Baryonification III: An accurate analytical model for the dispersion measure probability density function of fast radio bursts

We develop an analytical framework to predict the one-point probability distribution function (PDF) of dispersion measures (DMs) for fast radio bursts (FRBs) within the baryonification (BFC) model. BFC provides a computationally efficient alternative to expensive hydrodynamical simulations for modelling baryonic effects on cosmological scales. By applying the halo mass function and halo bias, we convolve contributions from individual halos across a range of masses and redshifts to derive the large-scale structure contribution to the DM PDF. We validate our analytical predictions against consistency-check simulations and compare them with the IllustrisTNG hydrodynamical simulation over the redshift range $ z = 0$ to $z = 5$, demonstrating excellent agreement. We demonstrate that our model produces consistent results when fitting gas profiles and predicting the PDF, and vice versa. We show that the BFC parameters controlling the gas profile, particularly the halo mass scale ($M_\mathrm{c}$), mass-dependent slope ($\mu$), and outer truncation ($\delta$), are the primary drivers of the PDF shape. Additionally, we investigate the validity of the log-normal approximation commonly used for DM distributions, finding that it provides a sufficient description for a few hundred FRBs. Our work provides a self-consistent model that links gas density profiles to integrated DM statistics, enabling future constraints on baryonic feedback processes from FRB observations.