The temperature of the ionised IGM, as inferred from the b-parameter in the Lya forest, has not previously been modelled correctly. Using a radiative transfer code coupled to a n-body code, the temperature structure of the IGM due to the passage of individual ionisation fronts can be modelled. I present the results of simulations performed with different spectra of ionising radiation: a quasar (power law), miniquasar, starburst, and a time-varying spectrum that evolves from a starburst spectrum to a quasar. The time-varying spectrum simulation reproduces not only the observed mean temperatures, but also the large fluctuations in the HeII to HI number density ratio observed, suggesting the measured values indicate the onset of the HeII reionisation epoch.
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