Correcting the z~8 Galaxy Luminosity Function for Gravitational Lensing Magnification Bias

We present a Bayesian framework to account for the magnification bias from both strong and weak gravitational lensing in estimates of high-redshift galaxy luminosity functions. We illustrate our method by estimating the $z\sim8$ UV luminosity function using a sample of 97 Y-band dropouts (Lyman break galaxies) found in the Brightest of Reionizing Galaxies (BoRG) survey and from the literature. We find the luminosity function is well described by a Schechter function with characteristic magnitude of $M^\star = -19.85^{+0.30}_{-0.35}$, faint-end slope of $\alpha = -1.72^{+0.30}_{-0.29}$, and number density of $\log_{10} \Psi^\star [\textrm{Mpc}^{-3}] = -3.00^{+0.23}_{-0.31}$. These parameters are consistent within the uncertainties with those inferred from the same sample without accounting for the magnification bias, demonstrating that the effect is small for current surveys at $z\sim8$, and cannot account for the apparent overdensity of bright galaxies compared to a Schechter function found recently by Bowler et al. (2014a,b) and Finkelstein et al. (2014). We estimate that the probability of finding a strongly lensed $z\sim8$ source in our sample is in the range $\sim 3-15 \%$ depending on limiting magnitude. We identify one strongly-lensed candidate and three cases of intermediate lensing in BoRG (estimated magnification $\mu>1.4$) in addition to the previously known candidate group-scale strong lens. Using a range of theoretical luminosity functions we conclude that magnification bias will dominate wide field surveys -- such as those planned for the Euclid and WFIRST missions -- especially at $z>10$. Magnification bias will need to be accounted for in order to derive accurate estimates of high-redshift luminosity functions in these surveys and to distinguish between galaxy formation models.