# Prospects for Constraining Interacting Dark Energy Models with 21 cm
Intensity Mapping Experiments

Ming Zhang  Department of Physics, College of Sciences, & MOE Key Laboratory
of Data Analytics and Optimization for Smart Industry, Northeastern
University, Shenyang 110819, People’s Republic of China;
<EMAIL_ADDRESS>Bo Wang  Department of Physics, College of Sciences,
& MOE Key Laboratory of Data Analytics and Optimization for Smart Industry,
Northeastern University, Shenyang 110819, People’s Republic of China;
<EMAIL_ADDRESS>Peng-Ju Wu  Department of Physics, College of
Sciences, & MOE Key Laboratory of Data Analytics and Optimization for Smart
Industry, Northeastern University, Shenyang 110819, People’s Republic of
China<EMAIL_ADDRESS>Jing-Zhao Qi  Department of Physics, College
of Sciences, & MOE Key Laboratory of Data Analytics and Optimization for Smart
Industry, Northeastern University, Shenyang 110819, People’s Republic of
China<EMAIL_ADDRESS>Yidong Xu  National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, People’s Republic
of China<EMAIL_ADDRESS>Jing-Fei Zhang  Department of Physics, College of
Sciences, & MOE Key Laboratory of Data Analytics and Optimization for Smart
Industry, Northeastern University, Shenyang 110819, People’s Republic of
China<EMAIL_ADDRESS>Xin Zhang  Department of Physics, College of
Sciences, & MOE Key Laboratory of Data Analytics and Optimization for Smart
Industry, Northeastern University, Shenyang 110819, People’s Republic of
China<EMAIL_ADDRESS>
###### Abstract

We forecast constraints on cosmological parameters in the interacting dark
energy models using the mock data generated for neutral hydrogen intensity
mapping (IM) experiments. In this work, we only consider the interacting dark
energy models with energy transfer rate $Q=\beta H\rho_{\rm c}$, and take
BINGO, FAST, SKA1-MID, and Tianlai as typical examples of the 21 cm IM
experiments. We find that the Tianlai cylinder array will play an important
role in constraining the interacting dark energy model. Assuming perfect
foreground removal and calibration, and using the Tianlai-alone data, we
obtain $\sigma(H_{0})=0.19$ km s-1 Mpc-1, $\sigma(\Omega_{\rm m})=0.0033$ and
$\sigma(\sigma_{8})=0.0033$ in the I$\Lambda$CDM model, which are much better
than the results of Planck+optical BAO (i.e. optical galaxy surveys). However,
the Tianlai-alone data cannot provide a very tight constraint on the coupling
parameter $\beta$ compared with Planck+optical BAO, while the Planck+Tianlai
data can give a rather tight constraint of $\sigma(\beta)=0.00023$ due to the
parameter degeneracies being well broken by the data combination. In the
I$w$CDM model, we obtain $\sigma(\beta)=0.00079$ and $\sigma(w)=0.013$ from
Planck+Tianlai. In addition, we also make a detailed comparison among BINGO,
FAST, SKA1-MID, and Tianlai in constraining the interacting dark energy
models. We show that future 21 cm IM experiments will provide a useful tool
for exploring the nature of dark energy and play a significant role in
measuring the coupling between dark energy and dark matter.

HI line emission; Dark energy; Cosmological parameters; Cosmology

##  1 Introduction

According to our understanding of contemporary cosmology, dark energy is
responsible for the accelerating expansion of the universe (Riess et al.,
1998). However, the nature of dark energy still remains a deep mystery. To
explore dark energy in depth and also test the possible deviation from general
relativity, it is necessary to precisely measure the late-time expansion
history of the universe. Among several ways of exploring the cosmic expansion
history, the baryon acoustic oscillations (BAOs; Blake & Glazebrook 2003; Seo
& Eisenstein 2003) have been proven to be a very useful tool to measure
cosmological distances and the Hubble expansion rate and explore the nature of
dark energy. Because the acoustic waves are frozen in the eras after
recombination, the BAO peak wavelength, as a cosmological standard ruler,
allows accurate measurement of the expansion history.

In addition to the oscillatory features on the power spectrum of the cosmic
microwave background (CMB), the BAO imprint signals as a preferred clustering
scale in the large-scale structure of the later universe. A traditional way of
mapping out the large-scale structure of the universe is to use galaxies as
tracers of the underlying dark matter distribution, and, by measuring the
galaxy distribution with galaxy surveys, one can reconstruct the three-
dimensional matter density field. However, a more efficient way to map out the
matter density field over a wide range of redshifts and scales is to use
neutral hydrogen (HI) as the tracer of dark matter. The HI pervades space from
the epoch of recombination all through to the present day, and with its
hyperfine-structure emission line at a rest-frame wavelength of 21 cm, we can
realize the tomography of the universe over a vast cosmological volume. In the
post-reionization epochs, the bulk of HI resides in dense gas clouds in
galaxies. Detecting a sufficiently large number of galaxies with HI 21 cm
emission could make it possible to provide a useful tool for cosmological
research, but actually it is unnecessary to perform a galaxy survey for the
study of large-scale structure. We can instead measure the total HI intensity
over large angular scales without needing to resolve individual galaxies.
Similar to the CMB map, this intensity mapping (IM) methodology makes it
possible to efficiently survey large volumes with modern radio telescopes
(Battye et al., 2004, 2013; McQuinn et al., 2006; Loeb & Wyithe, 2008;
Pritchard & Loeb, 2008; Wyithe & Loeb, 2008; Mao et al., 2008; Wyithe et al.,
2008; Bagla et al., 2010; Seo et al., 2010; Lidz et al., 2011; Ansari et al.,
2012; Bull et al., 2015c; Xu et al., 2015; Braun et al., 2015; Bull, 2016;
Bull et al., 2015a; Pourtsidou et al., 2017; Yohana et al., 2019; Tramonte et
al., 2019; Tramonte & Ma, 2020; Xu & Zhang, 2020; Zhang et al., 2019, 2020).

Using the HI IM method, Chang et al. (2010) reported the measurements of the
cross-correlation function between the HI map observed with the Green Bank
Telescope (GBT) and the galaxy map observed with the DEEP2 optical redshift
survey. The cross-power spectrum between the HI and optical galaxy survey was
also detected with the GBT 21 cm IM survey and the WiggleZ Dark Energy Survey
(Masui et al., 2013). Recently, Anderson et al. (2018) reported the results
from 21 cm IM acquired from the Parkes radio telescope and cross-correlated
with galaxy maps from the 2dF galaxy survey. So far, there are many current
and future HI IM experiments comprised of wide-field and high-sensitivity
radio telescopes or interferometers. Here we consider several typical
examples, namely, Baryon acoustic oscillations from Integrated Neutral Gas
Observations (BINGO; Battye et al. 2013; Dickinson 2014; Wuensche 2019;
Wuensche et al. 2020), the Five-hundred-meter Aperture Spherical radio
Telescope (FAST; Nan et al. 2011; Smoot & Debono 2017; Bigot-Sazy et al. 2016;
Li et al. 2013; Yu et al. 2017; Hu et al. 2020), the Square Kilometre Array
(SKA; Bull et al. 2015b; Santos et al. 2015; Braun et al. 2015, 2019), and the
Tianlai cylinder array (Chen, 2011, 2012; Xu et al., 2015; Wu et al., 2016; Li
et al., 2020; Wu et al., 2021). This work aims to forecast how future HI IM
experiments, including BINGO, FAST, the SKA Phase I mid-frequency array
(SKA1-MID), and Tianlai, can constrain the interacting dark energy (IDE)
models.

