# Finding the missing baryons in the intergalactic medium with localized fast
radio bursts

K. B. Yang School of Astronomy and Space Science, Nanjing University, Nanjing
210093, China  Q. Wu School of Astronomy and Space Science, Nanjing
University, Nanjing 210093, China  F. Y. Wang School of Astronomy and Space
Science, Nanjing University, Nanjing 210093, China  Key Laboratory of Modern
Astronomy and Astrophysics (Nanjing University), Ministry of Education,
Nanjing 210093, China F. Y. Wang<EMAIL_ADDRESS>
###### Abstract

The missing baryon problem is one of the major unsolved problems in astronomy.
Fast radio bursts (FRBs) are bright millisecond pulses with unknown origins.
The dispersion measure of FRBs is defined as the electron column density along
the line of sight, and accounts for every ionized baryon. Here we measure the
baryon content of the Universe using 22 localized FRBs. Unlike previous works
that fixed the value of dispersion measure of FRB host galaxies and ignored
the inhomogeneities of the intergalactic medium (IGM), we use the probability
distributions of dispersion measures contributed by host galaxies and IGM from
the state-of-the-art IllustrisTNG simulations. We derive the cosmic baryon
density of $\Omega_{b}=0.0490^{+0.0036}_{-0.0033}$ (1$\sigma$), with a
precision of 7.0%. This value is dramatically consistent with other
measurements, such as the cosmic microwave background and Big Bang
nucleosynthesis. Our work supports that the baryons are not missing, but
residing in the IGM.

Fast Radio Bursts, intergalactic medium

##  1 Introduction

The observation of cosmic microwave background (CMB) by Planck Collaboration
shows that more than 95% of our universe is composed of dark energy and dark
matter, while the rest part of less than 5% is baryonic matter (Planck
Collaboration et al., 2020). In the late-time Universe, approximately $17\%$
of the expected baryons are observed to reside in the collapsed phase
(Fukugita et al., 1998), including galaxies, groups, and clusters. Numerical
simulations (Cen & Ostriker, 1999) and observations (Fukugita & Peebles, 2004)
are performed to study the baryon distribution. However, the fraction of
baryons observed in the photoionized Ly$\alpha$ forest and warm-hot
intergalactic medium (WHIM) traced by highly ionized states oxygen is only
about 53%, which left a vacancy of about 30% (Shull et al., 2012). In recent
years, some baryons are found in various ways. de Graaff et al. (2019) found
additional $11\pm 7\%$ baryons with the Sunyaev-Zel’dovich effect from
filaments, and the still missing fraction became $18\pm 16\%$. Some missing
baryons had been found in the warm–hot intergalactic medium (Nicastro et al.,
2018). Most missing baryons may be residing in diffuse IGM, which is too faint
to detect (McQuinn, 2016). Finding the missing baryon is important for us to
understand many crucial problems, such as galaxy evolution.

Fast radio bursts (FRBs) are bright millisecond pulses (Petroff et al., 2019;
Cordes & Chatterjee, 2019; Xiao et al., 2021). Since the first burst, FRB
010724 was discovered in 2007 (Lorimer et al., 2007), over 600 FRBs have been
detected (CHIME/FRB Collaboration et al., 2021). When FRBs travel through cold
plasma, there will be a time difference between their components of different
frequencies, which can be used to define the dispersion measure (DM). DM is
defined as the integral of free electron number density along the line of
sight. Owing to their large DM in excess of the Galactic value, they are
suggested to be extragalactic or cosmological. It is estimated that the
majority of DM contribution comes from IGM. The IGM is comprised of highly
ionized plasma, which means that the baryon density can be derived from the
electron density. Based on the average DM-$z$ relation, short timescales and
cosmological origin make them a powerful probe to study the universe and
fundamental physics, including measuring Hubble constant (Li et al., 2018; Wu
et al., 2022; Hagstotz et al., 2022; James et al., 2022) and Hubble parameter
(Wu et al., 2020), measuring cosmic proper distance (Yu & Wang, 2017),
measuring dark energy (Zhou et al., 2014; Walters et al., 2018; Qiu et al.,
2022), bounding the photon rest mass (Wang et al., 2021), measuring
reionization history (Zheng et al., 2014; Zhang et al., 2021), testing the
weak equivalence principle (Wei et al., 2015), probing compact dark matter
(Muñoz et al., 2016; Wang & Wang, 2018), and finding “missing” baryons
(McQuinn, 2014; Walters et al., 2018, 2019; Macquart et al., 2020; Li et al.,
2020; Dai & Xia, 2021; Wang & Wei, 2022). Macquart et al. (2020) performed the
first observational constraint on the cosmic baryon density $\Omega_{b}$ using
five FRBs, and found $\Omega_{b}=0.051^{+0.021}_{-0.025}$ at 95% confidence
level, which is consistent with the value derived from the CMB and from Big
Bang nucleosynthesis (BBN).

