We have co-added the X-ray flux of all known Lya emitters (LAEs) in the 4 Ms Chandra Deep Field South (CDF-S) region, achieving the tightest upper limits yet on the X-ray to Lya ratio. We use the X-ray data to place sensitive upper limits on the average unobscured star formation rate (SFRX) in these galaxies. A very small fraction of Lya galaxies in the field are individually detected in the X-rays, implying a low fraction of active galactic nucleus activity. After excluding the few X-ray-detected LAEs, we stack the undetected LAEs located in the 4 Ms CDF-S data and 250 ks Extended CDF-S (ECDF-S) data, and compute a 1 sigma upper limit on SFRX < 1.6, 14, 28, 28, 140, 440, 880M(circle dot) yr(-1) for LAEs located at z similar to 0.3 and z = 2.1, 3.1, 3.2, 4.5, 5.7, and 6.5, respectively. The upper limit of SFRX in LAEs can be then compared to SFRLy alpha derived from Lya line and thus can constrain on the Lya escape fraction (f(Ly alpha)(Esc)). The f(Ly alpha)(Esc) from X-ray at z similar to 0.3 is substantially larger than that from UV or H alpha. Three X-ray-detected LAE galaxies at z similar to 0.3 show f(Ly alpha)(Esc) similar to 3%-22%, and the average Ly alpha escape fraction from stacking the X-ray-undetected LAEs show f(Ly alpha)(Esc) > 28% at 3 sigma significance level at the same redshift. We derive a lower limit on f(Ly alpha)(Esc) > 14% (84% confidence level, 1 sigma lower limit) for LAEs at redshift z similar to 2.1 and z similar to 3.1-3.2. At z > 4, the current LAE samples are not of sufficient size to constrain SFRX well. By averaging all the LAEs at z > 2, the X-ray non-detection constrains f(Ly alpha)(Esc)17% (84% confidence level, 1 sigma lower limit), and rejects f(Ly alpha)(Esc) < 5.7% at the 99.87% confidence level from 2.1 < z < 6.5.
2012. Vol. 746, no 1