The IDE model originates from a long-standing conjecture that there might be
some coupling between dark energy and cold dark matter (CDM; for a recent
review, see Wang et al. 2016). The model with an interaction between vacuum
energy (for convenience, dark energy with $w=-1$ is called the vacuum energy
in this work and denoted by $\Lambda$) and CDM is usually called the
I$\Lambda$CDM model. Besides, we also wish to consider a more general IDE
model in which the dark energy equation of state (EoS) parameter $w$ is a
constant, usually called the I$w$CDM model. In the I$w$CDM scenario, the
energy conservation equations for dark energy and CDM satisfy

$\displaystyle\dot{\rho}_{\rm de}=-3H(1+w)\rho_{\rm de}+Q,$ (1)
$\displaystyle\dot{\rho}_{\rm c}=-3H\rho_{\rm c}-Q,$ (2)

where $Q$ is the energy transfer rate; $\rho_{\rm de}$ and $\rho_{\rm c}$
denote the energy densities of dark energy and CDM, respectively;
$H=\dot{a}/a$ represents the Hubble parameter; $a$ is the scale factor of the
universe; and a dot represents the derivative with respect to the cosmic time
$t$. Here the case of $w=-1$ corresponds to the I$\Lambda$CDM model. Many
specific forms for $Q$ have been constructed and discussed in previous works
(Amendola, 2000; Zhang, 2005a, b; Zhang et al., 2006; Barrow & Clifton, 2006;
Zhang et al., 2008; He & Wang, 2008; He et al., 2009; Valiviita et al., 2010;
Boehmer et al., 2008; Kim et al., 2017; Xia, 2009; Wei, 2011; Li & Zhang,
2011; Clemson et al., 2012; Li & Zhang, 2014; Li et al., 2014a, b; Zhang,
2017; Cui et al., 2015; Wang et al., 2014; Geng et al., 2015; Väliviita &
Palmgren, 2015; Di Valentino et al., 2020a; Guo et al., 2017, 2018, 2019; Yang
et al., 2018; Feng et al., 2019, 2020; Li et al., 2019, 2020a; Di Valentino et
al., 2020b; Zhao et al., 2020; Li et al., 2020b; Xiao et al., 2019;
Mukhopadhyay et al., 2021; Liu et al., 2020). In this paper, we employ a
phenomenological form of $Q=\beta H\rho_{\rm c}$, where $\beta$ denotes a
dimensionless coupling parameter. From Eqs. (1) and (2), it can be seen that
$\beta>0$ indicates that CDM decays into dark energy, $\beta<0$ indicates that
dark energy decays into CDM, and $\beta=0$ means that there is no interaction
between dark energy and CDM.

In this work, we study what role the 21 cm IM experiments would play in
constraining cosmological parameters in the I$\Lambda$CDM and I$w$CDM models.
Combining with the Planck CMB data (Aghanim et al., 2020), we wish to forecast
how these 21 cm IM experiments will improve constraints on cosmological
parameters. We also make a comparison with optical galaxy surveys. Unless
otherwise stated, we employ the spatially flat $\Lambda$CDM model with
parameters fixed by fitting to the Planck 2018 data (Aghanim et al., 2020) as
a fiducial model to generate mock data.

The constraining power of 21 cm IM experiments for the IDE models was first
studied by Xu et al. (2018). However, our work differs from the previous work
in the following aspects. (i) Xu et al. (2018) considered a model with
$Q=3H(\xi_{1}\rho_{\rm c}+\xi_{2}\rho_{\rm de})$ with $\xi_{1}$ and $\xi_{2}$
being the dimensionless coupling coefficients, and in order to avoid the
curvature perturbation divergence in early times, they excluded a part of the
parameter space. In our work, we consider a different IDE model, and we do not
sacrifice any part of the parameter space. To avoid the perturbation
divergence, we employ an extended parameterized post-Friedmann (ePPF)
framework developed by some of the authors of this work (see Li et al. 2014a,
b). (ii) They used the angular power spectrum, while in this work we use the
power spectrum $P(k)$ (actually, the BAO and redshift space distortion (RSD)
information extracted from the power spectrum). (iii) They considered BINGO,
FAST, and SKA, while here we also consider the Tianlai cylinder array as an
additional facility, for which we will highlight a promising capability in the
following sections.

This paper is organized as follows. In Section 2, we give a detailed
description of the methodology. In Section 2.1, we introduce the signal power
spectrum and noise power spectrum of the 21 cm IM experiments and construct
the Fisher matrix. We further give a detailed description of the experimental
configurations in Section 2.2 and describe the methods and data employed in
this paper in Section 2.3. In Section 3, we present forecast constraints on
cosmological parameters and provide some relevant discussions. Finally, we
give our conclusions in Section 4.

##  2 Methodology

###  2.1 21 cm IM

The mean HI brightness temperature is given by (the detailed derivation can be
found in Battye et al. 2013)

$\displaystyle\overline{T}_{b}(z)=180\Omega_{\rm H{\tt
I}}(z)h\frac{(1+z)^{2}}{H(z)/H_{0}}\,{\rm mK},$ (3)

where $\Omega_{\rm H{\tt I}}(z)$ is the fractional density of HI, $H(z)$ is
the Hubble parameter as a function of redshift $z$, $H_{0}\equiv 100h~{}\rm
km~{}s^{-1}~{}Mpc^{-1}$ is its value today and $h$ is the dimensionless Hubble
constant. The fractional HI density can be written as

$\displaystyle\Omega_{\rm H{\tt I}}(z)\equiv(1+z)^{-3}\rho_{\rm H{\tt
I}}(z)/\rho_{c,0},$ (4)

where $\rho_{c,0}$ is the critical density today, and $\rho_{\rm H{\tt I}}(z)$
is the proper HI density at redshift $z$, given by

$\displaystyle\rho_{\rm H{\tt I}}(z)=\int_{M_{\rm min}}^{M_{\rm
max}}dM\frac{dn}{dM}M_{\rm H{\tt I}}(M,z).$ (5)

Here $M$ is the dark matter halo mass, $dn/dM$ is the proper halo mass
function, and $M_{\rm H{\tt I}}(M,z)$ is the HI mass in a halo of mass $M$ at
redshift $z$. Note that the $(1+z)^{-3}$ term appears in Equation (4) because
$dn/dM$ is the halo mass function in proper volume units. Once $M_{\rm H{\tt
I}}(M,z)$ is specified, we can then obtain $\Omega_{\rm H{\tt I}}(z)$. The
detailed calculations can be found in Bull et al. (2015c), and the resulting
$\Omega_{\rm H{\tt I}}(z)$ is shown in their Figure 20.

Considering the effect of RSDs (Kaiser, 1987), the signal $\rm H{\tt I}$ power
spectrum can be written as (Seo & Eisenstein, 2003)

$\begin{split}P^{\rm S}(k_{f},\mu_{f},z)=&\overline{T}^{2}_{b}\frac{D_{\rm
A}^{2}(z)_{f}H(z)}{D_{\rm A}^{2}(z)H(z)_{f}}b^{2}_{\rm H{\tt I}}(z)F_{\rm
RSD}(k_{f},\mu_{f})P(k,z),\end{split}$ (6)

where the subscript $f$ denotes the quantities calculated in the fiducial
cosmology, $D_{\rm A}(z)$ is the angular diameter distance, and $b_{\rm H{\tt
I}}(z)$ is the HI bias, calculated by

$\displaystyle b_{\rm H{\tt I}}(z)=\rho_{\rm H{\tt I}}^{-1}(z)\int_{M_{\rm
min}}^{M_{\rm max}}dM\frac{dn}{dM}M_{\rm H{\tt I}}(M,z)b(M,z),$ (7)

where $b(M,z)$ is the halo bias. The detailed calculations can also be found
in Xu et al. (2015) and Bull et al. (2015c).

The RSD effect is given by $F_{\rm RSD}(k_{f},\mu_{f})$ in Equation (6),

$\begin{split}F_{\rm RSD}(k_{f},\mu_{f})=[1+\beta_{\rm H{\tt
I}}(z)\mu^{2}]^{2}e^{-k^{2}\mu^{2}\sigma_{\rm NL}^{2}},\end{split}$ (8)

where the first term in the square bracket corresponds to the Kaiser formula,
the exponential term accounts for the “fingers of God” effect,
$\mu=\hat{k}\cdot\hat{z}$, $\beta_{\rm H{\tt I}}(z)$ is the RSD parameter
equal to $f/b_{\rm H{\tt I}}(z)$ ($f\equiv d{\rm ln}D/d{\rm ln}a$ is the
linear growth rate, with $a$ being the scale factor) in linear theory, and
$\sigma_{\rm NL}$ is the nonlinear dispersion scale. For the fiducial model,
we choose a value of $\sigma_{\rm NL}=7~{}{\rm Mpc}$ (Li et al., 2007).

The matter power spectrum $P(k,z)=D^{2}(z)P(k,z=0)$, with $D(z)$ being the
growth factor and $P(k,z=0)$ being the matter power spectrum at $z=0$ that can
be generated by CAMB (Lewis et al., 2000).