However, there are some obstacles in practice with cosmological applications
of FRBs. First, the number of FRBs with measured redshifts is too small, which
can be improved with increasing localized FRBs. Thanks to the high event rate,
i.e., $10^{3}-10^{4}$ per day all sky (Thornton et al., 2013; Caleb et al.,
2016), the increasing advanced telescopes will localize more FRBs. The second
one is that we know little about $\rm DM_{host}$, the DM contribution from the
host galaxy. It is also degenerated with $\rm DM_{IGM}$, the DM contribution
from IGM. How to model $\rm DM_{host}$ is a crucial problem. Third, the IGM
inhomogeneity causes the fluctuations of DMIGM along different lines of sight.
From cosmological simulations, the variance around the mean DMIGM can be up to
70% (McQuinn, 2014; Pol et al., 2019; Zhang et al., 2021). In most previous
works, the value of $\rm DM_{host}$ for all host galaxies is treated as a
constant, and the fluctuations of $\rm DM_{IGM}$ are ignored. A more
appropriate method to describe $\rm DM_{IGM}$ and $\rm DM_{host}$ is needed in
the cosmological application of FRBs.

In this letter, we measure the cosmic baryon density with 22 localized FRBs
through the DM-$z$ relation. Unlike previous works which assume the $\rm
DM_{host}$ as a constant, we use the probability distributions of it from the
state-of-the-art IllustrisTNG simulations (Zhang et al., 2020). The effect of
different properties of host galaxies on $\rm DM_{host}$ is also taken into
consideration. Meanwhile, the fluctuations of DMIGM along different lines of
sight are also taken into account (Zhang et al., 2021).

The letter is organized as follows. In section 2, we introduce the probability
distributions of $\rm DM_{host}$ and $\rm DM_{IGM}$, and the method to derive
$\Omega_{b}$. In section 3, we constrain the value of $\Omega_{b}$ through the
Monte Carlo Markov Chain (MCMC) analysis. Discussion and conclusions are given
in section 4.

##  2 Method

The observed DM of FRBs can be divided into four parts

${\rm DM}_{\rm obs}={\rm DM}_{\rm MW}+{\rm DM}_{\rm halo}+{\rm DM}_{\rm
IGM}+\frac{{\rm DM}_{\rm host}}{1+z},$ (1)

where $\rm DM_{MW}$ is contributed by the interstellar medium of the Milky
Way, $\rm DM_{halo}$ is contributed by the Milky Way halo, and $\rm DM_{IGM}$
refers to the contribution of IGM, and $\rm DM_{host}$ is contributed by the
host galaxy. For the first part $\rm DM_{MW}$, some models have been built,
including NE2001 and YMW16 models (Cordes & Lazio, 2002; Yao et al., 2017). In
this work, we use the NE2001 model to estimate the value of $\rm DM_{MW}$.
After deducing the $\rm DM_{MW}$ from $\rm DM_{obs}$, we can get $\rm
DM^{\prime}_{FRB}$ = $\rm DM_{IGM}$ \+ $\rm DM_{host}$ \+ $\rm DM_{halo}$.
Only the part $\rm DM_{IGM}$ contains the cosmological information, including
the cosmic baryon density $\Omega_{b}$. However, it’s difficult to separate it
from $\rm DM_{host}$ and $\rm DM_{halo}$, which is the key point for the
cosmological applications of FRBs. Besides, due to the matter inhomogeneity in
the IGM and the diverse properties of host galaxies, both of them can not be
accurately determined.

In a flat $\Lambda$CDM universe, the average $\rm DM_{IGM}$ is (Deng & Zhang,
2014; Macquart et al., 2020)

$\langle{\rm DM_{\rm IGM}}\rangle=\frac{3c\Omega_{b}H_{0}}{8\pi
Gm_{p}}\int_{0}^{z_{FRB}}\frac{f_{\rm
IGM}(z)f_{e}(z)(1+z)}{\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}}dz,$ (2)

where $m_{p}$ is the rest proton mass, $\Omega_{m}$ is the cosmic matter
density, and the electron fraction is
$f_{e}(z)=Y_{H}X_{e,H}(z)+\frac{1}{2}Y_{He}X_{e,He}(z)=7/8$. For a flat
universe, $\Omega_{m}=1-\Omega_{\Lambda}$. $f_{\rm IGM}(z)$ is the fraction of
baryons in the IGM. At present, we know little about the evolution model of
$f_{\rm IGM}(z)$ with redshift $z$. $f_{\rm IGM}(z)$ could increase with
redshifts as massive halos grow abundant with the universe evolving. It’s
estimated that the value of $f_{\rm IGM}$ is 0.82 at low redshifts and 0.9 at
$z>1.5$ (Meiksin, 2009; Shull et al., 2012). For the Hubble constant $H_{0}$,
a fiducial value of 70 km/s/Mpc is used.

