# A forecast of using fast radio burst observations to constrain holographic
dark energy

Xing-Wei Qiu  Ze-Wei Zhao  Ling-Feng Wang  Jing-Fei Zhang  Xin
Zhang11footnotetext: Corresponding author.

###### Abstract

Recently, about five hundred fast radio bursts (FRBs) detected by CHIME/FRB
Project have been reported. The vast amounts of data would make FRBs a
promising low-redshift cosmological probe in the forthcoming years, and thus
the issue of how many FRBs are needed for precise cosmological parameter
estimation in different dark energy models should be detailedly investigated.
Different from the usually considered $w(z)$-parameterized models in the
literature, in this work we investigate the holographic dark energy (HDE)
model and the Ricci dark energy (RDE) model, which originate from the
holographic principle of quantum gravity, using the simulated localized FRB
data as a cosmological probe for the first time. We show that the Hubble
constant $H_{0}$ can be constrained to about 2% precision in the HDE model
with the Macquart relation of FRB by using 10000 accurately-localized FRBs
combined with the current CMB data, which is similar to the precision of the
SH0ES value. Using 10000 localized FRBs combined with the CMB data can achieve
about 6% constraint on the dark-energy parameter $c$ in the HDE model, which
is tighter than the current BAO data combined with CMB. We also study the
combination of the FRB data and another low-redshift cosmological probe, i.e.
gravitational wave (GW) standard siren data, with the purpose of measuring
cosmological parameters independent of CMB. Although the parameter
degeneracies inherent in FRB and in GW are rather different, we find that more
than 10000 FRBs are demanded to effectively improve the constraints in the
holographic dark energy models.

##  1 Introduction

In 1998, two independent studies of distant supernovae discovered the
accelerated expansion of the universe [1, 2], which is one of the most
surprising astronomical discoveries in history. One can explain the
accelerated expansion by introducing a component with negative pressure,
usually called dark energy, and study its properties by cosmological
observations. The cosmic microwave background (CMB) data measured by the
_Planck_ satellite [3] provided precise constraints on the cosmological
parameters in the $\Lambda$ cold dark matter ($\Lambda$CDM) model, which is
usually regarded as the standard model of cosmology [4] with the origin of
dark energy being explained as the cosmological constant $\Lambda$.

The $\Lambda$CDM model fits the current cosmological observations quite well,
but it suffers from two theoretical puzzles: the fine-tuning problem and the
coincidence problem [5]. Some dynamical dark energy models may relieve these
problems [6, 7, 8, 9]. However, the extra dark-energy parameters are hard to
be precisely constrained by the CMB data alone due to the strong parameter
degeneracies [10]. The baryon acoustic oscillation (BAO) measurements from
galaxy redshift surveys [11, 12, 13] as a representative low-redshift
cosmological probe are usually combined with the CMB data to break the
cosmological parameter degeneracies. In the future, some other low-redshift
cosmological probes are expected to yield large amounts of data, such as fast
radio bursts (FRBs) and gravitational waves (GWs). Therefore, what precision
to cosmological parameters these two non-optical probes could measure and how
many events are needed for precise cosmological parameter estimation in
different dark energy models are important questions at present.

FRBs are extremely bright, short-duration radio signals. One of the important
characteristics of FRBs is the high dispersion measure (DM), which contains
valuable information on the cosmological distance they have traveled. In 2007,
the first FRB event, FRB010724, was observed by the 64-m Parkes Radio
Telescope in Australia [14]. Although FRB010724 has a high Galactic latitude,
it was previously thought to be caused by artificial interference due to its
low signal-to-noise ratio (SNR). The second FRB event, FRB010621, was reported
in ref. [15] in 2012. Later in 2013, Thornton et al. reported four new FRB
samples [16], which makes the study of FRBs become an important new direction
in radio astronomy. Recently, the Canadian Hydrogen Intensity Mapping
Experiment (CHIME)/FRB Project has released its first catalog of 535 FRBs,
including 61 bursts from 18 previously reported repeating sources [17]. Till
now, 14 FRBs’ host galaxies and cosmological redshifts have been identified
[18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].
These abundant data tremendously exceed the amount of current data of GWs with
electromagnetic (EM) counterparts and further prove that FRBs may become a
promising cosmological probe in the future.

The data of localized FRBs could be used to constrain cosmological parameters
with the Macquart relation, which provide a relationship between $\rm DM_{\rm
IGM}$ and $z$ [31]. There have been a series of works using FRBs as a
cosmological probe to study the expansion history of the universe, such as
estimating the cosmological parameters [37] and conducting cosmography [38] by
using FRBs and Gamma-Ray Bursts association, using FRBs to measure Hubble
parameter $H(z)$ [39] and the Hubble constant [40, 41], using the FRB data
combined with the BAO data or the type Ia supernovae (SN) data to break the
cosmological parameter degeneracies [42, 43], using FRB dispersion measures as
distance measures [44], probing compact dark matter [45, 46], testing
Einstein’s weak equivalence principle [47, 48, 49], and many other works [50,
51, 52, 53, 54]. For a recent review, see ref. [55].

In addition, there is also an idea of combining the FRB data with the GW data
[56]. Wei et al. noticed the fact that DM is proportional to the Hubble
constant $H_{0}$ while luminosity distance $d_{\rm L}$ is inversely
proportional to $H_{0}$, and they found that the product of DM from FRB and
$d_{\rm L}$ from GW can yield a quantity free of $H_{0}$, which may be useful
in cosmological parameter estimation. Inspired by this work, ref. [57]
introduced a cosmology-independent estimate of the fraction of baryon mass in
the intergalactic medium (IGM). In ref. [58], the authors (including three of
the authors in the present paper) also showed that combining the FRB data with
the GW data could be helpful in cosmological parameter estimation. GWs provide
a new method to measure the cosmic distance [59], dubbed as “standard sirens”
[60], which successfully avoids the possible systematics in the cosmic
distance ladder method. With the third-generation ground-based GW detectors,
such as the Einstein Telescope (ET), large amounts of GW-EM events from the
mergers of binary neutron stars (BNSs) are expected to be detected [61]. Thus,
GWs may become a new precise cosmological probe to determine various
cosmological parameters [62, 63, 64, 65, 66, 67, 68, 61]. Especially, the GW
data could be very helpful in breaking the cosmological parameter degeneracies
when combined with other cosmological probes [69, 70, 71, 72, 73, 74].

