Limits on Population III star formation in minihaloes implied by Planck
  Recently, Planck measured a value of the cosmic microwave background (CMB)
optical depth due to electron scattering of $\tau=0.066 \pm 0.016$. Here we
show that this low value leaves essentially no room for an early partial
reionisation of the intergalactic medium (IGM) by high-redshift Population III
(Pop III) stars, expected to have formed in low-mass minihaloes. We perform
semi-analytic calculations of reionisation which include the contribution from
Pop II stars in atomic cooling haloes, calibrated with high-redshift galaxy
observations, and Pop III stars in minihaloes with feedback due to Lyman-Werner
(LW) radiation and metal enrichment. We find that without LW feedback or prompt
metal enrichment (and assuming a minihalo escape fraction of 0.5) the Pop III
star formation efficiency cannot exceed $\sim{\rm a~few}\times 10^{-4}$,
without violating the constraints set by Planck data. This excludes massive Pop
III star formation in typical $10^6 M_\odot$ minihaloes. Including LW feedback
and metal enrichment alleviates this tension, allowing large Pop III stars to
form early on before they are quenched by feedback. We find that the total
density of Pop III stars formed across cosmic time is $\lesssim
10^{4-5}~M_\odot~{\rm Mpc^{-3}}$ and does not depend strongly on the feedback
prescription adopted. Additionally, we perform a simple estimate of the
possible impact on reionisation of X-rays produced by accretion onto black hole
remnants of Pop III stars. We find that unless the accretion duty cycle is very
low ($\lesssim 0.01$), this could lead to an optical depth inconsistent with
Planck.
