Asking Fast Radio Bursts for More than Reionization History

We propose different estimators to probe the intergalactic medium (IGM) during epoch of reionization (EoR) using the dispersion measure (${\rm DM}$) of the fast radio bursts. We consider three different reionization histories, which we can distinguish with a total of $\lesssim 1000\,{\rm DM}$ measurements during EoR if their redshifts are known. We note that the redshift derivatives of ${\rm DM}$ are also directly sensitive to the reionization history. The major point of this work is to explore the variance in the ${\rm DM}$ measurements and the information encoded in them. We find that the all-sky average $\overline{{\rm DM}}(z)$ gets biased from the line-of-sight (LoS) fluctuations in the ${\rm DM}$ measurements introduced by the ionization of IGM during EoR. We find that the ratio $\sigma_{\rm DM}/\overline{{\rm DM}}$ depends directly on the ionization bubble sizes as well as the reionization history. On the other hand, we also find that angular variance (coined as $\textit{structure function}$) of ${\rm DM}$ encodes the information about the duration of reionization and the typical bubble sizes as well. We establish the usefulness of variances in ${\rm DM}$ using toy models of reionization and later verify it with the realistic reionization simulations.