Possible bias of the constraints on the Hubble constant owing to the quasi-Gaussian distribution of DMIGM in fast radio bursts

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Gui-Yao Chen and Xin Li. The quasi-Gaussian distribution of DMIGM in fast radio bursts may bias the constraints on the Hubble constant[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad3942
Gui-Yao Chen and Xin Li. The quasi-Gaussian distribution of DMIGM in fast radio bursts may bias the constraints on the Hubble constant[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad3942 shu
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Possible bias of the constraints on the Hubble constant owing to the quasi-Gaussian distribution of DMIGM in fast radio bursts

  • 1. Department of Physics, Chongqing University, Chongqing 401331, China
  • 2. Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing 401331, China

Abstract: Fast radio bursts (FRBs) are useful cosmological probes with numerous applications in cosmology. The distribution of the dispersion measurement contribution from the intergalactic medium is a key issue. A quasi-Gaussian distribution has been used to replace the traditional Gaussian distribution, yielding promising results. However, this study suggests that there may be additional challenges in its application. We used 35 well-localized FRBs to constrain the Hubble constant H0 along with two FRB-related parameters, yielding H0=60.99+4.574.90 kms1Mpc1. The best-fitting Hubble constant H0 is smaller than the value obtained from the Cosmic Microwave Background (CMB), which may be caused by the small sample size of current FRB data. Monte Carlo simulations indicate that a set of 100 simulated FRBs provides a more precise fitting result for the Hubble constant. However, the precision of the Hubble constant does not improve when further enlarging the FRB sample. Additional simulations reveal a systematic deviation in the fitting results of H0, attributed to the quasi-Gaussian distribution of the dispersion measure in the intergalactic medium. Despite this, the results remain reliable within 1σ uncertainty, assuming that a sufficient number of FRB data points are available.

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    I.   INTRODUCTION
    • Fast radio bursts (FRBs) are bright and energetic radio pulses characterized by millisecond durations that primarily originate from extragalactic sources [14]. The first FRB signal, now named FRB 010724, was discovered by Lorimer et al. [5] in 2007 from the 2001 archive data of the Parkes 64-m telescope. In 2013, Thornton et al. [6] discovered several similar signals; thereafter, FRBs began to attract people's interest. FRBs can be generally categorized into two classes: repeaters and non-repeaters. However, it is unclear whether there are essential differences between them. The first discovered repeating FRB was FRB 121102, which is extremely active and can repeat over a thousand times in 60 hours [7]. Other repeating FRBs might repeat only two or three times [8]. FRBs also have a high occurrence rate across the full sky. To date, hundreds of FRBs have been discovered by various telescopes [1, 9]. FRB 200428 confirmed that at least some FRBs may originate from magnetars, but there is still no consensus on the origin of most FRBs. The large dispersion measures (DMs) show that the majority of FRBs have an extragalactic origin, with only one FRB confirmed to come from the Milky Way [10]. The localization of the host galaxy and the direct measurement of redshift confirm that they have an extragalactic origin [1113], thus making FRBs a useful tool to study the universe.

      Recently, the Hubble tension problem, i.e., the contradiction between the Hubble constant values measured from the cosmic microwave background (CMB) and Type Ia supernovae (SNe Ia), called people's attention [1417]. New probes for the universe are necessary to resolve the Hubble tension problem, and FRBs may potentially be such probes. The high energy of FRBs enables their detection at high redshift, and their high DM also makes them useful cosmological probes. For instance, Macquart et al. [18] used FRBs to constrain the missing baryons in the universe. Gao et al. [19] used FRB/GRB systems to constrain the dark energy. Zhao et al. [20] used simulated FRB data to break the parameter degeneracies inherent to the CMB data. Qiu et al. [21] combined simulated FRB data with current CMB data to constrain the holographic dark energy (HDE) and Ricci dark energy (RDE) models. Zhao et al. [22] used unlocalized FRB data to constrain the Hubble constant. Zhang et al. [23] investigated how the upcoming FRB observations with the Square Kilometre Array (SKA) can help advance cosmology, and showed that 106 FRB data can tightly constrain the dark-energy equation of state parameters and the baryon density Ωbh. Bhattacharya et al. [24] used FRBs to explore the reionization history of the universe. Lin and Sang [25] used FRBs to test the anisotropy of the universe. Wu et al. [26] used FRBs to constrain the Hubble constant. Li et al. [27] used strongly lensed FRBs to constrain cosmological parameters such as the Hubble constant and the curvature of the Universe. Pearson et al. [28] pointed out that strongly lensed repeating FRBs can be used to search for gravitational waves.

