Probing the baryon mass fraction in IGM and its redshift evolution with fast radio bursts using Bayesian inference method

We investigate the fraction of baryon mass in intergalactic medium ($f_{\mathrm{I}\mathrm{G}\mathrm{M}}$), using 18 well-localized FRBs in the redshift range $z\in (0.0039,0.66)$. We construct a five-parameter Bayesian inference model, with the probability distributions of dispersion measures (DM) of IGM and host galaxy properly taken into account. To check the possible redshift evolution, we parameterize $f_{\mathrm{I}\mathrm{G}\mathrm{M}}$ as a mildly evolving function of redshift, $f_{\mathrm{I}\mathrm{G}\mathrm{M}}=f_{\mathrm{I}\mathrm{G}\mathrm{M},0}[1+\alpha z\mathrm{/}(1+z)]$. By simultaneously constraining five parameters, we get $f_{\mathrm{I}\mathrm{G}\mathrm{M},0}=0.92_{-0.12}^{+0.06}$ and $\alpha =0.49_{-0.47}^{+0.59}$, and the median value of DM of host galaxy is $\exp (\mu )=72.49_{-25.62}^{+33.31}\mathrm{p}\mathrm{c}\mathrm{c}{\mathrm{m}}^{-3}$. By fixing two parameters which can be constrained independently with other observations, we obtain $\alpha =0.11_{-0.27}^{+0.24}$ in the three-parameter fit, which is consistent with zero within $1\sigma$ uncertainty. Monte Carlo simulations show that even 300 FRBs are not enough to tightly constrain five parameters simultaneously. This is mainly caused by the correlation between parameters. Only if two parameters are fixed, 100 FRBs are necessary to achieve unbiased constraints on the remaining parameters.