The noise power spectrum models the instrumental and sky noises for a given
experiment. The survey noise properties have been described in detail in
Battye et al. (2013) and Bull et al. (2015c). Here we summarize them for
completeness. The frequency resolution of IM surveys performs well, so we
ignore the instrument response function in the radial direction and only
consider the response due to the finite angular resolution,

$\displaystyle W^{2}(k)={\rm
exp}\left[-k_{\perp}^{2}r^{2}(z)\left(\frac{\theta_{\rm B}}{\sqrt{8{\rm
ln}2}}\right)^{2}\right],$ (9)

where $k_{\perp}$ is the transverse wavevector, $r(z)$ is the comoving radial
distance at redshift $z$, and $\theta_{\rm B}$ is the full width at half-
maximum of the beam of an individual dish.

Considering a redshift bin between $z_{1}$ and $z_{2}$, the survey volume can
be written as

$\displaystyle V_{\rm sur}=\Omega_{\rm
tot}\int_{z_{1}}^{z_{2}}dz\frac{dV}{dzd\Omega}=\Omega_{\rm
tot}\int_{z_{1}}^{z_{2}}dz\frac{cr^{2}(z)}{H(z)},$ (10)

where $\Omega_{\rm tot}=S_{\rm area}$ is the solid angle of the survey area.
The pixel volume $V_{\rm pix}$ is also calculated with a similar formula, with
$\Omega_{\rm tot}$ substituted by $\Omega_{\rm pix}\approx\theta^{2}_{\rm B}$.

For an experiment using single-dish mode, the pixel noise can be written as

$\displaystyle\sigma_{\rm pix}=\frac{T_{\rm sys}}{\sqrt{\Delta\nu\,t_{\rm
tot}(\theta_{\rm B}^{2}/S_{\rm area})}}\frac{\lambda^{2}}{A_{e}\theta_{\rm
B}^{2}}\frac{1}{\sqrt{N_{\rm dish}N_{\rm beam}}},$ (11)

and for an interferometer, the pixel noise can be written as

$\displaystyle\sigma_{\rm pix}=\frac{T_{\rm sys}}{\sqrt{\Delta\nu\,t_{\rm
tot}({\rm FoV}/S_{\rm area})}}\frac{\lambda^{2}}{A_{e}\sqrt{\rm
FoV}}\frac{1}{\sqrt{n(k_{\perp})N_{\rm beam}}},$ (12)

where $T_{\rm sys}$ is the system temperature, $N_{\rm dish}$ is the number of
dishes, $N_{\rm beam}$ is the number of beams, $t_{\rm tot}$ is the total
observing time, and $A_{e}$ is the effective collecting area of each element.
For a dish reflector, $A_{e}=\eta\pi(D_{\rm dish}/2)^{2}$ and $\theta_{\rm
B}\approx\lambda/D_{\rm dish}$, where $D_{\rm dish}$ is the diameter of the
dish, and $\eta$ is an efficiency factor for which we adopt 0.7 in this work.
For a cylindrical reflector, $A_{e}=\eta l_{\rm cyl}w_{\rm cyl}/N_{\rm feed}$
and ${\rm FoV}\approx 90^{\circ}\times\lambda/w_{\rm cyl}$, where $w_{\rm
cyl}$ and $l_{\rm cyl}$ are width and length of the cylinder, respectively,
and $N_{\rm feed}$ is the number of feeds per cylinder. Unlike a single dish,
we need to calculate the baseline density $n(k_{\perp})$ for the
interferometer. In this work, we consider all of the baselines of the Tianlai
cylinder array, as detailed in Xu et al. (2015).

For BINGO, FAST, and Tianlai, the system temperature is given by

$\displaystyle T_{\rm sys}=T_{\rm rec}+T_{\rm gal}+T_{\rm CMB},$ (13)

where $T_{\rm rec}$ is the receiver temperature for each of these experiments
with the values given in Table 1, $T_{\rm gal}\approx
25~{}\rm{K}(408~{}\rm{MHz}/\nu)^{2.75}$ is the contribution from the Milky Way
for a given frequency $\nu$, and $T_{\rm CMB}\approx 2.73~{}\rm K$ is the CMB
temperature. For the SKA1-MID array, the system temperature is calculated by

$\displaystyle T_{\rm sys}=T_{\rm rec}+T_{\rm spl}+T_{\rm gal}+T_{\rm CMB},$
(14)

where $T_{\rm spl}\approx 3$ K is the contribution from spillover. The
receiver temperature $T_{\rm rec}$ for SKA1-MID is assumed to be (Bacon et
al., 2020)

$\displaystyle T_{\rm rec}=15~{}\rm{K}+30~{}\rm{K}\left(\frac{\nu}{\rm
GHz}-0.75\right)^{2}.$ (15)

Finally, the noise power spectrum is then given by

$\displaystyle P^{\rm N}(k)=\sigma^{2}_{\rm pix}V_{\rm pix}W^{-2}(k).$ (16)

The Fisher matrix for a set of parameters $\\{p\\}$ is given by Tegmark
(1997),

$\displaystyle F_{ij}=\frac{1}{8\pi^{2}}\int^{1}_{-1}d\mu\int^{k_{\rm
max}}_{k_{\rm min}}k^{2}dk\;\frac{\partial{\rm ln}P^{\rm S}}{\partial
p_{i}}\frac{\partial{\rm ln}P^{\rm S}}{\partial p_{j}}V_{\rm eff},$ (17)

where we define the “effective volume” as (Bull et al., 2015c; Pourtsidou et
al., 2017)

$\displaystyle V_{\rm eff}=V_{\rm sur}\left(\frac{P^{\rm S}}{P^{\rm S}+P^{\rm
N}}\right)^{2}.$ (18)

Next, we have assumed that the bias $b_{\rm H{\tt I}}$ depends only on the
redshift $z$. This assumption is appropriate only for large scales, so we
impose a nonlinear cutoff at $k_{\rm max}\simeq 0.14(1+z)^{2/3}\,{\rm
Mpc}^{-1}$ (Smith et al., 2003). The largest scale the survey can probe
corresponds to a wavevector $k_{\rm min}\simeq 2\pi/V_{\rm sur}^{1/3}$ (Smith
et al., 2003). In this work, we choose the parameter set $\\{p\\}$ as
$\\{D_{\rm A}(z),H(z),[f\sigma_{8}](z),[b_{\rm H{\tt
I}}\sigma_{8}](z),\sigma_{\rm NL}\\}$. When inverting the Fisher matrix, we
can get the covariance matrix that gives us the forecast constraint on the
chosen parameter set. Note that we only use the forecast cosmological
observables $D_{A}(z)$, $H(z)$, and $[f\sigma_{8}](z)$ to constrain the
cosmological parameters.

###  2.2 Experimental configurations

In this paper, we focus on the Tianlai, BINGO, FAST, and SKA1-MID experiments.
These experiments are potentially suitable for the HI IM survey in the post-
reionization epochs of the universe. In this subsection, we give a brief
description of these experiments.

Table 1: Experimental configurations for Tianlai, BINGO, FAST, and SKA1-MID. | Tianlai | BINGO | FAST | SKA1-MID
---|---|---|---|---
$z_{\rm min}$ | 0 | 0.13 | 0 | 0.35
$z_{\rm max}$ | 2.55 | 0.48 | 0.35 | 3
$N_{\rm dish}$ | – | 1 | 1 | 197
$N_{\rm beam}$ | 1 | 50 | 19 | 1
$D_{\rm dish}~{}[\rm m]$ | – | 40 | 300 | 15
$S_{\rm area}~{}[\rm deg^{2}]$ | 10,000 | 3000 | 20,000 | 20,000
$t_{\rm tot}~{}[\rm hr]$ | 10,000 | 10,000 | 10,000 | 10,000
$T_{\rm rec}~{}[\rm K]$ | 50 | 50 | 20 | Eq. (15)

Tianlai. The Tianlai project111http://tianlai.bao.ac.cn is an HI IM experiment
aimed at measuring the dark energy EoS by detecting the BAO features in the
large-scale structure power spectrum. The full-scale Tianlai cylinder array
will consist of eight adjacent cylinders to be built in northwest China, with
each cylinder 15 m wide and 120 m long with 256 dual polarization feeds (Chen,
2011, 2012; Xu et al., 2015). Note that, currently, there is a Tianlai
pathfinder array commissioning, which uses a much smaller-scale cylinder
array, but in this work, we will only discuss the full-scale Tianlai cylinder
array that is to be built in the future.

BINGO. The BINGO experiment222http://bingotelescope.org is a project to build
a special-purpose radio telescope to map redshifted HI emission in the
redshift range of $z=0.13-0.48$. The design of BINGO is a dual-mirror compact
antenna telescope with a 40 m primary mirror and an offset focus, which is
proposed to have a receiver array containing 50–60 feed horns, with a 90 m
focal length. It will be built in eastern Brazil (Battye et al., 2013;
Dickinson, 2014).

