We develop an efficient method of studying the effects of reionization history on the temperature-density relation of the intergalactic medium in the low-density limit (overdensity δ≾5). It is applied to the study of photo-reionization models in which the amplitude, spectrum and onset epoch of the ionizing flux, as well as the cosmology, are systematically varied. We find that the mean temperature-density relation at z = 2−4 is well approximated by a power-law equation of state for uniform reionization models. We derive analytical expressions for its evolution and exhibit its asymptotic behaviour: it is found that for sufficiently early reionization, imprints of reionization history prior to z∼10 on the temperature-density relation are washed out. In this limit the temperature at the cosmic mean density is proportional to . While the amplitude of the radiation flux at the ionizing frequency of H i is found to have a negligible effect on the temperature-density relation as long as the universe reionizes before z∼5, the spectrum can change the overall temperature by about 20 per cent, through variations in the abundances of helium species. However, the slope of the mean equation of state is found to lie within a narrow range for all reionization models we study, where reionization takes place before z∼5. We discuss the implications of these findings for the observational properties of the Lyα forest. In particular, uncertainties in the temperature of the intergalactic medium, arising from the uncertain reionization history of our Universe, introduce a 30 per cent scaling in the amplitude of the column density distribution while the slope of the distribution is only affected by about 7 per cent. Finally, we discuss how a fluctuating ioiuzing field affects the above results. We argue that under certain conditions, the loss of memory of reionization history implies that at late times, the temperature-density relation of a gas in a fluctuating ionizing background can be approximated by one that results from a uniform radiation field, provided the universe reionizes sufficiently early.

Equation of state of the photoionized intergalactic medium