Inhomogeneous Helium Reionization and the Equation of State of the Intergalactic Medium

The temperature of the intergalactic medium (IGM) is set by the competition between photoheating and adiabatic cooling, which are usually assumed to define a tight equation of state in which the temperature increases monotonically with density. We use a semi-analytic model, accurate at low to moderate IGM densities (<5 times the mean), to show that inhomogeneous reionization can substantially modify these expectations. Because reionization is driven by biased sources, dense pockets of the IGM are likely to be ionized first. As a result, voids initially remain cool while dense regions are heated substantially. However, near the end of reionization, dense regions have already cooled from their initially large temperature while voids have only just been heated. Thus, near the end of helium reionization the equation of state can invert itself, with the hottest gas inside voids. The degree to which this happens depends on the magnitude of the density bias during reionization: if rare, bright sources dominate reionization, so that each ionized region contains a typical volume of the IGM, the equation of state will remain monotonic. We also show that the distribution of temperatures at a fixed density has significant scatter and evolves rapidly throughout and even after reionization. Finally, we show that the observed temperature jump at z ~3.2 is consistent with the behavior at the end of helium reionization, although it requires a somewhat larger temperature increase than expected.