The capability of future FRB data in improving the cosmological parameter
estimation in the $w\rm{CDM}$ and Chevallier-Polarski-Linder (CPL) models was
studied in ref. [58]. It has been shown that the FRB data could break the
parameter degeneracies in the CMB and GW data. But these two models and the
most considered models in the literature [37, 38, 39, 40, 41, 42, 43] are
parametred and phenomenological models, and thus it is important to consider
some dark energy models with deep and solid theoretical foundations and see
whether the FRB data are still useful in cosmological parameter estimation in
these dark-energy theoretical models. In this work, we investigate two dark
energy models, i.e., the holographic dark energy (HDE) model and the Ricci
dark energy (RDE) model. The HDE model is viewed to have a quantum gravity
origin, which is constructed by combining the holographic principle of quantum
gravity with the effective quantum field theory [75, 76]. The HDE model not
only can naturally explain the fine-tuning and coincidence problems [75], but
also can fit the observational data well [77, 78, 79, 80, 81]. Its theoretical
variant, the RDE model, uses the average radius of the Ricci scalar curvature
rather than the future event horizon of the universe as the infrared (IR)
cutoff within the theoretical framework of holographic dark energy [82, 83].
Although the RDE model is not favored by the current observations [81], we
still study it as a demonstration in the issue of forecasting the capability
of FRB data in cosmological parameter estimation.

In this paper, we shall first combine the future FRB data with the current CMB
data from _Planck_ , and then we will study the combination of two future big-
data low-redshift measurements, the FRB data and the GW standard siren data,
in cosmological parameter estimation in the HDE and RDE models.

This paper is organized as follows. A brief description of the HDE and RDE
models, the methods for simulating the FRB data and the GW data, and the
current cosmological data are introduced in Section 2. The constraint results
and relevant discussions are given in Section 3. We present our conclusions in
Section 4. Throughout this paper, we adopt the units in which the speed of
light equals 1.

##  2 Methods and data

###  2.1 Brief description of the HDE and RDE models

According to the Bekenstein entropy bound, an effective field theory
considered in a box of size $L$ with ultraviolet (UV) cutoff
${\varLambda}_{\rm uv}$ gives the total entropy $S=L^{3}\varLambda_{\rm
uv}^{3}\leq S_{\rm{BH}}\equiv\pi M^{2}_{\rm{Pl}}L^{2}$, where $S_{\rm{BH}}$ is
the entropy of a black hole with the same size $L$ and
$M_{\rm{Pl}}=1/\sqrt{8\pi G}$ is the reduced Planck mass. However, Cohen et
al. pointed out that in quantum field theory a short distance (i.e., UV)
cutoff is related to a long distance (i.e., IR) cutoff due to the limit set by
forming a black hole, and proposed a more restrictive bound, i.e., the energy
bound [84]. If the quantum zero-point energy density $\rho_{\rm{vac}}$ is
relevant to a UV cutoff, the total energy of a system with size $L$ would not
exceed the mass of a black hole of the same size, namely,
$L^{3}\rho_{\rm{vac}}\leq LM^{2}_{\rm{Pl}}$. Obviously, the IR cutoff size of
this effective quantum field theory is taken to be the largest length size
compatible with this bound.

If we take the whole universe into account, the vacuum energy related to this
holographic principle is viewed as dark energy, called “holographic dark
energy”. The dark energy density can be expressed as [75]

$\displaystyle\rho_{\rm{de}}=3c^{2}M^{2}_{\rm{Pl}}L^{-2},$ (2.1)

where $c$ is a dimensionless model parameter, which is used to characterize
all the theoretical uncertainties in the effective quantum field theory, and
this parameter is extremely important in phenomenologically determining the
evolution of holographic dark energy.

If $L$ is chosen to be the Hubble scale of the universe, then the dark energy
density will be close to the observational result. However, Hsu pointed out
that the equation of state (EoS) of dark energy in this case is not correct
[85]. Li subsequently suggested that $L$ should be chosen to be the size of
the future event horizon [75],

$\displaystyle
R_{\mathrm{eh}}(a)=a\int_{t}^{\infty}\frac{dt^{\prime}}{a}=a\int_{a}^{\infty}\frac{da^{\prime}}{H(a^{\prime})a^{\prime
2}},$ (2.2)

where $a$ is the scale factor of the universe and $H(a)$ is the Hubble
parameter as a function of $a$. In this case, the EoS of dark energy can
realize the cosmic acceleration, and the model with such a setting is usually
called the HDE model.

In the HDE model, the dynamical evolution of dark energy is governed by the
following differential equations,

$\displaystyle\frac{1}{E(z)}\frac{dE(z)}{dz}$
$\displaystyle=-\frac{\Omega_{\mathrm{de}}(z)}{1+z}\left(\frac{1}{2}+\frac{\sqrt{\Omega_{\mathrm{de}}(z)}}{c}-\frac{3}{2\Omega_{\mathrm{de}}(z)}\right),$
(2.3) $\displaystyle\frac{d\Omega_{\mathrm{de}}(z)}{dz}$
$\displaystyle=-\frac{2\Omega_{\mathrm{de}}(z)\left(1-\Omega_{\mathrm{de}}(z)\right)}{1+z}\left(\frac{1}{2}+\frac{\sqrt{\Omega_{\mathrm{de}}(z)}}{c}\right),$
(2.4)

where $E(z)\equiv H(z)/H_{0}$ is the dimensionless Hubble parameter and
$\Omega_{\rm{de}}(z)$ is the fractional density of dark energy. Solving these
two differential equations with the initial conditions $\Omega_{\rm
de}=1-\Omega_{\rm m0}$ and $E(0)=1$ will obtain the evolutions of
$\Omega_{\rm{de}}(z)$ and $E(z)$. Then from the energy conservation equations,

$\displaystyle\dot{\rho}_{\rm{m}}+3H\rho_{\rm{m}}$ $\displaystyle=0,$
$\displaystyle\dot{\rho}_{\rm{de}}+3H(1+w)\rho_{\rm{de}}$ $\displaystyle=0,$
(2.5)

where a dot represents the derivative with respect to the cosmic time $t$ and
$\rho_{\rm{m}}$ is the matter density, one can get the EoS of dark energy in
the HDE model,

$\displaystyle w=-\frac{1}{3}-\frac{2\sqrt{\Omega_{\rm{de}}}}{3c}.$ (2.6)

The RDE model is defined by choosing the average radius of the Ricci scalar
curvature as the IR cutoff length scale in the theory. In FRW cosmology, the
Ricci scalar takes

$\displaystyle R=-6\left(\dot{H}+2H^{2}+\frac{k}{a^{2}}\right),$ (2.7)

where $k=1$, $0$, and $-1$ stands for closed, flat, and open geometry,
respectively, and we take $k=0$ in the rest of this work. The dark energy
density in the RDE model can be expressed as [82, 83]

$\displaystyle\rho_{\mathrm{de}}=3\gamma
M_{\mathrm{pl}}^{2}\left(\dot{H}+2H^{2}\right),$ (2.8)

where $\gamma$ is a positive constant redefined in terms of $c$. The evolution
of the Hubble parameter is determined by the following differential equation,