As mentioned above, it is difficult to determine the exact value of $\rm
DM_{halo},DM_{host}$ and DMIGM. So we focus on the probability distributions
of them. The $\rm DM_{halo}$ contributed by the halo is estimated to be
$50\sim 80\rm\ pc\ cm^{-3}$ (Prochaska & Zheng, 2019). According to this
estimation, we assume that the distribution of $\rm DM_{halo}$ is a Gaussian
distribution with the standard deviation $\sigma_{\rm halo}=15~{}\rm pc\
cm^{-3}$ and the mean value $\mu_{\rm halo}=65~{}\rm pc\ cm^{-3}$ (Wu et al.,
2022), i.e.,

$P_{\rm halo}(\rm DM_{\rm halo}|\mu_{halo},\sigma_{halo})=\frac{1}{\sigma_{\rm
halo}\sqrt{2\pi}}\exp[-\frac{(DM_{\rm halo}-\mu_{\rm
halo})^{2}}{2\sigma^{2}_{\rm halo}}].$ (3)

It is worth noting that $\rm DM_{halo}$ is larger than $30\rm\ pc\ cm^{-3}$,
and should be smaller than $100\rm\ pc\ cm^{-3}$ for FRBs at high galactic
latitudes and longitudes (Yamasaki & Totani, 2020). Positions of FRBs with
known coordinates are shown in figure 1. All of them satisfy this condition.
For conservative estimations, the range of $\rm DM_{halo}$ is adopted as
$30\sim 100\rm\ pc\ cm^{-3}$ in our calculation (Yamasaki & Totani, 2020).

For localized FRBs, we can calculate the average $\langle{\rm DM_{\rm
IGM}}\rangle$ with equation (2). The probability distribution of DMIGM can be
described as a long-tailed quasi-Gaussian function of the average value
(McQuinn, 2014; Macquart et al., 2020; Zhang et al., 2021),

$P_{\rm
IGM}(\Delta)=A\Delta^{-\beta}\exp[-\frac{(\Delta^{-\alpha}-C_{0})^{2}}{2\alpha^{2}\sigma^{2}_{\rm
DM}}],\Delta>0,$ (4)

where $\Delta=\rm DM_{IGM}/\langle DM_{IGM}\rangle$, $\alpha$ and $\beta$ are
related to profile scales and their best fits are $\alpha=3$ and $\beta=3$
(Macquart et al., 2020). $\sigma_{DM}$ is the effective deviation caused by
the inhomogeneities of IGM. Zhang et al. (2021) have given the best fits of
$C_{0}$, $A$ and $\sigma_{DM}$ at several different redshifts derived from the
IllustrisTNG simulation. These redshifts do not include the redshifts of the
FRBs used in this work. Using the cubic spline interpolation method, we get
$C_{0}$, $A$ and $\sigma_{DM}$ as a function of redshift. Then, the values of
$C_{0}$, $A$ and $\sigma_{DM}$ at given redshifts are derived.

Figure 1:  The coordinates of 18 localized FRBs in the galactic coordinate
system. There is no FRB with both low galactic latitudes and low galactic
longitudes.

A log-normal function can well describe the distribution of DMhost (Macquart
et al., 2020; Zhang et al., 2020),

$P_{\rm host}({\rm DM_{host}}|\mu,\sigma)=\frac{1}{{\rm
DM_{host}}\sigma\sqrt{2\pi}}{{\exp}}[-\frac{({\rm ln}{\rm
DM_{host}}-\mu)^{2}}{2\sigma^{2}}],$ (5)

where $\mu$ and $\sigma$ are free parameters. For this distribution, the
expected value and standard deviation are $e^{\mu}$ and
$e^{2\mu+{\sigma}^{2}}[e^{\sigma^{2}}-1]$, respectively. Zhang et al. (2020)
have given the best fitting values of $\mu$ and $\sigma$ through the state-of-
the-art IllustrisTNG simulations. In their work, FRBs are divided into three
types according to the observational properties of host galaxies: repeating
FRBs like FRB 121102 (Type I), repeating FRBs like FRB 180916 (Type II) and
non-repeating FRBs (Type III). Repeating bursts may originate from newborn
neutron stars (Wang et al., 2020). FRB 121102 is localized to a dwarf galaxy
with a low star formation rate, hence repeating FRBs like FRB121102 may be
from dwarf galaxies with low metallicity (Chatterjee et al., 2017). The host
galaxy of FRB 180916 is a spiral galaxy, which is different from that of FRB
121102. It was born in a star-forming region (Marcote et al., 2020). Therefore
repeating FRBs like FRB 180916 are likely to reside in the star-forming
region. Non-repeating FRBs may originate from the merger of compact binaries
(Totani, 2013; Wang et al., 2016), and their locations are far from the galaxy
centers (Bannister et al., 2019; Prochaska et al., 2019). In our calculation,
as before, we use their results to get $\mu$ and $\sigma$ at given redshifts
by cubic spline interpolation.

For an FRB with a given redshift $z$ and DM${}^{\prime}_{\rm FRB,i}$, we can
calculate the total probability density function (PDF) through

$\begin{split}P_{i}\left({\rm DM}_{{\rm FRB}^{\prime},i}\mid
z_{i}\right)=&\int_{30}^{{100}}\int_{0}^{{\mathrm{DM}}^{\prime}_{{\rm
FRB},i}-\rm DM_{halo}}P_{\text{IGM}}(\rm
DM_{{IGM}})P_{\text{halo}}({DM}_{halo}|\mu_{halo},\sigma_{halo})\\\
&P_{\text{host }}({\text{DM}^{\prime}_{{\rm FRB},i}-{\rm DM}_{{\rm IGM}}-{\rm
DM}_{\rm halo}}|\mu,\sigma_{host})d{\rm DM_{\rm IGM}}d{\rm DM_{\rm
halo}}.\end{split}$ (6)

At last, we obtain the joint likelihood function by multiplying the PDF of
each FRB

