The [CII] 158 micron line is one of the strongest IR emission lines, which has been shown to trace the star-formation rate (SFR) of galaxies in the nearby Universe and up to $z sim 2$. Whether this is also the case at higher redshift and in the early Universe remains debated. The ALPINE survey, which targeted 118 star-forming galaxies at $4.4 < z< 5.9$, provides a new opportunity to examine this question with the first statistical dataset. Using the ALPINE data and earlier measurements from the literature we examine the relation between the [CII] luminosity and the SFR over the entire redshift range from $z sim 4-8$. ALPINE galaxies, which are both detected in [CII] and dust continuum, show a good agreement with the local L([CII])-SFR relation. Galaxies undetected in the continuum with ALMA are found to be over-luminous in [CII], when the UV SFR is used. After accounting for dust-obscured star formation, by an amount SFR(IR)$approx$SFR(UV) on average, which results from two different stacking methods and SED fitting, the ALPINE galaxies show an L([CII])-SFR relation comparable to the local one. When [CII] non-detections are taken into account, the slope may be marginally steeper at high-z, although this is still somewhat uncertain. When compared in a homogeneous manner, the $z>6 $ [CII] measurements (detections and upper limits) do not behave very differently from the $z sim 4-6$ data. We find a weak dependence of L([CII])/SFR on the Lyman-alpha equivalent width. Finally, we find that the ratio L([CII])/LIR $sim (1-3) times 10^{-3}$ for the ALPINE sources, comparable to that of normal galaxies at lower redshift. Our analysis, which includes the largest sample ($sim 150$ galaxies) of [CII] measurements at $z>4$ available so far, suggests no or little evolution of the L([CII])-SFR relation over the last 13 Gyr of cosmic time.
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