$\displaystyle
E^{2}=\Omega_{\mathrm{m}}e^{-3x}+\gamma\left(\frac{1}{2}\frac{dE^{2}}{dx}+2E^{2}\right),$
(2.9)

with $x\equiv\ln a$. The solution to this differential equation is found to be

$\displaystyle
E(z)=\left(\frac{2\Omega_{\mathrm{m}}}{2-\gamma}(1+z)^{3}+\left(1-\frac{2\Omega_{\mathrm{m}}}{2-\gamma}\right)(1+z)^{\left(4-\frac{2}{\gamma}\right)}\right)^{1/2}.$
(2.10)

###  2.2 Simulation of FRBs

When electromagnetic waves propagate in plasma, they will interact with free
electrons and generate dispersion. The group velocities of EM waves vary with
respect to frequency, resulting in lower frequency signal being delayed. By
measuring the time delay $(\Delta t)$ of the pulse signal between the highest
frequency $(\nu_{\rm h})$ and the lowest frequency $(\nu_{\rm l})$, we could
obtain the dispersion measure of an FRB,

$\displaystyle\mathrm{DM}=\Delta t\frac{2\pi
m_{\mathrm{e}}}{e^{2}}\frac{\left(\nu_{\rm
l}\nu_{\mathrm{h}}\right)^{2}}{\nu_{\mathrm{h}}^{2}-\nu_{\rm l}^{2}},$ (2.11)

where $m_{\rm e}$ is the electron mass and $e$ is the unit charge. The
physical interpretation of DM is the integral of the electron number density
along the line-of-sight, which is expressed as

$\displaystyle\mathrm{DM}=\int_{0}^{D}n_{\mathrm{e}}(l)\mathrm{d}l,$ (2.12)

where $n_{\mathrm{e}}$ is the electron number density, $l$ is the path length,
and $D$ is the distance to FRB. The DMs of current FRB observations are mainly
in the range of $100\sim 3000$ $\mathrm{pc}~{}\mathrm{cm}^{-3}$ [17],
exceeding the amount of the dispersion contributed by the Milky Way by 10 to
20 times.

Until now, the progenitors of FRBs have not been generally figured out
(excluding a Galactic magnetar), so the real redshift distribution of FRBs is
still an open issue. We assume that the comoving number density distribution
of FRBs is proportional to the cosmic star formation history (SFH) [86, 87],
and we thus obtain the SFH-based redshift distribution of FRBs [45],

$\displaystyle
N_{\rm{SFH}}(z)=\mathcal{N}_{\rm{SFH}}\frac{\dot{\rho}_{*}{d^{2}_{\rm
C}}(z)}{H(z)(1+z)}e^{-{d^{2}_{\rm{L}}}(z)/[2{d^{2}_{\rm{L}}}(z_{\rm cut})]},$
(2.13)

where $\mathcal{N}_{\rm{SFH}}$ is a normalization factor, $d_{\rm C}$ is the
comoving distance at redshift $z$, and $z_{\rm cut}=1$ is a Gaussian cutoff.
The number of detected FRBs decreases at $z>z_{\rm cut}$ due to the
instrumental signal-to-noise ratio threshold effect. The parameterized density
evolution $\dot{\rho}_{*}(z)$ reads

$\displaystyle\dot{\rho}_{*}(z)=l\frac{a+bz}{1+(z/n)^{d}},$ (2.14)

with $l=0.7$, $a=0.0170$, $b=0.13$, $n=3.3$, and $d=5.3$ [86, 88].

The total observed DM of an FRB consists of the contributions from the FRB’s
host galaxy, IGM, and the Milky Way [16, 37],

$\displaystyle\rm{DM}_{\rm{obs}}=\rm{DM}_{\rm{host}}+\rm{DM}_{\rm{IGM}}+\rm{DM}_{\rm{MW}}.$
(2.15)

The second term on the right hand side, $\rm{DM}_{\rm{IGM}}$, relates to
cosmology, and its average value is expressed as

$\displaystyle\langle\mathrm{DM}_{\mathrm{IGM}}\rangle=\frac{3H_{0}\Omega_{\rm
b}f_{\mathrm{IGM}}}{8\pi
Gm_{\mathrm{p}}}\int_{0}^{z}\frac{\chi(z^{\prime})(1+z^{\prime})dz^{\prime}}{E(z^{\prime})},$
(2.16)

with

$\displaystyle\chi(z)=Y_{\rm H}\chi_{\mathrm{{e,H}}}(z)+\frac{1}{2}Y_{\rm
He}\chi_{\mathrm{{e,He}}}(z),$ (2.17)

where the parameter $\Omega_{\mathrm{b}}$ is the present-day baryon fractional
density, $f_{\mathrm{IGM}}\simeq 0.83$ is the fraction of baryon mass in IGM
[89], $G$ is Newton’s constant, $m_{\mathrm{p}}$ is the proton mass, $Y_{\rm
H}=3/4$ is the hydrogen mass fraction, $Y_{\rm He}=1/4$ is the helium mass
fraction, and the terms $\chi_{\mathrm{{e,H}}}$ and $\chi_{\mathrm{{e,He}}}$
are the ionization fractions for H and He, respectively. We take
$\chi_{\mathrm{{e,H}}}=\chi_{\mathrm{{e,He}}}=1$ [90], since both H and He are
fully ionized when $z<3$.

From eq. (2.15), if we could determine $\rm{DM}_{\rm{obs}}$,
$\rm{DM}_{\rm{host}}$, and $\rm{DM}_{\rm{MW}}$, the last remaining term
$\rm{DM}_{\rm{IGM}}$ could be measured. The total uncertainty of
$\rm{DM}_{\rm{IGM}}$ is expressed as

$\displaystyle\sigma_{\rm{DM}_{\rm{IGM}}}=\left[\sigma_{\rm
obs}^{2}+\sigma_{\rm MW}^{2}+\sigma_{\rm IGM}^{2}+\left(\frac{\sigma_{\rm
host}}{1+z}\right)^{2}\right]^{1/2}.$ (2.18)

We take the observational uncertainty
$\sigma_{\rm{obs}}=1.5~{}{\rm{pc~{}cm^{-3}}}$ [91]. From the Australia
Telescope National Facility pulsar catalog [92], the average $\sigma_{\rm MW}$
is about $10~{}{\rm{pc~{}cm^{-3}}}$ for the sources at high Galactic
latitudes. Due to the inhomogeneity of the baryon matter in IGM, the deviation
of an individual event from the mean $\rm{DM}_{\rm{IGM}}$ is described by
$\sigma_{\rm{IGM}}$. Here we use the following formula [57],

$\displaystyle\sigma_{\rm
IGM}=\begin{cases}\frac{52-45z-263z^{2}+21z^{3}+582z^{4}}{1-4z+7z^{2}-7z^{3}+5z^{4}},&z\leq
1.03,\\\ -416+270z+480z^{2}+23z^{3}-162z^{4},&1.03<z\leq 1.13,\\\
38\arctan[0.6z+1]+17,&z>1.13,\end{cases}$ (2.19)

which is fitted from the simulations in refs. [93, 94]. $\sigma_{\rm{host}}$
is hard to estimate because it depends on the individual properties of an FRB,
for example, the type of the host galaxy, the location of FRB in the host
galaxy, and the near-source plasma. We take
$\sigma_{\rm{host}}=30~{}{\rm{pc~{}cm^{-3}}}$ as the uncertainty of ${\rm
DM_{host}}$.