$\mathcal{L}=\prod\limits_{i=1}^{N_{\mathrm{FRB}}}P_{i}\left({\rm
DM}^{\prime}_{{\rm FRB},i}\mid z_{i}\right).$ (7)

##  3 Data and Results

Figure 2:  The ${\rm DM}_{\rm IGM}-z$ relation for FRBs. The scatter star
points are 23 localized FRBs. $\rm DM_{IGM}$ values are derived by deducing
the $\rm DM_{MW}$ ,$\rm DM_{halo}$ and $\rm DM_{host}$ from $\rm DM_{obs}$.
$\rm DM_{MW}$ values are deduced from NE2001 model, and $\rm DM_{halo}$ values
are estimated to be $65\ \rm pc\ cm^{-3}$. We use the median value of $\rm
DM_{host}$ at different redshifts according to equation (5). The blue line is
the result derived from IllustrisTNG 300 cosmological simulation and the green
shaded region is the 95% confidence level.The red line represents the
theoretical model DMIGM (equation (2)) assuming $f_{\rm IGM}=0.82,\
\Omega_{b}=0.0488$ and $H_{0}=70~{}\rm km~{}\ s^{-1}~{}Mpc^{-1}$. Due to FRB
210117 (red point) deviates from the 95% confidence region of DMIGM, we
exclude it in the following analysis.

Though over 600 FRBs have been observed, only 24 of them have been localized.
19 of them are collected at http://frbhosts.org/#explore, including the
nearest one FRB 20200120E. This FRB is localized in a globular cluster in M81,
whose distance is only 3.6 Mpc (Bhardwaj et al., 2021a; Kirsten et al., 2021).
It is obvious that it contains little cosmological information and we exclude
it. The other five FRBs localized by the Australian Square Kilometre Array
Pathfinder (ASKAP) are given in James et al. (2022). The parameters of all 23
localized FRBs are shown in Table 1. It lists the redshifts, DMobs, DMMW,ISM,
host type and references of them. In figure 2, we plot the $\rm DM_{IGM}-z$
relation of 23 localized FRBs. The derived $\rm DM_{IGM}$ of FRBs are shown as
scatters. The red line shows the averaged value of $\rm DM_{IGM}$ from the
equation (2). The blue line is the $\rm DM_{IGM}$ estimated from the
IllustrisTNG simulation with 95% confidence region (Zhang et al., 2021). FRB
210117 deviates from the 95% confidence region, which may be caused by the DM
contributed by the plasma near the FRB source, similar as that of FRB 190520B
(Zhao & Wang, 2021; Katz, 2022). Therefore, we exclude this FRB in the
following analysis. We use the NE2001 model to calculate the DM contribution
by the ISM of Milky Way (Cordes & Lazio, 2002).

Figure 3: The normalized probability density distribution of $\Omega_{b}$ from
22 localized FRBs. The best fit is $\Omega_{b}=0.0490^{+0.0036}_{-0.0033}$ in
1$\sigma$ confidence level. The red solid line shows the best-fit value, and
dotted lines present the 1$\sigma$ and 2$\sigma$ confidence levels. The blue
shade region shows the constraint on $\Omega_{b}$ from CMB with 1$\sigma$
confidence level, and the yellow shade region represents the constraint from
BBN with 1$\sigma$ confidence level. The green dashed lines correspond to the
1$\sigma$ uncertainty range of $\Omega_{b}$ derived by Macquart et al. (2020).
Figure 4: Current observational measurements of the low redshift baryon
census. From the DM$-z$ relation of FRBs, 83% $\pm$ 6% of the cosmic baryons
in the diffuse form are derived. The values in collapsed forms, including in
galaxies, the CGM, the ICM, and cold gas, are adjusted from Shull et al.
(2012). Collapsed phases (galaxies, CGM, ICM, and cold neutral gas) contribute
17% $\pm$ 4% of the total cosmic baryons. From our study, almost 100% cosmic
baryons are found in the late-time Universe.

The MCMC Python module emcee is used to estimate $\Omega_{b}$ (Foreman-Mackey
et al., 2013). $\Omega_{m}$ and $H_{0}$ are also free parameters to be fitted
in our analysis. Cosmological observations have tightly constrained them. As
shown in below, they have little effect on the final constraints on
$\Omega_{b}$. The steps of MCMC analysis are as follows.

(a) Firstly, we get the (${\rm DM}_{{\rm FRB}^{\prime}}$,$z$) dataset by
deducing the $\rm DM_{MW}$ obtained from the NE2001 model from $\rm DM_{obs}$.
Then we calculate the probability density of ${\rm DM}_{{\rm FRB}^{\prime}}$
for each FRB using equations (3), (4), (5) and (6). The joint likelihood
function is obtained by multiplying the PDF of each FRB with equation (7).

