
establishing the current standard cosmological model, $\Lambda$CDM.

Although a remarkable fit to existing observations, $\Lambda$CDM requires 95%
of the energy content of the universe to be in the form of dark matter ($\sim$
25%), and a puzzling component responsible for the current accelerated
expansion of the universe, dark energy, with approximately 70%. Confidence in
the model requires the identification of these two components of the Universe
(Abdalla & Marins, 2020). Otherwise the $\Lambda$CDM model will be subject to
being challenged by alternative interpretations, many of which involving
modifications of our theory of gravitation on large scales.

The BAO feature, appearing as a fixed scale in both the CMB and large scale
structure (LSS) data (a geometrical probe), is originated in the early
Universe, when the temperature was high enough to keep photons and baryons
coupled. The BAO are the imprints left by acoustic waves traveling at
relativistic speed, generated by the gravitational infall of baryons (and dark
matter) into the potential wells of dark matter balanced by the radiation
pressure pushing out these baryons from overdense regions. The comoving
distance they traveled until recombination when the baryons were released from
the drag of the photons defines the BAO scale (Dodelson, 2003). This
characteristic scale is then imprinted in the LSS distribution and evolves
with structure formation as a standard ruler that can be measured as a
function of the redshift (Eisenstein et al., 2007b; Seo & Eisenstein, 2007).
This makes the BAO one of the most powerful and well established tool to
investigate the history of the acceleration of the expansion of the Universe
(Weinberg et al., 2013).

The first statistically significant measurements of the BAO feature imprinted
in the galaxy distribution were done by Eisenstein et al. (2005) analyzing the
Sloan Digital Sky Survey (SDSS) data, and by Cole et al. (2005), using the 2dF
Galaxy Survey (2dFGS). After that, several other detections of the BAO scale
have been performed using different biased tracers of the dark matter in
several redshift ranges. Among them we can cite the analyses of SDSS data at
various stages of data gathering (Anderson et al., 2012, 2014; Alam et al.,
2017; Ata et al., 2018; Alam et al., 2021), besides the WiggleZ Dark Energy
Survey (DES) (Hinton et al., 2016) and the 6dFGS (Carter et al., 2018)
analyses. More recently, data releases of the DES have been analyzed for the
detection of the BAO signature through different clustering measurement
methods: in configuration, in harmonic and Fourier spaces, and for 2- or
3-dimensional distribution of sources (Camacho et al., 2019; Abbott et al.,
2019, 2021). These results have used the common template-based fitting method
to study the BAO feature from clustering estimates. Alternative approaches for
measuring the BAO scale in the angular (2-dimensional) 2- and 3-point
statistics use an empirical parametric fit technique, for example, Sánchez et
al. (2011); Carnero et al. (2012); de Simoni et al. (2013); Carvalho et al.
(2016, 2017); De Carvalho et al. (2018, 2020) and de Carvalho et al. (2021).

A new and very promising way of measuring BAO is through the detection of
structures as traced by the redshifted diffuse 21 cm hyperfine transition line
of neutral hydrogen (Hi) through the so called intensity mapping (IM)
technique (Chang et al., 2008). Compared to radio surveys using the 21 cm
emission to detect individual galaxies, limited by the low luminosity of the
21 cm line emission, the IM technique can cover a larger volume of the
Universe in a much shorter time, using instruments with relatively small, and
consequently cheaper, collecting area, as discussed and evaluated in Battye et
al. (2013) and Abdalla et al. (2021a) (see also, Chang et al., 2008; Loeb &
Wyithe, 2008). The idea is to measure the overall Hi brightness temperature
field, similarly to what is done for CMB temperature fluctuations, but mapping
the Universe as a function of redshift. This is possible given the high
abundance of hydrogen, so that a hydrogen map is expected to be a powerful
tracer of the underlying total matter content of the Universe. In this sense,
through redshift surveys in radio frequencies, the 21 cm IM constitutes a new
window for observing the Universe, and for studying the dark sector.

To exploit this new observational window, several radio instruments, already
observing or still under construction, will survey a large volume of the
Universe through the 21 cm IM technique and measure the BAO feature. Among
them we can mention the Square Kilometer Array111https://www.skatelescope.org/
(SKA; SKA Cosmology SWG et al., 2020);
MeerKAT222https://www.ska.ac.za/science-engineering/meerkat (Santos et al.,
2017), a precursor of the SKA; the Five-Hundred-Meter Aperture Spherical Radio
Telescope, currently the largest single dish telescope in the world (FAST; Nan
et al., 2011); the Canadian Hydrogen Intensity Mapping
Experiment333https://chime-experiment.ca/ (CHIME; Bandura et al., 2014);
Tianlai444http://tianlai.bao.ac.cn/wiki/index.php/Main_Page (Chen, 2012);
HIRAX (Crichton et al., 2022); and the Baryon Acoustic Oscillations from
Integrated Neutral Gas Observations555https://www.bingotelescope.org/en/
(BINGO; Battye et al., 2013; Bigot-Sazy et al., 2015; Abdalla et al., 2021a),
described in the next section.

In this work we evaluate what to expect from future BINGO observations in
terms of measuring the transversal (angular) BAO signal. To perform this task
we employ and compare two clustering estimators, the two-point angular
correlation function (ACF), in configuration space, and the angular power
spectrum (APS), in harmonic space. To our knowledge, this is the first
forecasting study of the BAO detection from 21 cm IM simulations using ACF.
The anisotropic two-point correlation function estimator
$\xi(r_{\perp},r_{\parallel})$, was explored by Avila et al. (2021) for the
case of a SKA-like survey employing simulations, and by Kennedy & Bull (2021),
for a theoretical modeling approach focusing in MeerKAT survey. Both of them
focus on the impact of the instrumental beam smoothing and foreground removal
in recovering the BAO scale. Villaescusa-Navarro et al. (2017) also explore
the SKA case, assessing the BAO detection through the radial power spectrum,
under the presence of instrumental effects and foreground contamination, and
the impact of the angular resolution imposed by a single-dish instrument. They
find that the telescope beam will compromise the detection of the isotropic
BAO feature at redshifts $z\gtrsim 1$, while the radial BAO seems to be robust
against the foreground removal. These published results confirm the potential
of future 21 cm IM observations for detecting the BAO signature.

In the present work, we use a template-based method to model the APS and ACF
estimated from two types of mock realizations, mimicking future BINGO
observations, and extract the BAO information from each of them. We use the
covariance matrices calculated from the mocks to construct the likelihood
corresponding to each template and, using a maximum likelihood estimator, we
estimate the parameters of the model for each simulation. We apply this
template fitting procedure over three sets of consecutive bins, so that we can
evaluate the BAO detection at three different redshift intervals.

This paper is organized as follows. Section 2 briefly describes the main
aspects of the BINGO project. Section 3 introduces the Hi clustering theory
and the template model constructed for each estimator. Section 4 presents the
details about the cosmological 21 cm simulations, the characteristics of the
instrumental noise and the foreground contamination, as well as the foreground
cleaning process employed here. Section 5 summarizes our methodology for the
clustering measurements and the BAO fitting process, and the covariance matrix
construction. The results from all our analyses are discussed in Section 6,
while the conclusions are summarized in Section 7.

##  2 The BINGO telescope

BINGO is a radio telescope in construction in a site (see Peel et al. (2019)
for the site selection process) located in Paraíba State, northeastern Brazil
(latitude: 7∘ 2’ 27.6” S; longitude: 38∘ 16’ 4.8” W; altitude: 350 to 460 m).
Its main scientific goal is to measure the BAO signal imprinted in the 21 cm
distribution in the redshift range $0.127\leq z\leq 0.449$ (Abdalla et al.,
2021a). This goal will be achieved through an IM survey covering 5324 squares
degrees, using the telescope in a sky transit mode, with a
$14.75^{\circ}$-wide declination strip centered at a declination
$\delta=-15^{\circ}.$

The telescope has a crossed-Dragone design, with a 40-m diameter primary
paraboloid and a 34-m diameter secondary hyperboloid. During BINGO Phase 1,
the telescope will operate with a focal plane containing 28 horns (Wuensche et
al., 2021; Abdalla et al., 2021b). Each horn is sensitive to circular
polarization and is coupled to a correlation receiver with 4 amplifier chains,
and should perform at an expected system temperature $T_{sys}=70$ K. The
optical design will produce a suitable angular resolution for the BAO signal,
with a full width half maximum $\theta_{\mbox{\sc fwhm}}=40^{\prime}$ of the
beam in the central frequency $\nu=1120$ MHz. In a second phase, we intend for
BINGO to operate with 28 additional horns, totaling 56 horns.

The redshift range covered by BINGO corresponds to the frequency interval
$980-1260$ MHz, which is quite complementary to instruments like CHIME
($400-800$ MHz; Bandura et al., 2014). A more detailed description of the
BINGO project, its scientific goals and instrument status, are available in
previous publications of the collaboration (Abdalla et al., 2021a; Wuensche et
al., 2021).

##  3 Modeling the BAO signal

###  3.1 Hi clustering

The intensity mapping of the 21 cm signal measures the brightness temperature
of the average Hi intensity in a given volume of the Universe, which can be
written as function of the redshift, $z$, as (see, e.g., Battye et al., 2013;
Hall et al., 2013)

$\bar{T}_{\rm HI}(z)=188\,h\,\Omega_{\rm HI}(z)\frac{(1+z)^{2}}{E(z)}\,{\rm
mK}\,,$ (1)

where $H_{0}=100\,h\,\,{\rm km\,\,s}^{-1}{\rm Mpc}^{-1}$ is the Hubble
constant, $E(z)=H(z)/H_{0}$, and $\Omega_{\rm HI}$ is the Hi density
parameter. Fluctuations of the Hi brightness temperature, $\delta T_{\rm HI}$,
are a biased tracer of the dark matter fluctuations, $\delta$. Their Fourier
transform can be written, in terms of the growth function $D(z)$, as

$\delta T_{\rm HI}(\mathbf{k},z)=D(z)\,\bar{T}_{\rm HI}(z)\,b_{\rm
HI}(z)\,\delta(\mathbf{k},0)\,,$ (2)

where $b_{\rm HI}(z)$ is the bias factor as a function of redshift $z$, and
$\delta(\mathbf{k},0)$ is the underlying dark matter distribution at $z=0$. In
this way, the Hi power spectrum is given by $P_{\rm
HI}\approx[D(z)\,\bar{T}_{\rm HI}(z)\,b_{\rm HI}(z)]^{2}P(k)$, where $P(k)$ is
the matter power spectrum at $z=0$.

As detailed by Battye et al. (2013) and Seehars et al. (2016), since the 21 cm
IM surveys the Universe within tomographic bins of redshift, one can project
the 3D quantity $\delta T_{\rm HI}(\chi(z)\hat{n})=\bar{T}_{\rm
HI}(z)\delta_{\rm HI}(\chi(z)\hat{n})$ on the sky by integrating it along the
line of sight $\hat{n}$ as

$\delta T_{\rm HI}(\hat{{n}})=\int dz\,\phi(z)\,\bar{T}_{\rm
HI}(z)\,\delta_{\rm HI}(\chi(z)\,\hat{n})\,,$ (3)

where $\delta_{\rm HI}(\chi(z)\,\hat{n})$ is the density fluctuation of
neutral hydrogen at the comoving distance to redshift $z$, $\chi(z)$, and
$\phi(z)$ is the projection kernel666Here, we assume a top-hat kernel, that
is, $\phi(z)=1/(z_{max}-z_{min})$ for $z_{min}<z<z_{max}$ and $\phi(z)=0$
outside the redshift bin.  (a window function of observation). Decomposing the
Hi temperature fluctuations in spherical harmonics, we have (see also,
Sobreira et al., 2011; Costa et al., 2021)

$\delta T_{\rm HI}(\hat{n})=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}a_{\ell
m}Y_{\ell m}(\hat{n})\,,$ (4)

with the $a_{\ell m}$ harmonic coefficients written as

$a_{\ell m}=4\pi i^{\ell}\int dz\,\phi(z)\,\bar{T}_{\rm
HI}(z)\,\int\frac{d^{3}k}{(2\pi)^{3}}\,\delta_{\rm HI}(\@vec{\rm
k},z)\,j_{\ell}(k\chi(z))\,Y^{*}_{\ell m}(\@vec{\rm k})\,,$ (5)

where $j_{\ell}$ is the spherical Bessel function. Therefore, from the above
equations, we can define the APS of the temperature fluctuations,
$C_{\ell}=\langle\,|a_{\ell m}|^{2}\rangle$, as a function of the matter power
spectrum, such that (see also Loureiro et al., 2019, for the analogous case of
a galaxy distribution)