FAST. The FAST333https://fast.bao.ac.cn is a multibeam single-dish telescope
built in Guizhou province of southwest China. The aperture diameter is 500 m
with an effective illuminating diameter of 300 m. It uses an active surface
that adjusts shape to create parabolas in different directions. It will be
capable of covering the sky within a $40\degree$ angle from the zenith.
Nineteen beams are designed in one receiver array, which will greatly increase
the survey speed (Nan et al., 2011; Smoot & Debono, 2017).

Figure 1: Forecast fractional errors on $D_{\rm A}(z)$, $H(z)$, and
$[f\sigma_{8}](z)$, as a function of redshift.

SKA1-MID. The SKA project,444https://www.skatelescope.org currently under
construction, plans two stages of development. In this paper, we consider the
SKA1-MID array, based in the Northern Cape, South Africa. The SKA1-MID has 133
15 m SKA dishes and 64 13.5 m MeerKAT dishes. The SKA1-MID will perform an HI
IM survey over a broad range of frequencies and a large fraction of the sky
(Braun et al., 2015; Bull et al., 2015b; Santos et al., 2015; Bacon et al.,
2020). Here we consider only the Wide Band 1 Survey of the SKA1-MID and use
only its single-dish (autocorrelation) mode. In addition, in this work, for
simplicity, we consider SKA1-MID as an array with 197 15 m dishes.

The full instrumental parameters used for these experiments are listed in
Table 1.

###  2.3 Data and method

A method for forecasting cosmological constraints for HI IM surveys has been
presented in Bull et al. (2015c) and Witzemann et al. (2018). We will follow
the prescription given in Bull et al. (2015c) and Witzemann et al. (2018) to
perform the forecast for the 21 cm IM experiments. By performing measurements
of the full anisotropic power spectrum, we obtain constraints on the angular
diameter distance $D_{\rm A}(z)$, the Hubble parameter $H(z)$, and RSD
observable $[f\sigma_{8}](z)$, which are considered to be independent in each
redshift bin. We obtain covariance matrices for $\\{D_{\rm
A}(z_{j}),H(z_{j}),[f\sigma_{8}](z_{j});j=1...N\\}$ in a series of $N$
redshift bins $\\{z_{j}\\}$ by inverting the Fisher matrix. We perform the
Fisher matrix calculations by considering the aforementioned parameters. The
marginalized constraints on these parameters for these surveys are shown in
Figure 1.

These covariance matrices, plus the fiducial cosmology, generate the mock data
of these 21 cm IM experiments. Then we use these mock data to constrain the
cosmological parameters by performing a Markov Chain Monte Carlo (MCMC)
analysis. The cosmological parameters that we sample include $\beta$, $H_{0}$,
$\Omega_{\rm b}$, $\Omega_{\rm c}$, $\sigma_{8}$, and $w$. We consider flat
priors for these parameters with ranges of $\beta\in[-1,1]$,
$H_{0}\in[30,100]~{}\rm{km~{}s^{-1}~{}Mpc^{-1}}$, $\Omega_{\rm b}\in[0,1]$,
$\Omega_{\rm c}\in[0,1]$, $\sigma_{8}\in[0,2]$, and $w\in[-3,1]$. In the MCMC
analysis, we also employ the CMB angular power spectra data of Planck 2018
TT,TE,EE+lowE (Aghanim et al., 2020), and the BAO measurements from galaxy
redshift surveys, including SDSS-MGS (Ross et al., 2015), 6dFGS (Beutler et
al., 2011), and BOSS DR12 (Alam et al., 2017).

In this paper, we employ the ePPF framework to calculate the cosmological
perturbations in the IDE scenario (Li et al., 2014a, b). This is because we
need to avoid the perturbation divergence problem in the IDE cosmology.

It is well known that, in the IDE scenario, when calculating the cosmological
perturbations, it is found that for most cases, the curvature perturbation on
superhorizon scales at early times is divergent, which is a catastrophe for
the IDE cosmology. The underlying reason for this problem is that we actually
do not know how to consider the perturbations of dark energy. In the
traditional linear perturbation theory, for calculating the perturbations of
dark energy, we need to define a rest-frame sound speed for dark energy fluid
$c_{s}^{2}=\delta p/\delta\rho$ (with the gauge $v|_{\rm rf}=B|_{\rm rf}=0$)
to relate the dark energy density and pressure perturbations. This leads to
that in a general gauge, $\delta p$ has two parts, adiabatic and nonadiabatic,
and the interaction term appearing in the nonadiabatic part will occasionally
lead the nonadiabatic modes to be unstable.

Table 2: The 1$\sigma$ errors on the parameters in the I$\Lambda$CDM model from the different data combinations. Here, $H_{0}$ is in units of $\rm{km~{}s^{-1}~{}Mpc^{-1}}$. | $\beta$ | $H_{0}$ | $\Omega_{\rm m}$ | $\sigma_{8}$
---|---|---|---|---
Planck | 0.00240 | 1.80 | 0.0255 | 0.0150
Planck+BAO | 0.00120 | 0.69 | 0.0087 | 0.0110
Tianlai | 0.00715 | 0.19 | 0.0033 | 0.0033
Planck+Tianlai | 0.00023 | 0.16 | 0.0020 | 0.0018
Planck+BINGO | 0.00170 | 1.30 | 0.0180 | 0.0110
Planck+FAST | 0.00140 | 1.10 | 0.0150 | 0.0089
Planck+SKA1-MID | 0.00060 | 0.39 | 0.0054 | 0.0035
Table 3: The 1$\sigma$ errors on the parameters in the I$w$CDM model from the different data combinations. | $\beta$ | $w$
---|---|---
Planck | 0.00175 | 0.315
Planck+BAO | 0.00150 | 0.074
Tianlai | 0.01380 | 0.016
Planck+Tianlai | 0.00079 | 0.013
Planck+BINGO | 0.00160 | 0.048
Planck+FAST | 0.00160 | 0.036
Planck+SKA1-MID | 0.00120 | 0.021

In order to solve this problem, Li et al. (2014a, b) extended the original PPF
framework (Fang et al., 2008) to include the IDE scenario and used this method
to avoid the divergence of cosmological perturbations in the IDE models. The
ePPF method does not consider the dark energy pressure perturbation; it only
describes dark energy perturbations based on some basic facts of dark energy.
On large scales, far beyond the horizon, the relationship between the
velocities of dark energy and other components can be established through a
parameterization provided by a function $f_{\zeta}(a)$. On small scales, deep
inside the horizon, dark energy is smooth enough so that it can be viewed as a
pure background. We thus can use the Poisson equation to describe this limit.
In order to make these two limits compatible, we introduce a dynamical
function $\Gamma$ by which we can find an equation to describe the cases on
all scales. In the equation of motion of $\Gamma$, a parameter $c_{\Gamma}$ is
introduced, giving a transition scale in terms of the Hubble scale under which
dark energy is smooth enough. In this equation, there is no perturbation
variable of dark energy, and when the evolution of $\Gamma$ is derived, we can
directly obtain the density and velocity perturbations of dark energy. Hence,
this method avoids using the pressure perturbation of dark energy defined by
the sound speed. Here we only give a very brief description of the ePPF
method, and we refer the reader to Li et al. (2014a, b) for more details.

We employ the CosmoMC package (Lewis & Bridle, 2002) to perform the MCMC
calculations and insert the ePPF code as a part of it to treat the
cosmological perturbations in the IDE models.

##  3 Results

In this section, we will present the forecast results showing relative
constraining capabilities for the I$\Lambda$CDM and I$w$CDM models by
combining each of the 21 cm IM experiments with Planck. Tables 2 and 3 list
the 1$\sigma$ errors for the marginalized parameter constraints for the
I$\Lambda$CDM and I$w$CDM models, respectively. In Section 3.1, we will show
what role the 21 cm IM experiments, taking the example of Tianlai, could play
in constraining cosmological parameters and compare with the BAO measurements
from galaxy redshift surveys. In Section 3.2, we will compare the capabilities
of constraining cosmological parameters for the different 21 cm IM
experiments.

Figure 2: Constraints on cosmological parameters from Planck, Planck+BAO,
Tianlai, and Planck+Tianlai in the I$\Lambda$CDM model.