(b) Next, we need to set the priors for the parameters to be fit. For
$\Omega_{b}$, we assume a uniform prior of range [0.03, 0.065]. $H_{0}$ and
$\Omega_{b}$ are also set priors to estimate their impact. For the Hubble
constant $H_{0}$, the measurements of $H_{0}$, from SNe Ia and CMB, are mainly
distributed between 67 $\rm km\ s^{-1}Mpc^{-1}$ and 73 $\rm km\
s^{-1}Mpc^{-1}$ (Planck Collaboration et al., 2020; Riess et al., 2022).
Therefore, we assume a uniform prior of range [67, 73] $\rm km\
s^{-1}Mpc^{-1}$. A fiducial value of $H_{0}$=70 km/s/Mpc is used by Macquart
et al. (2020). Finally, we assume a uniform prior of [0.28, 0.35] for
$\Omega_{m}$ because most of the measurements of $\Omega_{m}$ are in this
range (Brout et al., 2022; Scolnic et al., 2018).

(c) At last, we run 10000 steps of MCMC using Python package emcee with the
likelihood function equation 7. The result is
$\Omega_{b}=0.0490^{+0.0036}_{-0.0033}$ (1$\sigma$) with a 7.0% precision.
This value is consistent with previous results in 1$\sigma$ confidence level,
such as CMB and BBN (Planck Collaboration et al., 2020; Cooke et al., 2018).
Macquart et al. (2020) used five FRBs to constrain $\Omega_{b}$, and found
$\Omega_{b}=0.051^{+0.021}_{-0.025}$ with 95 percent confidence by assuming
$H_{0}$=70 km/s/Mpc. This value is compatible with our result in 1$\sigma$
confidence level, but with large error.

In the calculation, the fraction of baryons in the IGM $f_{\rm IGM}=0.82$ is
used. It must be noted that $\Omega_{b}$ and $f_{\rm IGM}$ are degenerated. In
fact, only the product of $\Omega_{b}$, $H_{0}$ and $f_{IGM}$ can be derived.
Conversely, if the value of $\Omega_{b}$ from CMB is fixed, the constraint on
$f_{\rm IGM}$ should be close to 0.82. The same prior for $H_{0}$ and the best
constraint on $\Omega_{b}$ from CMB are used. Using the method as above, we
derive $f_{\rm IGM}=0.83\pm 0.06$.

Figure 4 shows the current census of baryons in the low-redshift Universe,
including diffuse form revealed by FRBs, and collapsed forms. Collapsed forms
contain galaxies, the circumgalactic medium (CGM), intercluster medium (ICM),
and cold neutral gas (H I and He I). These slices show the contributions,
$\Omega^{i}_{b}/\Omega_{\rm tot,b}$ to the total baryon content from
components $i$. Previous studies found that $5\%\pm 3\%$ may reside in
circumgalactic gas, 7%$\pm$2% in galaxies, 1.7%$\pm$0.4% in cold gas and
4%$\pm$1.4% in clusters (Shull et al., 2012). From our analysis, the
measurement of FRBs can account for 83% $\pm$ 6% of the baryons in the
Universe. Therefore, the missing baryons are found in the IGM through
localized FRBs.

##  4 Discussion and Conclusions

The statistical and systematic errors must be taken into consideration. The
statistical error refers to the small number of localized FRBs. As mentioned
above, this will be improved with the increase of localized FRBs. The
systematic error includes all the uncertainties except statistical errors,
such as cosmological parameters, observational errors of DM and redshift. It
cannot be eliminated by more data. More precise measurements of these
parameters are needed. Besides, YMW16 model gives a larger DMMW value than the
NE2001 model. This may lead to a similar but smaller result.

In equation (4), the best-fit parameters for $C_{0}$, $\sigma_{DM}$ and $A$
are taken from Zhang et al. (2021). Their uncertainties may affect the final
$\Omega_{b}$ constraint. We derive their uncertainties from the original data.
From equations (4) (6) and (7), it is obvious that the uncertainty of $A$ has
no effect on the result. To consider the effect of $C_{0}$ and $\sigma_{DM}$,
we derive $\Omega_{b}$ using the best-fit values of $C_{0}$ and $\sigma_{DM}$
plus or minus the uncertainties respectively, totaling four probability
distributions of DMIGM. The results of fits are
$\Omega_{b}=0.0492^{+0.0035}_{-0.0034}$, $0.0492^{+0.0035}_{-0.0034}$,
$0.0481^{+0.0036}_{-0.0034}$ and $0.0493^{+0.0036}_{-0.0033}$ in these four
case. These constraints are very close to the original result (approximately
0.25$\sigma$) and their errors are almost the same. Therefore we can conclude
that uncertainties of $C_{0}$, $\sigma_{DM}$ and $A$ have almost no effect on
final $\Omega_{b}$ constraint. Besides, our calculation is not entirely
independent of cosmological parameters, such as $\Omega_{m}$ and $H_{0}$. We
adopt their ranges limited by observations.

In summary, we estimate the cosmic baryon density, $\Omega_{b}$ using 22
localized FRBs by considering the probability distributions of DMs contributed
by host galaxies and IGM. We obtain the result
$\Omega_{b}=0.0492^{+0.0035}_{-0.0034}$ (1$\sigma$) with a 7.0% precision. The
result is consistent with those derived from CMB and BBN. So, FRBs can provide
an independent measurement of $\Omega_{b}$ with high precision. Our work shows
that the missing baryons are found in diffuse IGM by FRBs.

## acknowledgements

We thank the anonymous referee for helpful comments. This work was supported
by the National Natural Science Foundation of China (grant Nos. U1831207 and
12273009), the National Key Research and Development Program of China
(2022SKA0130100), the Fundamental Research Funds for the Central Universities
(No. 0201-14380045) and the China Manned Spaced Project (CMS-CSST-2021-A12).