$C^{ij}_{\ell}=\frac{2}{\pi}\int dk\,W^{i}_{{\rm HI},\ell}(k)\,W^{j}_{{\rm
HI},\ell}(k)\,k^{2}\,P(k)\,,$ (6)

where the indices $i$ and $j$ denote two tomographic bins. For $i=j$ and
$i\neq j$, we obtain auto- and cross-APS respectively (see also, Costa et al.,
2021). The redshift dependence is given by the window function

$W^{i}_{{\rm HI},\ell}(k)=\int dz\,b_{\rm HI}(z)\,\phi(z)\,\bar{T}_{\rm
HI}(z)\,D(z)\,j_{\ell}(k\chi(z))\,.$ (7)

In a similar way to the APS one can also define the ACF, $\omega(\theta)$,
defined as the probability to find galaxies separated by an angle $\theta$.
For a tomographic bin around $z$, the ACF can be defined as a function of the
matter 3D spatial correlation function $\xi(r)$ (Sobreira et al., 2011; Crocce
et al., 2011; Sánchez et al., 2011; de Simoni et al., 2013)

$\displaystyle\omega(\theta)$ $\displaystyle\equiv$
$\displaystyle\langle\delta T_{\rm HI}(\hat{{n}})\,\delta T_{\rm
HI}(\hat{n}+\@vec{{\theta}})\rangle$ (8) $\displaystyle=$ $\displaystyle\int
dz_{1}\,f(z_{1})\int dz_{2}\,f(z_{2})\,\xi(r(z_{1},z_{2},\theta),\bar{z})\,,$
(9)

where $f(z)=b_{\rm HI}(z)\phi(z)\bar{T}_{\rm HI}(z)$ and
$r(z_{1},z_{2},\theta)$ is the radial comoving distance between spatial
fluctuations $\delta_{\rm HI}$ at redshifts $z_{1}$ and $z_{2}$ separated by
an angle $\theta$. This equation neglects the time evolution of the matter
correlation function inside the bin, so that it can be evaluated at a given
$\bar{z}$ (the average redshift of the bin) as

$\xi(r,\bar{z})=\frac{1}{2\pi}\int dkk^{2}j_{0}(rk)P(k,\bar{z})\,,$ (10)

where $j_{0}$ is the spherical Bessel function of zeroth order and
$P(k,\bar{z})$ is the matter power spectrum at redshift $\bar{z}$.

Alternatively, following Sobreira et al. (2011) and Crocce et al. (2011) and
using Eq. (4), we can still obtain the ACF as

$\omega(\theta)=\bigg{\langle}\sum_{\ell=0}^{\infty}\sum_{\ell=-\ell}^{\ell}a_{\ell
m}a_{\ell^{\prime}m^{\prime}}Y_{\ell
m}(\hat{n})Y_{\ell^{\prime}m^{\prime}}(\hat{n}+\@vec{\theta})\bigg{\rangle}\,,$
(11)

and, as a function of the APS, as

$\displaystyle\omega(\theta)$ $\displaystyle=$ $\displaystyle\sum_{\ell
m}C_{\ell}Y_{\ell m}(\hat{n})Y_{\ell m}(\hat{n}+\@vec{\theta})$ (12)
$\displaystyle=$
$\displaystyle\sum_{\ell}C_{\ell}\frac{2\ell+1}{4\pi}P_{\ell}(\cos\theta)\,,$
(13)

where $P_{\ell}$ are the Legendre polynomials.

The theoretical auto- and cross-$C_{\ell}$’s employed in this paper, both as
input to the 21 cm log-normal simulations (see Section 4.1.1) and to construct
the BAO templates, are calculated using the Unified Cosmological Library for
$C_{\ell}$’s (UCLCL) code (Loureiro et al., 2019; McLeod et al., 2017). This
code implements Eqs. (6) and (7) using the CLASS Boltzmann Code (Lesgourgues,
2011; Blas et al., 2011) to estimate the primordial power spectra and transfer
function. To account for all processes involved in the evolution of the
Universe, we use the window function $W^{Tot,i}(k)=W^{i}_{{\rm
HI},\ell}(k)+W^{i}_{RSD,\ell}(k)$. In addition to Eq. (7), it also includes a
term describing the redshift space distortion (RSD) effect,
$W^{i}_{RSD,\ell}(k)$. A detailed description of how the two terms are
implemented in the UCLCl code can be found in Loureiro et al. (2019).

###  3.2 BAO template

The extraction of the BAO features from data clustering estimates is commonly
performed by fitting a template model, derived from a parameterization of the
matter power spectrum (see, e.g., Anderson et al., 2014, and references in the
Introduction). Following the same approach, we write this parameterization as
a function of the linear power spectrum, $P_{\rm lin}$, and the no-wiggle (no
BAO feature) power spectrum, $P^{nw}$,

$P^{temp}(k)=[P^{lin}(k)-P^{nw}(k)]\,e^{-k^{2}\Sigma_{nl}^{2}}+P^{nw}(k)\,.$
(14)

We employ the nbodykit777https://github.com/bccp/nbodykit code to obtain both
$P^{lin}(k)$ and $P^{nw}(k)$, using the transfer functions calculated by the
CLASS code and the analytic calculation by Eisenstein & Hu (1998),
respectively. The exponential term, $e^{-k^{2}\Sigma_{nl}^{2}}$, takes into
account the effect of non-linear structure growth by damping the signal around
the BAO scale. The damping scale, $\Sigma^{2}_{\rm
nl}=(\Sigma^{2}_{\perp}+\Sigma^{2}_{\parallel})/2$, is written in terms of the
components along ($\Sigma_{\parallel}$) and across ($\Sigma_{\perp}$) the line
of sight, taking into account the RSD effect. Seo & Eisenstein (2007) have
shown that these components can be written as
$\Sigma_{\perp}=10.4D(z)\sigma_{8}$ (prediction for real space) and
$\Sigma_{\parallel}=(1+f)\Sigma_{\perp}$, where $f$ is the growth rate of
cosmic structures (see also discussion by Chan et al., 2018; Ata et al.,
2018). Since we already account for the RSD in the window function
$W^{i}_{{\rm HI},\ell}$ when projecting $P^{temp}(k)$ into the APS (Eqs. (6)
and (7)), we can consider $\Sigma_{\rm nl}=\Sigma_{\perp}$. For our fiducial
cosmology (see Section 4.1.1), we then fix the damping scale at $\Sigma_{\rm
nl}=7.6,7.1$, and $6.7$ $h^{-1}$Mpc, appropriate values for the lower,
intermediate and higher redshift intervals in which we split the BAO fitting
analyses, as discussed later. Then, from the projection of $P^{temp}(k)$ into
$C^{temp}(\ell)$, we construct the template used for the APS analysis

$C(\ell)=B\,C^{temp}(\ell/\alpha)+\sum_{q}A_{q}\,\ell^{q}\,,$ (15)

where $\alpha$, $B$ and $A_{q}$ are free parameters, the last two of them
intended to absorb linear and non-linear bias effects, noise, uncertainties in
the RSD, besides any other difference between the data points and the full
shape template, such as those introduced by systematic effects.

For the ACF analyses, the template model is constructed substituting the same
$C^{temp}(\ell)$ in Eq. (12) to calculate $\omega^{temp}(\theta)$, so that we
guarantee the consistency among both templates, obtaining

$\omega(\theta)=B\omega^{temp}(\alpha\theta)+\sum_{q}\frac{A_{q}}{\theta^{q}}\,,$
(16)

where $\alpha$, $B$ and $A_{q}$ are free parameters, as in Eq. (15). The last
term in both templates, appearing as functions of $\ell$ and $\theta$, can be
written so that the number $q$ of $A_{q}$ parameters optimize our fitting
pipeline. Section 6.4 shows results from testing different degrees of freedom,
that is, different choices for minimum and maximum values to run the $q$
index, for both $C_{\ell}$ and $\omega(\theta)$ templates.

In both cases the $\alpha$ parameter, the most important parameter of the
analysis, is the so called shift parameter, associated to the change in the
BAO peak position with respect to a fiducial cosmology,

$\alpha=\frac{D_{A}(z)/r_{d}}{(D_{A}(z)/r_{d})_{fid}}\,,$ (17)

where $D_{A}$ is the angular diameter distance and $r_{d}$ is the sound
horizon scale at the drag epoch. Then, $\alpha$ characterizes any observed
deviation with respect to the model, so that $\alpha>1$ ($\alpha<1$) indicates
a shift of the acoustic peak to smaller (larger) scales. Since the fiducial
cosmology adopted to model the BAO templates and to generate the synthetic
data are the same, we expect to find $\alpha\approx 1$ when fitting them to
the APS and ACF clustering computed from the simulations, allowing to test the
methodology and to predict what to expect from the BAO analysis of the future
BINGO data.

##  4 Synthetic data preparation

In this section we describe the different sets of simulations employed in the
analyses. Sky maps are produced in the HEALPix pixelization scheme (Gorski et
al., 2005), with $N_{side}=256$. In most of the tests we use a large set of
(fast) log-normal simulations of the 21 cm signal generated by the FLASK888
http://www.astro.iag.usp.br/~flask/ code (Xavier et al., 2016), hereafter
FLASK mocks. We also test the BAO fitting pipeline over a smaller data set
based on the density contrast from N-body simulations, as described below,
hereafter N-body mocks.

Semi-realistic mock data sets (referred as BINGO-like simulations) include
instrumental noise, beams effects, the BINGO sky coverage scan and simulated
foreground signals that will contaminate 21 cm observations in the BINGO
frequency range. A foreground cleaning pipeline, outlined in Sec. 4.2, is
applied to a small set of BINGO-like simulations, constructed from a subset of
the FLASK mocks, to mitigate the effect of the foreground signal on the 21 cm
signal and estimate the residual contamination in the recovered maps.

###  4.1 BINGO-like simulations

####  4.1.1 Cosmological signal

  1. (a)

Fast log-normal distributions

The first set of 21 cm IM simulations consists of full-sky log-normal
realizations generated with FLASK. This is a publicly available code able to
produce two- or three-dimensional tomographic realizations of an arbitrary
number of random astrophysical fields and reproducing the desired cross-
correlations between them. FLASK takes as input the auto- and cross-APS,
$C^{ij}(\ell)$, previously calculated for each of the $i$ and $j$ redshift
slices and creates 2D HEALPix maps of correlated log-normal realizations of
the projected 21 cm signal in each redshift bin (z-bin). We use the fiducial
$C^{ij}(\ell)$ computed using the UCLCl code. Cosmological parameters match
those from WMAP 5-yr results (Table 2, Five-Year Mean values, of Dunkley et
al., 2009), $\Omega_{m}=0.26$, $\Omega_{b}=0.044$, $\Omega_{\Lambda}=0.74$,
and $H_{0}=72\,{\rm km}\,{\rm s}^{-1}{\rm Mpc}^{-1}$, for consistency with the
N-body mock simulations described later. Although more recent parameter
constraints have been obtained with the Planck satellite (Planck Collaboration
et al., 2020), our pipeline and results does not drastically depend on the
exact cosmological parameter values. Here we are not interested in
constraining these parameters, but to show the consistency of our method and
the detection of BAO.

We generate a total of 1500 log-normal realizations, each of them
corresponding to 30 Hi maps, one for each tomographic z-bin. Note that,
despite of our ad-hoc choice of 30 channels ($\delta\nu=9.33$ MHz width each),
the BINGO hardware and data analysis pipeline allows different choices for the
number of channels. Consequently, their width can be chosen according to what
is best suitable for cosmological analyses, for example, as function of the
efficacy of foreground cleaning procedure (Mericia et al., 2022). Tests on how
the BAO measurement is impacted by using a different number of channels are
left for future work.

  1. (b)

N-body simulations

A second set of mocks is generated using the FoF (Friend-of-Friends) halo
catalogs from HR4 N-body simulation (Kim et al., 2015). HR4 has a box size of
3150 Mpc/h and uses WMAP 5-yr cosmological parameters.

First, we select 100 random observer locations in the simulation box, to mimic
100 different realizations in just one simulation. We then construct the light
cone catalog from snapshots at $z=0.1,0.15,0.2,0.3,0.4,$ and $0.5$. According
to the designated frequency bin, the halos from the closest two snapshots are
selected. These halos are further filtered randomly, and the selection
probability is proportional to the radial distance of this bin and the
corresponding comoving distance at the redshift of the snapshot. Therefore,
the generated light cone halo catalog is a random mixture of the snapshots. We
validate the method by comparing the angular power spectrum and halo mass
function to the light cone halo catalog provided by HR4. Both results are
consistent and meet our requirements.