###  3.1 Constraints on cosmological parameters from the Tianlai cylinder
array

The 1$\sigma$ and 2$\sigma$ posterior distribution contours are shown in
Figure 2 for Planck, Planck+BAO, Tianlai, and Planck+Tianlai in the
I$\Lambda$CDM model. We find that the future full-scale Tianlai experiment can
give very tight constraints on $H_{0}$, $\Omega_{\rm m}$, and $\sigma_{8}$.
With the Tianlai data alone, we obtain $\sigma(H_{0})=0.19$ km s-1 Mpc-1,
$\sigma(\Omega_{\rm m})=0.0033$, and $\sigma(\sigma_{8})=0.0033$, which are
even much better than the results of $\sigma(H_{0})=0.69$ km s-1 Mpc-1,
$\sigma(\Omega_{\rm m})=0.0087$, and $\sigma(\sigma_{8})=0.0110$ from the
Planck+optical BAO. Comparing with the Planck data, the data combination
Planck+Tianlai can improve the constraint accuracies of $H_{0}$, $\Omega_{\rm
m}$, and $\sigma_{8}$ by $(1.80-0.16)/1.80=91.1\%$,
$(0.0255-0.0020)/0.0255=92.2\%$, and $(0.0150-0.0018)/0.0150=88.0\%$,
respectively.

Actually, the forecast analysis for some 21 cm IM experiments (including
Tianlai) was made in Bull et al. (2015c). In Table 4 of Bull et al. (2015c),
the results are shown for the $w_{0}$$w_{a}$CDM model with spatial curvature.
It can be seen that our constraint results (for the I$\Lambda$CDM model) are
stronger than theirs, although the models considered are different. The
reasons lie in the following facts. (i) Bull et al. (2015c) considered the
foreground residuals, but we considered a perfect removal of foreground in
this work. (ii) They ignored some baselines (those shorter than 15 m), but we
considered all of the baselines, especially for those short baselines formed
by the large number of close-by feeds on the Tianlai cylinder array, which add
a lot of sensitivity for low-redshift BAO signals. (ii) We employed the
experimental configuration for Tianlai based on Xu et al. (2015), which is
different from Bull et al. (2015c). For example, the redshift range and survey
area we use are $z=0$–2.55 and $S_{\rm area}=10,000$ deg2, while in Bull et
al. (2015c), they adopted $z=0.49$–1.58 and $S_{\rm area}=25,000$ deg2. In
addition, we considered the combination with Planck 2018 data, while they used
only a Planck prior from the Planck 2013 result. Also, Bull et al. (2015c)
considered two more cosmological parameters in their forecast than ours. These
differences lead to the stronger constraints we have than theirs.

For the coupling parameter $\beta$, we find that the Tianlai-alone data can
only give a relatively weak constraint, $\sigma(\beta)=0.0072$; as a
comparison, the Planck-alone data give a result of $\sigma(\beta)=0.0024$. In
fact, for the IDE model with $Q=\beta H\rho_{c}$, the CMB data usually could
provide a tight constraint on the coupling parameter $\beta$. This is because
in the early universe, both $H$ and $\rho_{c}$ take rather high values, and
the energy transfer rate $Q$ can take a moderate value even if $\beta$ is very
small. Thus, the CMB data as an early-universe probe can give a relatively
tight constraint on $\beta$. This is why the Planck CMB data can offer a much
tighter constraint on $\beta$, compared with the case of the Tianlai data, but
for the other cosmological parameters, the Tianlai data can give much better
constraints. Since the degeneracy directions of $\beta$ and other parameters
for Planck and Tianlai are rather different, we can eventually obtain a rather
tight constraint on $\beta$, i.e., $\sigma(\beta)=0.00023$, from the
Planck+Tianlai data combination, which is much better than the result of
$\sigma(\beta)=0.00120$ from the Planck+BAO data combination.

Figure 3: Constraints on $\beta$ and $w$ from Planck, Planck+BAO, Tianlai, and
Planck+Tianlai in the I$w$CDM model. Figure 4: Constraints on cosmological
parameters from Planck+BINGO, Planck+FAST, Planck+SKA1-MID, and Planck+Tianlai
in the I$\Lambda$CDM model.

In Figure 3, we show the constraint results of the I$w$CDM model. Here we only
show the posterior distribution contours in the $\beta$–$w$ plane that we are
most interested in. We can see that using only the Planck CMB data cannot give
a good constraint on the dark energy EoS parameter $w$, i.e.,
$\sigma(w)\approx 0.3$. It is necessary to use the late-universe measurements
to break the parameter degeneracies inherent in the CMB. As a contrast, the
Tianlai-alone data can provide a rather tight constraint on $w$, giving a
result of $\sigma(w)=0.016$.

However, as discussed above, the CMB data can tightly constrain the coupling
parameter $\beta$ in the I$w$CDM model with the interaction term $Q=\beta
H\rho_{c}$. So, we can see that the Planck-alone data give
$\sigma(\beta)=0.00175$, which is much better than the result given by the
Tianlai-alone data, $\sigma(\beta)=0.01380$.

Since the CMB data can tightly constrain $\beta$ and the Tianlai data can
tightly constrain $w$, their degeneracy directions of them are entirely
different. It is known that Planck alone, and even Planck+BAO, can only
provide a loose or moderate constraint on the I$w$CDM model, as shown in
Figure 3. Nevertheless, since the cosmological parameter degeneracies can be
broken by the Tianlai data, the parameter constraints are greatly improved by
adding the Tianlai data in the fit. We obtain $\sigma(\beta)=0.00079$ and
$\sigma(w)=0.013$ from the Panck+Tianlai data combination, and we find that
the constraints on $\beta$ and $w$ are improved by
$(0.00175-0.00079)/0.00175=54.9\%$ and $(0.315-0.013)/0.315=95.9\%$,
respectively, by adding the Tianlai data. Comparing with the constraint
results of $\sigma(\beta)=0.00150$ and $\sigma(w)=0.074$ from Planck+BAO, we
can see that the future 21 cm IM experiments will exhibit a powerful
capability of constraining cosmological parameters.

###  3.2 Comparison with constraints from different 21 cm IM experiments

In this subsection, we will discuss the ability to constrain cosmological
parameters for the different 21 cm IM experiments.

Figure 4 visualizes the constraint results for the I$\Lambda$CDM model from
each of these experiments, including BINGO, FAST, SKA1-MID, and Tianlai,
combined with Planck. We can clearly see from Figure 4 that the constraining
capabilities of the two arrays, i.e. Tianlai and SKA1-MID, are much better
than those of the single dishes, FAST and BINGO. Comparing Tianlai and
SKA1-MID, Tianlai is evidently better, and comparing FAST and BINGO, FAST is
much better. As shown in Figure 1, FAST and BINGO can only observe in low
redshifts and can only cover narrow redshift ranges, i.e., $0<z<0.35$ for FAST
and $0.13<z<0.48$ for BINGO. Although the redshift range coverages are
similar, comparing FAST and BINGO for the relative measurement errors on
$D_{\rm A}(z)$, $H(z)$, and $[f\sigma_{8}](z)$, we find that FAST is obviously
better than BINGO, mainly due to the much larger survey area and larger
aperture size. Comparing Tianlai and SKA1-MID, we find that both of them can
cover a wide redshift range, i.e., $0<z<2.55$ for Tianlai and $0.35<z<3$ for
SKA1-MID (Wide Band 1 Survey). However, for the redshift range of $z>0.5$, the
relative errors on $D_{\rm A}(z)$, $H(z)$, and $[f\sigma_{8}](z)$ of Tianlai
are much smaller than those of SKA1-MID. This highlights the advantage of a
compact interferometer array with a large number of receivers and explains why
Tianlai’s capability of constraining cosmological parameters is better than
that of SKA1-MID.

Concretely, for constraining the coupling parameter $\beta$, we obtain
$\sigma(\beta)=0.00170$, 0.00140, 0.00060, and 0.00023, from Planck+BINGO,
Planck+FAST, Planck+SKA1-MID, and Planck+Tianlai, respectively. This shows
that, for constraining $\beta$ in the I$\Lambda$CDM model, the capability of
Tianlai is the best, in the sense of combining with Planck, much better than
those of BINGO, FAST, and SKA1-MID. Comparing with the results of the Planck-
alone data, the Planck+BINGO, Planck+FAST, Planck+SKA1-MID, and Planck+Tianlai
data can improve the constraints on $\beta$ by $29.2\%$, $41.7\%$, $75.0\%$,
and $90.4\%$, respectively.

