Cosmic Dispersion Measure from Gamma-Ray Burst Afterglows: Probing the Reionization History and the Burst Environment

We show a possible way to measure the column density of free electrons along the light path, the so-called Dispersion Measure (DM), from the early $[\sim 415 (\nu/1 {\rm GHz})^{-2} ({\rm DM}/10^{5} {\rm pc} {\rm cm}^{-3}) {\rm s}]$ radio afterglows of the gamma-ray bursts. We find that the proposed Square Kilometer Array can detect bright radio afterglows around the time $\sim 10^{3}(\nu/160 {\rm MHz})^{-2}$ s to measure the intergalactic DM ($\simg 6000$ pc cm$^{-3}$ at redshift $z>6$) up to $z\sim 30$, from which we can determine the reionization history of the universe and identify the missing warm-hot baryons. At low $z$, DM in the host galaxy may reach $\sim 10^{5}$ pc cm$^{-3}$ depending on the burst environment, which may be probed by the current detectors. Free-free absorption and diffractive scattering may also affect the radio emission in a high density.