00footnotetext: Received 202x

# Cosmological Forecasts from the Baryon Acoustic Oscillations in 21cm
Intensity Mapping

Benjamin Ostergaard1 André A. Costa1,2,3 E-mail<EMAIL_ADDRESS>(Corresponding author) Yu Sang1 1Center for Gravitation and Cosmology, College
of Physical Science and Technology, Yangzhou University, Yangzhou 225009,
China
2College of Physics, Nanjing University of Aeronautics and Astronautics,
Nanjing 211106, China
3Instituto Federal de Educação, Ciência e Tecnologia de Minas Gerais, Campus
Itabirito, Itabirito 35450-000, Brazil

###### Abstract

We use the baryon acoustic oscillation (BAO) feature in the angular power
spectrum (APS) of the 21cm line of neutral hydrogen (HI) to constrain
cosmological parameters. As the BAO shift parameter can only constrain the
product of the Hubble constant and the sound horizon $H_{0}r_{s}$, we combine
our Fisher matrices with cosmic microwave background (CMB) Fisher matrices
from the covariance of $Planck$ telescope to break this degeneracy. In
particular, we find that the best constraints we can get with this method are
on the Hubble parameter $h$, and the dark energy parameters $w_{0}$ and
$w_{a}$. Assuming a noise level as for the BINGO telescope, we find
$\sigma_{h}=0.0055\;(0.8\%)$ in the $\Lambda$CDM model. For the $w$CDM model,
we find $\sigma_{h}=0.020\;(2.9\%)$ and $\sigma_{w_{0}}=0.075\;(7.5\%)$. In
the CPL parameterization, we find $\sigma_{h}=0.029\;(4.4\%)$,
$\sigma_{w_{0}}=0.40\;(40\%)$, and $\sigma_{w_{a}}=1.7$. Although these
constraints improve those from $Planck$ alone, we observe the BINGO data will
be more significant for the $w$CDM model. We also observe the constraints from
the BAO only is not as strong as using the whole 21cm angular power spectrum,
but it is less susceptible to systematic effects.

###### keywords:

cosmology, baryon acoustic oscillations, 21cm intensity mapping

00footnotetext: $\scriptstyle\copyright$2017 Chinese Physical Society and the
Institute of High Energy Physics of the Chinese Academy of Sciences and the
Institute of Modern Physics of the Chinese Academy of Sciences and IOP
Publishing Ltd

##  1 Introduction

Baryon acoustic oscillations (BAOs) are sinusoidal oscillations found in the
angular power spectra (APS) of both the cosmic microwave background (CMB) and
galaxy and matter distribution. They are the result of overdensities in the
primordial plasma. In the adiabatic model, the overdensity includes all
particle species [1, 2]. Photons overpressure the region and drive a spherical
sound wave carrying baryons along with it. The cold dark matter (CDM) does not
interact with any other species and thus remains confined in a small radius.
After decoupling, the sound wave freezes out as the sound speed decreases
rapidly. We are left with a spherical baryon overdensity with a radius $r_{s}$
and a CDM overdensity at the center of the sphere. These overdensities at both
the sphere and its center grow by gravitational instability and provide the
seeds for galaxy generation [1, 2].

Our universe is then composed of many superpositions of these spheres.
Fortunately, the perturbation behave linearly at high redshifts so we can
simply add them all together [2]. Of course, the sphere does not have zero
thickness. There are three factors that determine its width: Photons leak out
of the sound wave producing a phenomenon called Silk damping [3]; Over time
the inertia of the baryons relative to the photons increases thus changing the
sound speed and broadening the wave adiabatically [2]; Finally, the initial
perturbations can interfere with each other. Since galaxies are more likely to
form on these spheres, we see a peak in the galaxy 2-point correlation
function at about $\sim 150$Mpc. This peak in the correlation function
corresponds to an oscillation in the power spectrum, the Fourier transform of
the correlation function.

In this paper, we use intensity maps of 21cm line of neutral hydrogen as a
tracer for the matter angular power spectrum. The hydrogen atom is one of the
simplest objects in atomic physics, simply consisting of a proton and an
electron. In its ground 1S state, there are actually two “hyperfine” states
related to the interaction of the proton and electron’s magnetic moments. A
hydrogen atom in the excited hyperfine state, where both the spins of the
proton and electron are aligned, can spontaneously decay into the lower
hyperfine state where the electron’s spin “flips” and the two spins become
antiparallel. The energy difference between these two states is $\Delta
E=5.9\times 10^{-6}$eV. Energy conservation leads to the difference being
accounted for by a release of a photon with this energy. The corresponding
wavelength and frequency of this photon are 21cm and 1420 MHz, respectively.
Studies of hydrogen masers have provided very precise measurements of these
quantities [4]. Thus, the 21cm line makes a great, well-understood
astronomical probe.

