An 8.0% Determination of the Baryon Fraction in the Intergalactic Medium from Localized Fast Radio Bursts

Fast radio bursts (FRBs) are a class of brief (∼millisecond) and intense (∼Janksy) radio transients with large dispersion measures (DMs), well in excess of the expected contributions from the Milky Way (Lorimer et al. 2007; Thornton et al. 2013; Petroff et al. 2016; Platts et al. 2019; Xiao et al. 2021; Zhang 2022). Owing to their anomalously high DMs, FRBs are believed to be of extragalactic or even cosmological origin. To date, more than 600 FRBs have been detected, and over two dozen of them have been reported to repeat (CHIME/FRB Collaboration et al. 2021). There are more than 20 FRBs with definite host galaxies and redshift measurements. These observations suggest that FRBs are promising tools for studying cosmology. Some proposals include using localized FRBs to measure the baryon number density of the universe (Deng & Zhang 2014; McQuinn 2014; Macquart et al. 2020; Yang et al. 2022), constrain the dark energy equation of state (Gao et al. 2014; Zhou et al. 2014; Wei et al. 2018; Walters et al. 2018; Zhao et al. 2020; Qiu et al. 2022), constrain the cosmic reionization history (Deng & Zhang 2014; Zheng et al. 2014; Hashimoto et al. 2021), measure cosmological distances (Yu & Wang 2017; Kumar & Linder 2019), measure the Hubble constant (Hagstotz et al. 2022; Wu et al. 2022), using strongly lensed FRBs to probe the nature of compact dark matter (Muñoz et al. 2016; Wang & Wang 2018), or measure the Hubble constant and cosmic curvature (Li et al. 2018).

Another cosmological puzzle is the baryon distribution of the universe. While it is widely believed that more than three-quarters of the baryonic content of the universe resides in the diffuse intergalactic medium (IGM), with only a small fraction in galaxies and galaxy clusters (Fukugita et al. 1998; Cen & Ostriker 2006), gaining direct observational evidence of the baryon distribution is challenging. There have been many studies of detecting the baryon fraction in the IGM, f_IGM, through numerical simulations (Cen & Ostriker 1999, 2006; Meiksin 2009) or observations (Fukugita et al. 1998; Fukugita & Peebles 2004; Shull et al. 2012; Hill et al. 2016; Muñoz & Loeb 2018). For instance, Meiksin (2009) performed numerical simulations and suggested that ∼90% of the baryons produced by the Big Bang are contained within the IGM at redshifts of z ≥ 1.5 (i.e., f_IGM ≈ 0.9). It was observed that 18% ± 4% of the baryons exists in galaxies, circumgalactic medium, intercluster medium, and cold neutral gas at z ≤ 0.4, or equivalently f_IGM ≈ 0.82 (Shull et al. 2012). There is an ongoing debate about the value of f_IGM.

The observed DMs of cosmological FRBs are mainly contributed by the IGM. Since the DM contributed by the IGM (DM_IGM) carries important information on the location of baryons in the late universe, one may combine the DM_IGM and z information of FRBs to constrain the IGM baryon fraction f_IGM. Indeed, a number of methods have been proposed to estimate f_IGM by utilizing the DM(z) data of FRBs (Li et al. 2019; Walters et al. 2019; Wei et al. 2019; Li et al. 2020; Qiang & Wei 2020; Dai & Xia 2021; Lemos et al. 2022; Lin et al. 2022). However, one issue that restricts such studies is the strong degeneracy between cosmological parameters and f_IGM. It is hard to determine f_IGM directly only relying on FRB data. Moreover, there is another thorny issue that DMs contributed by FRB host galaxies (DM_host) and inhomogeneities in the IGM (DM_IGM) cannot be exactly extracted from observations (Macquart et al. 2020). Previous studies assumed fixed values for them, which would bring uncontrollable systematic uncertainties in the analysis. A more plausible approach is to treat them as probability distributions derived from cosmological simulations (Jaroszynski 2019; Macquart et al. 2020; Zhang et al. 2020, 2021; Wu et al. 2022).

In this paper, we present a high-precision measurement of the baryon fraction in the IGM with 17 localized FRBs through the DM_IGM–z relation. In order to break the degeneracy between cosmological parameters and f_IGM, we combine FRB data with current constraints from type Ia supernovae (SNe Ia), baryon acoustic oscillations (BAOs), and cosmic microwave background (CMB) radiation. In our f_IGM estimation, the reasonable probability distributions of DM_host and DM_IGM derived from the IllustrisTNG simulation (Zhang et al. 2020, 2021) are adopted to reduce the systematic errors. Additionally, to explore the possible evolution of f_IGM with redshift, we also consider two different parametric models, namely a constant model and a time-dependent model.

The rest of our paper is organized as follows. In Section 2, we review the FRB DM measurements and the probability distributions of DM_host and DM_IGM derived from the IllustrisTNG simulation. In Section 3, we give an introduction of the compilations of the three other cosmological probes, including SNe Ia, BAO, and CMB. Markov Chain Monte Carlo (MCMC) parameter inference results are presented in Section 4. Finally, conclusions are summarized in Section 5.