## References

  * Bannister et al. (2019) Bannister, K. W., Deller, A. T., Phillips, C., et al. 2019, Science, 365, 565, doi: 10.1126/science.aaw5903
  * Bhandari et al. (2020) Bhandari, S., Sadler, E. M., Prochaska, J. X., et al. 2020, ApJ, 895, L37, doi: 10.3847/2041-8213/ab672e
  * Bhandari et al. (2021) Bhandari, S., Heintz, K. E., Aggarwal, K., et al. 2021, arXiv e-prints, arXiv:2108.01282.  https://arxiv.org/abs/2108.01282
  * Bhardwaj et al. (2021a) Bhardwaj, M., Gaensler, B. M., Kaspi, V. M., et al. 2021a, ApJ, 910, L18, doi: 10.3847/2041-8213/abeaa6
  * Bhardwaj et al. (2021b) Bhardwaj, M., Kirichenko, A. Y., Michilli, D., et al. 2021b, ApJ, 919, L24, doi: 10.3847/2041-8213/ac223b
  * Brout et al. (2022) Brout, D., Scolnic, D., Popovic, B., et al. 2022, ApJ, 938, 110, doi: 10.3847/1538-4357/ac8e04
  * Caleb et al. (2016) Caleb, M., Flynn, C., Bailes, M., et al. 2016, MNRAS, 458, 718, doi: 10.1093/mnras/stw109
  * Cen & Ostriker (1999) Cen, R., & Ostriker, J. P. 1999, ApJ, 514, 1, doi: 10.1086/306949
  * Chatterjee et al. (2017) Chatterjee, S., Law, C. J., Wharton, R. S., et al. 2017, Nature, 541, 58, doi: 10.1038/nature20797
  * CHIME/FRB Collaboration et al. (2021) CHIME/FRB Collaboration, Amiri, M., Andersen, B. C., et al. 2021, ApJS, 257, 59, doi: 10.3847/1538-4365/ac33ab
  * Chittidi et al. (2021) Chittidi, J. S., Simha, S., Mannings, A., et al. 2021, ApJ, 922, 173, doi: 10.3847/1538-4357/ac2818
  * Cooke et al. (2018) Cooke, R. J., Pettini, M., & Steidel, C. C. 2018, ApJ, 855, 102, doi: 10.3847/1538-4357/aaab53
  * Cordes & Chatterjee (2019) Cordes, J. M., & Chatterjee, S. 2019, ARA&A, 57, 417, doi: 10.1146/annurev-astro-091918-104501
  * Cordes & Lazio (2002) Cordes, J. M., & Lazio, T. J. W. 2002, arXiv e-prints, astro.  https://arxiv.org/abs/astro-ph/0207156
  * Dai & Xia (2021) Dai, J.-P., & Xia, J.-Q. 2021, MNRAS, 503, 4576, doi: 10.1093/mnras/stab785
  * Day et al. (2020) Day, C. K., Deller, A. T., Shannon, R. M., et al. 2020, MNRAS, 497, 3335, doi: 10.1093/mnras/staa2138
  * de Graaff et al. (2019) de Graaff, A., Cai, Y.-C., Heymans, C., & Peacock, J. A. 2019, A&A, 624, A48, doi: 10.1051/0004-6361/201935159
  * Deng & Zhang (2014) Deng, W., & Zhang, B. 2014, ApJ, 783, L35, doi: 10.1088/2041-8205/783/2/L35
  * Fong et al. (2021) Fong, W.-f., Dong, Y., Leja, J., et al. 2021, ApJ, 919, L23, doi: 10.3847/2041-8213/ac242b
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Fukugita et al. (1998) Fukugita, M., Hogan, C. J., & Peebles, P. J. E. 1998, ApJ, 503, 518, doi: 10.1086/306025
  * Fukugita & Peebles (2004) Fukugita, M., & Peebles, P. J. E. 2004, ApJ, 616, 643, doi: 10.1086/425155
  * Hagstotz et al. (2022) Hagstotz, S., Reischke, R., & Lilow, R. 2022, MNRAS, 511, 662, doi: 10.1093/mnras/stac077
  * Heintz et al. (2020) Heintz, K. E., Prochaska, J. X., Simha, S., et al. 2020, ApJ, 903, 152, doi: 10.3847/1538-4357/abb6fb
  * James et al. (2022) James, C. W., Ghosh, E. M., Prochaska, J. X., et al. 2022, arXiv e-prints, arXiv:2208.00819.  https://arxiv.org/abs/2208.00819
  * Katz (2022) Katz, J. I. 2022, MNRAS, 510, L42, doi: 10.1093/mnrasl/slab128
  * Kirsten et al. (2021) Kirsten, F., Marcote, B., Nimmo, K., et al. 2021, arXiv e-prints, arXiv:2105.11445.  https://arxiv.org/abs/2105.11445
  * Law et al. (2020) Law, C. J., Butler, B. J., Prochaska, J. X., et al. 2020, ApJ, 899, 161, doi: 10.3847/1538-4357/aba4ac
  * Li et al. (2020) Li, Z., Gao, H., Wei, J. J., et al. 2020, MNRAS, 496, L28, doi: 10.1093/mnrasl/slaa070
  * Li et al. (2018) Li, Z.-X., Gao, H., Ding, X.-H., Wang, G.-J., & Zhang, B. 2018, Nature Communications, 9, 3833, doi: 10.1038/s41467-018-06303-0
  * Lorimer et al. (2007) Lorimer, D. R., Bailes, M., McLaughlin, M. A., Narkevic, D. J., & Crawford, F. 2007, Science, 318, 777, doi: 10.1126/science.1147532
  * Macquart et al. (2020) Macquart, J. P., Prochaska, J. X., McQuinn, M., et al. 2020, Nature, 581, 391, doi: 10.1038/s41586-020-2300-2
  * Marcote et al. (2020) Marcote, B., Nimmo, K., Hessels, J. W. T., et al. 2020, Nature, 577, 190, doi: 10.1038/s41586-019-1866-z
  * McQuinn (2014) McQuinn, M. 2014, ApJ, 780, L33, doi: 10.1088/2041-8205/780/2/L33
  * McQuinn (2016) —. 2016, ARA&A, 54, 313, doi: 10.1146/annurev-astro-082214-122355
