We use ultra-deep ultraviolet VLT/VIMOS intermediate-band and VLT/FORS1 narrowband imaging in the GOODS Southern field to derive limits on the distribution of the escape fraction (fesc) of ionizing radiation for L ≥ L*z =3 Lyman-break galaxies (LBGs) at redshift 3.4-4.5. Only one LBG, at redshift z = 3.795, is detected in its Lyman continuum (LyC; S/N ≃ 5.5), the highest redshift galaxy currently known with a direct detection. Its ultraviolet morphology is quite compact (Reff = 0.8 kpc physical). Three out of seven active galactic nuclei are also detected in their LyC, including one at redshift z = 3.951 and z850 = 26.1. From stacked data (LBGs), we set an upper limit to the average fesc in the range 5%-20%, depending on how the data are selected (e.g., by magnitude and/or redshift). We undertake extensive Monte Carlo simulations that take into account intergalactic attenuation, stellar population synthesis models, dust extinction, and photometric noise in order to explore the moments of the distribution of the escaping radiation. Various distributions (exponential, log-normal, and Gaussian) are explored. We find that the medianfesc is lower than ≃6% with an 84% percentile limit not larger than 20%. If this result remains valid for fainter LBGs down to current observational limits, then the LBG population might be not sufficient to account for the entire photoionization budget at the redshifts considered here, with the exact details dependent upon the assumed ionizing background and QSO contribution thereto. It is possible that fesc depends on the UV luminosity of the galaxies, with fainter galaxies having higher fesc, and estimates of f esc from a sample of faint LBGs from HUDF (i775 ≤ 28.5) are in broad quantitative agreement with such a scenario.
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