Was reionization complete by z ~ 5-6?

It is generally taken for granted that reionization has completed by z=6, due to the detection of flux in the Lyman alpha forest at redshifts z<6. However, since reionization is expected to be highly inhomogeneous, much of the spectra pass through just the ionized component of the intergalactic medium (IGM) even for non-negligible values of the volume-weighted mean neutral hydrogen fraction, x_HI. We study the expected signature of an incomplete reionization at z ~ 5--6, using very large-scale (2 Gpc) seminumeric simulations. We find that ruling out an incomplete reionization is difficult at these redshifts since: (1) quasars reside in biased regions of the ionization field, with fewer surrounding HI patches than implied by the global mean, x_H; (2) absorption from the residual neutral hydrogen inside the ionized IGM generally dominates over the absorption from the remaining HI regions; (3) modeling the Lyman alpha forest and its redshift evolution even in just the ionized IGM is very difficult, and nearly impossible to do a priori. We propose using the fraction of pixels which are dark as a simple, nearly model-independent upper limit on x_HI. Alternately, the size distribution of regions with no detectable flux (dark gaps) can be used to place a more model dependent constraint. Either way, the current sample of quasars is statistically insufficient to constrain x_HI at z~6 to even the 10 per cent level. At z~5, where there are more available sightlines and the forest is less dark, constraining x_HI < 0.1 might be possible given a large dynamic range from very deep spectra and/or the Lyman beta forest. We conclude with the caution against over-interpreting the observations. There is currently no direct evidence that reionization was complete by z ~ 5-6.