      There are some key issues in using FRBs as cosmological probes. First, although hundreds of FRBs have been measured, only a few of them have been subjected to a direct measurement of redshift. Fortunately, many more FRBs are expected to be detected in the future by telescopes such as the Canadian Hydrogen Intensity Mapping Experiment (CHIME) and the Five-hundred-meter Aperture Spherical Telescope (FAST), and some of them are expected to be subjected to a direct redshift measurement [29, 30]. Second, the DM contribution from the host galaxy (DMhost) is not fully understood, and it may significantly differ among FRBs. Therefore, it is difficult to precisely model the DMhost term. Some previous studies assumed that the DMhost term accommodates the evolution of the star formation rate history; hence, it evolves with redshift [31]. However, Lin et al. [32] found that there is no strong evidence for the redshift evolution of DMhost within the present data. A more reasonable approach is to consider the probability distribution of DMhost and marginalize over it. Based on theoretical considerations and numerical simulations, Macquart et al. [18] and Zhang et al. [33] showed that DMhost follows a log-normal distribution. Finally, owing to the fluctuation of the matter density, the DM contribution from the intergalactic medium (DMIGM) may significantly deviate from the mean value. Therefore, using the mean value to estimate DMIGM may strongly bias the results. A usual approach is to treat DMIGM as a Gaussian distribution centered on the mean value. The Gaussian distribution is useful owing to the Gaussianity of structures on large scales; however, a large skew may emerge from several large-scale structures. Considering this, Macquart et al. [18] proposed a quasi-Gaussian distribution of DMIGM based on hydrodynamic simulations and theoretical motivation. Different treatments of DMIGM may affect the constraint on cosmological parameters.

      In this study, we used 18 well-localized FRBs to constrain the Hubble constant H0. The probability distributions of DMIGM and DMhost were considered, and the free parameters that govern the probability distributions were constrained simultaneously with the Hubble constant. The rest of this paper is organized as follows. Sec. II introduces the Bayesian method used to constrain cosmological parameters. Sec. III reports on the observational data and constraining results. Sec. IV analyzes the Monte Carlo simulations performed to check the validity of the proposed method and compares the impacts of different probability distributions of DMIGM. Finally, a discussion and conclusions are provided in Sec. V.

    II.   METHODS
    • When an FRB signal travels from the host galaxy to the Earth, it experiences a time delay due to the interaction of electromagnetic waves with free electrons. This time delay depends on the frequency of the electromagnetic waves. Measuring the time delay between photons of different frequencies allows determining the number density of electrons integrated along the wave path; this is referred to as the dispersion measure (DM). The total dispersion measure of an extragalactic FRB can be generally separated into four components [18],

      DMobs=DMMW+DMhalo+DMIGM+DMhost1+z,

      (1)

      where DMMW comes from the contribution of Galactic interstellar medium, which can be estimated from Galactic electron density models such as the NE2001 model [34]. The DMhalo term is the contribution from the Galactic halo, which is not fully constrained yet; Prochaska et al. [35] provided an estimated value of approximately 50100 pc cm3. In this study, we followed Macquart et al. [18] and assumed DMhalo=50 pc cm3. The DMhost term is the contribution from the host galaxy, and the factor 1+z arises from the cosmic expansion. The DMIGM term is the contribution from the intergalactic medium; it carries the information of the universe. Other studies also mentioned the DMsource term [3, 4, 23], which represents the DM contribution from the immediate environment of the source. The DMsource term is highly dependent on the source environment of each FRB, which is not fully understood yet. The value of DMsource is usually smaller than the error of DMhost and DMIGM [36]. Therefore, we ignored the DMsource term in this study. The DMMW and DMhalo terms can be subtracted from the total observed DM, leaving behind the extragalactic DM, which is defined as

      DMEDMobsDMMWDMhalo=DMIGM+DMhost1+z.