The precise localization of FRBs to their host galaxies provides an ensemble of DM and z measurements. The DM measurement represents the integrated column density of free electrons along the line of sight. For an extragalactic FRB, its observed DM can be separated into the following components:

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{obs}}(z)={\mathrm{DM}}_{\mathrm{ISM}}^{\mathrm{MW}}+{\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}+{\mathrm{DM}}_{\mathrm{IGM}}(z)+\displaystyle \frac{{\mathrm{DM}}_{\mathrm{host}}}{1+z},\end{eqnarray} \tag{ 1 }$$

where ${\mathrm{DM}}_{\mathrm{ISM}}^{\mathrm{MW}}$, ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}$, DM_IGM, and DM_host represent the DM contributions from the Milky Way interstellar medium (ISM), the Milky Way halo, the IGM, and the FRB host galaxy, respectively. The (1 + z) factor converts the local DM_host to the observed one (Deng & Zhang 2014). Because of the inhomogeneity of the free electron distribution in the IGM, two sources at the same redshift but in different sightlines will likely have significant differences in the measured value of DM_IGM. Adopting a flat ΛCDM cosmological model, the average value of DM_IGM can be estimated as (Deng & Zhang 2014):

$$\begin{eqnarray}&&\left\langle {\mathrm{DM}}_{\mathrm{IGM}}\right\rangle (z)=\frac{3c{{\rm{\Omega }}}_{b}{H}_{0}}{8\pi {{Gm}}_{p}}{\int }_{0}^{z}\frac{(1+{z}^{^{\prime} }){f}_{\mathrm{IGM}}({z}^{^{\prime} }){\chi }_{e}({z}^{^{\prime} })}{\sqrt{{{\rm{\Omega }}}_{m}{\left(1+{z}^{^{\prime} }\right)}^{3}+1-{{\rm{\Omega }}}_{m}}}{{dz}}^{{\prime} },\end{eqnarray} \tag{ 2 }$$

where m_p is the proton mass, H₀ is the Hubble constant, f_IGM(z) is the baryon fraction in the IGM, and Ω_b and Ω_m are the present-day baryon and matter density parameters, respectively. The free electron number fraction per baryon is ${\chi }_{e}(z)\,=\tfrac{3}{4}{\chi }_{e,{\rm{H}}}(z)+\tfrac{1}{8}{\chi }_{e,\mathrm{He}}(z)$, where χ_e,H(z) and χ_e,He(z) are the ionization fractions of hydrogen and helium, respectively. Both hydrogen and helium are completely ionized at redshifts z < 3 (Meiksin 2009; Becker et al. 2011), allowing one to set χ_e,H = χ_e,He = 1, which gives χ_e = 7/8.

The DM_IGM value of a well-localized FRB can be extracted using ${\mathrm{DM}}_{\mathrm{IGM}}={\mathrm{DM}}_{\mathrm{obs}}-{\mathrm{DM}}_{\mathrm{ISM}}^{\mathrm{MW}}-{\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}-{\mathrm{DM}}_{\mathrm{host}}/(1+z)$. Here the ${\mathrm{DM}}_{\mathrm{ISM}}^{\mathrm{MW}}$ term can be well estimated from the NE2001 model of the ISM free electron distribution (Cordes & Lazio 2002). The ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}$ term is not well constrained, but is expected to contribute 50–80 pc cm⁻³ (Prochaska & Zheng 2019). Hereafter we assume that the probability distribution of ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}$ can be described by a Gaussian distribution with mean μ_halo = 65 pc cm⁻³ and standard deviation σ_halo = 15 pc cm⁻³ (Wu et al. 2022):

$$\begin{eqnarray}&&{P}_{\mathrm{halo}}({\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}})=\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{\mathrm{halo}}}\exp \left[-\displaystyle \frac{{\left({\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}-{\mu }_{\mathrm{halo}}\right)}^{2}}{2{\sigma }_{\mathrm{halo}}^{2}}\right].\end{eqnarray} \tag{ 3 }$$

Based on the state-of-the-art IllustrisTNG simulation (Springel et al. 2018), Zhang et al. (2020) selected a large sample of simulated galaxies with similar properties to observed FRB hosts to derive the distributions of DM_host of repeating and non-repeating FRBs. The distributions of DM_host can be well described by the log-normal function (Macquart et al. 2020; Zhang et al. 2020):

$$\begin{eqnarray}&&\begin{array}{l}{P}_{\mathrm{host}}\left({\mathrm{DM}}_{\mathrm{host}\ }\right)=\displaystyle \frac{1}{\sqrt{2\pi }{\mathrm{DM}}_{\mathrm{host}\ }{\sigma }_{\mathrm{host}\ }}\\ \times \,\exp \left[-\displaystyle \frac{{\left({\mathrm{lnDM}}_{\mathrm{host}\ }-{\mu }_{\mathrm{host}}\right)}^{2}}{2{\sigma }_{\mathrm{host}}^{2}}\right],\end{array}\end{eqnarray} \tag{ 4 }$$

