Where are all the Population III galaxies? We can now see back to 400 million years after the Big Bang.  Yet we still haven't seen any Population III galaxies.  Why is this?  How do metal-rich galaxies evolve in just 400 million years?
 A: As far as I know, the highest redshift galaxy confirmed is at $z=11.1$, corresponding to a mere 400 million years after the big bang (Oesch et al. 2016). This galaxy, and a few others like it, were discovered because of their extremely red colours and a sharp step in their spectral energy distribution in the near-infrared, known as the Lyman break. Photons shortward of the Lyman limit in the rest frame of the galaxy (91.2 nm) are heavily absorbed by neutral hydrogen in that galaxy. This absorption edge is then redshifted by a factor $1+z$ in the observed spectral energy distribution. However, in the early universe, prior to the main epoch of reionisation at $z \sim 6$, much of the intergalactic medium is also in the form of neutral hydrogen, and the presence of this, at a range of redshifts between $6<z<11$ means that the Lyman series effectively absorbs everything shortward of the Lyman alpha line at a rest wavelength of 121.6 nm.
The discoveries are confirmed with long spectroscopic exposures in the near infrared, to establish the reality of the Lyman break, measure its wavelength and hence determine the redshift of the galaxy. The only thing that is detectable is a (weak) continuum - no spectral emission or absorption lines. So there is no information about the metallicity of the galaxy and therefore one cannot call it a population III galaxy.
It could be that subsequent measurements, perhaps by JWST, were to find little evidence for metals in such galaxies. Indeed, there has been some evidence for extremely metal-poor galaxies (Sobral et al. 2015) at redshifts of $z \sim 7$ (about 800 million years after the big bang). They detect no emission lines except Lyman alpha and ionised helium, but even here, Sobral et al. suggest that the spectra are best described as a mixture of population III and more chemically evolved populations.
It would not be surprising to find a metal-enriched interstellar medium and/or stars with metals after only a few hundred million years. Some population III stars are expected to be very massive (100-1000$M_{\odot}$), with lifetimes of only a few million years before they explode in some type of supernova (still theoretically uncertain). The cycle of enrichment will thus begin after a galaxy has formed in only a fraction of the age of the universe at that time. How effective that is will depend on all sorts of uncertain factor like what the masses of population III stars are, what the yields are from their supernovae, how important mass loss is from population III stars and how effective are the mixing processes that churn the gas from which new stars are formed. Either way, if a galaxy is discovered on the basis of its Lyman alpha or Lyman continuum emission it is likely to be dominated by the light from hot, young, very massive stars that have recently formed and exhibit the metallicity of the material of the interstellar medium at that time, and not longer-lived, somewhat lower mass stars that may well have zero/little metallicity.