The first CHIME/FRB catalog suggests a sky rate of $818\pm 64~{}({\rm
stat.})^{+220}_{-200}~{}({\rm sys.})$ sky-1 day-1 above a fluence of 5 Jy ms
at 600 MHz. Currently, among hundreds of FRB sources, there are 14 FRBs
localized to host galaxies. Therefore, we expect that $\sim 10$ FRBs with
redshifts can be detected per day [54]. Thus, we take $N_{\rm FRB}=1000$ as a
normal expectation and $N_{\rm FRB}=10000$ as an optimistic expectation for a
few years observation, with $N_{\rm FRB}$ being the event number of FRBs.

###  2.3 Simulation of standard sirens

To simulate the GW standard siren data generated by BNS mergers from ET, we
first need to assume the redshift distribution of GWs [61, 67],

$\displaystyle P(z)\propto\frac{4\pi d_{\mathrm{C}}^{2}(z)R(z)}{H(z)(1+z)},$
(2.20)

where $R(z)$ is the time evolution of the burst rate with the form [95, 96,
67]

$\displaystyle R(z)=\left\\{\begin{array}[]{rcl}1+2z,&z\leq 1,\\\
\frac{3}{4}(5-z),&1<z<5,\\\ 0,&z\geq 5.\end{array}\right.$ (2.24)

By observing the GW waveform of a compact binary merger, one could
independently determine the luminosity distance $d_{\rm L}$ to the GW source.
If the redshift of the GW source is obtained through the observation of the EM
counterpart, then the $d_{\rm L}$–$z$ relation can be established to study the
expansion history of the universe [60].

An incoming GW signal $h(t)$ could be written as a linear combination of two
wave polarizations in the transverse traceless gauge,

$\displaystyle
h(t)=F_{+}(\theta,\phi,\psi)h_{+}(t)+F_{\times}(\theta,\phi,\psi)h_{\times}(t),$
(2.25)

where $\psi$ is the polarization angle, ($\theta$, $\phi$) are the location
angles of the GW source, and $+$ and $\times$ denote the plus and cross
polarizations, respectively. The antenna pattern functions of one Michelson-
type interferometer of ET are [61]

$\displaystyle F_{+}^{(1)}(\theta,\phi,\psi)=$
$\displaystyle~{}~{}\frac{{\sqrt{3}}}{2}\Big{[}\frac{1}{2}(1+{\cos^{2}}\theta)\cos(2\phi)\cos(2\psi)-\cos\theta\sin(2\phi)\sin(2\psi)\Big{]},$
(2.26) $\displaystyle F_{\times}^{(1)}(\theta,\phi,\psi)=$
$\displaystyle~{}~{}\frac{{\sqrt{3}}}{2}\Big{[}\frac{1}{2}(1+{\cos^{2}}\theta)\cos(2\phi)\sin(2\psi)+\cos\theta\sin(2\phi)\cos(2\psi)\Big{]}.$
(2.27)

Since the three interferometers of ET have inclined angles of $60^{\circ}$
with each other, the other two pattern functions are
$F_{+,\times}^{(2)}(\theta,\phi,\psi)=F_{+,\times}^{(1)}(\theta,\phi+2\pi/3,\psi)$
and
$F_{+,\times}^{(3)}(\theta,\phi,\psi)=F_{+,\times}^{(1)}(\theta,\phi+4\pi/3,\psi)$.

Applying the stationary phase approximation to the time domain signal $h(t)$,
we can get its Fourier transform $\tilde{h}(f)$ [61],

$\displaystyle\tilde{h}(f)=\mathcal{A}f^{-7/6}\exp\left\\{i\left(2\pi
ft_{c}-\pi/4+2\psi(f/2)-\varphi_{I,(2,0)}\right)\right\\},$ (2.28)

where “$\sim$” above a function denotes the Fourier transform and
$\mathcal{A}$ is the amplitude in the Fourier space,

$\displaystyle\mathcal{A}=$ $\displaystyle~{}~{}\frac{1}{d_{\rm
L}}\sqrt{F_{+}^{2}(1+\cos^{2}\iota)^{2}+4F_{\times}^{2}\cos^{2}\iota}\sqrt{5\pi/96}\pi^{-7/6}\mathcal{M}_{\rm
c}^{5/6},$ (2.29)

where $\iota$ is the inclination angle between the direction of binary’s
orbital angular momentum and the line of sight. The observed chirp mass is
defined as $\mathcal{M}_{\rm c}=(1+z)M\eta^{3/5}$, with $M=m_{1}+m_{2}$ being
the total mass of coalescing binary system with component masses $m_{1}$ and
$m_{2}$, and $\eta=m_{1}m_{2}/M^{2}$ being the symmetric mass ratio. $\psi(f)$
and $\varphi_{I,(2,0)}$ are given by [97, 98]

$\displaystyle\psi(f)$
$\displaystyle=-\psi_{c}+\frac{3}{256\eta}\sum_{i=0}^{7}\psi_{i}(2\pi
Mf)^{(i-5)/3},$ $\displaystyle\varphi_{I,(2,0)}$
$\displaystyle=\tan^{-1}\left(-\frac{2\cos(\iota)F_{\times}}{\left(1+\cos^{2}(\iota)\right)F_{+}}\right),$
(2.30)

where $\psi_{c}$ is the coalescence phase, and the detailed expressions of the
Post-Newtonian (PN) coefficients $\psi_{i}$ are given in ref. [97]. Notice
that here $\psi$’s are the PN coefficients, different from the one in Eqs.
(2.25)–(2.27), which is the polarization angle.