  * Meiksin (2009) Meiksin, A. A. 2009, Reviews of Modern Physics, 81, 1405, doi: 10.1103/RevModPhys.81.1405
  * Muñoz et al. (2016) Muñoz, J. B., Kovetz, E. D., Dai, L., & Kamionkowski, M. 2016, Phys. Rev. Lett., 117, 091301, doi: 10.1103/PhysRevLett.117.091301
  * Nicastro et al. (2018) Nicastro, F., Kaastra, J., Krongold, Y., et al. 2018, Nature, 558, 406, doi: 10.1038/s41586-018-0204-1
  * Petroff et al. (2019) Petroff, E., Hessels, J. W. T., & Lorimer, D. R. 2019, A&A Rev., 27, 4, doi: 10.1007/s00159-019-0116-6
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6, doi: 10.1051/0004-6361/201833910
  * Pol et al. (2019) Pol, N., Lam, M. T., McLaughlin, M. A., Lazio, T. J. W., & Cordes, J. M. 2019, ApJ, 886, 135, doi: 10.3847/1538-4357/ab4c2f
  * Prochaska & Zheng (2019) Prochaska, J. X., & Zheng, Y. 2019, MNRAS, 485, 648, doi: 10.1093/mnras/stz261
  * Prochaska et al. (2019) Prochaska, J. X., Macquart, J.-P., McQuinn, M., et al. 2019, Science, 366, 231, doi: 10.1126/science.aay0073
  * Qiu et al. (2022) Qiu, X.-W., Zhao, Z.-W., Wang, L.-F., Zhang, J.-F., & Zhang, X. 2022, J. Cosmology Astropart. Phys, 2022, 006, doi: 10.1088/1475-7516/2022/02/006
  * Ravi et al. (2019) Ravi, V., Catha, M., D’Addario, L., et al. 2019, Nature, 572, 352, doi: 10.1038/s41586-019-1389-7
  * Riess et al. (2022) Riess, A. G., Yuan, W., Macri, L. M., et al. 2022, ApJ, 934, L7, doi: 10.3847/2041-8213/ac5c5b
  * Scolnic et al. (2018) Scolnic, D. M., Jones, D. O., Rest, A., et al. 2018, ApJ, 859, 101, doi: 10.3847/1538-4357/aab9bb
  * Shull et al. (2012) Shull, J. M., Smith, B. D., & Danforth, C. W. 2012, ApJ, 759, 23, doi: 10.1088/0004-637X/759/1/23
  * Spitler et al. (2016) Spitler, L. G., Scholz, P., Hessels, J. W. T., et al. 2016, Nature, 531, 202, doi: 10.1038/nature17168
  * Thornton et al. (2013) Thornton, D., Stappers, B., Bailes, M., et al. 2013, Science, 341, 53, doi: 10.1126/science.1236789
  * Totani (2013) Totani, T. 2013, PASJ, 65, L12, doi: 10.1093/pasj/65.5.L12
  * Walters et al. (2019) Walters, A., Ma, Y.-Z., Sievers, J., & Weltman, A. 2019, Phys. Rev. D, 100, 103519, doi: 10.1103/PhysRevD.100.103519
  * Walters et al. (2018) Walters, A., Weltman, A., Gaensler, B. M., Ma, Y.-Z., & Witzemann, A. 2018, ApJ, 856, 65, doi: 10.3847/1538-4357/aaaf6b
  * Wang & Wei (2022) Wang, B., & Wei, J.-J. 2022, arXiv e-prints, arXiv:2211.02209.  https://arxiv.org/abs/2211.02209
  * Wang et al. (2020) Wang, F. Y., Wang, Y. Y., Yang, Y.-P., et al. 2020, ApJ, 891, 72, doi: 10.3847/1538-4357/ab74d0
  * Wang et al. (2021) Wang, H., Miao, X., & Shao, L. 2021, Physics Letters B, 820, 136596, doi: 10.1016/j.physletb.2021.136596
  * Wang et al. (2016) Wang, J.-S., Yang, Y.-P., Wu, X.-F., Dai, Z.-G., & Wang, F.-Y. 2016, ApJ, 822, L7, doi: 10.3847/2041-8205/822/1/L7
  * Wang & Wang (2018) Wang, Y. K., & Wang, F. Y. 2018, A&A, 614, A50, doi: 10.1051/0004-6361/201731160
  * Wei et al. (2015) Wei, J.-J., Gao, H., Wu, X.-F., & Mészáros, P. 2015, Phys. Rev. Lett., 115, 261101, doi: 10.1103/PhysRevLett.115.261101
  * Wu et al. (2020) Wu, Q., Yu, H., & Wang, F. Y. 2020, ApJ, 895, 33, doi: 10.3847/1538-4357/ab88d2
  * Wu et al. (2022) Wu, Q., Zhang, G.-Q., & Wang, F.-Y. 2022, MNRAS, 515, L1, doi: 10.1093/mnrasl/slac022
  * Xiao et al. (2021) Xiao, D., Wang, F., & Dai, Z. 2021, Science China Physics, Mechanics, and Astronomy, 64, 249501, doi: 10.1007/s11433-020-1661-7
  * Yamasaki & Totani (2020) Yamasaki, S., & Totani, T. 2020, ApJ, 888, 105, doi: 10.3847/1538-4357/ab58c4
  * Yao et al. (2017) Yao, J. M., Manchester, R. N., & Wang, N. 2017, ApJ, 835, 29, doi: 10.3847/1538-4357/835/1/29
  * Yu & Wang (2017) Yu, H., & Wang, F. Y. 2017, A&A, 606, A3, doi: 10.1051/0004-6361/201731607
  * Zhang et al. (2020) Zhang, G. Q., Yu, H., He, J. H., & Wang, F. Y. 2020, ApJ, 900, 170, doi: 10.3847/1538-4357/abaa4a
  * Zhang et al. (2021) Zhang, Z. J., Yan, K., Li, C. M., Zhang, G. Q., & Wang, F. Y. 2021, ApJ, 906, 49, doi: 10.3847/1538-4357/abceb9
  * Zhao & Wang (2021) Zhao, Z. Y., & Wang, F. Y. 2021, ApJ, 923, L17, doi: 10.3847/2041-8213/ac3f2f
  * Zheng et al. (2014) Zheng, Z., Ofek, E. O., Kulkarni, S. R., Neill, J. D., & Juric, M. 2014, ApJ, 797, 71, doi: 10.1088/0004-637X/797/1/71
  * Zhou et al. (2014) Zhou, B., Li, X., Wang, T., Fan, Y.-Z., & Wei, D.-M. 2014, Phys. Rev. D, 89, 107303, doi: 10.1103/PhysRevD.89.107303

