The ALPINE-ALMA [C II] Survey: [C II]158micron Emission Line Luminosity
  Functions at $z \sim 4-6$
  We present the [CII]158$\mu$m line luminosity functions (LFs) at $z\sim4-6$
using the ALMA observations of 118 sources, which are selected to have UV
luminosity $M_{1500A}<-20.2$ and optical spectroscopic redshifts in COSMOS and
ECDF-S. Of the 118 targets, 75 have significant [CII] detections and 43 are
upper limits. This is by far the largest sample of [CII] detections which
allows us to set constraints to the volume density of [CII] emitters at
$z\sim4-6$. But because this is a UV-selected sample, we are missing
[CII]-bright but UV-faint sources making our constraints strict lower limits.
Our derived LFs are statistically consistent with the $z\sim0$ [CII] LF at
$10^{8.25} - 10^{9.75}L_\odot$. We compare our results with the upper limits of
the [CII] LF derived from serendipitous sources in the ALPINE maps (Loiacono et
al. 2020). We also infer the [CII] LFs based on published far-IR and CO LFs at
$z\sim4-6$. Combining our robust lower limits with these additional estimates,
we set further constraints to the true number density of [CII] emitters at
$z\sim 4 - 6$. These additional LF estimates are largely above our LF at
$L_{[CII]}>10^9L_{\odot}$, suggesting that UV-faint but [CII]-bright sources
likely make a significant contributions to the [CII] emitter volume density.
When we include all the LF estimates, we find that available model predictions
underestimate the number densities of [CII] emitters at $z\sim4-6$. Finally, we
set a constraint on the molecular gas mass density at $z\sim4-6$, with
$\rho_{mol} \sim (2-7)\times10^7M_\odot$\,Mpc$^{-3}$. This is broadly
consistent with previous studies.