The main difference between intensity mapping and galaxy surveys, the other
major low-redshift observation technique, is that while galaxy surveys must
resolve each individual galaxy and their redshifts, intensity mapping simply
counts the number of photons that land on each pixel of the observation
volume. Essentially, intensity mapping surveys are simply digital cameras. The
instruments, data, and computing resources required for galaxy surveys such as
the Sloan Digital Sky Survey [5] are astronomical. Conversely, the instruments
required for intensity maps are relatively cheap and the data processing is
easier. If we want to understand galaxy distributions from an intensity map,
all we have to do is calculate and apply the bias to the power spectrum, a
much easier task than individually resolving and calculating the redshift of
every galaxy in the survey volume.

Because of these many advantages, there are several experiments under way to
observe the 21cm signal specifically for the purpose of measuring BAOs. We
mention a few: the Canadian Hydrogen Intensity Mapping Experiment (CHIME) [6],
MeerKAT [7], the Square Kilometer Array (SKA) [8], and Baryon acoustic
oscillations from Integrated Neutral Gas Observations (BINGO) [9, 10, 11].

We use specifications for the BINGO telescope in our analysis. A set of
companion papers have been published describing the BINGO project. BINGO I
[11] provides an overview of the project. BINGO II [12] provides a detailed
instrument description. BINGO III [13] discusses the optical design and
optimization of the focal plane. BINGO IV [14] simulates the mission and the
preliminary steps for component separation. BINGO V [15] describes the full
component separation and bispectrum analysis of the data. BINGO VI [16]
provides a mock to test our data analysis on. BINGO VII [17] makes
cosmological forecasts using the Fisher matrix method for the total angular
power spectrum and assesses the BINGO setup and how it compares to other
telescopes. BINGO VIII [18] discusses the recoverability of the BAO signal
from HI intensity maps using the Markov Chain Monte Carlo method.

This paper is organized as follows. In Section 2, we derive the 21cm angular
power spectrum we use in our analysis. In particular, we explain how we
rewrite the APS given in BINGO VII [17] in terms of the $3D$ power spectrum
$P(k)$ and extract the BAO shift parameter. In Section 3, we explain how we
get cosmological parameters from the APS using a Fisher matrix analysis. We
present our constraints from the BAO feature for three different models in
Section 4 and compare our results with previous works. We discuss some errors
with degeneracies and how we patch them with cosmic microwave background data.
We make our conclusions and potential future work in Section 5.

##  2 BAO feature from the 21cm signal

The 21cm brightness temperature perturbation in direction $\hat{n}$ is given
in Ref. [19]

$\begin{split}\Delta_{T_{b}}(z,\hat{n})&=\delta_{n}-\frac{1}{\mathcal{H}}\left[\hat{n}\cdot(\hat{n}\cdot\vec{\nabla})\vec{v}\right]\\\
&+\left(\frac{d\ln(a^{3}\bar{n}_{\text{HI}})}{d\eta}-\frac{\dot{\mathcal{H}}}{\mathcal{H}}-2\mathcal{H}\right)\delta\eta+\frac{1}{\mathcal{H}}\dot{\Phi}+\Psi,\end{split}$
(1)

where $\delta_{n}$ is the HI density perturbation, $\mathcal{H}$ is the Hubble
parameter in conformal time, $\vec{v}$ is the peculiar velocity,
$\bar{n}_{\text{HI}}$ is the average HI number density, and $\Psi$ and $\Phi$
are the gravitational perturbation functions. Overdots indicate derivatives
with respect to conformal time.

We then decompose this into spherical harmonics [19, 17]

$\Delta_{T_{b}}(z,\hat{n})=\sum\limits_{lm}\Delta_{T_{b},lm}(z)Y_{lm}(\hat{n}).$
(2)

We write the coefficients $\Delta_{T_{b},lm}(z)$ in terms of the Fourier
transforms of the perturbations

