The [ C ii ] 158 \mu m 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 [ C ii ] luminosity and the SFR over the entire redshift range from z \sim 4 - 8 .
ALPINE galaxies , which are both detected in [ C ii ] 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 [ C ii ] , 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 [ C ii ] 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 [ C ii ] 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 Ly \alpha equivalent width .
Finally , we find that the ratio L ( { [ CII ] } ) / L _ { IR } \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 [ C ii ] measurements at z > 4 available so far , suggests no or little evolution of the [ C ii ] –SFR relation over the last 13 Gyr of cosmic time .