Absorption Line Spectroscopy of Gravitationally-Lensed Galaxies: Further
  Constraints on the Escape Fraction of Ionizing Photons at High Redshift
  The fraction of ionizing photons escaping from high-redshift star-forming
galaxies remains a key obstacle in evaluating whether galaxies were the primary
agents of cosmic reionization. We previously proposed using the covering
fraction of low-ionization gas, measured via deep absorption line spectroscopy,
as a proxy. We now present a significant update, sampling seven
gravitationally-lensed sources at $4<z<5$. We show that the absorbing gas in
our sources is spatially inhomogeneous with a median covering fraction of 66\%.
Correcting for reddening according to a dust-in-cloud model, this implies an
estimated absolute escape fraction of $\simeq19\pm6$\%. With possible biases
and uncertainties, collectively we find the average escape fraction could be
reduced to no less than 11\%, excluding the effect of spatial variations. For
one of our lensed sources, we have sufficient signal/noise to demonstrate the
presence of such spatial variations and scatter in its dependence on the
Ly$\alpha$ equivalent width, consistent with recent simulations. If this source
is typical, our lower limit to the escape fraction could be reduced by a
further factor $\simeq$2. Across our sample, we find a modest anti-correlation
between the inferred escape fraction and the local star formation rate,
consistent with a time delay between a burst and leaking Lyman continuum
photons. Our analysis demonstrates considerable variations in the escape
fraction consistent with being governed by the small-scale behavior of
star-forming regions, whose activities fluctuate over short timescales. This
supports the suggestion that the escape fraction may increase toward the
reionization era when star formation becomes more energetic and burst-like.