$\Delta_{T_{b},lm}(z)=4\pi
i^{l}\int\frac{d^{3}\vec{k}}{(2\pi)^{3/2}}\Delta_{T_{b},l}(\vec{k},z)Y_{lm}^{*}(\hat{k}),$
(3)

where

$\begin{split}\Delta_{T_{b},l}(\vec{k},z)&=\delta_{n}j_{l}(k\chi)+\frac{kv}{\mathcal{H}}j^{\prime\prime}_{l}(k\chi)+\biggl{(}\frac{1}{\mathcal{H}}\dot{\Phi}+\Psi\biggr{)}j_{l}(k\chi)\\\
&-\biggl{[}\frac{1}{\mathcal{H}}\frac{d\ln(a^{3}\bar{n}_{\text{HI}})}{d\eta}-\frac{\dot{\mathcal{H}}}{\mathcal{H}^{2}}-2\biggr{]}\\\
&\times\biggl{[}\Psi
j_{l}(k\chi)+vj^{\prime}_{l}(k\chi)+\int_{0}^{\chi}(\dot{\Psi}+\dot{\Phi})j_{l}(k\chi^{\prime})d\chi^{\prime}\biggr{]},\end{split}$
(4)

where $j_{l}(k\chi)$ are the spherical Bessel functions and primes on the
Bessel functions denote derivatives with respect to the argument.

To get the data over some redshift bin we integrate over the redshift-
normalized window function $W(z)$ [17]

$\Delta_{T_{b},l}^{W}(\vec{k})=\int_{0}^{\infty}dz\bar{T}_{b}(z)W(z)\Delta_{T_{b},l}(\vec{k},z).$
(5)

The angular power spectrum is then defined as [19]

$C_{l}^{WW^{\prime}}=4\pi\int d\ln
k\mathcal{P}_{\mathcal{R}}(k)\Delta_{T_{b},l}^{W}(\vec{k},z)\Delta_{T_{b},l}^{W^{\prime}}(\vec{k},z)\,,$
(6)

where $\mathcal{P}_{\mathcal{R}}$ is the dimensionless power spectrum of the
primordial curvature perturbation $\mathcal{R}$.

The biggest advantage of the 21cm $C_{l}$’s over CMB $C_{l}$’s is that because
they are defined over different redshift bins, they allow for a tomographic
analysis. More specifically we can explicitly see how the matter distribution
of the universe evolves over time. Figure 2 shows the 21cm $C_{l}$’s
calculated for the 30 redshift bins used by the BINGO telescope. Going from
higher redshifts (red) to lower redshifts (blue) we can observe three
characteristics: first, as expected, matter structures grow and the power
spectra increase especially at lower multipoles; second, the BAO scale moves
to lower multipoles (larger angles), which means it increases with redshift;
third, there is a suppression at higher multipoles. The BAO feature is easily
identified as the “Wiggle” present across all redshifts.

Angular power spectra plotted over 30 redshift bins from $z=0.13$ (blue) to
$z=0.44$ (red).

While Eq. (6) does give us the angular power spectrum for the 21cm
distribution, we want a form of the APS that is given in terms of the matter
power spectrum $P(k)$. Fortunately, this is easy to do with a few
substitutions.

First, as noted in Hall et al. (2013) [19], the first two terms in Eq. (4) are
several orders of magnitude larger than the other components. Therefore, we
will restrict our calculations to these two contributions only. Second, we can
solve the velocity in terms of the density, such that we rewrite the
brightness temperature perturbation as [20]

$\Delta_{T_{b},l}(\vec{k},z)=\left(b_{\text{HI}}j_{l}(k\chi)-f(z)j^{\prime\prime}_{l}(k\chi)\right)D(z)\delta_{m}(0,k)\,,$
(7)

where $D(z)$ and $f(z)\equiv d\ln{D(z)}/d\ln{a}$ are the growth function and
growth rate, respectively. Taking the density component today outside
$\Delta_{T_{b},l}(\vec{k},z)$, we write the angular power spectra as [18]

$C_{l}^{WW^{\prime}}=\frac{2}{\pi}\int
dk\Delta_{T_{b},l}^{W}(\vec{k},z)\Delta_{T_{b},l}^{W^{\prime}}(\vec{k},z)k^{2}P(k)\,,$
(8)

where $P(k)$ is the $3D$ matter power spectrum today, which is related to the
transfer function $T(k)$ by [21]

$P(k)=\delta_{H}^{2}\frac{2\pi^{2}}{k^{3}}\left(\frac{ck}{H_{0}}\right)^{3+n}T^{2}(k),$
(9)

where $\delta_{H}$ is the density perturbation at horizon crossing.

