ner_tags: O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O B-Wavelength B-Tag O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O B-Wavelength I-Wavelength O O O O B-Wavelength I-Wavelength O O O O O O O B-Wavelength I-Wavelength O O B-Wavelength B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope B-Survey O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O B-Formula I-Formula I-Formula B-Wavelength O O O B-Telescope O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O
tokens: where we expected to be incomplete, and fewer at bright magnitudes, where we expected to be very complete. Flux uncertainties and missing VISTA fluxes were assigned in the same way as during the contamination test in Section 3.3 . Mock galaxies were not assigned photometric flags. This likely results in an incompleteness of only a couple percent, as the fraction of all sources affected by flags is small. All sources with an S/N of greater than 3.5 in any one band are flagged in another band less than 2% of the time on average (never exceeding 4%). In addition, all images have ∼0.5% of all pixels flagged on average (never exceeding 2%). The photometric catalog of 600,000 mock galaxies was then run through EAZY to generate z phot PDFs, and our high- z galaxy selection was applied. The completeness was defined as the number of mock galaxies recovered divided by the number of input mock galaxies, as a function of input absolute magnitude and redshift. Figure 7 shows the results of our simulation. We define the 50%-completeness limit as the absolute magnitude where the area under the curve falls to less than 50% of the areas under the average of the M UV, i ′ = −25 to M UV, i ′ = −28 curves, which we find to be at M UV, i ′ = −22 ( m = 24 for z = 4). 4. Results 4.1. The Rest-frame UV <italic>z</italic> = 4 Luminosity Function We utilize the effective volume method to correct for incompleteness in deriving our luminosity function. The effective volume ( V eff ) can be estimated aswhere is the comoving volume element, which depends on the adopted cosmology, and C ( M UV, i ′ , z ) is the completeness as calculated in Section 3.4 . The integral was evaluated over z = 3–5. To calculate the luminosity function, we convert the apparent i -band AB magnitudes ( m i ′ ) to the absolute magnitude at rest-frame 1500 Å ( M UV, i ′ ) using the following formula:where d L is the luminosity distance in pc. The second and third terms on the right-hand side are the distance modulus. Our UV luminosity function is shown as red diamonds in Figure 8 . We note that we do not include the luminosity function data points in bins M UV, i ′ ≥ −22 in our analysis, as these bins are below our 50%-completeness limit as discussed in Section 3.4 , where the completeness corrections are unreliable owing to the low S/N of our data. This is confirmed by comparing to the HST CANDELS results in these same magnitude bins, which are more reliable owing to their higher S/N. In Figure 8 we also include the UV luminosity function of z = 4 UV-selected galaxies from deeper HST imaging by Finkelstein et al. ( 2015 ) as green squares. In the bin where we overlap with this data set ( M UV, i ′ = −22.5), both results are consistent; however, if the luminosity function derived from HST imaging is extrapolated to brighter magnitudes, it would fall off more steeply than our luminosity function. While our luminosity function declines from M UV, i ′ = −22 to M UV, i ′ = −24, it flattens out at