We make the full sky 21 cm brightness temperature map from the light cone halo
catalog following the method described in Zhang et al. (2021). By further
considering redshift distortion effects, we construct 100 realizations of full
sky mock maps, each containing 30 redshift bins. Since BINGO exploits only a
strip of the sky, we get independent realizations by randomly rotating the
full sky maps before including realistic characteristics of the BINGO
observations. Using 4 independent rotations999Spherical rotations as performed
by the healpy package., we generate another 400 mock maps in the BINGO
footprint using the 100 original full sky maps. After performing these
simulations, we have 1500 FLASK and 500 N-body mocks for testing our BAO
detection pipelines.

####  4.1.2 Foreground signals, instrumental effects and sky coverage

Following previous BINGO papers by Fornazier et al. (2021) and Liccardo et al.
(2021), our simulations include the contribution of seven foreground
components, including Galactic synchrotron, free-free, thermal dust and
anomalous microwave (AME) emissions, extragalactic thermal and kinetic
Sunyaev-Zel’dovich (SZ) effects, and unresolved radio point sources, all
generated using a recent version of the Planck Sky Model software (PSM;
Delabrouille et al., 2013).

The specific configuration of the PSM code that was used to generate our
foreground emissions is as follows. Synchrotron is based on the 408 MHz all
sky map produced by Remazeilles et al. (2015), extrapolated to BINGO
frequencies with a spatially variable spectral index map following the model
derived by Miville-Deschênes et al. (2008). Free-free emission is simulated
using a template given by the $H\alpha$ emission map from Dickinson et al.
(2003), with a uniform frequency scaling over the sky and slowly varying with
frequency. For the AME, usually described as a radio emission produced by the
rapid rotation of electric dipoles associated to small dust grains, we use a
high resolution thermal dust template from Planck observations, scaled to low
frequency according to the ratio between AME and thermal dust as found by Ade
et al. (2016), and extrapolated it to BINGO frequencies using a single
emission law. A template for the thermal dust emission is obtained by applying
Generalized Needlet Internal Linear Combination101010This is the component
separation code also employed here for cleaning the 21 cm maps; see Section
4.2 for references and a brief description of the code. (GNILC) code to Planck
2015 data. Dust spectral index and temperature maps, used to extrapolate
thermal dust emission across frequencies using a single modified blackbody in
each pixel, are obtained from fits of GNILC dust maps at different
frequencies. Although AME and thermal dust emission are subdominant at BINGO
frequencies, both are taken into account here for completeness.

The extragalactic emission from thermal and kinetic SZ effects are modeled
according to prior knowledge of existing galaxy clusters for a number density
as a function of mass and redshift predicted by the cosmological model of
interest, given by Tinker et al. (2008). The unresolved radio point sources
component is simulated using catalogs of observed sources from 850 MHz and
4.85 GHz. A population of sources below detection threshold is simulated on
the basis of theoretical number counts. A more complete discussion of
foreground simulation for BINGO frequency can be found in Fornazier et al.
(2021) and Liccardo et al. (2021).

We also take into account the expected thermal (white) noise level per pixel,
employing a system temperature of 70 K. This level is estimated considering 28
horns, each of them observing at a constant elevation, achieving a survey area
of 5324 deg2 (a sky fraction of $f_{sky}\sim 0.13$). We assume 5 years of
observation and the horn arrangement designed for the Phase 1 of observations,
as discussed in previous BINGO publications (Wuensche et al., 2021; Abdalla et
al., 2021b; Liccardo et al., 2021). For good sampling of the map in
declination, the position of the horns will be shifted to displace their
pointing on the sky by a fraction of a beam width in elevation. This will
homogenize the observation time over the innermost BINGO area. A detailed
description of the noise simulations employed here can be found in Fornazier
et al. (2021).

Note that, although this paper does not account for the low frequency ($1/f$)
noise, we are aware it is an important complication in real data and can be
very detrimental to Hi IM data if not efficiently removed (Bigot-Sazy et al.,
2015; Li et al., 2021). The $1/f$ noise is correlated across the frequency
band and, when projecting the observed signal into a map, it appears as
stripes, that is, as large scale spatial fluctuations (Harper et al., 2018).
In this sense, since such instrumental effect has the potential to impact our
analyses at lowest redshifts, where the BAO scale is comparable to the
$15^{\circ}$ stripe of the BINGO coverage, it is being addressed in various
working groups across the collaboration and our findings will be presented in
upcoming works of BINGO series. Moreover, it is also worth remembering that
another instrumental effect relevant for intensity mapping experiments is the
leakage of polarized foregrounds into the intensity data (Alonso et al., 2015;
Cunnington et al., 2021). Unlike the foregrounds aforementioned, this
polarization leakage is not expected to have a smooth dependence with
frequency, which makes the cleaning process more complex (Carucci et al.,
2020). In this sense, although BINGO is designed to minimize the impact of
such effect (Wuensche et al., 2021; Abdalla et al., 2021b), future work will
investigate its impact on the foreground cleaning efficiency, as well as on
the BAO recovery.

Since all simulations are generated as full sky maps, we select the BINGO
region using an appropriate mask, which accounts not only for the expected sky
coverage, but also cuts out a region with strong Galactic foreground signal.
To avoid the impact of sharp edges in the masked region, the mask is apodized
with a cosine square transition of 5 degrees using the NaMaster
code.111111Details about this method can be found in Alonso et al. (2019). See
also Section 5.1.1. A detailed description of the procedure to create this
mask is provided in Mericia et al. (2022). Fig. 1 shows an illustrative
example of a 21 cm IM simulation, in the lowest redshift bin $0.127<z<0.138$,
in the BINGO region.

We also account for the angular smoothing effect introduced by the BINGO beam,
with $\theta_{\mbox{\sc fwhm}}=40$ arcmin, which we approximate by a Gaussian
of same width for all the frequency bins. Each simulated sky map is convolved
with such a Gaussian beam before thermal noise is added. In addition, we
investigate the impact of $\theta_{\mbox{\sc fwhm}}$ scaled with the frequency
$\nu$ according to $\theta_{\mbox{\sc fwhm}}(\nu)=\theta_{\mbox{\sc
fwhm}}(\nu_{0})\times\nu_{0}/\nu$, where $\nu_{0}=1120$ MHz and
$\theta_{\mbox{\sc fwhm}}(\nu_{0})=40$ arcmin (Bigot-Sazy et al., 2015). Note
that, although in both cases we approximate the BINGO beam by a Gaussian,
there will be contribution from structures such as the sidelobes, discussed in
the companion papers Wuensche et al. (2021) and Abdalla et al. (2021b), as
well as the possible coupling of the horns, still to be investigated.

Figure 1: Illustrative example of a FLASK simulation, at the smallest redshift
bin ($0.127<z<0.138$), convolved with a Gaussian beam of $\theta_{\mbox{\sc
fwhm}}=40$ arcmin, after selecting the sky region (apodized mask) considered
in all the analyses. The color indicates the brightness temperature in $\mu$K.
The Mollweide projection corresponds to celestial coordinates.

###  4.2 Foreground cleaning process

One of the main challenges of 21 cm IM observations is the efficient
subtraction of the foreground contamination, whose amplitude can be up to
$10^{4}$ larger than the Hi signal. Indeed, even employing an efficient
cleaning technique, it is expected a residual contamination to remain in the
observations, so it is important to evaluate their impact on the cosmological
analyses. In this work, we use the GNILC method (Remazeilles et al., 2011), a
non-parametric component separation technique, to perform the foreground
cleaning process. GNILC has been tested in previous work and showed an
excellent performance when applied to 21 cm IM observations (Olivari et al.,
2016) and, more specifically, to the BINGO simulated data (see papers Liccardo
et al., 2021; Fornazier et al., 2021; Mericia et al., 2022).

The temperature signal, $\mathbf{x}(p)$, represented by a vector of dimension
$n_{\rm ch}$ (where $n_{\rm ch}$ is the number of BINGO channels) for each sky
pixel $p$, can be modeled as

$\mathbf{x}(p)=\mathbf{s}(p)+\mathbf{n}(p)+\mathbf{f}(p)\,,$ (18)

where $\mathbf{s}(p)$ is the 21 cm signal, $\mathbf{n}(p)$ is the instrumental
noise, and $\mathbf{f}(p)$ is the total foreground signal. All contributions
to $\mathbf{x}(p)$ have the same dimension $n_{\rm ch}$. To distinguish
foreground emission from 21 cm signals, GNILC exploits the fact that
foreground emissions are highly correlated between frequencies channels and
thus effectively rely on a few independent spectral degrees of freedom, while
the 21 cm signal, probing shells at different redshifts, is only weakly
correlated between frequency bands. Using a prior (theoretical template) of
the 21 cm and thermal noise power spectra, GNILC evaluates, locally in pixel
and harmonic space, the effective dimension of the foreground subspace, as
identified by a principal component analysis of empirical channel data
covariance matrices, computed on a redundant basis (a tight frame) of a type
of wavelets on the sphere (called “needlets”). With this approach GNILC does
not use any prior information on the foreground components, except the safe
assumption of a strong correlation of their emission between frequencies.
GNILC then projects the data $\mathbf{x}(p)$ out of the foreground subspace.

The 21 cm plus noise signal is then reconstructed, for each wavelet scale,
applying an ILC filter $\mathbf{W}$ to the data,
$\mathbf{\hat{s}}=\mathbf{Wx}$, in the subspace orthogonal to that of the
foregrounds. The ILC filter is constructed in such a way to preserve the 21 cm
signal while filtering out the foreground contamination. The complete
reconstructed 21 cm plus noise map, at each frequency, is finally obtained by
adding the maps corresponding to each wavelet scale. Detailed discussion on
the use of the GNILC methodology for 21 cm foreground cleaning can be found in
Olivari et al. (2016) and Fornazier et al. (2021).

Finally, it is worth emphasizing that the reconstructed 21 cm maps contain,
along with noise, a residual contribution from the foreground components,

$\mathbf{\hat{s}}\simeq\mathbf{s}+\mathbf{Wn}+\mathbf{Wf}\,.$ (19)

where $\mathbf{Wn}$ and $\mathbf{Wf}$ are, respectively, the residual noise
and foreground contribution to the reconstructed 21 cm signal. Then, in the
case of simulations for which we know the exact input foreground
contamination, as is the case here, one can estimate its residual contribution
over the recovered cosmological signal. Here, we use this facility to include
a realistic foreground residual contribution to the simulations to evaluate
its impact on the BAO fitting process, avoiding the need for applying GNILC to
each one of them (see Section 6.3).

##  5 Methodology

This section summarizes the methodology employed to extract the BAO
information from BINGO simulated Hi maps. We describe: 1) how the APS and ACF
are measured from the simulated maps; 2) how the covariance matrices are
calculated; and 3) the fitting method used to estimate the $\alpha$ parameter
from each of the simulations.

###  5.1 Clustering measurements

####  5.1.1 Angular power spectrum

The partial-sky coverage is a common situation for CMB and galaxy surveys,
whether due to the observation strategy or cuts of regions with strong
foreground contamination. The APS computation in these cases is not as
straightforward as it would be in full-sky analyses, where $C_{\ell}$’s are
computed by expanding the radiation field in spherical harmonics and averaging
the $a_{\ell m}$ coefficients:

$C_{\ell}=\frac{1}{2\ell+1}\sum_{m=-\ell}^{m=+\ell}|a_{\ell m}|^{2}\,.$ (20)

The effect of a mask when calculating the $a_{\ell m}$ coefficients over
partial sky is to mix (or couple) different multipoles so that the estimated
APS, $\hat{C}_{\ell}$, can be written in terms of the true one, $C_{\ell}$,
through

$\hat{C}_{\ell}=\sum_{\ell^{\prime}}\mathcal{M}_{\ell\ell^{\prime}}C_{\ell^{\prime}}\,,$
(21)

where $\mathcal{M}_{\ell\ell^{\prime}}$ is the multipole mixing matrix, which
depends on the mask geometry (Hivon et al., 2002).

Here we estimate the APS using the so-called pseudo-$C_{\ell}$ formalism, as
implemented in the NaMaster code121212https://namaster.readthedocs.io (Alonso
et al., 2019). The mixing matrix is estimated analytically, according to the
shape of the mask, and is used to obtain an unbiased estimate of $C_{\ell}$
(for examples of application of this formalism and/or this code to different
cases see, e.g., Brown et al., 2005; Moura-Santos et al., 2016; Loureiro et
al., 2019; Nicola et al., 2020).

