  Context: Population III (Pop III) stars may be fast rotating. An expected
consequence of fast rotation is strong internal mixing that deeply affects
their evolutionary tracks in the Hertzsprung-Russell diagram and hence their
ionising power. aims: We investigate the impact on the ionising power of Pop
III stars in an extreme case of internal mixing, the one leading to chemically
homogeneous evolution (CHE). In that situation, during the main sequence phase,
the star keeps the same chemical composition from its center to its surface.
Homogeneous stars have larger effective temperatures and luminosities than
stars evolving non-homogeneously and thus are stronger ionising sources.
Methods: The stellar evolution models are based on $n=3$ polytropes with a time
varying hydrogen mass fraction. The ionisation model employs the self-similar
champagne flow solution from Shu et al. (2002), as well as numerical
simulations for the stochastic treatment of star clusters over a grid of
redshifts and halo masses. Results: We find that haloes containing chemically
homogeneous stars have an escape fraction of ionising photons about twice that
of haloes containing classical Pop III stars. By extrapolating the high-$z$
ionisation history powered by Pop III stars (at $z\gtrsim 15$) to the
post-reionisation epoch, we derive the Thomson scattering optical depth $\tau$,
which is compared with the value measured by $\textit{Planck}$. We find that
$\tau$ is overproduced by $\sim1.5- 5\sigma$, when all Pop III stars evolve
homogeneously. This indicates that CHE is unlikely to be realised in the
majority of Pop III stars, although the present study cannot exclude that a
fraction of them undergoes CHE. Conclusions: Fast rotation might have a
significant impact on the ionising budget of Pop III stars, and thus on early
cosmic reionisation.