We use a numerical fit for the transfer function as given by Eisenstein and Hu
(1998) [22]. This fit works in the linear regime, thus the power spectra is
linear as well. To account for the growth of nonlinear structure, we make a
template power spectrum [18, 23]

$P_{\text{temp}}(k)=[P_{\text{lin}}(k)-P_{\text{nw}}(k)]e^{-k^{2}\Sigma_{nl}^{2}}+P_{\text{nw}}(k),$
(10)

where $\Sigma_{nl}=5.2$Mpc/h is the nonlinear damping scale given by [24]. We
use this equation for $P(k)$ in Eq. (8). Figure 2 shows the $C_{l}$’s obtained
from inserting $P_{\text{temp}}(k)$, $P_{\text{lin}}$, and $P_{\text{nw}}$
into Eq. 8, and also the effect of the template on the BAO scale alone.

(Upper panel) Comparison of $C_{l}^{\text{temp}}$ (blue), $C_{l}^{\text{nw}}$
(red), and $C_{l}^{\text{lin}}$ (green). The difference between
$C_{l}^{\text{temp}}$ and $C_{l}^{\text{lin}}$ is the nonlinear damping which
suppresses the oscillations at higher multipoles. (Lower panel) BAO scale
alone obtained by subtracting the $C_{l}$s by the $C_{l}^{\text{nw}}$.

We intend to extract from the 21cm intensity maps the scale associated with
the BAO feature, which is encoded in the angular diameter distance $d_{A}(z)$.
Following [23], we use a template method and write the power spectrum signal
as

$C(l)=B_{0}C_{\text{temp}}(l/\alpha)+A_{0}+A_{1}l+A_{2}/l^{2},$ (11)

where $B_{0}$ and $A_{i}$ are nuisance parameters. Specifically, the amplitude
$B_{0}$ is related to the linear bias squared and the parameters $A_{i}$
account for scale-dependent bias, shot noise, uncertainties in the redshift
space distortion, etc. We set the fiducial values as $B_{0}=1$ and $A_{i}=0$,
and allow them to vary with redshift.

Most importantly, Eq. (11) has a much more powerful feature than just allowing
us to account for more sources of error. The shift parameter $\alpha$ measures
the shift of the BAO peak positions with respect to our fiducial cosmology. It
is defined as [23]

$\alpha\equiv\frac{(d_{A}(z)/r_{s})}{(d_{A}(z)/r_{s})}_{\text{fid}},$ (12)

where $d_{A}(z)$ is the angular diameter distance and $r_{s}$ is the sound
horizon at drag epoch. Measuring $\alpha$ is a very good test for how accurate
a fiducial cosmology is. Not only that, but $\alpha$ only depends on the BAO
effect, it is completely independent of bias and other parameters meaning that
we only have to understand BAO physics well to get accurate cosmological
constraints.

##  3 Fisher Matrix Analysis

For a set of parameters $\theta_{i}$ and a model $\mathcal{M}$, the Fisher
information matrix can be calculated as [25]

$F_{ij}\equiv\biggl{\langle}\frac{\partial^{2}\mathcal{L}}{\partial\theta_{i}\partial\theta_{j}}\biggl{\rangle}=\frac{1}{2}\mathrm{Tr}\left[\mathrm{\bf
C}^{-1}\frac{\partial\mathrm{\bf C}}{\partial\theta_{i}}\mathrm{\bf
C}^{-1}\frac{\partial\mathrm{\bf C}}{\partial\theta_{j}}\right]\,,$ (13)

where $\mathcal{L}=-\ln{L}$ and $L$ is the likelihood function.

We write the covariance C as the sum of the template $C_{l}$s and the noise
spectra

$\textbf{C}=C(l)+C_{l}^{\text{shot}}(z_{i},z_{j})+N_{l}(z_{i},z_{j}),$ (14)

where $C_{l}^{\text{shot}}(z_{i},z_{j})$ and $N_{l}(z_{i},z_{j})$ are
respectively the shot and thermal noise given in [17].

We follow a similar approach to the CMB case, where we decompose the
brightness temperature perturbation in spherical harmonics [18].

