The Doppler parameter distribution of Lyα absorption lines is calculated for a set of numerical simulations with different re-ionization histories. The differences in temperature between different re-ionization histories are as large as a factor of 3–4 depending on the spectrum of the ionizing sources and the redshift of helium re-ionization. These temperature differences result in observable differences in the Doppler parameter distribution. The best agreement with the observed Doppler parameter distribution between redshifts of 2 and 4 is found if hydrogen and helium are re-ionized simultaneously at or before a redshift of 5 with a quasar-like spectrum.
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Haehnelt et al. (1998) studied this question.