where ${e}^{{\mu }_{\mathrm{host}}}$ and ${e}^{2{\mu }_{\mathrm{host}}+{\sigma }_{\mathrm{host}}^{2}}({e}^{{\sigma }_{\mathrm{host}}^{2}}-1)$ are the mean and variance of the distribution, respectively. Due to the diversity of host galaxies, Zhang et al. (2020) computed the DM_host distributions for repeating FRBs in dwarf galaxies (like FRB 121102, FRB 180301, FRB 181030, and FRB 190711), repeating FRBs in spiral galaxies (like FRB 180916 and FRB 201124), and non-repeating FRBs separately. Here we divide the localized FRBs into these three types according to their host properties. The evolution of the median of DM_host (μ_host) can be fitted by μ_host(z) = κ(1 + z)^γ , where κ and γ are given by Zhang et al. (2020). The propagated uncertainty σ_host of DM_host is calculated from the uncertainties of κ and γ. With this expression of redshift evolution, we can derive the DM_host distributions at any redshift of a localized FRB.

To date, more than 20 FRBs have already been localized. Nonetheless, some of them are not available for our analysis. For example, the DM of FRB 181030 is only 103.396 pc cm⁻³ (Bhardwaj et al. 2021a), which will be reduced to a negative value after subtracting DM contributions from the Milky Way ISM and halo. That is, the integral upper limit (${\mathrm{DM}}_{{\rm{E}}}\,\equiv {\mathrm{DM}}_{\mathrm{obs}}-{\mathrm{DM}}_{\mathrm{ISM}}^{\mathrm{MW}}-{\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}$) in the probability of the external DM contribution outside our galaxy (see Equation (7)) will become negative. FRB 190520B is co-located with a compact, persistent radio source and associated with a dwarf host galaxy at a redshift of 0.241 (Niu et al. 2022). It is a clear outlier from the general trend of the extragalactic DM_E–z relation, with an unprecedented DM contribution from its host galaxy. Thus its DM_host term cannot be accurately deducted. FRB 200110E is located in a globular cluster in the direction of the nearby galaxy M81 (Bhardwaj et al. 2021b; Kirsten et al. 2022). The distance of FRB 200110E is only 3.6 Mpc, and the IGM between the Milky Way and M81 contributes of the order of DM_IGM ∼ 1 pc cm⁻³. Thus the cosmological information carried by FRB 200110E is too little. Additionally, the peculiar velocity effect is significant, which makes it unuseful for cosmological studies. After excluding these FRBs, we use a sample of 17 FRBs in the redshift range 0.0337 ≤ z ≤ 0.66 to constrain f_IGM(z). ⁴ Table 1 lists the redshifts, DM_obs, and ${\mathrm{DM}}_{\mathrm{ISM}}^{\mathrm{MW}}$ of our sample. The estimated DM_IGM and measured z values for the 17 localized FRBs are shown in Figure 1. We have estimated DM_IGM by subtracting the following from the observed DM_obs value: ${\mathrm{DM}}_{\mathrm{ISM}}^{\mathrm{MW}}$ from the Galactic ISM model; a median of 65 pc cm⁻³ contributed by ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}};$ and a median value of DM_host at different redshifts estimated from the IllustrisTNG simulation (Zhang et al. 2020).

Zoom In Zoom Out Reset image size

Table 1. Properties of the 17 Localized FRBs

Name	Redshift	DMobs	${\mathrm{DM}}_{\mathrm{ISM}}^{\mathrm{MW}}$	References
		(pc cm−3)	(pc cm−3)
FRB 121102	0.19273	557 ± 2	188.0	1
FRB 180301	0.3304	536 ± 0.2	152.0	2
FRB 180916	0.0337	349.349 ± 0.005	200.0	3
FRB 180924	0.3214	361.42 ± 0.06	40.5	4
FRB 181112	0.4755	589.27 ± 0.03	102.0	5
FRB 190102	0.291	363.6 ± 0.3	57.3	6
FRB 190523	0.66	760.8 ± 0.6	37.0	7
FRB 190608	0.1178	338.7 ± 0.5	37.2	8
FRB 190611	0.378	321.4 ± 0.2	57.83	9
FRB 190614	0.6	959.2 ± 0.5	83.5	10
FRB 190711	0.522	593.1 ± 0.4	56.4	9
FRB 190714	0.2365	504 ± 2	38.0	9
FRB 191001	0.234	506.92 ± 0.04	44.7	9
FRB 191228	0.2432	297.5 ± 0.05	33.0	2
FRB 200430	0.16	380.1 ± 0.4	27.0	9
FRB 200906	0.3688	577.8 ± 0.02	36.0	2
FRB 201124	0.098	413.52 ± 0.05	123.2	11

Note. References: (1) Chatterjee et al. (2017); (2) Bhandari et al. (2022); (3) Marcote et al. (2020); (4) Bannister et al. (2019); (5) Prochaska et al. (2019); (6) Bhandari et al. (2020); (7) Ravi et al. (2019); (8) Chittidi et al. (2021); (9) Heintz et al. (2020); (10) Law et al. (2020); and (11) Ravi et al. (2022).