      (2)

      In the flat ΛCDM model, the average value of the DMIGM term can be expressed as [37, 38]

      DMIGM(z)=3cH0ΩbfIGMfe8πGmpz01+zΩm(1+z)3+ΩΛ dz,

      (3)

      where H0 is the Hubble constant, Ωb is the cosmic baryon mass density, Ωm is the matter density, and ΩΛ is the vacuum energy density of the Universe; c, G, and mp are three constants that represent the speed of light in vacuum, the Newtonian gravitational constant, and the mass of proton, respectively; fe=YHXe,H+YHeXe,He/2 denotes the extent of ionization progress of hydrogen and helium, where YH=0.75 and YHe=0.25 are the hydrogen and helium mass fractions, respectively; Xe,H and Xe,He are the corresponding ionization fractions. Given that both hydrogen and helium are expected to be fully ionized at z3 [39, 40], we set Xe,H=Xe,He=1; fIGM is the fraction of baryon mass in IGM.

      Note that Eq. (3) only describes the mean value of DMIGM. Its real value is associated with large-scale matter density fluctuations and may vary around the mean value. Usually, a Gaussian distribution is used to describe DMIGM [4143]. However, theoretical results and numerical simulations show that the probability distribution of DMIGM can be modeled using a quasi-Gaussian distribution [18, 44],

      pIGM(Δ)=AΔβexp[(ΔαC0)22α2σ2IGM],Δ>0,

      (4)

      where ΔDMIGM/DMIGM, σIGM is the effective standard deviation, and α and β are related to the inner density profile of gas in haloes. Hydrodynamic simulations show that α=β=3 provides the best fit to the model [18, 44]; hence, we fixed these two parameters. We also followed Macquart et al. [18] to parameterize the effective standard deviation as σIGM=Fz1/2, where F is a free parameter and z is the redshift of the FRB source. Here, A is a normalization constant, and C0 is chosen to ensure that the mean of this distribution is unity. Note that the distribution of DMIGM is a function of z, implying that A and C0 vary with z.

      The DMhost term may range from tens to hundreds of pc cm3. For example, Xu et al. [45] estimated that the DMhost of FRB20201124A can range from 10 to 310 pc cm3, while Niu et al. [46] estimated that the value of DMhost for FRB20190520B can reach 900 pc cm3. To account for the large variation of DMhost, it is often modeled with a log-normal distribution [18, 33],

      phost(DMhost|μ,σhost)=12πDMhostσhostexp[(lnDMhostμ)22σ2host],

      (5)

      where μ and σhost are the mean and standard deviation of lnDMhost, respectively. This log-normal distribution allows for the appearance of large values of DMhost. Generally, both μ and σhost might be redshift-dependent. However, Zhang et al. [33] showed that, for non-repeating bursts, they do not vary significantly with redshift. Lin et al. [32] also proved that there is no strong evidence for redshift evolution of DMhost within the present data. Hence, we followed Macquart et al. [18] and treated them as two constants.

      Given that it is challenging to separate the terms DMIGM and DMhost, we introduce the probability distribution of the extragalactic DM as [18]

      pE(DME|z)=(1+z)DME0pIGM(DMEDMhost1+z|H0,F)phost(DMhost|μ,σhost)dDMhost.

      (6)

      If a large sample of well-localized FRBs is observed, the joint likelihood function can be expressed as

      L(FRBs|H0,μ,σhost,F)=Ni=1pE(DME,i|zi),

      (7)

      where N is the number of FRBs. According to the Bayesian theorem, the posterior probability density function of the free parameters is given by

      P(H0,μ,σhost,F|FRBs)L(FRBs|H0,μ,σhost,F)P0(H0,μ,σhost,F),

      (8)

      where P0 is the prior probability function of the parameters.