We then show the results for the constraints on the other cosmological
parameters in the I$\Lambda$CDM model. We obtain $\sigma(H_{0})=1.30$ km s-1
Mpc-1, $\sigma(\Omega_{\rm m})=0.0180$, and $\sigma(\sigma_{8})=0.0110$ from
Planck+BINGO and $\sigma(H_{0})=1.10$ km s-1 Mpc-1, $\sigma(\Omega_{\rm
m})=0.0150$, and $\sigma(\sigma_{8})=0.0089$ from Planck+FAST. We can see
that, for constraining $H_{0}$, $\Omega_{\rm m}$, and $\sigma_{8}$, FAST
performs better than BINGO. But we also notice that neither FAST nor BINGO is
as powerful as Tianlai and SKA1-MID. We obtain $\sigma(H_{0})=0.39$ km s-1
Mpc-1, $\sigma(\Omega_{\rm m})=0.0054$, and $\sigma(\sigma_{8})=0.0035$ from
Planck+SKA1-MID and $\sigma(H_{0})=0.16$ km s-1 Mpc-1, $\sigma(\Omega_{\rm
m})=0.0020$, and $\sigma(\sigma_{8})=0.0018$ from Planck+Tianlai. We can
clearly see that in the future, the full-scale Tianlai cylinder array
experiment will play a significant role in precisely measuring cosmological
parameters.

Here we note that, in this work, we only discuss the Wide Band 1 Survey of the
SKA1-MID. Actually, SKA1 has two other surveys targeting cosmology, i.e., the
Medium-Deep Band 2 Survey (with SKA1-MID) and the Deep SKA-LOW Survey (with
SKA1-LOW). The Medium-Deep Band 2 Survey covers the redshift range of
$0<z<0.5$ (with $S_{\rm area}=5000~{}{\rm deg}^{2}$ and $t_{\rm total}=10,000$
hr), and the Deep SKA-LOW Survey covers the redshift range of $3<z<6$ (with
$S_{\rm area}=100~{}{\rm deg}^{2}$ and $t_{\rm total}=5000$ hr). It is of
great interest to use the combination of the three surveys of SKA1 in
measuring the expansion history of the post-reionization epoch of the universe
to explore various cosmological issues. We will leave this work for the
future.

Figure 5: Constraints on $\beta$ and $w$ from Planck+BINGO, Planck+FAST,
Planck+SKA1-MID, and Planck+Tianlai in the I$w$CDM model.

In order to compare their constraint abilities in the I$w$CDM model, we show
the 1$\sigma$ and 2$\sigma$ measurement error contours for Planck+BINGO,
Planck+FAST, Planck+SKA1-MID, and Planck+Tianlai in Figure 5. We obtain
$\sigma(\beta)=0.00160$ and $\sigma(w)=0.048$ from Planck+BINGO, which are
improved by $8.6\%$ and $84.8\%$, respectively, when combining the BINGO data
in the cosmological fit to the Planck CMB data. This shows that, although
BINGO’s constraining capability is the weakest among the four 21 cm IM
experiments considered in this work, it still can play an important role in
breaking the parameter degeneracies in CMB and improving the measurement
accuracies of the cosmological parameters (especially $w$) in the I$w$CDM
model. FAST performs slightly better than BINGO, and we obtain
$\sigma(\beta)=0.00160$ and $\sigma(w)=0.036$ from Planck+FAST. Evidently,
SKA1-MID performs much better than BINGO and FAST, and we obtain
$\sigma(\beta)=0.00120$ and $\sigma(w)=0.021$ from Planck+SKA1-MID. The most
stringent constraints on the I$w$CDM model are from Planck+Tianlai, and in
this case, we have $\sigma(\beta)=0.00079$ and $\sigma(w)=0.013$. Compared
with the Planck result, we find that FAST, SKA1-MID, and Tianlai can improve
the constraints on $\beta$ by 8.6%, 31.4%, and 54.9%, respectively.

##  4 Conclusion

In this work, we investigate the constraint capabilities of the future 21 cm
IM experiments for the IDE models. We consider BINGO, FAST, SKA1-MID, and
Tianlai as typical examples of 21 cm IM experiments and find that among them,
a compact interferometer array like the full-scale Tianlai cylinder array
would be the best one in constraining the IDE model.

We find that the 21 cm observations with the full-scale Tianlai cylinder array
can tightly constrain $H_{0}$, $\Omega_{\rm m}$, and $\sigma_{8}$. For
example, in the I$\Lambda$CDM model, the Tianlai-alone data can give the
constraint accuracies of $\sigma(H_{0})=0.19$ km s-1 Mpc-1,
$\sigma(\Omega_{\rm m})=0.0033$, and $\sigma(\sigma_{8})=0.0033$, even much
better than those of Planck+optical BAO. But, relatively speaking, the Tianlai
data cannot provide a tight constraint on the coupling parameter $\beta$ as
much as constraints on other cosmological parameters. However, it is also
found that the parameter degeneracy directions from Planck and Tianlai are
entirely different; thus, the combination of Planck and Tianlai can well break
the parameter degeneracies and give a rather tight constraint on $\beta$. In
the I$\Lambda$CDM and I$w$CDM models, we obtain $\sigma(\beta)=2.3\times
10^{-4}$ and $7.9\times 10^{-4}$, respectively, from Planck+Tianlai. This
shows that the constraints on $\beta$ can be improved by 90.4% and 54.9% in
the two models by adding the Tianlai data in the cosmological fit, compared
with the case of using only the Planck data.

We also make a detailed comparison for BINGO, FAST, SKA1-MID, and Tianlai in
the cosmological-fit study of the IDE models. We find that, for the constraint
capability, Tianlai is the best one, and SKA1-MID is slightly less powerful
than Tianlai, but both are much better than FAST and BINGO. Our goal is not to
show the superiority or inferiority of these experiments against each other
but rather to give a global picture of their relative prospects. Our results
show that the 21 cm IM experiments will provide a promising tool for exploring
the nature of dark energy, and, in particular, a compact interferometer array
will play a significant role in measuring the coupling between dark energy and
dark matter.

## Acknowledgments

We thank Xin Wang, Xue-Lei Chen, Ze-Wei Zhao, Ling-Feng Wang, Li-Yang Gao, and
Yue Shao for helpful discussions. This work was supported by the MoST-BRICS
Flagship Project (Grant No. 2018YFE0120800), the National Natural Science
Foundation of China (Grant Nos. 11975072, 11875102, 11835009, 11690021,
11973047, and 11633004), the National SKA Program of China (Grant No.
2020SKA0110401), the Chinese Academy of Sciences (CAS) Strategic Priority
Research Program (Grant No. XDA15020200), the Liaoning Revitalization Talents
Program (Grant No. XLYC1905011), the Fundamental Research Funds for the
Central Universities (Grant No. N2005030), and the Top-Notch Young Talents
Program of China (Grant No. W02070050).