Download table as:  ASCIITypeset image

To construct a likelihood function ${ \mathcal L }$ from the FRB measurements, we build a model for DM_IGM. The model probability distribution for DM_IGM has been derived from theoretical treatments of the IGM and galaxy halos with a standard deviation σ_DM dominated by the variance in DM_IGM. The DM_IGM distributions derived in both semi-analytic models and cosmological simulations can be well fitted by a quasi-Gaussian function with a long tail (McQuinn 2014; Prochaska & Zheng 2019; Macquart et al. 2020):

$$\begin{eqnarray}&&{P}_{\mathrm{IGM}}({\rm{\Delta }})=A{{\rm{\Delta }}}^{-\beta }\exp \left[-\displaystyle \frac{{\left({{\rm{\Delta }}}^{-\epsilon }-{C}_{0}\right)}^{2}}{2{\epsilon }^{2}{\sigma }_{\mathrm{DM}}^{2}}\right],\,\,\,{\rm{\Delta }}\gt 0,\end{eqnarray} \tag{ 5 }$$

where ${\rm{\Delta }}\equiv {\mathrm{DM}}_{\mathrm{IGM}}/\left\langle {\mathrm{DM}}_{\mathrm{IGM}}\right\rangle$, A is a normalization coefficient, and the indices and β are related to the inner density profile of gas in halos. Here we take = 3 and β = 3, as Macquart et al. (2020) did in their treatment. C₀ is a free parameter, which can be fitted when the mean 〈Δ〉 = 1. The motivation for this analytic form (Equation (5)) is that in the limit of small σ_DM, the DM_IGM distribution should approach a Gaussian owing to the more diffuse halo gas and the Gaussianity of structure on large scales. Conversely, when the variance is large, this probability distribution captures the large skew due to a few large structures that contribute to the DM of many sightlines. Recently, Zhang et al. (2021) used the IllustrisTNG simulation to estimate the probability distributions of DM_IGM at different redshifts realistically. Following Wu et al. (2022), the best-fit parameters (A, C₀, and σ_DM) of the DM_IGM distributions at the different redshifts presented by Zhang et al. (2021) are used for our purpose. The uncertainties of these best-fit parameters may impact our final f_IGM constraint. To investigate whether the uncertainties of these parameters affect the Ω_b constraint (similar to our f_IGM constraint), Yang et al. (2022) derived Ω_b using the best-fit values of these parameters plus or minus the uncertainties. They found that these uncertainties have almost no effect on the final Ω_b constraint. Given the fact that our f_IGM constraint is almost the same as that of Yang et al. (2022), we can come to the same conclusion. Note that since the DM_IGM distributions are given in discrete redshifts (Zhang et al. 2021), we extrapolate them to the redshifts of the localized FRBs through cubic spline interpolation.

Given the model for DM_IGM, we estimate the likelihood function by computing the joint likelihoods of the 17 FRBs (Macquart et al. 2020):

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{FRB}}=\displaystyle \prod _{i=1}^{{N}_{\mathrm{FRB}}}{P}_{i}\left({\mathrm{DM}}_{{\rm{E}},i}\right),\end{eqnarray} \tag{ 6 }$$

where P_i (DM_E,i ) is the probability of the total observed DM_obs corrected for our galaxy, i.e., ${\mathrm{DM}}_{{\rm{E}}}\equiv {\mathrm{DM}}_{\mathrm{obs}}-{\mathrm{DM}}_{\mathrm{ISM}}^{\mathrm{MW}}\,-{\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}={\mathrm{DM}}_{\mathrm{IGM}}+{\mathrm{DM}}_{\mathrm{host}}/(1+z)$. For a burst at redshift z_i , we have:

$$\begin{eqnarray}\begin{array}{rcl}{P}_{i}\left({\mathrm{DM}}_{{\rm{E}},i}\right) & = & {\displaystyle \int }_{0}^{{\mathrm{DM}}_{{\rm{E}},i}}{P}_{\mathrm{host}}\left({\mathrm{DM}}_{\mathrm{host}}\right)\\ & \times & {P}_{\mathrm{IGM}}\left({\mathrm{DM}}_{{\rm{E}},i}-{\mathrm{DM}}_{\mathrm{host},{z}_{i}}\right){\mathrm{dDM}}_{\mathrm{host}},\end{array}\end{eqnarray} \tag{ 7 }$$

where the probability density functions for P_host(DM_host) and P_IGM(DM_IGM) are obtained from Equations (4) and (5), respectively. Note that the Milky Way halo DM distribution (${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}$) will be considered as a free parameter in our analysis. We will marginalize ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}$ using a Gaussian prior of ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}=65\pm 15$ pc cm⁻³ over the range of [μ_halo − 3σ_halo, μ_halo + 3σ_halo], where μ_halo = 65 pc cm⁻³ and σ_halo = 15 pc cm⁻³ (see Equation (3)).