    III.   DATA AND RESULTS
    • To date, over 30 published well-localized extragalactic FRBs have identified host galaxies and well-measured redshifts 1. Among them, FRB20200120E, FRB20190614D, FRB20190520B, and FRB 20220319D are excluded. FRB20200120E is so close to the Milky Way that the peculiar velocity dominates the Hubble flow, resulting in a negative spectroscopic redshift of z=0.001 [47, 48]. FRB20190614D only has a photometric redshift of z0.6 [49], with no available spectroscopic redshift. The value of DMhost for FRB20190520B is estimated to be as large as 900 pc cm3 [46], which is much larger than the value of normal FRBs. FRB20220319D is also excluded owing to its total DM, which is lower than DMMW [50]. The remaining 35 FRBs have well-measured spectroscopic redshifts, and their main properties are listed in Table 1. These FRBs were used to constrain the cosmological parameters in this study.

      FRBs RA Dec DMobs/() DMMW/pccm3 DME/pccm3 zsp/pccm3 Reference
      20121102A 82.99 33.15 557.00 157.60 349.40 0.1927 Chatterjee et al. [12]
      20171020A 22.15 −19.40 114.10 38.00 26.10 0.0087 Li et al. [51]
      20180301A 93.23 4.67 536.00 136.53 349.47 0.3305 Bhandari et al. [52]
      20180916B 29.50 65.72 348.80 168.73 130.07 0.0337 Marcote et al. [53]
      20180924B 326.11 −40.90 362.16 41.45 270.71 0.3214 Bannister et al. [54]
      20181030A 158.60 73.76 103.50 40.16 13.34 0.0039 Bhardwaj et al. [55]
      20181112A 327.35 −52.97 589.00 41.98 497.02 0.4755 Prochaska et al. [56]
      20190102C 322.42 −79.48 364.55 56.22 258.33 0.2913 Macquart et al. [18]
      20190523A 207.06 72.47 760.80 36.74 674.06 0.6600 Ravi et al. [57]
      20190608B 334.02 −7.90 340.05 37.81 252..24 0.1178 Macquart et al. [18]
      20190611B 320.74 −79.40 332.63 56.60 226.03 0.3778 Macquart et al. [18]
      20190711A 329.42 −80.36 592.60 55.37 487.23 0.5217 Macquart et al. [18]
      20190714A 183.98 −13.02 504.13 38.00 416.13 0.2365 Heintz et al. [58]
      20191001A 323.35 −54.75 507.90 44.22 413.68 0.2340 Heintz et al. [58]
      20191228A 344.43 −29.59 297.50 33.75 213.75 0.2432 Bhandari et al. [52]
      20200430A 229.71 12.38 380.25 27.35 302.90 0.1608 Bhandari et al. [52]
      20200906A 53.50 −14.08 577.80 36.19 491.61 0.3688 Bhandari et al. [52]
      20201124A 77.01 26.06 413.52 126.49 237.03 0.0979 Fong et al. [59]
      20210405I 255.34 −48.48 565.17 468.18 46.99 0.066 Driessen et al. [60]
      20210410D 326.09 −78.68 572.62 56.20 466.45 0.1415 Caleb et al. [61]
      20210603A 10.27 21.23 500.15 40.00 410.15 0.1772 Cassanelli et al. [62]
      20211127A 199.81 −18.84 234.83 41.75 143.08 0.0496 Khrykin et al. [63]
      20211212A 157.35 1.36 206.00 38.29 117.71 0.0713 Khrykin et al. [63]
      20220207C 310.20 72.88 262.38 74.99 137.39 0.0430 Law et al. [64]
      20220307B 350.87 72.19 499.27 119.82 329.45 0.2481 Law et al. [64]
      20220310F 134.72 73.49 462.24 45.07 367.17 0.4780 Law et al. [64]
      20220418A 219.10 70.10 623.25 36.49 536.76 0.6220 Law et al. [64]
      20220506D 318.04 72.83 396.97 82.88 264.09 0.3004 Law et al. [64]
      20220509G 282.67 70.24 269.53 55.36 164.17 0.0894 Law et al. [64]
      20220825A 311.98 72.58 651.24 77.26 523.98 0.2414 Law et al. [64]
      20220912A 347.27 48.71 220.7 120.44 50.26 0.0771 Zhang et al. [65]
      20220914A 282.06 73.34 631.28 54.38 526.9 0.1139 Law et al. [64]
      20220920A 240.26 70.92 314.99 39.62 225.37 0.1582 Law et al. [64]
      20221012A 280.80 70.52 441.08 53.69 337.39 0.2847 Law et al. [64]
      20221022A 48.63 86.87 116.84 60.12 6.72 0.0149 Mckinven et al. [66]