However, the loss of information due to the mask usually makes the mixing
matrix ill-conditioned and, for this reason, singular, with no direct
inversion. To deal with that, NaMaster allows the usage of bandbowers,
calculating a binned and invertible version of
$\mathcal{M}_{\ell\ell^{\prime}}$. Here, we consider a linear binning,
calculating $C^{i}_{\ell}$ at bins of width $\Delta\ell$, centered at $\ell$,
a procedure that also helps making the corresponding covariance matrix more
diagonal. Different $\Delta\ell$ widths are tested so that we can find the
more appropriate one to be used in the more realistic analyses, composing the
fiducial configuration, defined when running the several robustness tests.

####  5.1.2 Angular correlation function

We measure the ACF from each of the simulations using the following equation
(de Simoni et al., 2013):

$\omega^{i}(\theta)=\frac{\sum_{pp^{\prime}}\delta T_{p}\delta
T_{p^{\prime}}wt_{p}wt_{p^{\prime}}}{\sum_{pp^{\prime}}wt_{p}wt_{p^{\prime}}}\,,$
(22)

with $\delta T_{p}=T_{b}-\langle T_{b}\rangle$, where $T_{b}$ is the Hi
brightness temperature at each pixel $p$, $\theta$ is the angular separation
between pixels $p$ and $p^{\prime}$. The parameter $wt_{p}$ is associated to
the observed sky region (mask information; Fig. 1), receiving value 1 in case
pixel $i$ contains valid data and 0 in case its area has not been observed or
if it is located in the Galactic region and is removed from the analyses. The
$\omega(\theta)$, as given by Eq. (22), were calculated at equally spaced
$\theta$ values, with bin width $\Delta\theta$, using the public code
TreeCorr131313https://rmjarvis.github.io/TreeCorr (Jarvis et al., 2004). As
well as for the APS analyses, the best $\Delta\theta$ width for our fiducial
configuration will be defined after testing few different values.

###  5.2 Covariance matrix

We use $N=1500$ FLASK and $N=500$ N-body mock realizations to estimate the
covariance matrices as

$\mathtt{C}^{ij}_{km}=\frac{1}{N-1}\sum_{n=1}^{N}(X^{i,n}_{k}-\bar{X}^{i}_{k})\,(X^{j,n}_{m}-\bar{X}^{j}_{m})\,,$
(23)

where $k$ and $m$ indices run over the different $\ell$ or $\theta$ bins, and
$i$ and $j$ indicate the z-bin; $X^{i}_{k}$ is the clustering measurement,
$C^{i}_{\ell}$ or $\omega^{i}(\theta)$, and $\bar{X}^{i}_{k}$ is the
respective average over the simulations.

It is well known that the statistical noise in the covariance matrix estimated
from simulations introduces a bias on its inverse,
$[\mathtt{C}^{ij}_{km}]^{-1}$, the precision matrix. The bias is mainly
dependent of the number $N$ of simulations used to construct
$\mathtt{C}^{ij}_{km}$ and the number of entries $N_{p}$ of the data vectors
$X$. However, as proposed by Hartlap et al. (2007) and used, for example, in
the analyses of Camacho et al. (2019); Ata et al. (2018) and Anderson et al.
(2014), an unbiased version of the precision matrix can be obtained by
rescaling it as

$[\mathtt{C}^{ij}_{km}]^{-1}\rightarrow\frac{N-N_{p}-2}{N-1}[\mathtt{C}^{ij}_{km}]^{-1}\,.$
(24)

The rescaled version of the precision matrix is employed in all our analyses.

###  5.3 Parameter inference

The fit of the template models to the estimated clustering measurement goes
through two (recursive) steps: one of them is to fit the nuisance parameters,
$B$ and $A_{q}$, appearing linearly in both template models, and the other is
to use the maximum likelihood estimator (MLE) to evaluate $\alpha$, as
described in Eqs. 15 and 16. Following Chan et al. (2018) and Camacho et al.
(2019), we perform a least-square fit to find the best-fit $A_{q}$ values,
and, fixing them, we repeat the procedure to obtain the best-fit $B$ value, in
both cases by minimizing the $\chi^{2}$ defined as

$\chi^{2}(\lambda)=\sum_{k,m}[X^{i}_{k}-x^{i}_{k}(\lambda)]\,[\mathtt{C}^{ij}_{km}]^{-1}\,[X^{j}_{m}-x^{j}_{m}(\lambda)]\,,$
(25)

where $\lambda=\\{\alpha,B,A_{q}\\}$ and $x^{i}_{k}$ is the template model for
the $i$th z-bin and the $k$th $\ell$/$\theta$ bin. Having done that, we
maximize the likelihood function,

$L(\alpha)\propto\exp(-\chi^{2}/2)\,,$ (26)

now dependent on only one parameter, since $\chi(\lambda)$ is reduced to
$\chi(\alpha)$, allowing to estimate the best-fit $\alpha$ value (see also,
Anderson et al., 2014). In practice, this value is obtained through a grid
search, so that the least-square fit of the nuisance parameters, $A_{q}$ and
$B$, is repeated for each $\alpha$ value evaluated.

Taking advantage of a tomographic approach, we are able to combine redshift
bins so that the significance of the BAO detection can increase. We fit one
$\alpha$ parameter through a joint analysis applying the MLE over a set of
$N_{z}$ consecutive z-bins. In particular, we fit the nuisance parameters, $B$
and $A_{q}$, to each z-bin individually, and, fixing their values, we find one
best-fit $\alpha$ for each set of $N_{z}$ z-bins.

The $1\sigma$ error for the $\alpha$ estimates, $\sigma_{\alpha}$, is defined
as the deviation from the maximum likelihood point, at $\chi_{min}^{2}$, by
$\Delta\chi^{2}=1$. It is worth noting that this $\Delta\chi^{2}=1$ rule is
valid in the case of a Gaussian likelihood, $L(p)$, for only one parameter, in
our case $p=\alpha$ (see Chan et al., 2018, and references therein, for
details). In principle, this can be employed in our analyses, as commonly seen
in literature (e.g., Ata et al., 2018; Carter et al., 2018; Abbott et al.,
2019, 2021), but one has to be careful with deviations from Gaussian
distribution, which can lead to $\sigma_{\alpha}$ values that underestimate
the error bars. In the next section we discuss and evaluate the validity of
this rule for each clustering statistic used here.

##  6 BAO fitting results

In this section we present our findings from analyzing FLASK and N-body mock
realizations. We start with an ideal case, analyzing pure 21 cm simulations,
only accounting for the BINGO sky coverage and fixed 40 arcmin beam (Section
6.1). The contribution of thermal noise and foreground residuals are then
considered one at a time, so that we can investigate the impact of each one
separately, as well as the impact of a redshift dependent beam size (Sections
6.2 and 6.3). We also present the results from an extensive (but not
exhaustive) list of robustness tests evaluating several aspects of the
methodology employed here to extract the BAO signal from BINGO-like
simulations, using FLASK mocks as the cosmological signal (Section 6.4). Such
tests allow finding the most appropriate configuration (hereafter called
fiducial configuration), optimizing the analysis. In particular, the fiducial
configuration is defined by:

  1. (a)

angular and multipole binning: $\Delta\theta=0.5^{\circ}$ and $\Delta\ell=10$,

  2. (b)

$A_{q}$ parameters for $C_{\ell}$ and $\omega(\theta)$ templates: $q=-1,0,1,2$
and $q=0,1,2$, respectively.

The 500 N-body mocks are analyzed using only this fiducial configuration
(Section 6.5).

Also important is the choice for the range of scales, $\theta$ and $\ell$
intervals. Since BINGO coverage is limited to a declination strip of $\sim
15^{\circ}$, the $C_{\ell}$ determination at the smallest multipoles
($\ell\lesssim 15$) is compromised. Then, applying the BAO fitting to the
(high signal-to-noise ratio) mean $C_{\ell}$ from the 1500 FLASK mocks we
chose $\ell_{min}\approx 32$, for which $\alpha$ estimate is the least biased
with the smallest $\sigma_{\alpha}$ error. This way the choice of multipole
range can also account for any artifact introduced by the mask geometry, which
would be more evident in the mean clustering measurement. Details about the
fitting procedure over the mean APS and ACF are presented in the next
subsections.

For all the z-bins we consider the same minimum multipole, but the maximum one
is chosen according to the multipole ranges at which the BAO wiggles are
concentrated, which are more spread out over higher multipoles the higher the
redshift (Fig. 2). In this sense, after few tests over the mean $C_{\ell}$, we
chose $141\lesssim\ell_{max}\lesssim 401$ for lower to higher redshifts, so
that we use only the range containing the BAO wiggles, avoiding to increase
the number of data points with multipoles that do not bring information on the
BAO signal (see discussion in Sec. 5.2).

A similar procedure is used to choose the $\theta$ range for the ACF analyses.
Applying the BAO fitting pipeline over the mean $\omega(\theta)$ measurements
from the FLASK mocks we chose, for each z-bin, the range of scales large
enough to encompass a bit more than the full width of the BAO peak. Our
choices for the $\theta$ ranges goes from
$10.5^{\circ}\lessapprox\theta\lessapprox 21.0^{\circ}$, for the smallest
z-bin, to $2.5^{\circ}\lessapprox\theta\lessapprox 8.0^{\circ}$, for the
highest z-bin.

The $\theta$ and $\ell$ ranges used for each z-bin in all our analyses are
presented in Table 1. The large number of z-bins demanded a large number of
tests to find the more appropriate angular and mutipole ranges for each
estimator, reason why we only show our final choice in each case. We use the
approximate ($\approx$) symbol to refer to minimum and maximum $\theta$ and
$\ell$ values because, although these numbers are exact for the fiducial
values of $\Delta\theta$ and $\Delta\ell$, they are slightly different when
testing other binning schemes. For such cases we choose the range of scales as
close as possible to the fiducial configuration, shown in Table 1. Note that
all tests to choose the best angular and multipole ranges are applied to the
BINGO-like simulations, since our goal is optimize our analyses for future
BINGO observations.

Figure 2: Theoretical APS calculated for three of the 30 tomographic bins linearly spaced in frequency, whose average redshifts are $\bar{z}_{5}=0.166$, $\bar{z}_{15}=0.263$ and $\bar{z}_{25}=0.377$. Note the BAO wiggles appearing at higher multipoles the higher the redshift. The black dashed line shows the white noise amplitude. The expected contribution from foreground residual at the highest z-bin ($\bar{z}_{30}=0.442$; the one with the highest amplitude of foreground residual) is represented by the gray line. The dotted lines show the effect introduced by the instrumental beam with $\theta_{\mbox{\sc fwhm}}=40^{\prime}$.  Table 1: Maximum multipole, $\ell_{max}$, and minimum and maximum angular scales, $\theta_{min}$ and $\theta_{max}$, in degrees, considered for the template fitting over APS and ACF clustering estimates from each z-bin. For the APS, the minimum multipole is the same for all the z-bins, $\ell_{min}\approx 32$. The tomographic bins, from lower to higher redshifts, are enumerated from 1 to 30. The central redshift of each z-bin, $\bar{z}$, is also shown. # | $\bar{z}$ | $\ell_{max}$ | $[\theta_{min},\theta_{max}]$ | # | $\bar{z}$ | $\ell_{max}$ | $[\theta_{min},\theta_{max}]$
---|---|---|---|---|---|---|---
1 | 0.131 | 141 | [10.5, 21.0] | 16 | 0.274 | 301 | [5.0, 10.5]
2 | 0.140 | 141 | [10.5, 20.0] | 17 | 0.284 | 311 | [4.5, 10.5]
3 | 0.148 | 151 | [9.0, 18.5] | 18 | 0.295 | 321 | [4.5, 10.0]
4 | 0.157 | 161 | [8.5, 17.5] | 19 | 0.306 | 331 | [4.0, 10.0]
5 | 0.166 | 181 | [8.0, 17.0] | 20 | 0.318 | 341 | [4.0, 10.0]
6 | 0.175 | 201 | [7.5, 16.0] | 21 | 0.329 | 361 | [4.0, 9.5]
7 | 0.184 | 211 | [7.0, 15.5] | 22 | 0.341 | 371 | [3.5, 9.5]
8 | 0.194 | 231 | [7.0, 15.0] | 23 | 0.353 | 371 | [3.5, 9.5]
9 | 0.203 | 251 | [6.5, 14.0] | 24 | 0.365 | 381 | [3.5, 9.0]
10 | 0.213 | 251 | [6.5, 13.5] | 25 | 0.377 | 391 | [3.0, 9.0]
11 | 0.222 | 271 | [6.0, 13.0] | 26 | 0.390 | 391 | [3.0, 9.0]
12 | 0.232 | 271 | [6.0, 12.5] | 27 | 0.403 | 401 | [3.0, 8.0]
13 | 0.242 | 271 | [5.5, 11.5] | 28 | 0.416 | 401 | [3.0, 8.0]
14 | 0.252 | 281 | [5.5, 11.5] | 29 | 0.429 | 401 | [2.5, 8.0]
15 | 0.263 | 291 | [5.0, 11.0] | 30 | 0.442 | 401 | [2.5, 8.0]

###  6.1 Fitting clustering measurements from 21 cm only simulations

Employing the fiducial configuration, the BAO signal is measured by jointly
fitting three sets of $N_{z}=10$ consecutive $z$-bins, the lower, intermediate
and higher $z$-bins (redshift ranges of $\Delta z\approx 0.09,0.11,0.12$
widths), estimating only one $\alpha$ parameter for each set. The advantage of
a tomographic approach, combining different z-bins, is to improve the error
bars and statistical significance of the measurements. Notice that the
nuisance parameters, one parameter $B$ and 4 (3) parameters $A_{q}$, are
fitted to the $C_{\ell}$ ($\omega(\theta)$) estimated for each of the 10
z-bins individually, so that, for the fiducial configuration, we have a total
of $10\times 5(4)=50(40)$ nuisance parameters. The $\alpha$ fitting results
obtained from analyzing the 21 cm only simulations (accounting for BINGO sky
coverage and beam) are shown in the first part of Tables 2 and 3 for the ACF
and APS estimators, respectively.