As we discussed in Section 1, in order to break the degeneracy between the IGM baryon fraction f_IGM and other cosmological parameters (Ω_m , Ω_b , and H₀; see Equation (2)), we use up-to-date cosmological data compilations, including SNe Ia, BAO, and CMB data. The exact compilations and the details for the likelihoods are described separately in what follows.

3.1. SNe Ia

The SNe Ia data set that we use in this work is the Pantheon sample, which consists of 1048 SNe Ia in the redshift range 0.01 < z < 2.3 (Scolnic et al. 2018). The observed distance modulus of each SN is given as:

$$\begin{eqnarray}&&{\mu }_{\mathrm{SN}}={m}_{\mathrm{corr}}-{M}_{B},\end{eqnarray} \tag{ 8 }$$

where m_corr is the corrected apparent magnitude and M_B is the absolute magnitude.

The theoretical distance modulus μ_th(z) is defined as:

$$\begin{eqnarray}&&{\mu }_{\mathrm{th}}=5{\mathrm{log}}_{10}\left[\displaystyle \frac{{d}_{L}(z)}{\mathrm{Mpc}}\right]+25,\end{eqnarray} \tag{ 9 }$$

where ${d}_{L}(z)=(1+z)\tfrac{c}{{H}_{0}}{\int }_{0}^{z}\tfrac{{dz}^{\prime} }{\sqrt{{{\rm{\Omega }}}_{m}{\left(1+z^{\prime} \right)}^{3}+1-{{\rm{\Omega }}}_{m}}}$ is the luminosity distance in the flat ΛCDM model. Thus, the ${\chi }_{\mathrm{SN}}^{2}$ function for the Pantheon data is:

$$\begin{eqnarray}&&{\chi }_{\mathrm{SN}}^{2}={\rm{\Delta }}{\hat{{\boldsymbol{\mu }}}}^{{\text{}}T}\cdot {C}_{\mathrm{SN}}^{-1}\cdot {\rm{\Delta }}\hat{{\boldsymbol{\mu }}},\end{eqnarray} \tag{ 10 }$$

where ${\rm{\Delta }}\hat{\mu }={\hat{\mu }}_{\mathrm{SN}}-{\hat{\mu }}_{\mathrm{th}}$ is the data vector, defined by the difference between the SN distance modulus ${\mu }_{\mathrm{SN}}$ and the theoretical distance modulus μ_th, and ${{\boldsymbol{C}}}_{\mathrm{SN}}$ is the covariance matrix that contains both the statistical and systematic uncertainties of the SNe.

3.2. BAOs

Primordial perturbations in the early universe excite acoustic waves in the plasma, known as BAOs. After the recombination period, the propagation of acoustic waves was frozen. Thus, there is a characteristic scale called the comoving sound horizon r_s , which can be approximated as (Aubourg et al. 2015):

$$\begin{eqnarray}\begin{array}{rcl}{r}_{s}({z}^{* }) & = & {\displaystyle \int }_{{z}^{* }}^{\infty }\displaystyle \frac{{c}_{s}(z)}{H(z)}{dz}\\ & \approx & \displaystyle \frac{55.154\exp \left[-72.3{\left({{\rm{\Omega }}}_{v}{h}^{2}+0.0006\right)}^{2}\right]}{{\left({{\rm{\Omega }}}_{b}{h}^{2}\right)}^{0.12807}{\left({{\rm{\Omega }}}_{m}{h}^{2}\right)}^{0.25351}}\,\mathrm{Mpc},\end{array}\end{eqnarray} \tag{ 11 }$$

where c_s (z) is the sound speed of the photon–baryon fluid, z is the redshift at the drag epoch, Ω_v is the present-day neutrino density, and h ≡ H₀/(100 km s⁻¹ Mpc⁻¹) is the reduced Hubble constant. Here we use a combination of 11 BAO measurements from Ryan et al. (2019). Six of these BAO measurements are correlated, in which case χ² is given by:

$$\begin{eqnarray}&&{\chi }_{\mathrm{BAO}1}^{2}={\left({{\boldsymbol{A}}}_{\mathrm{obs}}-{{\boldsymbol{A}}}_{\mathrm{th}}\right)}^{{\text{}}T}\cdot {{\boldsymbol{C}}}_{\mathrm{BAO}}^{-1}\cdot \left({{\boldsymbol{A}}}_{\mathrm{obs}}-{{\boldsymbol{A}}}_{\mathrm{th}}\right),\end{eqnarray} \tag{ 12 }$$

where A _obs ( A _th) is the vector that contains all of the six measured (theoretical) values and C _BAO is the covariance matrix for the BAO data sets. The other five BAO measurements are uncorrelated, so:

$$\begin{eqnarray}&&{\chi }_{\mathrm{BAO}2}^{2}=\displaystyle \sum _{i=1}^{5}\displaystyle \frac{{\left[{A}_{\mathrm{th}}({z}_{i})-{A}_{\mathrm{obs}}({z}_{i})\right]}^{2}}{{\sigma }_{i}^{2}},\end{eqnarray} \tag{ 13 }$$

where σ_i is the standard deviation of the i-th BAO measurement A_obs(z_i ). The data are combined into a χ²-statistic as ${\chi }_{\mathrm{BAO}}^{2}={\chi }_{\mathrm{BAO}1}^{2}+{\chi }_{\mathrm{BAO}2}^{2}$.