## References

  * Aghanim et al. (2020) Aghanim, N., et al. 2020, Astron. Astrophys., 641, A6, doi: 10.1051/0004-6361/201833910
  * Alam et al. (2017) Alam, S., et al. 2017, Mon. Not. Roy. Astron. Soc., 470, 2617, doi: 10.1093/mnras/stx721
  * Amendola (2000) Amendola, L. 2000, Phys. Rev. D, 62, 043511, doi: 10.1103/PhysRevD.62.043511
  * Anderson et al. (2018) Anderson, C., et al. 2018, Mon. Not. Roy. Astron. Soc., 476, 3382, doi: 10.1093/mnras/sty346
  * Ansari et al. (2012) Ansari, R., Campagne, J., Colom, P., et al. 2012, Astron. Astrophys., 540, A129, doi: 10.1051/0004-6361/201117837
  * Bacon et al. (2020) Bacon, D. J., et al. 2020, Publ. Astron. Soc. Austral., 37, e007, doi: 10.1017/pasa.2019.51
  * Bagla et al. (2010) Bagla, J., Khandai, N., & Datta, K. K. 2010, Mon. Not. Roy. Astron. Soc., 407, 567, doi: 10.1111/j.1365-2966.2010.16933.x
  * Barrow & Clifton (2006) Barrow, J. D., & Clifton, T. 2006, Phys. Rev. D, 73, 103520, doi: 10.1103/PhysRevD.73.103520
  * Battye et al. (2013) Battye, R., Browne, I., Dickinson, C., et al. 2013, Mon. Not. Roy. Astron. Soc., 434, 1239, doi: 10.1093/mnras/stt1082
  * Battye et al. (2004) Battye, R. A., Davies, R. D., & Weller, J. 2004, Mon. Not. Roy. Astron. Soc., 355, 1339, doi: 10.1111/j.1365-2966.2004.08416.x
  * Beutler et al. (2011) Beutler, F., Blake, C., Colless, M., et al. 2011, Mon. Not. Roy. Astron. Soc., 416, 3017, doi: 10.1111/j.1365-2966.2011.19250.x
  * Bigot-Sazy et al. (2016) Bigot-Sazy, M.-A., Ma, Y.-Z., Battye, R. A., et al. 2016, ASP Conf. Ser., 502, 41.  https://arxiv.org/abs/1511.03006
  * Blake & Glazebrook (2003) Blake, C., & Glazebrook, K. 2003, Astrophys. J., 594, 665, doi: 10.1086/376983
  * Boehmer et al. (2008) Boehmer, C. G., Caldera-Cabral, G., Lazkoz, R., & Maartens, R. 2008, Phys. Rev. D, 78, 023505, doi: 10.1103/PhysRevD.78.023505
  * Braun et al. (2019) Braun, R., Bonaldi, A., Bourke, T., Keane, E., & Wagg, J. 2019.  https://arxiv.org/abs/1912.12699
  * Braun et al. (2015) Braun, R., Bourke, T., Green, J. A., Keane, E., & Wagg, J. 2015, PoS, AASKA14, 174, doi: 10.22323/1.215.0174
  * Bull (2016) Bull, P. 2016, Astrophys. J., 817, 26, doi: 10.3847/0004-637X/817/1/26
  * Bull et al. (2015a) Bull, P., Camera, S., Raccanelli, A., et al. 2015a, PoS, AASKA14, 024, doi: 10.22323/1.215.0024
  * Bull et al. (2015b) Bull, P., Camera, S., Raccanelli, A., et al. 2015b.  https://arxiv.org/abs/1501.04088
  * Bull et al. (2015c) Bull, P., Ferreira, P. G., Patel, P., & Santos, M. G. 2015c, Astrophys. J., 803, 21, doi: 10.1088/0004-637X/803/1/21
  * Chang et al. (2010) Chang, T.-C., Pen, U.-L., Bandura, K., & Peterson, J. B. 2010, Nature, 466, 463, doi: 10.1038/nature09187
  * Chen (2011) Chen, X. 2011, Scientia Sinica Physica, Mechanica & Astronomica, 41, 1358, doi: 10.1360/132011-972
  * Chen (2012) Chen, X. 2012, in International Journal of Modern Physics Conference Series, Vol. 12, International Journal of Modern Physics Conference Series, 256–263, doi: 10.1142/S2010194512006459
  * Clemson et al. (2012) Clemson, T., Koyama, K., Zhao, G.-B., Maartens, R., & Valiviita, J. 2012, Phys. Rev. D, 85, 043007, doi: 10.1103/PhysRevD.85.043007
  * Cui et al. (2015) Cui, J.-L., Yin, L., Wang, L.-F., Li, Y.-H., & Zhang, X. 2015, JCAP, 09, 024, doi: 10.1088/1475-7516/2015/09/024
  * Di Valentino et al. (2020a) Di Valentino, E., Melchiorri, A., Mena, O., & Vagnozzi, S. 2020a, Phys. Rev. D, 101, 063502, doi: 10.1103/PhysRevD.101.063502
  * Di Valentino et al. (2020b) —. 2020b, Phys. Dark Univ., 30, 100666, doi: 10.1016/j.dark.2020.100666
  * Dickinson (2014) Dickinson, C. 2014, in 49th Rencontres de Moriond on Cosmology, 139–142.  https://arxiv.org/abs/1405.7936
  * Fang et al. (2008) Fang, W., Hu, W., & Lewis, A. 2008, Phys. Rev. D, 78, 087303, doi: 10.1103/PhysRevD.78.087303
  * Feng et al. (2020) Feng, L., Li, H.-L., Zhang, J.-F., & Zhang, X. 2020, Sci. China Phys. Mech. Astron., 63, 220401, doi: 10.1007/s11433-019-9431-9
  * Feng et al. (2019) Feng, L., Zhang, J.-F., & Zhang, X. 2019, Phys. Dark Univ., 23, 100261, doi: 10.1016/j.dark.2018.100261
  * Geng et al. (2015) Geng, J.-J., Li, Y.-H., Zhang, J.-F., & Zhang, X. 2015, Eur. Phys. J. C, 75, 356, doi: 10.1140/epjc/s10052-015-3581-8
  * Guo et al. (2017) Guo, R.-Y., Li, Y.-H., Zhang, J.-F., & Zhang, X. 2017, JCAP, 05, 040, doi: 10.1088/1475-7516/2017/05/040
  * Guo et al. (2018) Guo, R.-Y., Zhang, J.-F., & Zhang, X. 2018, Chin. Phys. C, 42, 095103, doi: 10.1088/1674-1137/42/9/095103
  * Guo et al. (2019) —. 2019, JCAP, 02, 054, doi: 10.1088/1475-7516/2019/02/054
  * He & Wang (2008) He, J.-H., & Wang, B. 2008, JCAP, 06, 010, doi: 10.1088/1475-7516/2008/06/010
  * He et al. (2009) He, J.-H., Wang, B., & Jing, Y. 2009, JCAP, 07, 030, doi: 10.1088/1475-7516/2009/07/030
  * Hu et al. (2020) Hu, W., Wang, X., Wu, F., et al. 2020, Mon. Not. Roy. Astron. Soc., 493, 5854, doi: 10.1093/mnras/staa650
  * Kaiser (1987) Kaiser, N. 1987, Mon. Not. Roy. Astron. Soc., 227, 1
  * Kim et al. (2017) Kim, J.-S., Kim, C.-J., Hwang, S. C., & Ko, Y. H. 2017, Phys. Rev. D, 96, 043507, doi: 10.1103/PhysRevD.96.043507
  * Lewis & Bridle (2002) Lewis, A., & Bridle, S. 2002, Phys. Rev. D, 66, 103511, doi: 10.1103/PhysRevD.66.103511
  * Lewis et al. (2000) Lewis, A., Challinor, A., & Lasenby, A. 2000, Astrophys. J., 538, 473, doi: 10.1086/309179
  * Li et al. (2007) Li, C., Jing, Y. P., Kauffmann, G., et al. 2007, Mon. Not. Roy. Astron. Soc., 376, 984, doi: 10.1111/j.1365-2966.2007.11518.x
  * Li et al. (2013) Li, D., Nan, R., & Pan, Z. 2013, IAU Symp., 291, 325, doi: 10.1017/S1743921312024015
  * Li et al. (2019) Li, H.-L., Feng, L., Zhang, J.-F., & Zhang, X. 2019, Sci. China Phys. Mech. Astron., 62, 120411, doi: 10.1007/s11433-019-9439-8
  * Li et al. (2020a) Li, H.-L., He, D.-Z., Zhang, J.-F., & Zhang, X. 2020a, JCAP, 06, 038, doi: 10.1088/1475-7516/2020/06/038
  * Li et al. (2020b) Li, H.-L., Zhang, J.-F., & Zhang, X. 2020b, Commun. Theor. Phys., 72, 125401, doi: 10.1088/1572-9494/abb7c9
  * Li et al. (2020) Li, J., Zuo, S., Wu, F., et al. 2020, Science China Physics, Mechanics, and Astronomy, 63, 129862, doi: 10.1007/s11433-020-1594-8
  * Li et al. (2014a) Li, Y.-H., Zhang, J.-F., & Zhang, X. 2014a, Phys. Rev. D, 90, 063005, doi: 10.1103/PhysRevD.90.063005
