The evolution of the effective equation of state of the intergalactic medium

We develop a method to extract the "effective equation of state" of the int ergalactic medium from the Doppler (b) parameter distribution of the low-de nsity Ly alpha forest. We test the method on numerical simulations and then apply it to published observations of the Ly alpha forest at redshifts z s imilar or equal to 0-4. We find that the effective equation of state is clo se to isothermal at redshift z similar to 3, indicating that a second rehea ting of the intergalactic medium took place at z similar to 3. This reheati ng can plausibly be identified with the reionization of He II observed to o ccur at z similar to 3.