We start testing our BAO fitting pipeline over the high signal-to-noise ratio
mean clustering measurements, that is, the mean $C_{\ell}$ and
$\omega(\theta)$ from the 1500 FLASK mocks. Our aim in fitting such high
signal-to-noise measurement is not only to help decide about angular and
multipole ranges and their binning widths, $\Delta\ell$ and $\Delta\theta$,
but also to be able to identify any possible systematic bias on $\alpha$
estimates. Hereafter, we use $\alpha^{m}$ and $\sigma^{m}_{\alpha}$ to denote
the results of fitting the mean $C_{\ell}$ or $\omega(\theta)$, and $\alpha$
and $\sigma_{\alpha}$ the results for individual mocks. The fitting over the
mean clustering measurement requires the original covariance matrix to be
divided by the total number of simulations, but such scaling would not provide
a realistic estimate of $\sigma_{\alpha}$ for one mock realization. Therefore,
since likelihoods can be well approximated by Gaussians for high signal-to-
noise ratio measurements of the BAO feature, we follow Abbott et al. (2019)
and divide the original covariance matrix by 10 to obtain the likelihood and
fit the $\alpha^{m}$ parameter. The corresponding $1\sigma$ uncertainty is
calculated as described in the previous section and, then, normalized to
obtain $\sigma^{m}_{\alpha}\rightarrow\sqrt{10}\sigma^{m}_{\alpha}$. Notice
that using a different normalization factor, for example, 20 instead of 10,
leads to similar error amplitude. The $\alpha^{m}\pm\sigma^{m}_{\alpha}$
results are presented in the last column of Tables 2 and 3. Using the fiducial
configuration to fit the mean clustering measurements from the 21 cm only
simulations we find a prediction of 24%, 10%, and 5% errors for the lower,
intermediate and higher z-bins, respectively. Note that the same procedure is
employed to all tests fitting the mean clustering measurements, namely,
varying the fiducial configuration or/and including contaminant signals, whose
results are shown in the last column of Tables 2 to 5 and discussed in the
next subsections. In general, $\alpha^{m}$ estimates are consistent with
unity, although presenting small bias, which, in all the cases, is well below
the statistical uncertainty.

In addition, Tables 2 to 5 present the same summary statistics obtained
applying the BAO fitting pipeline to each mock for each of the three redshift
ranges. In particular, there we show the average shift parameter
$\langle\alpha\rangle$, the average error in $\alpha$ estimates
$\langle\sigma_{\alpha}\rangle$, and two measures of the dispersion of the
$\alpha$ distribution, the symmetric error around $\langle\alpha\rangle$
encompassing 68% of the mocks (less sensitive to the tails) $\sigma_{68}$, and
the common standard deviation of the distribution $\sigma_{std}$, as well as,
the average $\langle\chi^{2}\rangle$ (and degrees of freedom, dof). Notice
that all these quantities are calculated from the fraction $N_{s}$ of the
total 1500 mocks, which correspond to the mocks fulfilling our selection
criteria, namely, those whose $\alpha\pm\sigma_{\alpha}$ interval are inside
the range $[0.6,1.4]$ and that have $\chi^{2}>\chi^{2}_{nw}$, where
$\chi^{2}_{nw}$ is obtained from fitting a non-BAO template model to the
clustering estimates (see Section 6.6). This fraction $N_{s}$ is also
presented in the tables, as well as the fraction $N_{d}$ of the mock whose
$\alpha$ estimate is inside the range $[0.8,1.2]$. We consider these as the
mocks with a BAO detection. Figures 3 and 4 show, for ACF and APS,
respectively, the distribution of the $\alpha$ parameter, the
$\sigma_{\alpha}$ error and the $\chi^{2}$, estimated from the $N_{s}$
fraction of the mocks; the blue curves show the 21 cm only results.

It is worth mentioning that it is commonly employed in literature the
detection criterion of having the full interval $\alpha\pm\sigma_{\alpha}$
inside the prior range $[0.8,1.2]$ (e.g., Ata et al., 2018; Chan et al.,
2018). However, one can observe from these analyses that the distribution of
the $\alpha$ parameter in general seems to be more Gaussian than what we find
in our analyses (left panels in Figs. 3 and 4). A possible reason for this is
the smaller redshift we consider here, where the contribution from non-linear
effects is stronger. However, results from Villaescusa-Navarro et al. (2017),
investigating BAO detection from 21 cm signal for the SKA case, for the
redshift range $0.35<z<3.05$, also show $\alpha$ distributions with clear
deviations from a perfect Gaussian (but, note the smaller number of
simulations employed there, namely, 100). As pointed by Chan et al. (2018) and
Abbott et al. (2021), a natural consequence from having a (approximate)
Gaussian distribution is a reasonable concordance among the three different
error measurements, that is,
$\langle\sigma_{\alpha}\rangle\sim\sigma_{68}\sim\sigma_{std}$. This is not
our case, as can be seen from Tables 2 and 3 (as well as from Tables 4 and 5),
indicating that $\langle\sigma_{\alpha}\rangle$ is not meaningful or
representative of the error in the $\alpha$ measurements for individual mock
realizations (see also Figs. 3 and 4). Our results show that, compared to
$\langle\sigma_{\alpha}\rangle$, the errors given by $\sigma_{68}$ are
overestimated by $\sim$ 18% to 33%, from the smallest to the highest z-bins,
when using the ACF, while underestimated by $\sim$ 50% for APS. Also,
comparing the expected $\sigma^{m}_{\alpha}$ obtained fitting the mean
$C_{\ell}$ and $\omega(\theta)$ to the average
$\langle\sigma_{\alpha}\rangle$, we find a reasonable agreement for
intermediate and higher z-bins, but not for the lower z-bins, in particular
for the ACF estimator. In addition, although $\sigma_{std}$ is in better
agreement with $\sigma_{68}$, we still find non negligible differences among
them, which confirms our $\alpha$ distributions as non-Gaussian (a larger
$\sigma_{std}$ indicate the presence of non-Gaussian tails). Such reasons
motivated our choice of using the $\alpha$ values, instead of
$\alpha\pm\sigma_{\alpha}$ ranges, belonging to the interval $[0.8,1.2]$ as
criterion for a BAO detection (defining the $N_{d}$ fraction), as well as our
choice of using the 68% spread of the $\alpha$ distributions, $\sigma_{68}$,
as the representative error in our measurements.

Figure 3: Histogram distribution of the $\alpha$ parameter, $\sigma_{\alpha}$
error, and $\chi^{2}$ obtained applying the BAO fitting pipeline to each of
the FLASK simulations for the ACF estimator using the fiducial configuration.
Rows from top to bottom show results for each redshift range, the lower (1-10
z-bins), the intermediate (11-20 z-bins), and the higher redshifts (21-30
z-bins). The average redshift for each range is presented in the first panel
of each row. In all the plots, the solid blue, dashed red and dotted green
lines show the distribution of the parameters resulting from analyzing,
respectively, the 21 cm only simulations (21 cm), the noisy 21 cm simulations
(21 cm + WN), and the BINGO-like simulations (adding also the foreground
residual; 21 cm + WN + FG). All these results correspond to the $N_{s}$
fraction of the mocks. See Table 2.

Figure 4: Same as Fig. 4 but for the APS estimator. Again all distributions
are constructed from the corresponding $N_{s}$ fraction of the mocks. See
Table 3.

Comparing the results obtained for the three redshift ranges we confirm both
clustering estimators to perform better at higher redshifts, as expected.
Remember that the lower the redshift the larger the BAO angular scale,
$\theta_{BAO}$, which for the BINGO redshift range, $0.127<z<0.449$,
corresponds to $18^{\circ}\gtrsim\theta_{BAO}\gtrsim 7^{\circ}$, for the
fiducial cosmology. It means that, at the smallest redshifts the
$\theta_{BAO}$ is larger than the $15^{\circ}$ stripe of the BINGO coverage,
leading to worse statistics at such z-bins. Figure 5 compares APS and ACF
results obtained from 21 cm only simulations (first row), showing the
improvement in their performance for intermediate and higher redshifts, as
well as a high correlation among their results. In fact, comparing the
performance from each of the clustering estimators we found that they show
different sensitivities to specific redshift ranges. Our results show that
$C_{\ell}$ measurements lead to smaller $\sigma_{68}$ errors for intermediate
and higher redshift ranges, namely, 20% and 25% smaller $\sigma_{68}$ when
compared to $\omega(\theta)$. Also, both $N_{s}$ and $N_{d}$ fractions of the
mocks are larger for the APS at lower and intermediate redshifts, and
comparable at higher redshifts. For the $N_{d}$ fraction the APS has $\sim$1.8
higher probability of detecting BAO at lower redshifts at 21 cm only mocks
compared to the ACF.

Finally, comparing the two estimators we find, in summary, an apparent better
performance of the APS over the ACF. Such conclusion is mainly based on the
following aspects of the ACF results: the slightly larger $\sigma_{68}$
uncertainties at intermediate and higher redshifts; the quite larger bias on
the average $\langle\alpha\rangle$, especially at lower z-bins, corresponding
to 0.95$\sigma_{68}$; and the smaller $N_{s}$ and $N_{d}$ fractions at small
and intermediate z-bins.

Notice that we also evaluate the z-bin width and the way the 30 z-bins are
combined to estimate the shift parameter. First, the BAO fitting pipeline was
applied to three ranges of redshift composed by $N_{z}=12,10,8$ consecutive
z-bins with $\delta\nu=9.33$ MHz, so that $\alpha$ is estimated over three
redshift ranges of same total width each ($\Delta z\approx 0.10$). Then, we
test a tomographic binning of the BINGO frequency band into 30 z-bins linearly
spaced in redshift ($\delta z\approx 0.01$), instead of frequency, but still
estimating $\alpha$ for three redshift ranges of $N_{z}=10$ z-bins each (same
total redshift width). We find that, in general, both cases lead to results
comparable to those from Tables 2 and 3, although each redshift range and
clustering estimator has its own most advantageous configuration. In this
sense, for simplicity, and because previous BINGO papers employ a redshift
binning linear in frequency (Liccardo et al., 2021; Fornazier et al., 2021;
Costa et al., 2021), we keep using z-bins with $\delta\nu=9.33$ MHz width and
the BAO fitting process considering three sets of $N_{z}=10$ z-bin for both
estimators.