The total SNR of ET is

$\rho=\sqrt{\sum\limits_{i=1}^{3}(\rho^{\left(i\right)})^{2}},$ (2.31)

where
$\rho_{i}=\sqrt{\left\langle\tilde{h}^{(i)},\tilde{h}^{(i)}\right\rangle}$,
with the inner product being defined as

$\left\langle{a,b}\right\rangle=4\int_{f_{\rm lower}}^{f_{\rm
upper}}\frac{\tilde{a}(f)\tilde{b}^{\ast}(f)+\tilde{a}^{\ast}(f)\tilde{b}(f)}{2}\frac{df}{S_{\rm
n}(f)},$ (2.32)

where $S_{\rm n}(f)$ is the one-sided noise power spectral density. We use the
interpolation method to fit the sensitivity data of ET to get the fitting
function of $S_{\rm n}(f)$ [99]. We choose $f_{\rm lower}=1$ Hz as the lower
cutoff frequency and $f_{\rm upper}=2/(6^{3/2}2\pi M_{\rm obs})$ as the upper
cutoff frequency with $M_{\rm obs}=(m_{1}+m_{2})(1+z)$ [61].

The luminosity distance $d_{\rm L}$ from GW observation is directly used to
constrain cosmological parameters. The total error of $d_{\rm L}$ is expressed
as

$\displaystyle\sigma_{d_{\mathrm{L}}}=\sqrt{\left(\sigma_{d_{\mathrm{L}}}^{\text{inst
}}\right)^{2}+\left(\sigma_{d_{\mathrm{L}}}^{\text{lens
}}\right)^{2}+\left(\sigma_{d_{\mathrm{L}}}^{\mathrm{pv}}\right)^{2}},$ (2.33)

where $\sigma^{\rm inst}_{d_{\rm L}}$ , $\sigma^{\rm lens}_{d_{\rm L}}$, and
$\sigma^{\rm pv}_{d_{\rm L}}$ are the instrumental error, the weak-lensing
error, and the peculiar velocity error of luminosity distance, respectively.
In our previous work [58], we adopted an approximation to the instrumental
error, $\sigma_{d_{\rm L}}^{\rm inst}\simeq 2d_{\rm L}/\rho$, in which the
factor of 2 accounts for the maximal effect of the correlation between the
luminosity distance and the inclination angle [100]. Indeed, this
approximation does not exactly show the covariance between the luminosity
distance and other parameters and may lead to biased results. Hence, in this
work we employ a full Fisher matrix analysis to correctly account for the
parameter correlations. The Fisher information matrix of the GW signal can be
expressed as

$\displaystyle\boldsymbol{F}_{ij}=\left\langle\frac{\partial\boldsymbol{h}(f)}{\partial\theta_{i}},\frac{\partial\boldsymbol{h}(f)}{\partial\theta_{j}}\right\rangle,$
(2.34)

where $\boldsymbol{h}$ is given by three interferometers of ET,

$\displaystyle\boldsymbol{h}(f)=\left[\frac{\tilde{h}_{1}(f)}{\sqrt{S_{\mathrm{n}}(f)}},\frac{\tilde{h}_{2}(f)}{\sqrt{S_{\mathrm{n}}(f)}},\frac{\tilde{h}_{3}(f)}{\sqrt{S_{\mathrm{n}}(f)}}\right]^{\mathrm{T}},$
(2.35)

and $\theta_{i}$ denotes the set of nine parameters $(\mathcal{M}_{\rm
c},\eta,d_{\rm L},\theta,\phi,\psi,\iota,t_{c},\psi_{c})$ for a GW event. We
then could estimate $\sigma^{\rm inst}_{d_{\rm L}}$ as

$\displaystyle\sigma^{\rm inst}_{d_{\rm L}}=\Delta\theta_{d_{\rm
L}}=\sqrt{\left(F^{-1}\right)_{d_{\rm L}d_{\rm L}}},$ (2.36)

where $F_{ij}$ is the total Fisher information matrix for the ET with three
interferometers.

The GW data used in this work are “bright sirens” and thus need the redshift
information. The redshifts of GW sources are measured by their EM
counterparts. In this work, we consider the short gamma ray bursts (SGRBs) as
EM counterparts. Since SGRBs are supposed to be strongly beamed, we limit the
observations of inclination angle from SGRBs within $20^{\circ}$. Therefore,
in the Fisher matrix for each GW event, the sky location $(\theta,\phi)$, the
inclination angle $\iota$, the mass of each NS ($m_{1}$, $m_{2}$), the
coalescence phase $\psi_{\rm c}$, and the polarization angle $\psi$ are evenly
sampled in the ranges of [0, $\pi$], [0, $2\pi$], [0, $\pi/9$], $[1,2]\
M_{\odot}$, $[1,2]\ M_{\odot}$, [0, $2\pi$], and [0, $2\pi$], respectively,
where $M_{\odot}$ is the solar mass. To eliminate the scatter of mock data,
these parameters are randomly selected for 100 times to perform the Fisher
matrix analysis, and we adopt the average results.

We then turn to the other two errors. The weak-lensing error is given by [101,
102]

$\displaystyle\sigma_{d_{\mathrm{L}}}^{\text{lens
}}(z)=d_{\mathrm{L}}(z)\times
0.066\left[\frac{1-(1+z)^{-0.25}}{0.25}\right]^{1.8}.$ (2.37)

Here we introduce a delensing factor, i.e. the use of dedicated matter surveys
along the line of sight of the GW event in order to estimate the lensing
magnification distribution and thus remove part of the weak lensing
uncertainty. Following refs. [103, 104], the delensing could be achieved 30%
at $z=2$ and the delensing factor can be expressed as

$\displaystyle F_{\text{delens
}}(z)=1-\frac{0.3}{\pi/2}\arctan\left(z/z_{*}\right),$ (2.38)

with $z_{*}=0.073$. The final lensing uncertainty on $d_{\rm L}$ is

$\displaystyle\sigma_{d_{\mathrm{L}}}^{\text{wl}}(z)=F_{\text{delens
}}(z)\sigma_{d_{\mathrm{L}}}^{\text{lens }}(z)$ (2.39)

Here we use $\sigma_{d_{\mathrm{L}}}^{\text{wl}}$ to replace
$\sigma_{d_{\mathrm{L}}}^{\text{lens }}$ in eq. (2.33).

The peculiar velocity of the source relative to the Hubble flow introduces an
additional source of error. For the peculiar velocity error, we take the form
in ref. [105]

$\displaystyle\sigma_{d_{\mathrm{L}}}^{\mathrm{pv}}(z)=d_{\mathrm{L}}(z)\times\left[1+\frac{(1+z)^{2}}{H(z)d_{\mathrm{L}}(z)}\right]\sqrt{\left\langle
v^{2}\right\rangle},$ (2.40)

where $H(z)$ is the Hubble parameter and $\sqrt{\left\langle
v^{2}\right\rangle}$ is the peculiar velocity of the GW source, set to be
$500\,{\rm km\ s^{-1}}$.

