  The Doppler parameter distribution of Lyman-alpha absorption is calculated
for a set of different reionization histories. The differences in temperature
between different reionization histories are as large as a factor three to four
depending on the spectrum of the ionizing sources and the redshift of helium
reionization. These temperature differences result in observable differences in
the Doppler parameter distribution. Best agreement with the observed Doppler
parameter distribution between redshift two and four is found if hydrogen and
helium are reionized simultaneously at or before redshift five with a
quasar-like spectrum.
