  The observed number density of Lyman-alpha sources implies a minimum volume
of the inter-galactic medium that must be ionized, in order to allow the
Lyman-alpha photons to escape attenuation. We estimate this volume by assigning
to each Lyman-alpha emitter the minimum Stromgren sphere that would allow half
its Lyman-alpha photons to escape. This implies a lower limit to ionized gas
volume fraction of 20-50% at redshift z=6.5. This is a lower limit in two ways:
First, we conservatively assume that the Lyman-alpha sources seen (at a
relatively bright flux limit) are the only ones present; and second, we assume
the smallest Stromgren sphere volume that will allow the photons to escape.
This limit is completely independent of what ionizing photon sources produced
the bubbles. Deeper Lyman-alpha surveys are possible with present technology,
and can strengthen these limits by detecting a higher density of Lyman-alpha
galaxies.
