The ALPINE-ALMA [CII] Survey: the population of [CII]-undetected
  galaxies and their role in the $\mathrm{L_{[CII]}}$-SFR relation
  The [CII] 158$~\mu$m emission line represents so far one of the most
profitable tools for the investigation of the high-redshift galaxies in the
early Universe. Being one of the brightest cooling lines in the rest-frame
far-infrared regime of star-forming galaxies, it has been successfully
exploited as a tracer of star-formation rate (SFR) in local sources. The
picture is more complex at higher redshifts, where its usability in this
context is still under investigation. Recent results from the ALMA Large
Program to INvestigate [CII] at Early times (ALPINE) survey suggest that there
is no (or weak) evolution of the L$\mathrm{_{[CII]}}$-SFR relation up to
$z\sim6$ but their reliability is hampered by the presence of a large
population of [CII] non-detected galaxies. In this work, we characterize the
population of [CII] non-detections in ALPINE. By stacking their ALMA spectra,
we obtain a signal detected at $\sim5.1\sigma$, resulting in a [CII] luminosity
of $\mathrm{log(L_\mathrm{[CII]}}/\mathrm{L_{\odot}})$ $\sim7.8$. When
combining this value with those from the [CII] detections, we find a
$\mathrm{L_{[CII]}}$-SFR relation with a slope $b=1.14\pm0.11$, in agreement
within the uncertainties both with the linear relation found in the local
Universe, and with the previous findings from ALPINE at $z\sim5$. This suggests
that the [CII] line can be considered a good tracer of star formation up to the
distant Universe. Finally, we show that the galaxies of our sample that most
deviate from the observed L$_\mathrm{[CII]}$-SFR relation could suffer from a
less precise redshift estimation, perhaps artificially reducing their [CII]
luminosity. In this respect, we claim that there is no evidence in favour of a
deficit of [CII] content in high-z galaxies, in contrast with earlier studies.