      Table 1.  Properties of the Host/FRB catalog. Column 1: FRB name; Columns 2 and 3: right ascension and declination of the FRB source on the sky; Column 4: observed DM; Column 5: DM of the Milky Way ISM calculated using the NE2001 model; Column 6: extragalactic DM calculated by subtracting DMMW and DMhalo from the observed DMobs, assuming DMhalo=50pccm3for the Milky Way halo; Column 7: spectroscopic redshift; Column 8: references.

      We employed Monte Carlo Markov Chain (MCMC) analysis to constrain H0 and three free parameters (eμ, σhost,F) from the probability functions of DMhost and DMIGM. We used eμ instead of μ as a free parameter because eμ directly represents the median value of DMhost. For fIGM, there is no precise constraint yet. It may slowly increase with redshift [67]; Li et al. [31] parameterized it as fIGM=fIGM,0(1+αz/(1+z)). However, there is no strong evidence showing that fIGM increases with redshift in the current FRB data [68]. To be conservative, we assumed that fIGM follows a uniform distribution U (0.747, 0.913) and marginalized over it [26]. Additionally, we set Ωbh2=0.0224 and Ωm=0.315 according to Planck 2018 results [69]. The posterior probability density functions of the free parameters were calculated using the publicly available Python code emcee [70]. Based on previous results [18, 71], flat priors were applied to the four parameters as follows: H0U(0,100)kms1Mpc1, eμU(20,200) pc cm3, σhostU(0.2,2), and FU(0.01,0.5).

      The 2D marginalized posterior distributions and 13 σ confidence contours of the four parameters are plotted in the left panel of Fig. 1. It is evident that only the parameters H0 and σhost are tightly constrained, while the other two parameters, i.e., eμ and F, are not well-constrained. A possible reason is that the FRB sample is not large enough to simultaneously constrain a model with too many free parameters. A notable feature is that the best-fitting value of the Hubble constant is H0=49.34+4.393.83kms1Mpc1, which is much lower than the Planck 2018 value. This discrepancy may be caused by the correlation between parameters. As can be seen from the contour plot, H0 seems to be positively correlated with eμhost and negatively correlated with F.

      Figure 1.  Constraints on the four free parameters (H0, eμ, σhost, F) and three free parameters (H0, eμ, σhost) using the 35 FRBs samples. The black-dashed lines from left to right in each subfigure represents the 16%, 50%, and 84% quantiles of the distribution, respectively. The contours from the inner to outer represent 1σ, 2σ, and 3σ confidence regions, respectively. For simplicity, units are omitted in this figure.

      Several previous studies suggest that a suitable value for the parameter F is approximately F=0.2, or even smaller for z1 [72, 73]. However, our four-parameter fit from the FRB sample indicates that a larger value of F may be more suitable, although it cannot be tightly constrained. This discrepancy might introduce bias to the other parameters. To address this issue, we fixed F=0.2 while keeping the other three parameters (H0, eμ, σhost) free. The corresponding 2D marginalized posterior distributions and 13 σ confidence contours of the three parameters are presented in the right panel of Fig. 1. Note that both H0 and σhost are well-constrained, and the parameter eμhost is also constrained but with a relatively small value. The best-fitting Hubble constant, H0=60.99+4.574.90kms1Mpc1, is notably larger than that from the four-parametric fit. However, the median value of the Hubble constant is still relatively lower than the Planck 2018 value. A possible explanation is that the FRB sample is not large enough to provide a robust result. The results for both parameters from the probability functions of DMhost are eμhost=48.35+20.8115.00 pc cm3 and σhost=1.63+0.230.20, both of which have relatively large uncertainties.