  * Li et al. (2014b) —. 2014b, Phys. Rev. D, 90, 123007, doi: 10.1103/PhysRevD.90.123007
  * Li & Zhang (2011) Li, Y.-H., & Zhang, X. 2011, Eur. Phys. J. C, 71, 1700, doi: 10.1140/epjc/s10052-011-1700-8
  * Li & Zhang (2014) —. 2014, Phys. Rev. D, 89, 083009, doi: 10.1103/PhysRevD.89.083009
  * Lidz et al. (2011) Lidz, A., Furlanetto, S. R., Oh, S., et al. 2011, Astrophys. J., 741, 70, doi: 10.1088/0004-637X/741/2/70
  * Liu et al. (2020) Liu, X.-W., Heneka, C., & Amendola, L. 2020, JCAP, 2020, 038, doi: 10.1088/1475-7516/2020/05/038
  * Loeb & Wyithe (2008) Loeb, A., & Wyithe, S. 2008, Phys. Rev. Lett., 100, 161301, doi: 10.1103/PhysRevLett.100.161301
  * Mao et al. (2008) Mao, Y., Tegmark, M., McQuinn, M., Zaldarriaga, M., & Zahn, O. 2008, Phys. Rev. D, 78, 023529, doi: 10.1103/PhysRevD.78.023529
  * Masui et al. (2013) Masui, K., et al. 2013, Astrophys. J. Lett., 763, L20, doi: 10.1088/2041-8205/763/1/L20
  * McQuinn et al. (2006) McQuinn, M., Zahn, O., Zaldarriaga, M., Hernquist, L., & Furlanetto, S. R. 2006, Astrophys. J., 653, 815, doi: 10.1086/505167
  * Mukhopadhyay et al. (2021) Mukhopadhyay, U., Majumdar, D., & Datta, K. K. 2021, Phys. Rev. D, 103, 063510, doi: 10.1103/PhysRevD.103.063510
  * Nan et al. (2011) Nan, R., Li, D., Jin, C., et al. 2011, Int. J. Mod. Phys. D, 20, 989, doi: 10.1142/S0218271811019335
  * Pourtsidou et al. (2017) Pourtsidou, A., Bacon, D., & Crittenden, R. 2017, Mon. Not. Roy. Astron. Soc., 470, 4251, doi: 10.1093/mnras/stx1479
  * Pritchard & Loeb (2008) Pritchard, J. R., & Loeb, A. 2008, Phys. Rev. D, 78, 103511, doi: 10.1103/PhysRevD.78.103511
  * Riess et al. (1998) Riess, A. G., et al. 1998, Astron. J., 116, 1009, doi: 10.1086/300499
  * Ross et al. (2015) Ross, A. J., Samushia, L., Howlett, C., et al. 2015, Mon. Not. Roy. Astron. Soc., 449, 835, doi: 10.1093/mnras/stv154
  * Santos et al. (2015) Santos, M. G., et al. 2015, PoS, AASKA14, 019, doi: 10.22323/1.215.0019
  * Seo et al. (2010) Seo, H.-J., Dodelson, S., Marriner, J., et al. 2010, Astrophys. J., 721, 164, doi: 10.1088/0004-637X/721/1/164
  * Seo & Eisenstein (2003) Seo, H.-J., & Eisenstein, D. J. 2003, Astrophys. J., 598, 720, doi: 10.1086/379122
  * Smith et al. (2003) Smith, R., Peacock, J., Jenkins, A., et al. 2003, Mon. Not. Roy. Astron. Soc., 341, 1311, doi: 10.1046/j.1365-8711.2003.06503.x
  * Smoot & Debono (2017) Smoot, G. F., & Debono, I. 2017, Astron. Astrophys., 597, A136, doi: 10.1051/0004-6361/201526794
  * Tegmark (1997) Tegmark, M. 1997, Phys. Rev. Lett., 79, 3806, doi: 10.1103/PhysRevLett.79.3806
  * Tramonte & Ma (2020) Tramonte, D., & Ma, Y.-Z. 2020, Mon. Not. Roy. Astron. Soc., 498, 5916, doi: 10.1093/mnras/staa2727
  * Tramonte et al. (2019) Tramonte, D., Ma, Y.-Z., Li, Y.-C., & Staveley-Smith, L. 2019, Mon. Not. Roy. Astron. Soc., 489, 385, doi: 10.1093/mnras/stz2146
  * Valiviita et al. (2010) Valiviita, J., Maartens, R., & Majerotto, E. 2010, Mon. Not. Roy. Astron. Soc., 402, 2355, doi: 10.1111/j.1365-2966.2009.16115.x
  * Väliviita & Palmgren (2015) Väliviita, J., & Palmgren, E. 2015, JCAP, 07, 015, doi: 10.1088/1475-7516/2015/07/015
  * Wang et al. (2016) Wang, B., Abdalla, E., Atrio-Barandela, F., & Pavon, D. 2016, Rept. Prog. Phys., 79, 096901, doi: 10.1088/0034-4885/79/9/096901
  * Wang et al. (2014) Wang, S., Wang, Y.-Z., Geng, J.-J., & Zhang, X. 2014, Eur. Phys. J. C, 74, 3148, doi: 10.1140/epjc/s10052-014-3148-0
  * Wei (2011) Wei, H. 2011, Commun. Theor. Phys., 56, 972, doi: 10.1088/0253-6102/56/5/29
  * Witzemann et al. (2018) Witzemann, A., Bull, P., Clarkson, C., et al. 2018, Mon. Not. Roy. Astron. Soc., 477, L122, doi: 10.1093/mnrasl/sly062
  * Wu et al. (2016) Wu, F., Wang, Y., Zhang, J., Shi, H., & Chen, X. 2016, in 51st Rencontres de Moriond on Cosmology (ARISF), 315–318
  * Wu et al. (2021) Wu, F., et al. 2021, Mon. Not. Roy. Astron. Soc., 506, 3455, doi: 10.1093/mnras/stab1802
  * Wuensche (2019) Wuensche, C. 2019, J. Phys. Conf. Ser., 1269, 012002, doi: 10.1088/1742-6596/1269/1/012002
  * Wuensche et al. (2020) Wuensche, C., et al. 2020, Exper. Astron., 50, 125, doi: 10.1007/s10686-020-09666-9
  * Wyithe & Loeb (2008) Wyithe, S., & Loeb, A. 2008, Mon. Not. Roy. Astron. Soc., 383, 606, doi: 10.1111/j.1365-2966.2007.12568.x
  * Wyithe et al. (2008) Wyithe, S., Loeb, A., & Geil, P. 2008, Mon. Not. Roy. Astron. Soc., 383, 1195, doi: 10.1111/j.1365-2966.2007.12631.x
  * Xia (2009) Xia, J.-Q. 2009, Phys. Rev. D, 80, 103514, doi: 10.1103/PhysRevD.80.103514
  * Xiao et al. (2019) Xiao, L., An, R., Zhang, L., et al. 2019, Phys. Rev. D, 99, 023528, doi: 10.1103/PhysRevD.99.023528
  * Xu et al. (2018) Xu, X., Ma, Y.-Z., & Weltman, A. 2018, Phys. Rev. D, 97, 083504, doi: 10.1103/PhysRevD.97.083504
  * Xu et al. (2015) Xu, Y., Wang, X., & Chen, X. 2015, Astrophys. J., 798, 40, doi: 10.1088/0004-637X/798/1/40
  * Xu & Zhang (2020) Xu, Y., & Zhang, X. 2020, Sci. China Phys. Mech. Astron., 63, 270431, doi: 10.1007/s11433-020-1544-3
  * Yang et al. (2018) Yang, W., Pan, S., Di Valentino, E., et al. 2018, JCAP, 09, 019, doi: 10.1088/1475-7516/2018/09/019
  * Yohana et al. (2019) Yohana, E., Li, Y.-C., & Ma, Y.-Z. 2019, doi: 10.1088/1674-4527/19/12/186
  * Yu et al. (2017) Yu, H.-R., Pen, U.-L., Zhang, T.-J., Li, D., & Chen, X. 2017, Res. Astron. Astrophys., 17, 049, doi: 10.1088/1674-4527/17/6/49
  * Zhang et al. (2008) Zhang, J., Liu, H., & Zhang, X. 2008, Phys. Lett. B, 659, 26, doi: 10.1016/j.physletb.2007.10.086
  * Zhang et al. (2019) Zhang, J.-F., Gao, L.-Y., He, D.-Z., & Zhang, X. 2019, Phys. Lett. B, 799, 135064, doi: 10.1016/j.physletb.2019.135064
  * Zhang et al. (2020) Zhang, J.-F., Wang, B., & Zhang, X. 2020, Sci. China Phys. Mech. Astron., 63, 280411, doi: 10.1007/s11433-019-1516-y
  * Zhang (2005a) Zhang, X. 2005a, Mod. Phys. Lett. A, 20, 2575, doi: 10.1142/S0217732305017597
  * Zhang (2005b) —. 2005b, Phys. Lett. B, 611, 1, doi: 10.1016/j.physletb.2005.02.022
  * Zhang (2017) —. 2017, Sci. China Phys. Mech. Astron., 60, 050431, doi: 10.1007/s11433-017-9013-7
  * Zhang et al. (2006) Zhang, X., Wu, F.-Q., & Zhang, J. 2006, JCAP, 01, 003, doi: 10.1088/1475-7516/2006/01/003
  * Zhao et al. (2020) Zhao, M., Guo, R., He, D., Zhang, J., & Zhang, X. 2020, Sci. China Phys. Mech. Astron., 63, 230412, doi: 10.1007/s11433-019-1474-8