We note however that the CMB signal comes from one specific redshift and
yields a 2-dimensional map, while 21cm surveys spans a whole volume and are
3-dimensional with multiple redshifts. Then, the CMB diagonal matrix is
transformed into the block diagonal matrix [17]

$C=\begin{bmatrix}A_{l=2}&0&\dots&0\\\ 0&A_{3}&\dots&0\\\
\vdots&\vdots&\dots&\vdots\\\ 0&0&\dots&A_{n}\end{bmatrix},$ (15)

where

$A_{l}=(2l+1)\begin{bmatrix}C_{l}(z_{1},z_{1})&C_{l}(z_{1},z_{2})&\dots&C_{l}(z_{1},z_{n})\\\
C_{l}(z_{2},z_{1})&C_{l}(z_{2},z_{2})&\dots&C_{l}(z_{2},z_{n})\\\
\vdots&\vdots&\dots&\vdots\\\
C_{l}(z_{n},z_{1})&C_{l}(z_{n},z_{2})&\dots&C_{l}(z_{n},z_{n})\end{bmatrix}.$
(16)

Finally, if we have a Fisher matrix, $F_{ij}$, defined in terms of variables
$\vec{\Theta}=(\theta_{1},\theta_{2},\dots,\theta_{n})$, we can transform to a
new Fisher matrix, $F^{\prime}_{kl}$, defined in terms of variables
$\vec{\Theta}^{\prime}=(\theta^{\prime}_{1},\theta^{\prime}_{2},\dots,\theta^{\prime}_{m})$
for $m\leq n$ using the equation [26]

$F^{\prime}_{kl}=\sum_{ij}\frac{\partial\theta_{i}}{\partial\theta^{\prime}_{k}}\frac{\partial\theta_{j}}{\partial\theta^{\prime}_{l}}F_{ij}.$
(17)

##  4 Results

Our fiducial cosmological parameters come from Planck 2018 [27] and are given
in Table 4.

Fiducial values for cosmological parameters. $b_{\text{HI}}$ and
$\Omega_{\text{HI}}$ are related to 21cm physics. $B_{0}$, $A_{i}$, and
$\alpha$ are the $C_{l}$ template parameters.  Parameter Fiducial Value
$\Omega_{b}h^{2}$ 0.022383 $\Omega_{c}h^{2}$ 0.12011 $h$ 0.6732 $n_{s}$
0.96605 $A_{s}$ $2.1\times 10^{-9}$ $w_{0}$ -1 $w_{a}$ 0 $b_{\text{HI}}$ 1
$\Omega_{\text{HI}}$ $6.2\times 10^{-4}$ $B_{0}$ 1 $A_{i}$ 0 $\alpha$ 1

We make our analysis in two steps. First, we calculate the Fisher matrix using
the covariance for the template angular power spectra given by Eqs. (14) and
(11). We take the BAO shift parameter $\alpha$ and the nuisance parameters
$B_{0}$ and $A_{i}$, and allow them to assume different values by assuming an
independent constant every three redshift bins over the 30 bins of our survey.
This gives us a total of 50 parameters, 10 useful $\alpha$s and 40 nuisance
parameters. We do this for two reasons. Firstly, allowing the $\alpha$ to
assume different values in different bins makes it possible to study its
evolution with redshift. The second reason is that we ultimately want to
transform our Fisher matrix of template variables into a Fisher matrix of
physical variables given in Table 4. Since we can only transform Fisher
matrices into other Fisher matrices of the same size or smaller, we must
ensure we start out with enough parameters. Splitting the parameters across
redshift bins not only gives us more parameters. It also allows us to
investigate how these parameters vary over time.

We use these 50 parameters to calculate a $50\times 50$ Fisher matrix, which
can be done using Eq. (13), setting all $\alpha$s, $B_{0}$s, and $A_{i}$s to
their fiducial values, and calculating the derivatives numerically. This is
the most computationally intensive step in our analysis by far. Since $B_{0}$
and the $A_{i}$s are nuisance parameters, we marginalize over them. That means
we allow these parameters to assume any possible values and take into account
their effect into the uncertainties of $\alpha$s. We are left with a $10\times
10$ Fisher matrix consisting of 10 $\alpha$s split over 30 redshift bins from
BINGO (so the bins have to share some $\alpha$s). Finally, we then take the
inverse of the Fisher matrix to get the covariance matrix. Taking the square
root of the diagonal entries gives us the standard deviation of all 10
$\alpha$s. We then average these 10 $\sigma_{\alpha}$s across 3 redshift bins
to get the results shown in Table 4. For comparison, we also include
$\sigma_{\alpha}$s from the Markov Chain Monte Carlo method from BINGO VIII
[18].

Standard deviations for $\alpha$ across 3 redshift bins. Markov Chain Monte
Carlo results are from BINGO VIII [18].  z-bins
$\langle\sigma_{\alpha}\rangle$ Fisher $\langle\sigma_{\alpha}\rangle$ MCMC
1-10 0.0816 0.0653 11-20 0.0636 0.0410 21-30 0.0508 0.0322

The second step in our analysis consists in transforming our constraints in
the BAO shift parameters into constraints in the final cosmological
parameters. As we saw in Eq. (12), the shift parameter $\alpha$ depends on the
angular diameter distance and the sound horizon at the drag epoch, which are
ultimately functions of $\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, $h$, and the
dark energy equation of state parameters $w_{0}$ and $w_{a}$. Therefore, the
uncertainties in $\alpha$s can be related to uncertainties in these
parameters, which can be done by transforming our $10\times 10\;\alpha$ Fisher
matrix into a Fisher matrix for our final cosmological parameters.