ET is expected to detect about $10^{5}$ BNS mergers per year but in which only
$\sim 10^{-3}$ of the events with $\gamma$-ray bursts toward us [106]. So a
few $\times 10^{2}$ GW events could be treated as “bright sirens” per year.
Recently, Chen et al. forecasted that 910 GW standard siren events could be
detected by CE and Swift++ with a 10-year observation [107]. In this work, we
simulate 1000 standard siren events generated by BNS mergers within a 10-year
observation of ET. In figure 1, we make a comparison between the FRB mock data
in the normal expectation scenario and the GW mock data. We can see that in
our simulations, the FRB data are mainly distributed below $z\approx 1.5$, but
the GW data could extend to $z\approx 3$.

Figure 1: The mock FRB data (panel a) and the mock GW data from ET (panel b).
The FRB data are simulated from the normal expectation scenario with 1000 FRB
events, and the GW data are simulated from the 10-year observation of ET with
1000 BNS merger events.

###  2.4 Cosmological data

For the current mainstream cosmological data, we use the _Planck_ CMB
“distance priors” derived from the _Planck_ 2018 data release [108], and the
BAO measurements from 6dFGS at $z_{\rm eff}=0.106$ [11], SDSS-MGS at $z_{\rm
eff}=0.15$ [12], and BOSS-DR12 at $z_{\rm eff}=0.38$, 0.51, and 0.61 [13]. In
the generation of the FRB and GW mock data, the fiducial values of
cosmological parameters are taken to be the best-fit values of CMB+BAO+SN from
ref. [62]. We use the Markov-chain Monte Carlo analysis [109] to obtain the
posterior probability distribution of the cosmological parameters.

##  3 Results and discussion

###  3.1 CMB+FRB

Model | Error | CMB | CMB+BAO | FRB1 | CMB+FRB1 | FRB2 | CMB+FRB2
---|---|---|---|---|---|---|---
HDE | $\sigma(\Omega_{\rm m})$ | $0.032$ | $0.012$ | $0.062$ | $0.022$ | $0.026$ | $0.012$
$\sigma(h)$ | $0.035$ | $0.013$ | $-$ | $0.022$ | $-$ | $0.012$
$\sigma(c)$ | $0.18$ | $0.087$ | $0.55$ | $0.11$ | $0.44$ | $0.057$
$10^{2}\sigma(\Omega_{\rm b}h^{2})$ | $0.015$ | $0.015$ | $0.56$ | $0.014$ | $0.53$ | $0.0097$
RDE | $\sigma(\Omega_{\rm m})$ | $0.060$ | $-$ | $0.087$ | $0.043$ | $0.032$ | $0.020$
$\sigma(h)$ | $0.057$ | $-$ | $-$ | $0.045$ | $-$ | $0.021$
$\sigma(\gamma)$ | $0.019$ | $-$ | $0.24$ | $0.014$ | $0.058$ | $0.0081$
$10^{2}\sigma(\Omega_{\rm b}h^{2})$ | $0.015$ | $-$ | $0.67$ | $0.013$ | $0.51$ | $0.0076$
Table 1: Absolute errors ($1\sigma$) of the cosmological parameters in the HDE
and RDE models by using the CMB, CMB+BAO, FRB1, CMB+FRB1, FRB2, and CMB+FRB2
data. FRB1 and FRB2 denote the FRB data in the normal expectation scenario
(i.e., $N_{\rm FRB}=1000$) and optimistic expectation scenario (i.e., $N_{\rm
FRB}=10000$), respectively.

Figure 2: Two-dimensional marginalized contours (68.3% and 95.4% confidence
levels) in the $\Omega_{\rm m}$–$c$ plane (panel a) and the
$H_{0}$–$\Omega_{\rm b}h^{2}$ plane (panel b) for the HDE model, by using the
FRB, CMB, and CMB+FRB data. Here, the FRB data are simulated based on the
optimistic expectation scenario.

Figure 3:  Two-dimensional marginalized contours (68.3% and 95.4% confidence
levels) in the $\Omega_{\rm m}$–$\gamma$ plane (panel a) and the
$H_{0}$–$\Omega_{\rm b}h^{2}$ plane (panel b) for the RDE model, by using the
FRB, CMB, and CMB+FRB data. Here, the FRB data are simulated based on the
optimistic expectation scenario.

In this subsection, we investigate the capability of the FRB data of breaking
the parameter degeneracies inherent in the CMB data. We shall first discuss
the improvement of the precision of cosmological parameters with the addition
of the FRB data. Then we compare the capabilities of FRB and of BAO data in
breaking the parameter degeneracies. The absolute errors (1$\sigma$) of the
cosmological parameters in the HDE and RDE models are listed in table 1. We
use FRB1 and FRB2 to denote the FRB data in the normal expectation (i.e.,
$N_{\rm FRB}=1000$) and the optimistic expectation (i.e., $N_{\rm
FRB}=10000$), respectively. Here, for a cosmological parameter $\xi$, we use
$\sigma(\xi)$ to denotes its absolute error, and we also use the relative
error $\varepsilon(\xi)=\sigma(\xi)/\xi$ in the following discussions.

From table 1 we see that the cosmological constraints from the FRB1 and FRB2
data are looser than those from the CMB data (except for $\Omega_{\rm m}$ from
FRB2). However, with combining the CMB and FRB data, the constraints from the
CMB+FRB1 and CMB+FRB2 data are both evidently improved compared with those
from the CMB data alone. As shown in figure 2, in the HDE model, $\Omega_{\rm
m}$ and $c$ are positively correlated in the CMB data, but they are anti-
correlated in the FRB data. Combining the CMB and FRB data breaks the
parameter degeneracies, and thus improves the constraints on the EoS
parameters of dark energy and the matter density parameter. For the HDE model,
the CMB data combined with the FRB1 and FRB2 data give the results
$\varepsilon(c)=11.7\%$ and $\varepsilon(c)=6.3\%$, respectively. The absolute
errors of $c$ are reduced by about 40.0% and 67.7% by combining the FRB1 and
FRB2 data with the CMB data, respectively. It is worth emphasizing that the
constraint $\varepsilon(c)=6.3\%$ is close to the constraint precision given
by the CMB+BAO+SN data [110]. That is to say, using only the future FRB data
combined with the CMB data could provide precise cosmological constraints
comparable with the current mainstream data. For the RDE model, it is shown in
figure 3 that the FRB data also break the parameter degeneracies inherent in
the CMB data. The CMB data combined with the FRB1 and FRB2 data give
$\varepsilon(\gamma)=5.9\%$ and $\varepsilon(\gamma)=3.5\%$, respectively. The
absolute errors of $\gamma$ are reduced by about 24.3% and 56.2%, by adding
the FRB1 and FRB2 data to the CMB data, respectively.