    IV.   MONTE CARLO SIMULATION
    • With the progress of observational techniques, more and more FRBs are expected to be detected, and a fraction of them can be well-localized. Therefore, it is interesting to assess the constraining capability of a large sample of FRBs across a wide redshift range. To this end, we performed Monte Carlo simulations to investigate the efficiency of the proposed method.

      The intrinsic redshift distribution of FRBs is still unclear because of the small well-localized sample. Li et al. [31] assumed that FRBs have a constant comoving number density but with a Gaussian cutoff. Zhang et al. [38] argued that the redshift distribution of FRBs is expected to be related with cosmic star formation rate (SFR), or influenced by the compact star merger but with an additional time delay. In this study, we adopted the SFR-related model, in which the probability density function takes the form [38]

      P(z)4πD2c(z)SFR(z)(1+z)H(z),

      (9)

      where Dc(z)=z0cdzH(z) represents the comoving distance, c denotes the speed of light, H(z) represents the Hubble expansion rate, and SFR takes the form [74]

      SFR(z)=0.02[(1+z)aη+(1+zB)bη+(1+zC)cη]1/η,

      (10)

      where a=3.4, b=0.3, c=3.5, B=5000, C=9, and η=10.

      We simulated a set of mock FRB samples to constrain the parameters. The simulations were performed according to the flat ΛCDM model with Planck 2018 results [69]. The fiducial parameters were set as F=0.2, fIGM=0.83, eμ=100 pc cm3, and σhost=1.0. The simulation procedures were as follows. First, a certain number of redshifts were randomly drawn from Eq. (9), with zmax set to 3.0. Subsequently, the same number of Δ values were randomly drawn from Eq. (4). The average DMIGM(z) was calculated using Eq. (3), and DMIGM was obtained for each redshift as Δ×DMIGM(z). Next, the same number of DMhost values were randomly drawn from Eq. (5), and DME was calculated from Eq. (2). A sample of mock FRBs (zi,DME,i) was also part of the simulations.

      We used these mock FRBs to constrain the four parameters (H0, eμ, σhost, F) using the same method described above. The corresponding contour plots constrained from N=100 mock FRBs are shown in the left panel of Fig. 2. Note that only the parameter H0 can be tightly constrained. Note also that the parameter H0 fails to recover the fiducial value within 1σ uncertainty.

      Figure 2.  (color online) Constraints on the four free parameters (H0, eμ, σhost, F) using 100 mock FRBs with quasi-Gaussian (left panel) and Gaussian (right panel) distributions of DMIGM. The blue lines represent the fiducial values. The black-dashed lines from left to right in each subfigure represents the 16%, 50%, and 84% quantiles of the distribution, respectively. The contours from the inner to outer represent 1σ, 2σ, and 3σ confidence regions, respectively. For simplicity, units are omitted in this figure.

      For comparison, we also conducted MCMC analysis using a Gaussian distribution of DMIGM. In this case, the method is similar to that of the quasi-Gaussian distribution, with the primary difference being the replacement of the distribution function of DMIGM (i.e., Eq. (4)) with the Gaussian distribution G(DMIGM,σIGM) for the generation of mock FRBs samples and in the joint likelihood function. In this simulation, the fiducial value of σIGM was set to 100 pc cm3 [41, 42]. In the MCMC analysis, σIGM was treated as a free parameter, replacing the parameter F of the quasi-Gaussian case. The prior of σIGM was U(0,200) pc cm3. We employed mock FRBs generated with a Gaussian distribution of DMIGM to constrain the four parameters (H0, eμ, σhost, σIGM). The corresponding contour plots constrained from N=100 mock FRBs are shown in the right panel of Fig. 2. Note that with the Gaussian distribution of DMIGM, the parameters H0 and σhost can properly recover their fiducial values. However, the other two parameters (eμ, σIGM), especially σIGM, cannot be tightly constrained, showing a strong bias with respect to the fiducial value.

