Probing the Cosmic Reionization History and Local Environment of Gamma-Ray Bursts through Radio Dispersion

We discuss the effect of dispersion delay due to intervening ionized media in the radio emission of gamma-ray bursts (GRBs). For high redshift GRBs ($z \ga 3$), the ionized intergalactic medium (IGM) should be the dominant source of dispersion without substantial local or foreground contamination, offering a unique probe of the cosmic reionization epoch through measures of the free electron column density out to different redshifts. The delay times from $z \sim 10$ can be $\sim$ 1 hour at 100 MHz and $\sim$ 10 hours at 30 MHz. On the other hand, dispersion by local ionized material may be important for GRBs at lower redshifts if they occur inside or behind dense molecular clouds, providing clues to the GRB environment; free-free absorption may also be significant in this case. Detecting dispersion delay in the known radio afterglow emission should be extremely challenging due to the low fluxes at the relevant frequencies, but may be marginally possible for rare, bright afterglows by the {\it Square Kilometer Array}. If GRBs also emit prompt, coherent radio emission as predicted by Sagiv & Waxman (2002), the observational prospects can be much better; detection of a sufficient sample of high $z$ GRBs by the {\it Low Frequency Array} may allow a discrimination of different reionization histories. Interesting constraints on the low redshift, warm-hot IGM may also be obtainable through dispersion.