Then we turn our attention to the constraints on the baryon density
$\Omega_{\rm b}$ and the Hubble constant $H_{0}$. Using big bang
nucleosynthesis (BBN) and CMB can precisely determine the value of
$\Omega_{\rm b}h^{2}$, however, in the nearby universe, the observed baryons
in stars, the cold interstellar medium, residual Ly$\alpha$ forest gas, O vi,
broad H i Ly$\alpha$ absorbers, and hot gas in clusters of galaxies account
for only $\sim 50\%$ of the baryons [55]. This is called the missing baryon
problem. On the other hand, for the measurements of the Hubble constant, in
recent years there appeared the puzzle of “Hubble tension”. The values of the
Hubble constant derived from different observations show strong tension, which
actually reflects the inconsistency of measurements between the early universe
and the late universe [111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121].
One may expect to precisely measure $\Omega_{\rm b}h^{2}$ and $H_{0}$ with
powerful low-redshift probes.

In figure 2(b), we show the marginalized posterior probability distribution
contours in the $H_{0}$–$\Omega_{\rm b}h^{2}$ plane for the HDE model. We see
that $H_{0}$ and $\Omega_{\rm b}h^{2}$ are strong positively correlated when
using the FRB data alone, because $\mathrm{DM}_{\mathrm{IGM}}$ is proportional
to $H_{0}\Omega_{\rm b}$ [see eq. (2.16)]. It is found that solely using the
FRB2 data cannot constrain $H_{0}$, but can effectively constrain $\Omega_{\rm
b}h^{2}$, giving the result of $\sigma(\Omega_{\rm b}h^{2})\approx 0.005$,
which is an about 22.7% constraint. Although the precision is still low, the
method of using the FRB observation has been proven to provide a potential way
to solve the missing baryon problem. The CMB data can place tight constraint
on $\Omega_{\rm b}h^{2}$, and thus the data combination FRB+CMB can give a
precise measurement on $H_{0}$. For the HDE model, the simulated FRB1 data
combined with the current CMB data could give the result
$\varepsilon(h)=3.3\%$, which is reduced by 37.1% compared to using the CMB
data alone. The data combination CMB+FRB2 can give the result
$\varepsilon(h)=1.8\%$, which roughly equals the measurement precision of the
SH0ES value [122]. For CMB+FRB2 in the RDE model, the FRB2 data combined with
the CMB data could give $\varepsilon(h)=2.9\%$, and the error of $h$ are
reduced by about 64% by adding the FRB2 data to the CMB data.

Figure 4:  Two-dimensional marginalized contours (68.3% and 95.4% confidence
levels) in the $\Omega_{\rm m}$–$c$ plane for the HDE model, by using the
CMB+FRB1, CMB+BAO, and CMB+FRB2 data.

Moreover, we also compare the capabilities of the CMB+FRB and CMB+BAO data of
constraining cosmological parameters. From figure 4 we can see that the
constraints from the CMB+FRB2 data are comparable to those from the CMB+BAO
data, even slightly better. In the HDE model, both the CMB+FRB2 and CMB+BAO
data give a constraint of $\sigma(\Omega_{\rm m})=0.012$. For the parameter
$c$, the CMB+FRB2 and CMB+BAO data give the results $\sigma(c)=0.056$ and
$\sigma(c)=0.087$, respectively. It indicates that the FRB2 data have similar
capability of breaking parameter degeneracies as the BAO data.

###  3.2 GW+FRB

| Error | GW | GW+FRB | CMB+GW | CMB+GW+FRB
---|---|---|---|---|---
HDE | $\sigma(\Omega_{\rm m})$ | 0.024 | 0.016 | 0.0059 | 0.0052
$\sigma(h)$ | 0.010 | 0.0095 | 0.0064 | 0.0054
$\sigma(c)$ | 0.21 | 0.16 | 0.045 | 0.035
$10^{2}\sigma(\Omega_{\rm b}h^{2})$ | $-$ | 0.011 | 0.015 | 0.0074
RDE | $\sigma(\Omega_{\rm m})$ | 0.012 | 0.011 | 0.0090 | 0.0082
$\sigma(h)$ | 0.015 | 0.014 | 0.0093 | 0.0086
$\sigma(\gamma)$ | 0.028 | 0.020 | 0.0067 | 0.0059
$10^{2}\sigma(\Omega_{\rm b}h^{2})$ | $-$ | 0.011 | 0.014 | 0.0069
Table 2: Absolute errors ($1\sigma$) of the cosmological parameters in the HDE
and RDE models by using the GW, GW+FRB, CMB+GW, and CMB+GW+FRB data. Here, the
FRB data are simulated based on the optimistic expectation scenario.

Figure 5:  Two-dimensional marginalized contours (68.3% and 95.4% confidence
levels) in the $\Omega_{\rm m}$–$H_{0}$ plane (panel a) and the $c$–$H_{0}$
plane (panel b) for the HDE model, by using the FRB, GW, and GW+FRB data.
Here, the FRB data are simulated based on the optimistic expectation scenario.

In addition to the FRB observation, the GW observation is another powerful
cosmological probe. As demonstrated in ref. [58], the FRB and GW observation
could help each other to break the parameter degeneracies in the $w$CDM and
CPL models. In this subsection, we further study these two cosmological probes
in the HDE and RDE models, and the FRB data are simulated based on the
optimistic expectation scenario.

From table 2, we see that the constraints from the data combination GW+FRB are
slightly improved compared with the GW data alone. But from figure 5 we find
that the parameter degeneracy orientations formed by GW and by FRB are nearly
orthogonal. The inclusion of the FRB data reduces the relative error on
$\Omega_{\rm m}$ from $8.4\%$ to $5.2\%$ in the HDE model. The constraints on
$\Omega_{\rm m}$ and $\Omega_{\rm b}h^{2}$ from the GW+FRB data are comparable
to those from the CMB+BAO data. However, neither the GW data nor the FRB data
can give tight constraints on the parameter $c$. The data combination GW+FRB
gives $\varepsilon(c)=18.0\%$, which is only slightly better than the
constraint with CMB alone. For the Hubble constant $H_{0}$, the FRB data
actually contribute a little to the joint constraint of GW+FRB, since the GW
data alone are able to precisely constrain $H_{0}$ but FRB alone cannot
constrain it. Compared with the results in the $w\rm{CDM}$ model [58], the
improvement of parameter constraint by adding 10000 FRB data into the GW data
is weaker in the HDE model. For example, the improvements of the constraints
on $\Omega_{\rm m}$ are $53\%$ and $33\%$ in the $w\rm{CDM}$ and HDE models,
respectively. Therefore, we find that, for constraining the HDE model, much
more FRB data are needed to be combined with the GW data.

