Hubble constant constraint using 117 FRBs with a more accurate probability density function for ${\rm DM}_{\rm diff}$

Fast radio bursts (FRBs) are among the most mysterious astronomical transients. Due to their short durations and cosmological distances, their dispersion measure (DM) - redshift ($z$) relation is useful for constraining cosmological parameters and detecting the baryons in the Universe. The increasing number of localized FRBs in recent years has provided more precise constraints on these parameters. However, the larger dataset reveals limitations in the widely used probability density function ($p_{\rm diff}$) for ${\rm DM}_{\rm diff}$, which refers to the diffuse electron term of FRB DM. In this project, we collect 117 of the latest, localized FRBs, discuss the effect of a more accurate $\sigma_{\rm diff}$, which is a parameter in $p_{\rm diff}$ and once thoughts as ``effective standard deviation'', and more clearly rewrite their likelihood to better constrain the parameters above. We find that the widely used approximation $\sigma_{\rm diff} \sim F/\sqrt{z}$ only works under contrived assumptions and shows the greatest deviation from the true standard deviation in low redshift. In general, one should use an accurate method to derive this parameter from $p_{\rm diff}$. Our method yields better constraints on $H_0\Omega_b f_{\rm diff} = 2.813_{-0.258}^{+0.250}\;{\rm km/s/Mpc}$ or $H_0 = 66.889_{-5.459}^{+6.754} \;{\rm km/s/Mpc}$ when combining the FRB data with CMB measurements and taking $f_{\rm diff} = 0.84$. This fully analytical correction helps us better constrain cosmological parameters with the increasing number of localized FRBs available today.