And it is insensitive to $\ln(10^{10}A_{s})$, $n_{s}$, $b_{\text{HI}}$, and
$\Omega_{\text{HI}}$. The fact that the BAO shift parameter is insensitive to
the bias $b_{\text{HI}}$ and the neutral hydrogen density $\Omega_{\text{HI}}$
is the main benefit of this approach over considering the full angular power
spectra in our Fisher matrix analysis.

Table 1: Cosmological constraints for $\Lambda$CDM. | $Planck$ | BINGO VII + $Planck$ | BINGO BAO + $Planck$
---|---|---|---
Parameter | $\pm 1\sigma\;(100\%\times\sigma/\theta^{fid}_{i})$ | $\pm 1\sigma\;(100\%\times\sigma/\theta^{fid}_{i})$ | $\pm 1\sigma\;(100\%\times\sigma/\theta^{fid}_{i})$
$\Omega_{b}h^{2}$ | 0.00015 (0.7%) | 0.00013 (0.6%) | 0.00014 (0.6%)
$\Omega_{c}h^{2}$ | 0.0014 (1.1%) | 0.0010 (0.8%) | 0.0012 (1.0%)
$h$ | 0.0061 (0.9%) | 0.0045 (0.7%) | 0.0055 (0.8%)
$\ln(10^{10}A_{s})$ | 0.016 (0.5%) | 0.015 (0.5%) | 0.016 (0.5%)
$n_{s}$ | 0.0043 (0.4%) | 0.0039 (0.4%) | 0.0041 (0.4%)
Table 2: Cosmological constraints for $w$CDM. | $Planck$ | BINGO VII + $Planck$ | BINGO BAO + $Planck$
---|---|---|---
Parameter | $\pm 1\sigma\;(100\%\times\sigma/\theta^{fid}_{i})$ | $\pm 1\sigma\;(100\%\times\sigma/\theta^{fid}_{i})$ | $\pm 1\sigma\;(100\%\times\sigma/\theta^{fid}_{i})$
$\Omega_{b}h^{2}$ | 0.00015 (0.7%) | 0.00014 (0.6%) | 0.00015(0.7%)
$\Omega_{c}h^{2}$ | 0.0014 (1.2%) | 0.0011 (0.9%) | 0.0014 (1.2%)
$h$ | 0.089 (13%) | 0.0073 (1.1%) | 0.020 (2.9%)
$\ln(10^{10}A_{s})$ | 0.016 (0.5%) | 0.016 (0.5%) | 0.0016 (0.5%)
$n_{s}$ | 0.0044 (0.5%) | 0.0040 (0.4%) | 0.0043 (0.4%)
$w_{0}$ | 0.26 (25%) | 0.033 (3.3%) | 0.075 (7.5%)
Table 3: Cosmological constraints for CPL. | $Planck$ | BINGO VII + $Planck$ | BINGO BAO + $Planck$
---|---|---|---
Parameter | $\pm 1\sigma\;(100\%\times\sigma/\theta^{fid}_{i})$ | $\pm 1\sigma\;(100\%\times\sigma/\theta^{fid}_{i})$ | $\pm 1\sigma\;(100\%\times\sigma/\theta^{fid}_{i})$
$\Omega_{b}h^{2}$ | 0.00016 (0.7%) | 0.00014 (0.6%) | 0.00015 (0.7%)
$\Omega_{c}h^{2}$ | 0.0013 (1.1%) | 0.0011 (0.9%) | 0.0013 (1.1%)
$h$ | 0.088 (13%) | 0.019 (2.9%) | 0.029 (4.4%)
$\ln(10^{10}A_{s})$ | 0.016 (0.5%) | 0.016 (0.5%) | 0.016 (0.5%)
$n_{s}$ | 0.0044 (0.5%) | 0.0041 (0.4%) | 0.0044 (0.5%)
$w_{0}$ | 0.46 (46%) | 0.30 (30%) | 0.40 (40%)
$w_{a}$ | 1.8 | 1.2 | 1.7

Unfortunately, as discussed in [28], there is a large degeneracy between
$H_{0}$ and $r_{d}$ using the BAO data only, and we can only determine the
combination $H_{0}r_{s}$. In order to break this degeneracy, we need prior
knowledge about $\Omega_{b}h^{2}$. This can be done by combining the BAO data
with some external data which properly constrain this parameter. Therefore, we
combine our Fisher matrix results from the 21cm intensity mapping with the
Fisher matrix obtained from CMB data by the Planck telescope. We do this for
three cosmic acceleration models $\Lambda$CDM, $w$CDM, and CPL
($w_{0}w_{a}$CDM). Our results are shown in Tables 3, 4, and 5 for
$\Lambda$CDM, $w$CDM, and CPL, respectively.