Table 2: BAO fitting results for the ACF using 1500 FLASK mocks. The three first parts of the table correspond to analyses of 21 cm only, noisy 21 cm (+ white noise) and BINGO-like (+ white noise + foreground residuals) mocks, using in both cases the fiducial configuration. The other parts of the table show results from the robustness tests, where the first column points out the aspects varying with respect to the fiducial configuration, namely the $\Delta\theta$ binning width and the $q$ index from $\omega(\theta)$ template. All robustness tests performed over the BINGO-like mocks. The last part of the table corresponds to analyses of the BINGO-like mocks, employing the fiducial configuration, when varying the beam size with frequency. The last column shows $\alpha^{m}\pm\sigma^{m}_{\alpha}$ obtained from fitting the mean clustering from the mocks. See text for details. Tests | z-bins | $\langle\alpha\rangle$ | $\langle\sigma_{\alpha}\rangle$ | $\sigma_{68}$ | $\sigma_{std}$ |  $\langle\chi^{2}\rangle$/dof ($\chi^{2}_{red}$) |  $N_{s}$ (%) |  $N_{d}$ (%) | Mean
---|---|---|---|---|---|---|---|---|---
21 cm only
Fid. config. | 1–10 | 1.1158 | 0.1427 | 0.1209 | 0.1166 | 122.49/123 (1.00) | 65.27 | 45.73 | 1.0067 $\pm$ 0.2433
11–20 | 1.0426 | 0.0977 | 0.0775 | 0.0923 | 76.84/70 (1.10) | 81.00 | 75.00 | 1.0082 $\pm$ 0.1014
21–30 | 1.0060 | 0.0565 | 0.0425 | 0.0514 | 69.54/61 (1.14) | 92.60 | 92.00 | 1.0076 $\pm$ 0.0507
\+ white noise
| 1–10 | 1.0906 | 0.1619 | 0.1042 | 0.1066 | 122.05/123 (0.99) | 58.73 | 48.60 | 1.0070 $\pm$ 0.2915
Fid. Config. | 11–20 | 1.0524 | 0.1213 | 0.0924 | 0.0948 | 76.34/70 (1.09) | 80.53 | 73.80 | 1.0107 $\pm$ 0.1470
| 21–30 | 1.0267 | 0.0932 | 0.0807 | 0.0883 | 67.88/61 (1.11) | 89.60 | 85.13 | 1.0070 $\pm$ 0.0989
\+ white noise + foreground residuals
Fid. Config. ($\Delta\theta=0.50^{\circ}$; $q=0,1,2$) | 1–10 | 1.0897 | 0.1624 | 0.1123 | 0.1118 | 122.42/123 (1.00) | 57.07 | 46.27 | 0.9952 $\pm$ 0.3270
11–20 | 1.0362 | 0.1267 | 0.0969 | 0.1070 | 76.60/70 (1.09) | 70.20 | 63.93 | 0.9989 $\pm$ 0.1546
21–30 | 1.0074 | 0.1006 | 0.0809 | 0.0930 | 68.37/61 (1.12) | 78.00 | 73.53 | 0.9965 $\pm$ 0.1039
| 1–10 | 1.0901 | 0.1649 | 0.1018 | 0.1080 | 238.66/287 (0.83) | 53.40 | 44.67 | 0.9670 $\pm$ 0.6540
$\Delta\theta=0.25^{\circ}$ | 11–20 | 1.0370 | 0.1307 | 0.0999 | 0.1054 | 166.28/180 (0.92) | 71.67 | 64.87 | 1.0135 $\pm$ 0.2560
| 21–30 | 1.0063 | 0.1034 | 0.0842 | 0.0970 | 154.15/164 (0.94) | 76.67 | 72.33 | 0.9826 $\pm$ 0.1394
| 1–10 | 1.0888 | 0.1647 | 0.1100 | 0.1157 | 155.02/165 (0.94) | 57.73 | 46.60 | 1.0034 $\pm$ 0.3245
$\Delta\theta=0.40^{\circ}$ | 11–20 | 1.0375 | 0.1298 | 0.0969 | 0.1054 | 100.85/98 (1.03) | 72.00 | 65.20 | 0.9981 $\pm$ 0.1597
| 21–30 | 1.0160 | 0.0998 | 0.0833 | 0.0982 | 90.00/86 (1.05) | 78.80 | 73.07 | 0.9976 $\pm$ 0.1065
| 1–10 | 1.1687 | 0.1299 | 0.1092 | 0.1117 | 130.45/133 (0.98) | 37.40 | 18.73 | 0.9894 $\pm$ 0.3321
$q=0,1$ | 11–20 | 1.0364 | 0.1294 | 0.1022 | 0.1070 | 85.43/80 (1.07) | 70.00 | 63.60 | 0.9955 $\pm$ 0.1521
| 21–30 | 1.0068 | 0.1005 | 0.0823 | 0.0951 | 78.36/71 (1.10) | 85.80 | 80.67 | 1.0023 $\pm$ 0.0989
| 1–10 | 1.0790 | 0.1813 | 0.1432 | 0.1269 | 114.55/113 (1.01) | 34.93 | 26.47 | 1.0015 $\pm$ 0.3270
$q\leavevmode\nobreak\ =\leavevmode\nobreak\ -1,\leavevmode\nobreak\ 0,\leavevmode\nobreak\ 1,\leavevmode\nobreak\ 2$ | 11–20 | 1.0344 | 0.1307 | 0.0963 | 0.1018 | 67.38/60 (1.12) | 70.53 | 65.20 | 1.0047 $\pm$ 0.1521
| 21–30 | 1.0200 | 0.0954 | 0.0801 | 0.0895 | 58.92/51 (1.14) | 85.87 | 82.20 | 1.0096 $\pm$ 0.1014
| 1–10 | 1.0995 | 0.1657 | 0.1046 | 0.1039 | 216.71/123 (1.76) | 14.20 | 11.33 | 0.9952 $\pm$ 0.7275
$q=-2,0,1$ | 11–20 | 1.0444 | 0.1214 | 0.0839 | 0.0904 | 146.87/70 (2.10) | 39.87 | 37.73 | 0.9989 $\pm$ 0.1648
| 21–30 | 1.0178 | 0.0963 | 0.0790 | 0.0891 | 101.07/61 (1.67) | 63.87 | 61.07 | 0.9965 $\pm$ 0.1065
\+ white noise + foreground residual (varying beam size)
Fid. config. | 1–10 | 1.0929 | 0.1620 | 0.1162 | 0.1157 | 121.86/123 (0.99) | 58.20 | 46.33 | 0.9994 $\pm$ 0.3118
11–20 | 1.0419 | 0.1245 | 0.1003 | 0.1113 | 77.35/70 (1.10) | 72.87 | 64.27 | 1.0004 $\pm$ 0.1546
21–30 | 1.0164 | 0.0900 | 0.0665 | 0.0871 | 72.09/61 (1.18) | 80.53 | 76.27 | 1.0030 $\pm$ 0.0887
Table 3: Same as Table 2 but for the APS estimator. Tests | z-bins | $\langle\alpha\rangle$ | $\langle\sigma_{\alpha}\rangle$ | $\sigma_{68}$ | $\sigma_{std}$ |  $\langle\chi^{2}\rangle$/dof |  $N_{s}$ (%) |  $N_{d}$ (%) | Mean
---|---|---|---|---|---|---|---|---|---
21 cm only
Fid. config. | 1–10 | 1.0061 | 0.0594 | 0.1200 | 0.1243 | 102.14/100 (1.02) | 93.73 | 84.20 | 1.0012 $\pm$ 0.0608
11–20 | 1.0083 | 0.0316 | 0.0616 | 0.0889 | 185.70/207 (0.90) | 92.87 | 87.73 | 1.0062 $\pm$ 0.0304
21–30 | 1.0024 | 0.0178 | 0.0318 | 0.0399 | 244.17/295 (0.83) | 92.93 | 92.27 | 1.0028 $\pm$ 0.0152
\+ white noise
Fid. config. | 1–10 | 1.0041 | 0.0653 | 0.1311 | 0.1311 | 102.12/100 (1.02) | 92.33 | 79.20 | 0.9915 $\pm$ 0.0684
11–20 | 1.0057 | 0.0410 | 0.0929 | 0.1092 | 185.46/207 (0.91) | 90.87 | 83.27 | 1.0036 $\pm$ 0.0406
21–30 | 0.9965 | 0.0322 | 0.0707 | 0.0897 | 242.42/295 (0.82) | 91.60 | 86.87 | 0.9999 $\pm$ 0.0304
\+ white noise + foreground residual
Fid. Config. ($\Delta\ell=10$; $q=-1,0,1,2$) | 1–10 | 1.0009 | 0.0659 | 0.1257 | 0.1274 | 102.23/100 (1.02) | 90.47 | 78.87 | 0.9925 $\pm$ 0.0684
11–20 | 1.0012 | 0.0418 | 0.0888 | 0.1063 | 185.53/207 (0.90) | 92.47 | 84.77 | 0.9985 $\pm$ 0.0431
21–30 | 0.9975 | 0.0323 | 0.0701 | 0.0890 | 242.88/295 (0.82) | 88.13 | 83.73 | 1.0012 $\pm$ 0.0304
| 1–10 | 1.0253 | 0.0697 | 0.1226 | 0.1205 | 45.23/36 (1.26) | 94.07 | 82.67 | 1.0182 $\pm$ 0.0710
$\Delta\ell=15$ | 11–20 | 1.0070 | 0.0474 | 0.1002 | 0.1154 | 108.96/108 (1.01) | 92.87 | 83.00 | 1.0016 $\pm$ 0.0482
| 21–30 | 1.0015 | 0.0345 | 0.0685 | 0.0867 | 157.06/168 (0.93) | 94.47 | 90.40 | 1.0020 $\pm$ 0.0355
| 1–10 | 1.0299 | 0.0724 | 0.1293 | 0.1128 | 27.24/17 (1.60) | 96.87 | 87.80 | 1.0346 $\pm$ 0.0760
$\Delta\ell=20$ | 11–20 | 1.0199 | 0.0476 | 0.1088 | 0.1117 | 76.90/71 (1.08) | 94.60 | 86.93 | 1.0117 $\pm$ 0.0456
| 21–30 | 1.0009 | 0.0366 | 0.0726 | 0.0912 | 116.66/117 (1.00) | 96.20 | 91.47 | 0.9975 $\pm$ 0.0355
| 1–10 | 1.0523 | 0.0682 | 0.1307 | 0.1334 | 119.73/120 (1.00) | 62.73 | 50.80 | 1.0985 $\pm$ 0.0735
$q=0,1$ | 11–20 | 1.0558 | 0.0436 | 0.0788 | 0.0870 | 199.91/227 (0.88) | 73.33 | 68.47 | 1.0626 $\pm$ 0.0431
| 21–30 | 1.0279 | 0.0333 | 0.0517 | 0.0556 | 254.90/315 (0.81) | 76.27 | 75.67 | 1.0294 $\pm$ 0.0330
| 1–10 | 0.9995 | 0.0660 | 0.1236 | 0.1279 | 110.52/110 (1.00) | 71.87 | 62.40 | 0.9935 $\pm$ 0.0684
$q=0,1,2$ | 11–20 | 0.9912 | 0.0419 | 0.0858 | 0.1091 | 192.63/217 (0.89) | 82.73 | 74.80 | 0.9960 $\pm$ 0.0406
| 21–30 | 0.9958 | 0.0324 | 0.0637 | 0.0818 | 248.84/305 (0.82) | 84.33 | 81.53 | 0.9987 $\pm$ 0.0304
| 1–10 | 1.0161 | 0.0664 | 0.1324 | 0.1278 | 110.95/110 (1.01) | 83.73 | 72.40 | 1.0113 $\pm$ 0.0684
$q=-2,0,1$ | 11–20 | 1.0162 | 0.0422 | 0.0864 | 0.1020 | 193.18/217 (0.89) | 88.33 | 82.13 | 1.0164 $\pm$ 0.0431
| 21–30 | 1.0097 | 0.0326 | 0.0691 | 0.0870 | 249.10/305 (0.82) | 84.47 | 80.27 | 1.0131 $\pm$ 0.0304
\+ white noise + foreground residual (varying beam size)
Fid. config. | 1–10 | 1.0037 | 0.0642 | 0.1315 | 0.1305 | 102.23/100 (1.02) | 91.87 | 79.00 | 0.9946 $\pm$ 0.0659
11–20 | 0.9984 | 0.0408 | 0.0922 | 0.1088 | 185.68/207 (0.90) | 92.13 | 83.67 | 0.9975 $\pm$ 0.0406
21–30 | 0.9860 | 0.0338 | 0.0524 | 0.0670 | 337.97/295 (1.15) | 75.87 | 73.93 | 0.9939 $\pm$ 0.0330

###  6.2 Impact of including instrumental noise

Taking into account all specifications summarized in Section 4.1.2, we
estimate the white noise level at each pixel in the BINGO region,
$\sigma_{pix}$, considering the same resolution used to generate the 21 cm
simulations, $N_{side}=256$. From this $\sigma_{pix}$ map we can generate as
many realizations of the corresponding white noise map as necessary by
multiplying it by random values defined by a Gaussian distribution of zero
mean and unitary standard deviation. We add a different noise realization to
each of the 1500 FLASK mocks and repeat the same process of calculating
$C_{\ell}$ and $\omega(\theta)$ from each of them.