3.3. CMB

For the CMB measurements, we use the derived parameters, including the acoustic scale l_A , the shift parameter R, and Ω_b h² from the Planck analysis of the CMB (TT, TE, EE + lowE) (Chen et al. 2019; Planck Collaboration et al. 2020). The acoustic scale is:

$$\begin{eqnarray}&&{l}_{A}({z}^{* })=(1+{z}^{* })\displaystyle \frac{\pi {d}_{A}({z}^{* })}{{r}_{s}({z}^{* })},\end{eqnarray} \tag{ 14 }$$

where r_s is the comoving sound horizon at the recombination and d_A = d_L (1 + z)⁻² is the angular diameter distance. The shift parameter is:

$$\begin{eqnarray}&&R({z}^{* })=(1+{z}^{* })\displaystyle \frac{{d}_{A}(z* )\sqrt{{{\rm{\Omega }}}_{m}}{H}_{0}}{c}.\end{eqnarray} \tag{ 15 }$$

The redshift at decoupling z* is given by:

$$\begin{eqnarray}&&{z}^{* }=1048\,\left[1+0.00124{\left({{\rm{\Omega }}}_{b}{h}^{2}\right)}^{-0.738}\right]\left[1+{g}_{1}{\left({{\rm{\Omega }}}_{m}{h}^{2}\right)}^{{g}_{2}}\right],\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{g}_{1}=\displaystyle \frac{0.0738{\left({{\rm{\Omega }}}_{b}{h}^{2}\right)}^{-0.238}}{1+39.5{\left({{\rm{\Omega }}}_{b}{h}^{2}\right)}^{0.763}},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{g}_{2}=\displaystyle \frac{0.560}{1+21.1{\left({{\rm{\Omega }}}_{b}{h}^{2}\right)}^{1.81}}.\end{eqnarray} \tag{ 18 }$$

By setting x = (R, l_A , Ω_b h²), the ${\chi }_{\mathrm{CMB}}^{2}$ value for the CMB data is then:

$$\begin{eqnarray}&&{\chi }_{\mathrm{CMB}}^{2}={\left({{\boldsymbol{x}}}_{\mathrm{obs}}-{{\boldsymbol{x}}}_{\mathrm{th}}\right)}^{{\text{}}T}\cdot {{\boldsymbol{C}}}_{\mathrm{CMB}}^{-1}\cdot \left({{\boldsymbol{x}}}_{\mathrm{obs}}-{{\boldsymbol{x}}}_{\mathrm{th}}\right),\end{eqnarray} \tag{ 19 }$$

where x _obs = (1.7502, 301.471, 0.02236), and x _th contain the observed and theoretical values of the derived parameters, and C _CMB is the covariance matrix for the CMB data. Note that the distance priors derived from the CMB data are dependent on the specific cosmological model. Here we adopt the values of x _obs and C _CMB inferred from the flat ΛCDM model.

To assess how well-localized FRBs may help to constrain the evolution of the IGM baryon fraction f_IGM(z), we consider two different parametric models. First, a simple constant model:

$$\begin{eqnarray}&&{f}_{\mathrm{IGM}}={f}_{\mathrm{IGM},0}.\end{eqnarray} \tag{ 20 }$$

And second, a time-dependent model given by:

$$\begin{eqnarray}&&{f}_{\mathrm{IGM}}(z)={f}_{\mathrm{IGM},0}\left(1+\alpha \displaystyle \frac{z}{1+z}\right),\end{eqnarray} \tag{ 21 }$$

where f_IGM,0 is the present value of f_IGM and α quantifies any possible evolution of f_IGM. As massive halos are more abundant in the late universe, f_IGM is believed to grow with redshift (McQuinn 2014; Prochaska & Zheng 2019). Therefore, here we require α ≥ 0.

The quantities f_IGM(z), Ω_m , Ω_b h², H₀, M_B , and ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}$ are fitted to the FRB, SN Ia, BAO, and CMB data simultaneously using the Python MCMC module emcee (Foreman-Mackey et al. 2013). Given the relation ${ \mathcal L }\propto \exp [-{\chi }^{2}/2]$, the final log-likelihood sampled by emcee is a sum of the separated likelihoods of FRBs, SNe Ia, BAO, and CMB:

$$\begin{eqnarray}&&\mathrm{ln}{{ \mathcal L }}_{\mathrm{tot}}=\mathrm{ln}{{ \mathcal L }}_{\mathrm{FRB}}-{\chi }_{\mathrm{SN}}^{2}/2-{\chi }_{\mathrm{BAO}}^{2}/2-{\chi }_{\mathrm{CMB}}^{2}/2.\end{eqnarray} \tag{ 22 }$$

In our baseline analysis, we set flat priors on f_IGM,0 ∈ [0, 1] and α ∈ [0, 2], and a Gaussian prior on ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}=65\pm 15$ pc cm⁻³ over the 3σ range of [20, 110] pc cm⁻³.