Marginalized $68\%$ and $95\%$ Confidence Ellipses for $w_{0}$ and $h$ in the
$w$CDM model.

Marginalized $68\%$ and $95\%$ Confidence Ellipses for $w_{0}$ and $w_{a}$ in
the CPL model.

In most cases, The combined BINGO BAO + Planck constraints are not
statistically any stronger than the constraints from Planck alone. However,
the constraints for $h$, $w_{0}$, and $w_{a}$ do get improved significantly
over Planck. This makes sense as more data related to cosmic expansion is
encoded in late-time 21cm data than in early-time CMB data. Unfortunately,
these constraints are not as strong as those obtained by calculating the
Fisher matrix from the entire 21cm angular power spectrum as done in BINGO VII
[17]. Figure 4 shows the confidence ellipses for $w_{0}$ and $h$ in the $w$CDM
model for both Planck and BINGO BAO + Planck. Figure 4 shows the confidence
ellipses for $w_{0}$ and $w_{a}$ in the CPL model.

##  5 Discussions and Conclusions

We have found that calculating a Fisher matrix of a template power spectrum is
capable of putting constraints on the BAO peak shift parameter $\alpha$ with
results comparable to, but slightly less optimistic than, the MCMC analysis in
BINGO VIII [18]. It is not clear for us why the MCMC result produces better
results in this case, but this makes our analysis more conservative. Unlike
the MCMC analysis, the Fisher matrix method does not calculate the best fit
values for the parameters, however, it is significantly less computationally
intensive.

We have also found that transforming this Fisher matrix into a Fisher matrix
of cosmological parameters is far to degenerate to give any useful constraints
on its own. It has actually been demonstrated, e.g. in the DESI collaboration
[28], that the only cosmological constraints that can be obtained from BAO
alone in a flat $\Lambda$CDM model are in the $\Omega_{m}$ and $H_{0}r_{d}$.
Because of this, we combine the BAO information with CMB data to introduce a
prior in $\Omega_{b}h^{2}$ and break those degeneracies. Our results show that
the BAO data will mainly constrain the hubble parameter $h$ and the dark
energy equation of state paramters $w_{0}$ and $w_{a}$. In fact, the best
results were obtained for the $w$CDM model, which improved the constraints of
the Planck satellite from $h=0.089\,(13\%)$ and $w_{0}=0.26\,(25\%)$ to
$h=0.020\,(2.9\%)$ and $w_{0}=0.075\,(7.5\%)$. This result is consistent with
what was obtained by Olivari et al. [29], although they used a different
methodology to extract the BAO information.

We also compare our analysis with the results from the full angular power
spectra obtained in BINGO VII [17]. As expected, the full power spectra
contains more information than the BAO wiggles alone. Therefore, our
constraints were at the same level or worse than the previous analysis.
However, the BAO do not depend on the overall amplitude of the 21cm angular
power spectra, which makes it less susceptible to systematic effects from
foregrounds removal, bias, and non-linearities.

## Acknowledgements

Supported by the National Natural Science Foundation of China (12175192,
12005184).

