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 \sim 8 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 \sim 8 , and can not account for the apparent overdensity of bright galaxies compared to a Schechter function found recently by and .
We estimate that the probability of finding a strongly lensed z \sim 8 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 .