For the constant case, there are six free parameters, including the IGM baryon fraction f_IGM,0, the cosmological parameters (Ω_m , Ω_b h², and H₀), the SN absolute B-band magnitude M_B , and the Milky Way halo DM contribution ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}$. The 1D marginalized posterior distributions and 2D plots of the 1–2σ confidence regions for these six parameters are displayed in Figure 2. These contours show that, at the 1σ confidence level, the inferred parameter values are f_IGM,0 = 0.927 ± 0.075, Ω_m = 0.309 ± 0.006, Ω_b h² = 0.02245 ± 0.00013, H₀ =67.78 ± 0.44 km s⁻¹ Mpc⁻¹, and ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}=47\pm 10$ pc cm⁻³. The corresponding results are summarized in Table 2. Figure 1 also shows the theoretical curve for 〈DM_IGM〉 versus z for the constant case and a model estimate of the scatter (95% interval) due to the uncertainties of the inferred parameters. The theoretical curve reflects the trend of the data well. Moreover, the inferred value of the IGM baryon fraction is compatible with previous results obtained from observations (e.g., Fukugita et al. 1998; Fukugita & Peebles 2004; Shull et al. 2012; Hill et al. 2016; Muñoz & Loeb 2018) and simulations (e.g., Cen & Ostriker 1999, 2006). The constraint accuracy of f_IGM,0 is about 8.0%.

Zoom In Zoom Out Reset image size

Table 2. Constraints on All Parameters for the Two Different Parametric Models of f_IGM

Model	Constant	Time-dependent
Parameter	Estimation with 68% Limits
fIGM,0	0.927 ± 0.075	0.837 ± 0.089
α	⋯	<0.882
Ωm	0.309 ± 0.006	0.309 ± 0.006
Ωb h2	0.02245 ± 0.00013	0.02245 ± 0.00013
H0/[km s−1 Mpc−1]	67.78 ± 0.44	67.71 ± 0.42
MB	−19.415 ± 0.012	−19.417 ± 0.012
${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}/[\mathrm{pc}\,{\mathrm{cm}}^{-3}]$	47 ± 10	49 ± 10
$-2\mathrm{ln}{{ \mathcal L }}_{\max }$	1262.855	1263.099
ΔAIC	⋯	2.244

Download table as:  ASCIITypeset image

For the time-dependent case, the free parameters are $\{{f}_{\mathrm{IGM},0},\,\alpha ,\,{{\rm{\Omega }}}_{m},\,{{\rm{\Omega }}}_{b}{h}^{2},\,{H}_{0},\,{M}_{B},\,{\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}\}$. These seven parameters are constrained to be f_IGM,0 = 0.837 ± 0.089, α < 0.882, Ω_m = 0.309 ± 0.006, Ω_b h² = 0.02245 ± 0.00013, H₀ = 67.71 ±0.42 km s⁻¹ Mpc⁻¹, and ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}=49\pm 10$ pc cm⁻³, which are displayed in Figure 3 and summarized in Table 2. Note that with the requirement that f_IGM grows with redshift (α ≥ 0), only an upper limit on α can be estimated, which implies that there is no strong evidence for a redshift dependence of f_IGM. This is consistent with the cosmology-insensitive result obtained from five localized FRBs (Li et al. 2020). The comparison between columns 2 and 3 of Table 2 suggests that the nuisance parameters (Ω_m , Ω_b h², H₀, M_B , and ${\mathrm{DM}}_{\mathrm{halo}}^{\mathrm{MW}}$) are almost identical and have little effect on the adopted parametric model of f_IGM.

Zoom In Zoom Out Reset image size

Because the constant model and the time-dependent model do not have the same number of free parameters, a comparison of the likelihoods for either being closer to the correct model must be based on model selection criteria. We use the Akaike information criterion (AIC; Akaike1974, 1981) to test the statistical performance of the models, $\mathrm{AIC}=-2\mathrm{ln}{ \mathcal L }+2p$, where p is the number of free parameters. With AIC₁ and AIC₂ characterizing models ${{ \mathcal M }}_{1}$ (the constant model) and ${{ \mathcal M }}_{2}$ (the time-dependent model), respectively, the difference ΔAIC ≡ AIC₂ − AIC₁ determines the extent to which ${{ \mathcal M }}_{1}$ is favored over ${{ \mathcal M }}_{2}$. The evidence of ${{ \mathcal M }}_{1}$ being correct is judged "weak" when the outcome Δ ≡ AIC₂ − AIC₁ is in the range 0 < Δ < 2, "positive" when 2 < Δ < 6, and "strong" when Δ > 6. Therefore, the outcome ΔAIC = 2.244 shows a positive evidence in favor of the constant model (${{ \mathcal M }}_{1}$) with respect to the time-dependent model (${{ \mathcal M }}_{2}$). Nevertheless, we hold the opinion that this positive evidence may be due to the relatively low redshifts in the FRB data. To distinguish between the constant and time-dependent models better, a larger number of FRBs localized at higher redshifts is required in the future.