Table 1: Properties of localized FRBs Name | Redshift | DMobs | DMMW,ISM | Host Type | Reference
---|---|---|---|---|---
|  | $(\rm pc\ cm^{-3})$ | $(\rm pc\ cm^{-3})$ |  |
FRB 121102 | 0.19273 | 557 | 188.0 | I | Spitler et al. (2016)
FRB 180301 | 0.3304 | 534 | 152 | I | Bhandari et al. (2021)
FRB 180916 | 0.0337 | 348.8 | 200.0 | II | Marcote et al. (2020)
FRB 180924 | 0.3214 | 361.42 | 40.5 | III | Bannister et al. (2019)
FRB 181030 | 0.0039 | 103.5 | 41 | I | Bhardwaj et al. (2021b)
FRB 181112 | 0.4755 | 89.27 | 102.0 | III | Prochaska et al. (2019)
FRB 190102 | 0.291 | 363.6 | 57.3 | III | Bhandari et al. (2020)
FRB 190523 | 0.66 | 760.8 | 37.0 | III | Ravi et al. (2019)
FRB 190608 | 0.1178 | 338.7 | 37.2 | III | Chittidi et al. (2021)
FRB 190611 | 0.378 | 321.4 | 57.8 | III | Day et al. (2020)
FRB 190614 | 0.6 | 959.2 | 83.5 | III | Law et al. (2020)
FRB 190711 | 0.522 | 593.1 | 56.4 | I | Heintz et al. (2020)
FRB 190714 | 0.2365 | 504.13 | 38.0 | III | Heintz et al. (2020)
FRB 191001 | 0.234 | 507.9 | 44.7 | III | Heintz et al. (2020)
FRB 191228 | 0.2432 | 297.5 | 33 | III | Bhandari et al. (2021)
FRB 200430 | 0.16 | 380.25 | 27.0 | III | Heintz et al. (2020)
FRB 200906 | 0.3688 | 577.8 | 36 | III | Bhandari et al. (2021)
FRB 201124 | 0.098 | 413.52 | 123.2 | II | Fong et al. (2021)
FRB 210117 | 0.2145 | 730 | 34.4 | III | James et al. (2022)
FRB 210320 | 0.2797 | 384.8 | 42 | III | James et al. (2022)
FRB 210807 | 0.12927 | 251.9 | 121.2 | III | James et al. (2022)
FRB 211127 | 0.0469 | 234.83 | 42.5 | III | James et al. (2022)
FRB 211212 | 0.0715 | 206 | 27.1 | III | James et al. (2022)