White noise adds to the APS a constant term $N_{\ell}$ at all multipoles,
which for BINGO dominates over the 21 cm signal, as a function of redshift
bin, for $\ell\geq 200-350$ (Fig. 2). Then, we debias the $C_{\ell}$ of each
noisy 21 cm simulation by subtracting from it the expected $N_{\ell}$
amplitude, computed from the theoretical noise level. In contrast, white noise
affects the ACF over all angular scales, so that this debiasing cannot be
considered. However, since the $\omega(\theta)$ is calculated by averaging
over $\delta T_{p}\delta T_{p^{\prime}}$ from all pairs of pixels $p$ and
$p^{\prime}$ separated by $\theta$ (Eq. (16)), a natural consequence is a
reduction of the noise.

Our results show that the presence of thermal noise increase the $\sigma_{68}$
uncertainty by 19% and 90% for intermediate and higher z-bins when using the
ACF, while at lower z-bins this error surprisingly diminishes by 14%
(similarly to the respective bias on $\langle\alpha\rangle$). The noise impact
is more expressive for APS analyses, with 9%, 51% and 122% increase in
$\sigma_{68}$ from lower to higher z-bins. However, although APS results are
more affected by the noise, it still has a smaller $\sigma_{68}$ uncertainty
for higher z-bins, while for intermediate z-bins the two estimators have
comparable errors. The second row of Fig. 5 shows a comparison of the best-fit
$\alpha$ from $\omega(\theta)$ and $C_{\ell}$ estimators, clearly much more
spread due to the presence of noise. In addition, for both estimators the
$N_{s}$ and $N_{d}$ fractions decrease due to the presence of thermal noise,
with the greatest impact over $N_{s}$, decreasing by 10% for the ACF estimator
at lower z-bins. Again surprising, $N_{d}$ increases by 6% for ACF at small
z-bins after including noise. Apart from this, the number of simulations with
BAO detection has the larger decrease the higher the redshift. In summary, the
presence of thermal noise has the major impact on higher z-bins, for which the
BAO feature appears at smaller angular scales (spread to larger multipoles;
Fig. 2), where noise starts dominating.

###  6.3 Foreground residual contamination

The last step is to take into account the presence of residual foreground as
expected for BINGO. For this we add, along with the thermal noise, the most
important foreground signals contributing to the BINGO frequency range
(Section 4.1.2) to a 21 cm realization. Using the GNILC code, we perform
component separation to reconstruct the noisy 21 cm signal as discussed in
Section 4.2. Also, from Eq. (19), using the ILC filter and the known
foreground contamination we can estimate the amplitude of the foreground
residual in each z-bin, $\mathbf{Wf}$. Such process is repeated for 10
different realizations of the 21 cm signal, each of them contaminated by a
different realization of the thermal noise, but all with the same foreground
signal contribution. These 10 different estimates of the foreground residual
is used to include the expected contaminant signal to each of the 1500 mock
realizations (the BINGO-like simulations). Note that the noise realizations
are still different from one mock to another, while the foreground residual
will be repeated for each 10 mocks. This avoids the (computationally
expensive) component separation process to be applied to all the simulations.

Given the 1500 semi-realistic BINGO-like simulations, we calculate the
$C_{\ell}$ and $\omega(\theta)$ from each of them and repeat the BAO fitting
process using the fiducial configuration. Note that, again we debias the
$C_{\ell}$ by subtracting the expected $N_{\ell}$ term. The results are shown
in Tables 2 and 3, as well as in Figs. 3 and 4, for ACF and APS estimators,
respectively. The last row of Fig. 5 also shows how the foreground residual
can affect the concordance among the two estimators. In general, one can see
that such contamination has less significant impact, with negligible changes
on both $\langle\sigma_{\alpha}\rangle$, $\sigma_{68}$ and $\sigma_{std}$
uncertainties compared to including noise (similar conclusions are drawn by
Villaescusa-Navarro et al., 2017). Still, both $N_{s}$ and $N_{d}$ fractions
decrease, especially at higher z-bins using ACF, with another $\sim$ 10% of
the 1500 mocks no longer satisfying the selection and detection criteria. Note
also that, for all the redshift ranges, the best-fit $\alpha^{m}$ from the
mean ACF shows a slight shift to values smaller than those found including
only noise. Since this shift persists at some redshift ranges when changing
the fiducial configuration (Table 2) and also when using the N-body mocks
(Table 4), it suggests the foreground residual as the possible responsible for
such systematic bias. For the APS, on the other hand, the results do not lead
to the same conclusion. In any case, even though such shift is observed from
$\omega(\theta)$ results after including the foreground residual contribution,
it is quite small with respect to the error estimates, that is, small enough
to not be a concern.

In order to evaluate the validity of including the expected foreground
residual to the mocks, instead of applying the GNILC code to each of them, we
add to the last row of Fig. 5 the BAO fitting results from the 10
reconstructed maps. As one can see, the red dots appear to be in good
agreement with the black ones. The middle and last panels do not show all 10
red dots because not all the reconstructed maps provided best-fit $\alpha$
values in the range $[0.6,1.4]$, for (at least) one of the estimators.

Figure 5: Comparison of the $\alpha$ values estimated using the ACF and APS
estimators (black dots). The columns from left to right show estimates from
lower to higher redshift ranges, with average $\langle z\rangle=0.172$,
$\langle z\rangle=0.270$, $\langle z\rangle=0.386$. The first row shows
results for the 21 cm only analyses, while the middle and bottom rows show
results including white noise only and white noise + foreground residual,
respectively. The red dots on the bottom row show $\alpha$ estimates from the
reconstructed (foreground cleaned) maps.

Additionally, we test the impact of a frequency dependent beam size, with the
BINGO $\theta_{\mbox{\sc fwhm}}$ taking values from $\sim\,35^{\prime}$ to
$\sim\,45^{\prime}$ from small to high z-bins. In this case, the component
separation is applied after each frequency map (21 cm, along with thermal
noise and foreground signals) has its resolution converted to the common
$\theta_{\mbox{\sc fwhm}}\sim 45^{\prime}$, the largest beam size. The
foreground residual for each z-bin is estimated as before. Note that, the
foreground residual is added to a realization after each frequency map,
containing the 21 cm signal (FLASK mocks), convolved with the corresponding
beam size, and the thermal noise, is converted to the common
$\theta_{\mbox{\sc fwhm}}$. The BAO fitting results from this set of 1500 mock
realizations are presented in the last part of Tables 2 and 3. These results
show that, for both clustering estimators, a frequency dependent beam size
have negligible impact on the BAO analyses for lower and intermediate redshift
ranges. At higher z-bins one can observe a slightly larger bias on
$\langle\alpha\rangle$ for both, $\omega(\theta)$ and $C_{\ell}$, but still
quite smaller than the error estimates, besides a decrease and slight increase
in both $N_{s}$ and $N_{d}$ fractions for $C_{\ell}$ and $\omega(\theta)$,
respectively. In fact, we find that the foreground residual have a larger
amplitude on the higher z-bins when compared to the case with a fixed beam
size (Figure 2), contributing mainly at larger multipoles (a deeper
investigation on the impact of frequency dependent beam size in the foreground
cleaning is necessary but out of the scope of this paper and will be
considered in future work). Still, even with larger contribution, the
foreground residual does not seem to have a significant impact on BAO results,
allowing us to keep the previous conclusions.

###  6.4 Robustness tests

Here we test the aspects of the fiducial configuration, that is, we consider
variations of (a) the angular ($\Delta\theta$) and multipole ($\Delta\ell$)
binning and (b) the $q$ index in the templates, Eqs. (15) and (16). Such
robustness tests are performed over the BINGO-like simulations because they
are the closest to what BINGO will observe. Notice that we vary only one
aspect of the fiducial configuration at a time, the one pointed in the first
column of Tables 2 and 3, where the results of all robustness tests are
summarized.

From testing $\Delta\theta$ width, Table 2 shows that the different binning
lead to small changes at distinct statistics. The most discrepant result
concerns the larger bias on $\alpha^{m}$ obtained from the mean
$\omega(\theta)$, at all the redshift ranges, for $\Delta\theta=0.25^{\circ}$.
Also, although $\Delta\theta=0.40^{\circ}$ shows slightly better results for
some statistics, for example, larger $N_{s}$ and $N_{d}$ fractions at some
redshift ranges, we find a smaller bias on $\langle\alpha\rangle$ at higher
z-bins for $\Delta\theta=0.50^{\circ}$. In this sense, and in order to have a
more diagonal covariance matrix, we choose to use $\Delta\theta=0.50^{\circ}$
as the fiducial configuration. Similar conclusions can be drawn from the tests
on the $\omega(\theta)$ template, showing that some choices of $q$ can help
improving different statistics. However, by comparing all template tests, one
can notice the large reduced $\langle\chi^{2}\rangle$, for all redshift
ranges, using $q=-2,0,1$, and the large bias on $\langle\alpha\rangle$ and
$\alpha^{m}$ for the ACF at small z-bins for $q=0,1$. In general, we see that
cases $q=0,1,2$ and $q=-1,0,1,2$ present the better results, especially in
terms of the fraction $N_{d}$ of mocks with BAO detection. Indeed, the former
one produces larger $N_{s}$ and $N_{d}$ for lower z-bins, while the latter
provides larger fractions at higher z-bins. Given the smaller bias amplitude
in $\langle\alpha\rangle$ at higher z-bins, we choose $q=0,1,2$ as part of the
fiducial configuration.

Results from testing $\Delta\ell$ width, presented in Table 3, show a good
concordance between the different uncertainties from all three cases
considered, although a clear worse performance for $\Delta\ell=20$. While this
larger bin width gives larger $N_{s}$ and $N_{d}$ fractions, it also
introduces large bias on $\langle\alpha\rangle$ and $\alpha^{m}$ for most of
the redshift ranges. Comparing $\Delta\ell=10$ and $15$, the second one leads
to slightly larger $N_{s}$ and $N_{d}$ fractions, but also a larger bias on
$\langle\alpha\rangle$ and $\alpha^{m}$, as well as a larger
$\langle\chi^{2}\rangle$, for the lower z-bins. Then, we find more appropriate
to use $\Delta\ell=10$ as part of the fiducial configuration. Finally, testing
the different templates to fit $C_{\ell}$ measurements, we look again for the
one with the smallest bias and uncertainty, combined with the largest $N_{s}$
and $N_{d}$. In this sense, one can see that $q=0,1,2$ and $q=-1,0,1,2$ have
the best results, but the latter gives a larger fraction of the mocks with BAO
detection, $N_{d}$, and also larger $N_{s}$, motivating the choice of using it
as part of the fiducial configuration.

From all the robustness tests, considering each redshift range individually,
one could say that each of them would require a slightly different
configuration. Still, just to make things simpler, we chose to use the same
$\theta$/$\ell$ binning and $\omega(\theta)$/$C_{\ell}$ templates for all
three redshift ranges, that is, the (a) and (b) aspects of the fiducial
configuration, as discussed.