      Given that the best-fitting value of F is much larger than the expected one, we fixed F=0.2 to explore whether the results could be improved. The corresponding contour plots constrained from N=100 mock FRBs with quasi-Gaussian distribution of DMIGM are shown in the left panel of Fig. 3. With 100 mock FRBs, we obtained a well-constrained Hubble constant, providing a more precise result compared to that with 35 real FRBs. The other two parameters, i.e., eμ and σhost, were also well-constrained but exhibited a relatively large uncertainty. Regardless of this large uncertainty, the three parameters can properly recover the fiducial values within 1σ uncertainty. However, with a thousand repetitions, each involving a sample of 100 mock FRBs, nearly half of the estimates for eμ and σhost could not be well constrained. Therefore, caution is advised when discussing the results of eμ and σhost in real-world situations.

      Figure 3.  (color online) Constraints on the three free parameters (H0, eμ, σhost) using 100 mock FRBs with quasi-Gaussian (left panel) and Gaussian (right panel) distributions of DMIGM. The blue lines represent the fiducial values. The black-dashed lines from left to right in each subfigure represents the 16%, 50% and 84% quantiles of the distribution, respectively. The contours from the inner to outer represent 1σ, 2σ, and 3σ confidence regions, respectively. For simplicity, units are omitted in this figure.

      For comparison, the contour plots corresponding to a Gaussian distribution of DMIGM constrained from N= 100 mock FRBs are also presented in the right panel of Fig. 3. Note that the quasi-Gaussian distribution of DMIGM may influence the results for eμ and σhost. However, concerning the Hubble constant, the quasi-Gaussian distribution of DMIGM does not affect its reliability, yielding consistent results. We also compared with N=500 mock FRBs; the results are presented in Fig. 4. For the Gaussian distribution of DMIGM, the three parameters are well-constrained and can recover the fiducial values. In the case of the quasi-Gaussian distribution, the parameter eμ is not well-constrained, as mentioned before, owing to the influence of the DMIGM distribution. The Hubble constant H0 can still be constrained, and the fiducial value is recovered within 1σ uncertainty, albeit with a relatively small value. These results indicate that further enlarging the FRB sample size does not significantly improve the precision of the constraint on H0, primarily owing to the uncertainty associated with fIGM.

      Figure 4.  (color online) Constraints on the three free parameters (H0, eμ, σhost) using 500 mock FRBs with quasi-Gaussian (left panel) and Gaussian (right panel) distributions of DMIGM. The blue lines represent the fiducial values. The black-dashed lines from left to right in each subfigure represents the 16%, 50%, and 84% quantiles of the distribution, respectively. The contours from the inner to outer represent 1σ, 2σ, and 3σ confidence regions, respectively. For simplicity, units are omitted in this figure.

      To mitigate simulation fluctuations, we repeated the simulation 100 times. Specifically, we randomly generated 100 FRB samples, with each sample containing 100 mock FRBs. The 100 mock samples were then used to constrain the three free parameters (H0, eμ, σhost) using the method described above, resulting in 100 sets of best-fitting parameters. The distributions of the best-fitting parameter H0 in the 100 simulations are shown in the left panel of Fig. 5. The fiducial values are also shown as red-solid lines. Note that the Hubble constant can be tightly constrained in all cases. For comparison, we also present the results with a Gaussian distribution of DMIGM in the right panel of Fig. 5. It is evident that with the quasi-Gaussian distribution of DMIGM, the best-fitting value of H0 is systematically lower than the fiducial value, although it is still consistent with the fiducial value within 1σ uncertainty. On the contrary, if DMIGM is modeled with a Gaussian distribution, the best-fitting value of H0 properly recovers the fiducial value. This implies that the choice of the distribution of DMIGM may cause bias on the estimation of the Hubble constant.

