At redshifts z greater than or similar to2, most of the baryons reside in t
he smooth intergalactic medium which is responsible for the low column dens
ity Ly alpha forest. This photoheated gas follows a tight temperature-densi
ty relation which introduces a cut-off in the distribution of widths of the
Ly alpha absorption lines (b-parameters) as a function of column density.
We have measured this cut-off in a sample of nine high-resolution, high sig
nal-to-noise ratio quasar spectra and determined the thermal evolution of t
he intergalactic medium in the redshift range 2.0-4.5. At a redshift z simi
lar to3, the temperature at the mean density shows a peak and the gas becom
es nearly isothermal. We interpret this as evidence for the reionization of
He II.