Table 4: BAO fitting results for the ACF using 500 N-body mocks. The first part of the table show results obtained analyzing 21 cm only mock, while the second part correspond to BINGO-like simulations. In both cases the fiducial configuration is employed. Tests | z-bins | $\langle\alpha\rangle$ | $\langle\sigma_{\alpha}\rangle$ | $\sigma_{68}$ | $\sigma_{std}$ |  $\langle\chi^{2}\rangle$/dof |  $N_{s}$ (%) |  $N_{d}$ (%) | Mean
---|---|---|---|---|---|---|---|---|---
21 cm only | 1–10 | 1.1339 | 0.1411 | 0.1178 | 0.1085 | 98.31/123 (0.80) | 49.60 | 34.80 | 1.0105 $\pm$ 0.2991
11–20 | 1.0354 | 0.0973 | 0.0958 | 0.1001 | 67.63/70 (0.97) | 79.40 | 73.40 | 1.0045 $\pm$ 0.1217
21–30 | 1.0091 | 0.0560 | 0.0554 | 0.0642 | 61.79/61 (1.01) | 87.40 | 86.40 | 1.0035 $\pm$ 0.0558
\+ white noise \+ foreground residual | 1–10 | 1.1082 | 0.1605 | 0.1014 | 0.1074 | 97.59/123 (0.79) | 46.00 | 36.40 | 1.0194 $\pm$ 0.3752
11–20 | 1.0424 | 0.1317 | 0.1082 | 0.1106 | 66.97/70 (0.96) | 68.00 | 59.60 | 0.9903 $\pm$ 0.1749
21–30 | 0.9974 | 0.1024 | 0.1160 | 0.1209 | 61.04/61 (1.00) | 75.60 | 66.20 | 0.9698 $\pm$ 0.1242
Table 5: Same as Fig. 4 but for the APS estimator.  Tests | z-bins | $\langle\alpha\rangle$ | $\langle\sigma_{\alpha}\rangle$ | $\sigma_{68}$ | $\sigma_{std}$ |  $\langle\chi^{2}\rangle$/dof |  $N_{s}$ (%) |  $N_{d}$ (%) | Mean
---|---|---|---|---|---|---|---|---|---
21 cm only | 1–10 | 1.0075 | 0.0385 | 0.1373 | 0.1402 | 84.81/100 (0.85) | 84.00 | 68.60 | 1.0110 $\pm$ 0.0380
11–20 | 1.0128 | 0.0190 | 0.0682 | 0.0873 | 119.81/207 (0.58) | 78.40 | 74.60 | 1.0062 $\pm$ 0.0177
21–30 | 1.0096 | 0.0130 | 0.0359 | 0.0549 | 117.95/295 (0.40) | 78.20 | 76.60 | 1.0086 $\pm$ 0.0127
\+ white noise \+ foreground residual | 1–10 | 1.0017 | 0.0475 | 0.1413 | 0.1406 | 85.02/100 (0.85) | 84.00 | 70.40 | 1.0117 $\pm$ 0.0482
11–20 | 0.9917 | 0.0307 | 0.0971 | 0.1017 | 120.99/207 (0.58) | 76.20 | 70.60 | 0.9926 $\pm$ 0.0279
21–30 | 1.0080 | 0.0272 | 0.0974 | 0.1174 | 115.83/295 (0.39) | 57.80 | 50.80 | 1.0031 $\pm$ 0.0253

###  6.5 Tests over N-body mocks

An additional test of the BAO fitting pipeline concerns its application over
the set of 500 N-body mocks, generated through a completely different
methodology, as described in Section 4.1.1. We employ the fiducial
configuration to analyze the 21 cm only simulations and the BINGO-like
simulations, constructed using the N-body mocks. We use the same 10 foreground
residual maps as before and a fixed beam size $\theta_{\mbox{\sc fwhm}}=40$
arcmin. The summary statistics from these analyses are presented in Tables 4
and 5, for ACF and APS, respectively.

From the 21 cm only analyses, we find small differences with respect to what
is obtained from the FLASK mocks, more expressive for the APS estimator,
especially the smaller amplitude of $\langle\chi^{2}\rangle$ and fractions
$N_{s}$ and $N_{d}$ for all the redshift ranges. However, such differences can
be partially explained by the number of N-body mocks available, one third of
the number of FLASK mocks. In fact, the smaller number of simulations implies,
for example, a smaller signal-to-noise ratio from the mean clustering
measurements, providing a worse fitting over them (note the slightly large
bias on $\alpha^{m}$ from small and intermediate z-bins for both estimators).
Additionally, for the APS (ACF), the number of $dof$, increases (decreases)
with the redshift, making the covariance matrix and its inverse the more
biased the higher (smaller) the redshift, explaining the discrepancy in the
$\langle\chi^{2}\rangle$ and the uncertainty estimates. In this sense, we can
argue that the slight differences between other summary statistics from N-body
and FLASK mocks seems to be of statistical origin, although we also need to
take into account that the two types of mocks are generated following very
different methodologies.

The impact of introducing noise and foreground residuals to the N-body mocks
is very similar to that obtained using the FLASK mocks. The greater impact
appears on intermediate and high z-bins, for the same reasons previously
discussed (see Sections 6.2 and 6.3).

###  6.6 Wiggle versus no-wiggle template fitting

In order to evaluate the goodness-of-fit of the $\omega(\theta)$ and
$C_{\ell}$ templates to the corresponding measurements from each mock, we
compare the minimum $\chi^{2}$, evaluated at the best-fit $\alpha_{bf}$ value,
to that obtained by using a no-BAO template, $\chi^{2}_{nw}$. For this, we
construct a template model for each estimator (the same Eqs. (15) and (16))
using $C^{temp}(\ell)$ obtained by imposing $P^{temp}(k)=P^{nw}$, instead of
using the parameterization from Eq. (14). The square root of the difference
among the two quantities,
$S=\sqrt{\Delta\chi^{2}}=\sqrt{\chi^{2}-\chi^{2}_{nw}(\alpha_{bf})}$, given in
term of number of $\sigma$, is commonly used to evaluate the statistical
significance of the BAO signal detection (see, e.g., Villaescusa-Navarro et
al., 2017; Camacho et al., 2019; Abbott et al., 2021). It means one can say
that the BAO model (baryon signature) is preferred by $\Delta\chi^{2}$
compared to a model with no BAO.

To evaluate the effective significance of our BAO detection over BINGO-like
simulations we consider the complete BINGO redshift range, fitting all the 30
z-bins together instead of three subsets of them. Since the number of dof
increases significantly in this case, namely, 256 and 604 for ACF and APS,
respectively, an unbiased estimate of the covariance matrix and its inverse
requires a larger set of simulations, which directly influences the $\chi^{2}$
estimate (see discussion in Section 6.5). For this we rotate six times the
original 1500 FLASK mocks, similarly to the procedure used to enlarge the
number of N-body mocks, as described in section 4.1.1, achieving a total of
10500 mocks. The clustering statistics measured from the 10500 BINGO-like
simulations are used to estimate the covariance matrix, while the
$\Delta\chi^{2}$ quantity is calculated only for the 1500 original mocks. The
distribution of $\sqrt{\Delta\chi^{2}}$ from the corresponding $N_{s}\sim 90$%
and $94$% of the mocks for ACF and APS, with average significance of $\langle
S\rangle=1.48\sigma$ and $1.61\sigma$, respectively, are presented in Fig. 6.
From the $N_{s}$ fraction of mocks, we find $\sim 14$% and $\sim 24$% of them
with $S=\sqrt{\Delta\chi^{2}}>2\sigma$, for ACF and APS, respectively.
Confirming the slightly better performance of the APS over the ACF estimator,
the former one seems to provide BAO detection with higher significance than
the last one. Also, although the $\sqrt{\Delta\chi^{2}}$ distribution is shown
only for the BINGO-like simulations, notice that it is impacted by systematic
effects similarly to other summary statistics. Compared to 21 cm only case,
$\sqrt{\Delta\chi^{2}}$ is significantly shifted to smaller values due to the
inclusion of thermal noise, while the contribution of foreground residual has
a negligible impact.

Figure 6: Histogram distribution of the significance,
$S=\sqrt{\Delta\chi^{2}}$, of the BAO signal, in terms of number of $\sigma$,
estimated for each of the FLASK simulations (that is, the corresponding
$N_{s}$ fraction of mocks) considering each clustering estimator. These
results are obtained evaluating the BINGO-like simulations combining all the
30 z-bins.

##  7 Conclusions

The BAO feature measured from galaxy distribution is currently recognized as
one of the most important cosmological probes. Its dependence with the
evolution and the components of the Universe makes it a powerful tool to
investigate the physical nature of dark energy. Since its first detection, the
BAO scale has been measured using different matter tracers in several redshift
ranges, and now the radio 21 cm IM experiments, such as BINGO, plans to
provide a complementary route to study BAO.

Here we present the assessment of the BINGO telescope potential, in its Phase
1 operation, to detect the transversal BAO signal. For this we developed a
template fitting pipeline with two clustering estimators, $\omega(\theta)$ and
$C_{\ell}$, to extract the BAO information from two different sets of
simulations, FLASK and N-body mocks. Our analyses account for the BINGO sky
coverage, beam effect and evaluate the impact of including thermal noise and
foreground residual. The fitting procedure to measure the shift parameter,
$\alpha$, from the mocks is performed over three sets of consecutive z-bins
using a maximum likelihood estimator. Most of our analyses are performed
employing the fiducial configuration (aspects (a) and (b) defined in Section
6), optimized to be applied to all redshift ranges. Our main results can be
summarized as follows:

  * •

We found that both estimators perform better at higher redshifts, but show
different sensitivities to specific redshift ranges. The bias amplitude in
$\langle\alpha\rangle$, the $\sigma_{68}$ uncertainty, and the selection and
detection fractions, $N_{s}$ and $N_{d}$, suggest that the APS slightly
outperforms the ACF estimator, although the ACF presents slightly smaller
$\sigma_{68}$ at lower z-bins. Also, the APS leads to greater average
significance, $\langle S\rangle$, than the ACF. This behavior is observed
analyzing 21 cm only simulations and remains after including contaminant
signals and redshift dependent beam size (see Tables 2 to 5).

  * •

Accounting for the impact of including thermal noise and foreground residual
one at a time, we found that the former one has the greater impact in both
clustering estimators, affecting results more significantly the greater the
redshift. It enlarges significantly the $\sigma_{68}$ error, in particular for
the APS estimator, reaching 122% at higher z-bins. The reason for that is that
the BAO wiggles appear at higher multipoles (smaller angular scales) the
higher the redshift, scales at which the thermal noise starts dominating. The
inclusion of the foreground residual, on the other hand, does not seem to have
a significant impact in our results.

  * •

Robustness tests of the fiducial configuration, considering different
$\Delta\theta$ and $\Delta\ell$ binning and alternative template models ($q$
indexes), suggest that a more appropriate choice would be to select an
”optimal configuration” for each of the three redshift ranges.

  * •

Using the fiducial configuration, the BAO fitting pipeline applied to 500
N-body mocks (Tables 4 and 5) show a few aspects differing from those
analyzing FLASK mocks, caused mainly by the smaller number of N-body mocks.

  * •

We also evaluate how preferred a BAO model is when compared to a fit of a non-
BAO model, and find an average significance for the BAO detection of $\sim
1.48\sigma$, for ACF, and $\sim 1.48\sigma$, for APS, combining all 30 z-bins.
A fraction of 14% and 24% of the FLASK mocks, for ACF and APS, respectively,
provide a $>2\sigma$ BAO detection.

It is worth emphasizing that some aspects of the analysis presented here could
benefit from the inclusion of more realistic choices of instrumental issues,
such as the structure of the beam and instrumental effects like $1/f$ noise
and polarization leakage, or even aspects of the methodology that can be
improved and deserve further study. Among them we can mention the test of
alternative component separation procedures, a different number of z-bins for
the BAO fitting (see results from the BINGO paper, Mericia et al., 2022), and
a fitting method alternative to MLE.

Moreover, it is well known (Eisenstein et al., 2007a) that the non-linear
gravitational evolution of the Universe smears out the acoustic signature by
inducing a damping and broadening on the BAO peak, besides shifting its
position so that, the lower the redshift, the more difficult is to achieve an
accurate measurement of the BAO features. Since BINGO will survey a large
volume of the Universe at relative low redshifts, we also expect to improve
our BAO fitting results, in particular at lower redshifts, by considering a
reconstruction procedure (Obuljen et al., 2017).

In summary, we conclude that intermediate and higher redshift intervals are
the most promising in measuring the BAO scale, with a probability of detection
of more than $\sim 80\%$ ($\sim 70\%$) with the APS (ACF) estimator. Such
numbers are obtained using a fixed fiducial configuration and can be
significantly improved by the choice of an ”optimal configuration” for each
redshift interval. Indeed, although showing that systematic effects have a
non-negligible impact on $\alpha$ estimates, our analyses of a semi-realistic
scenario (BINGO-like mocks) confirm that BINGO should be able to successfully
detect the BAO signal in radio frequencies.

###### Acknowledgements.

The BINGO project is supported by São Paulo Research Foundation (FAPESP) grant
2014/07885-0. C.P.N. would like to thank Edilson de Carvalho, Armando Bernui,
Henrique Xavier, and Hugo Camacho for very enlightening and useful
discussions. C.P.N. also acknowledges FAPESP for financial support through
grant 2019/06040-0. J.Z acknowledges support from the Ministry of Science and
Technology of China (grant Nos. 2020SKA0110102). R.G.L. thanks CAPES (process
88881.162206/2017-01) and the Alexander von Humboldt Foundation for the
financial support. L.S. is supported by the National Key R&D Program of China
(2020YFC2201600). A.A.C acknowledges financial support from the National
Natural Science Foundation of China (grant 12175192). L.B., F.A.B, A.R.Q., and
M.V.S. acknowledge PRONEX/CNPq/FAPESQ-PB (Grant no. 165/2018). E.F.: The Kavli
IPMU is supported by World Premier International Research Center Initiative
(WPI), MEXT, Japan. We thank an anonymous referee for her/his very insightful
comments. This research made use of astropy (Price-Whelan et al., 2018),
healpy (Zonca et al., 2019), numpy (Van Der Walt et al., 2011), scipy
(Virtanen et al., 2020) and matplotlib (Hunter, 2007).

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