      Figure 5.  (color online) Distribution of the best-fitting value for the Hubble constant shown for quasi-Gaussian (left panel) and Gaussian (right panel) distributions of DMIGM in 100 simulations, with N=100 FRBs in each simulation. The black dots and blue lines represent the median values of the Hubble constant and their corresponding 1σ uncertainties, respectively. The red lines represent the fiducial values.

    V.   DISCUSSION AND CONCLUSIONS
    • In this study, we employed the Bayesian inference method to constrain the Hubble constant H0, together with three FRB-related parameters (eμ, σhost, F) using 35 well-localized FRBs. We found that 35 FRBs could not tightly constrain four parameters simultaneously. Specifically, although the Hubble constant could be constrained, the best-fitting value H0=49.34+4.393.83kms1Mpc1 is significantly lower than that value measured by CMB. Additionally, the FRB-related parameter F was found to be much larger than expected. Subsequently, we addressed this discrepancy by fixing the parameter F=0.2 according to other observations. The three-parameter fit yielded H0=60.99+4.574.90 km s1 Mpc1, a value that is still smaller than the results from CMB measurements. The relatively small median value of the Hubble constant may be attributed to the limited size of the FRB sample or the correlation between H0 and FRB-related parameters.

      Considering the anticipated growth in the number of well-located FRBs in the future, we conducted Monte Carlo simulations to assess the efficiency of the proposed method. We found that for 100 mock FRBs, H0 was effectively constrained; however, the best-fitting value of H0 is still lower than the fiducial value. This situation persists even if the number of FRBs is increased to 500, and the precision of the results did not significantly improve owing to the uncertainty of fIGM. Further simulations revealed a systematic bias in the estimation of the Hubble constant using FRBs, although the best-fitting value of H0 is consistent with the fiducial value within 1σ uncertainty. This is mainly due to the quasi-Gaussian distribution of DMIGM, which is correlated with the Hubble constant. The bias in DMIGM causes a biased estimation of H0.

      The quasi-Gaussian distribution of DMIGM accounts for the large skew resulting from a few large-scale structures in the universe. However, the non-Gaussianity may impact the fitting results of cosmological parameters, especially the Hubble constant. To prove our hypothesis, we simulated a set of FRB samples whose DMIGM values were drawn from a Gaussian distribution. We then used a mock sample to constrain the Hubble constant, and compared the results with the quasi-Gaussian case, as shown in Fig. 5. We found that in the Gaussian case, the best-fitting value of H0 correctly recovers the fiducial value. However, in the quasi-Gaussian case, the estimation of H0 is significantly systematically biased toward a lower value than the fiducial value. Recognizing and addressing this bias is crucial for refining the fitting results of the Hubble constant.

      Recently, Wu et al. [26] also used well-localized FRBs to constrain H0 employing a quasi-Gaussian distribution of DMIGM. They obtained values of H0=68.81+4.994.33kms1Mpc1 for a fixed value of fIGM=0.83, and H0=69.31+6.216.63kms1Mpc1 if fIGM follows a uniform distribution U(0.747,0.913). The median values of both results are relatively higher than ours. The reasons for this difference are diverse. First, the FRB sample employed in the present study was much larger than theirs. Second, for other cosmological parameters, such as Ωbh2 and Ωm, Wu et al. [26] treated them as free parameters but with a uniform prior limited to a very small range. This is approximately equivalent to fixing these two parameters, as done in our study. Third, for the DMhalo term, we set it as a constant, while Wu et al. [26] described it using a Gaussian distribution. Finally, the main difference is the treatment of FRB-related parameters. For parameters (A, C0, eμ, σhost, F) in the probability functions of DMhost and DMIGM, Wu et al. [26] set their values according to the results of the state-of-the-art IllustrisTNG simulation. This approach may introduce a loop problem due to the fiducial parameter settings in the IllustrisTNG simulation. The method employed by Wu et al. [26] is equivalent to fixing the FRB-related parameters in the MCMC analysis. In the present study, the FRB-related parameters were set free. Given that the FRB-related parameters are correlated with the Hubble constant, the estimation on H0 in this study is biased and has large uncertainty.

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