The results from the data combination CMB+GW+FRB further confirm this
statement. The inclusion of the FRB data in the data combination CMB+GW+FRB
tinily improves the constraints on $\Omega_{\rm m}$ and $H_{0}$, and the
contributions to the constraints mainly come from CMB+GW. Nevertheless, the
CMB+GW data can only constrain $c$ to the precision of around 5%, which is
still far away from the standard of precision cosmology. Compared to the GW
data, the FRB data are more likely to be enlarged to a larger sample.
Recently, Hashimoto et al. pointed out that large amounts of FRBs could be
detected by the Square Kilometre Array (SKA), with the number at least 2–3
orders of magnitude larger than the sample size we have considered [123]. With
the accumulation of more abundant and precise data, FRBs would provide a tight
constraint on the EoS of dark energy, so we expect that in the future the
large amounts of FRBs observed by SKA may have the potential to help the joint
constraint on $c$ achieve the precision around 1%.

###  3.3 Effect of $\rm DM_{host}$ uncertainty

In the above analyses, we have assumed that the $\rm DM_{host}$ uncertainty is
$30\,{\rm pc\,cm^{-3}}$. The progenitors of FRBs are actually not clear and
some factors are still open issues. The treatment of $\sigma_{\rm
host}=30\,{\rm pc\,cm^{-3}}$ thus could be regarded as an optimistic scenario.
In this subsection, we perform an analysis based on a more conservative
scenario with $\sigma_{\rm host}=150\,{\rm pc\,cm^{-3}}$. The FRB data of this
conservative scenario is represented by FRB3. The constraint results of
cosmological parameters are shown in table 3.

Model | Error | FRB3 | CMB+FRB3
---|---|---|---
HDE | $\sigma(\Omega_{\rm m})$ | $0.025$ | $0.013$
$\sigma(h)$ | $-$ | $0.013$
$\sigma(c)$ | $0.44$ | $0.058$
$10^{2}\sigma(\Omega_{\rm b}h^{2})$ | $0.51$ | $0.010$
RDE | $\sigma(\Omega_{\rm m})$ | $0.040$ | $0.024$
$\sigma(h)$ | $-$ | $0.023$
$\sigma(\gamma)$ | $0.076$ | $0.0089$
$10^{2}\sigma(\Omega_{\rm b}h^{2})$ | $0.54$ | $0.0080$
Table 3: Absolute errors ($1\sigma$) of the cosmological parameters in the HDE
and RDE models by using the FRB3 and CMB+FRB3 data. Here, FRB3 denotes the FRB
data with the ${\rm DM_{host}}$ uncertainty $\sigma_{\rm host}=150\,{\rm
pc\,cm^{-3}}$.

From table 3, we see that the constraints from FRB3 are slightly looser than
those from FRB2. Taking the HDE model as an example, we can see that FRB3 and
CMB+FRB3 give the results $\varepsilon(c)=37.3\%$ and $\varepsilon(c)=6.7\%$,
respectively, which are 2.8% and 6.0% larger than those from FRB2 and
CMB+FRB2, respectively. The influence on the constraints in the RDE model is
slightly larger. The relative errors in the cases of FRB3 and CMB+FRB3 are
$\varepsilon(\gamma)=27.9\%$ and $\varepsilon(\gamma)=3.8\%$, which are 20.4%
and 8.7% larger than those from FRB2 and CMB+FRB2, respectively. From eq.
(2.18), we see that $\sigma_{\rm host}$ contributes to the total uncertainty
of $\rm{DM}_{\rm{IGM}}$ with a factor $1/(1+z)$, thus even though $\sigma_{\rm
host}$ varies a lot, its impact on FRBs’ capability of constraining
cosmological parameters and breaking parameter degeneracies is slight. So,
there may be some other factors affecting the constraints, but the main
conclusions will still hold. This confirms the discussion in ref. [58].

##  4 Conclusion

In this paper, we investigate the capability of future FRB data in improving
the cosmological parameter estimation and how many FRB data are required to
effectively constrain the cosmological parameters in the HDE and RDE models.
We consider two FRB scenarios: the normal expectation scenario with 1000
localized FRB data and the optimistic expectation scenario with 10000
localized FRB data.

We find that, in the HDE model, combining the FRB data and the CMB data could
break the parameter degeneracies, and the joint constraints on $H_{0}$ and
dark-energy parameters are quite tight. To achieve the constraint precision of
$H_{0}$ comparable with the SH0ES result [122], around 10000 FRB data are need
to be combined with the CMB data. For the EoS of dark energy, 10000 FRB data
combined with the CMB data would give a $\sim 6\%$ constraint, which is close
to the precision given by the CMB+BAO+SN data. The data combination FRB+CMB
also give a tight constraint on $\Omega_{\rm b}h^{2}$. The results in the RDE
model also support the conclusion above. These results confirm the conclusion
in ref. [58] that the future $10^{4}$ FRB data can become useful in
cosmological parameter estimation when combined with CMB.

We also consider another powerful low-redshift cosmological probe, the GW
standard sirens observation, and investigate the capability of constraining
cosmological parameters when combining the FRB data with the GW data, which is
independent of CMB. We use a full Fisher matrix analysis to account for the
parameter correlations and avoid biased results. The inclusion of the FRB data
in the data combination GW+FRB only improves the constraints slightly, but we
show that the orientations of the parameter degeneracies formed by FRB and by
GW are nearly orthogonal. We show that in some dark energy models such as the
HDE model, 10000 FRB events are possibly not enough to be used to constrain
cosmological parameters when combined with the GW data. However, we can expect
that, in the near future, large amounts of localized FRBs will be observed by
SKA. If we could observe much more FRBs with higher precision in the future,
the combination with the GW data could significantly improve the constraints.
It can be expected that FRBs observed by SKA may have potential to help the
jiont constraint on $c$ achieve the precision around 1%, reaching the standard
of precision cosmology.

Finally, we investigate the effect of the ${\rm DM_{host}}$ uncertainty on the
constraint results. A larger ${\rm DM_{host}}$ uncertainty surely looses the
constraints, but the main conclusions still hold.

## Acknowledgments

We thank Zheng-Xiang Li, He Gao, Jing-Zhao Qi, Shang-Jie Jin, Li-Yang Gao,
Peng-Ju Wu, Hai-Li Li, Yi-Chao Li, and Di Li for helpful discussions. This
work was supported by the National Natural Science Foundation of China (Grants
Nos. 11975072, 11835009, 11875102, and 11690021), the Liaoning Revitalization
Talents Program (Grant No. XLYC1905011), the Fundamental Research Funds for
the Central Universities (Grant No. N2005030), and the National Program for
Support of Top-Notch Young Professionals (Grant No. W02070050).

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