In our analysis, the ${\mathrm{DM}}_{\mathrm{ISM}}^{\mathrm{MW}}$ values are estimated from the Galactic electron density model of NE2001 (Cordes & Lazio 2002). We also perform a parallel comparative analysis of the FRB data using the YMW16 model (Yao et al. 2017). The resulting constraints now turn to be f_IGM,0 = 0.901 ± 0.081 (${f}_{\mathrm{IGM},0}={0.788}_{-0.100}^{+0.082}$) for the constant (time-dependent) case. Comparing these inferred f_IGM,0 with those obtained using the NE2001 model, we see that the adoption of a different electron distribution model has a minimal influence on the results. Due to the larger ${\mathrm{DM}}_{\mathrm{ISM}}^{\mathrm{MW}}$ contribution at low Galactic latitudes in the YMW16 Model, the derived f_IGM,0 values are slightly smaller than those in the NE2001 model.

To make a direct comparison with previous works, in Figure 4 we plot some typical f_IGM,0 constraints from different FRB samples, as well as our constraints from both the NE2001 and YMW16 models. One can see from Figure 4 that our f_IGM,0 constraints are well consistent with previous results at the 1σ confidence level.

Zoom In Zoom Out Reset image size

The DM_IGM–z relation of FRBs has been used for probing the baryon fraction in the IGM, f_IGM. However, such studies have been restricted by the strong degeneracy between cosmological parameters and f_IGM. Moreover, the DM contribution from the IGM (DM_IGM) cannot be effectively distinguished from other DMs contributed by the Milky Way or host galaxy. In this work, we investigate precise constraints on f_IGM from the DM measurements of 17 localized FRBs. In order to break the parameter degeneracy, we combine FRB data with three other cosmological probes (including SNe Ia, BAO, and CMB) to infer cosmological parameters and f_IGM simultaneously. To avoid uncontrollable systematic errors induced by the DM_IGM deduction, we handle the DM contributions of the host galaxies and IGM as probability distributions derived from the the IllustrisTNG simulation.

Following the analysis method described in Section 2, we explore the possible redshift dependence of f_IGM(z) considering two different parametric models, which are expressed as the constant and time-dependent parameterizations given by Equations (20) and (21). A MCMC analysis is used to constrain f_IGM(z) and other cosmological parameters. For the constant model, we infer that f_IGM,0 = 0.927 ± 0.075, representing a precision of 8.0%. This constraint from FRB observations is roughly consistent with those obtained from other probes (Fukugita et al. 1998; Fukugita & Peebles 2004; Shull et al. 2012; Hill et al. 2016). For the time-dependent model, whereas only an upper limit on the evolution index α can be set (α < 0.882), we can obtain a good limit on the local f_IGM,0 = 0.837 ± 0.089, which is slightly looser but still consistent with previous results derived from different methods. According to the AIC model selection criteria, there is a mild evidence suggesting that the constant model is preferred over the time-dependent model. However, due to the fact that the number of current localized FRBs is small and their redshift measurements are relatively low, we cannot safely exclude the possibility of an evolving f_IGM(z).

Redshift measurements of a larger sample of FRBs are essential for using the method presented here to constrain f_IGM and its possible redshift evolution. Forthcoming radio telescopes such as the Deep Synoptic Array 2000-dish prototype (Hallinan et al. 2019) and the Square Kilometre Array (Dewdney et al. 2009), with improved detection sensitivity and localization capability, will be able to increase the current localized FRB sample size by orders of magnitude. With the rapid progress in localizing FRBs, the constraints on f_IGM will be significantly improved, and the baryon distribution of the universe will be better understood.

We are grateful to the anonymous referee for their helpful comments. This work is partially supported by the National Key Research and Development Program of China (2022SKA0130100), the National Natural Science Foundation of China (grant Nos. 11725314 and 12041306), the Key Research Program of Frontier Sciences (grant No. ZDBS-LY-7014) of Chinese Academy of Sciences, International Partnership Program of Chinese Academy of Sciences for Grand Challenges (114332KYSB20210018), the CAS Project for Young Scientists in Basic Research (grant No. YSBR-063), the CAS Organizational Scientific Research Platform for National Major Scientific and Technological Infrastructure: Cosmic Transients with FAST, the Natural Science Foundation of Jiangsu Province (grant No. BK20221562), and the Young Elite Scientists Sponsorship Program of Jiangsu Association for Science and Technology.