  We investigate the effect of helium on hydrogen reionisation using a
hydrodynamical simulation combined with the cosmological radiative transfer
code CRASH. The simulations are run in a 35.12/h comoving Mpc box using a
variety of assumptions for the amplitude and power-law extreme-UV (EUV)
spectral index, alpha, of the ionising emissivity. We use an empirically
motivated prescription for ionising sources which ensures all of the models are
consistent with constraints on the Thomson scattering optical depth and the
hydrogen photo-ionisation rate at z=6. The inclusion of helium slightly delays
reionisation due to the small number of ionising photons which reionise neutral
helium instead of hydrogen. However, helium has a significant impact on the
thermal state of the IGM. Models with alpha=3 produce IGM temperatures at the
mean density at z=6 which are about 20 % higher compared to models without
helium photo-heating. Harder EUV indices produce even larger IGM temperature
boosts. A comparison to recent observational estimates of the IGM temperature
at z=5 - 6 suggests that hydrogen reionisation was primarily driven by pop-II
stellar sources with a soft EUV index, alpha<3. We also find that faint, as yet
undetected galaxies, characterised by a luminosity function with a steepening
faint-end slope and an increasing Lyman continuum escape fraction (fesc=0.5),
are required to reproduce the ionising emissivity used in our simulations at
z>6. Finally, we note there is some tension between recent observational
constraints which indicate the IGM is > 10% neutral by volume z=7, and
estimates of the ionising emissivity at z=6 which indicate only between 1 and 3
ionising photons are emitted per hydrogen atom over a Hubble time. This tension
may be alleviated by either a lower neutral fraction at z=7 or an IGM which
still remains a few % neutral by volume at z=6.