## References

  * [1] P. J. E. Peebles and J. T. Yu.  Primeval adiabatic perturbation in an expanding universe.  Astrophysical Journal, 162:815–836, 1970.
  * [2] Daniel J. Eisenstein, Hee-Jong Seo, and Martin White.  On the robustness of the acoustic scale in the low-redshift clustering of matter.  The Astrophysical Journal, 664(2):660–674, August 2007.
  * [3] Joseph Silk.  Cosmic black body radiation and galaxy formation.  Astrophys. J., 151:459–471, 1968.
  * [4] H. M. Goldenberg, D. Kleppner, and N. F. Ramsey.  Atomic hydrogen maser.  Phys. Rev. Lett., 5:361–362, 10 1960.
  * [5] Donald G. York et al.  The Sloan Digital Sky Survey: Technical Summary.  Astron. J., 120:1579–1587, 2000.
  * [6] M. Amiri et al.  The CHIME Fast Radio Burst Project: System Overview.  3 2018.
  * [7] Mario G. Santos et al.  MeerKLASS: MeerKAT Large Area Synoptic Survey.  In MeerKAT Science: On the Pathway to the SKA, 9 2017.
  * [8] A. Weltman et al.  Fundamental physics with the Square Kilometre Array.  Publ. Astron. Soc. Austral., 37:e002, 2020.
  * [9] R. A. Battye, I. W. A. Browne, C. Dickinson, G. Heron, B. Maffei, and A. Pourtsidou.  Hi intensity mapping: a single dish approach.  Monthly Notices of the Royal Astronomical Society, 434(2):1239–1256, July 2013.
  * [10] M.-A. Bigot-Sazy, C. Dickinson, R. A. Battye, I. W. A. Browne, Y.-Z. Ma, B. Maffei, F. Noviello, M. Remazeilles, and P. N. Wilkinson.  Simulations for single-dish intensity mapping experiments.  Monthly Notices of the Royal Astronomical Society, 454(3):3240–3253, 10 2015.
  * [11] Elcio Abdalla et al.  The BINGO project - I. Baryon acoustic oscillations from integrated neutral gas observations.  Astron. Astrophys., 664:A14, 2022.
  * [12] Carlos A. Wuensche et al.  The BINGO project - II. Instrument description.  Astron. Astrophys., 664:A15, 2022.
  * [13] Filipe B. Abdalla et al.  The BINGO Project - III. Optical design and optimization of the focal plane.  Astron. Astrophys., 664:A16, 2022.
  * [14] Vincenzo Liccardo et al.  The BINGO project - IV. Simulations for mission performance assessment and preliminary component separation steps.  Astron. Astrophys., 664:A17, 2022.
  * [15] Karin S. F. Fornazier et al.  The BINGO project - V. Further steps in component separation and bispectrum analysis.  Astron. Astrophys., 664:A18, 2022.
  * [16] Jiajun Zhang et al.  The BINGO project - VI. H I halo occupation distribution and mock building.  Astron. Astrophys., 664:A19, 2022.
  * [17] Andre A. Costa et al.  The BINGO project - VII. Cosmological forecasts from 21 cm intensity mapping.  Astron. Astrophys., 664:A20, 2022.
  * [18] Camila Paiva Novaes et al.  The BINGO project - VIII. Recovering the BAO signal in HI intensity mapping simulations.  Astron. Astrophys., 666:A83, 2022.
  * [19] Alex Hall, Camille Bonvin, and Anthony Challinor.  Testing general relativity with 21-cm intensity mapping.  Physical Review D, 87(6), March 2013.
  * [20] Scott Dodelson.  Modern cosmology.  Academic Press, 2003.
  * [21] Emory F. Bunn and Martin White.  The 4 yearcobenormalization and large-scale structure.  The Astrophysical Journal, 480(1):6–21, May 1997.
  * [22] Daniel J. Eisenstein and Wayne Hu.  Baryonic features in the matter transfer function.  The Astrophysical Journal, 496(2):605–614, April 1998.
  * [23] H. Camacho et al.  Dark Energy Survey Year 1 Results: Measurement of the Galaxy Angular Power Spectrum.  Mon. Not. Roy. Astron. Soc., 487(3):3870–3883, 2019.
  * [24] K. C. Chan et al.  BAO from Angular Clustering: Optimization and Mitigation of Theoretical Systematics.  Mon. Not. Roy. Astron. Soc., 480(3):3031–3051, 2018.
  * [25] Max Tegmark, Andy N. Taylor, and Alan F. Heavens.  Karhunen-loeve eigenvalue problems in cosmology: How should we tackle large data sets?  The Astrophysical Journal, 480(1):22–35, May 1997.
  * [26] Dan Coe.  Fisher matrices and confidence ellipses: A quick-start guide and software, 2009.
  * [27] N. Aghanim et al.  Planck 2018 results. VI. Cosmological parameters.  Astron. Astrophys., 641:A6, 2020.  [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * [28] A. G. Adame et al.  DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations.  4 2024.
  * [29] L. C. Olivari, C. Dickinson, R. A. Battye, Y-Z. Ma, A. A. Costa, M. Remazeilles, and S. Harper.  Cosmological parameter forecasts for HI intensity mapping experiments using the angular power spectrum.  Mon. Not. Roy. Astron. Soc., 473(3):4242–4256, 2018.