The interstellar medium of high-redshift galaxies: Gathering clues from C III] and [C II] lines

Context. A tight relation between [C II] 158 mu m line luminosity and the star formation rate (SFR) has been observed for local galaxies. At high redshift (z > 5), galaxies instead deviate downwards from the local Sigma([C II]) - Sigma(SFR) relation. This deviation might be caused by different interstellar medium (ISM) properties in galaxies at early epochs. Aims. To test this hypothesis, we combined the [C II] and SFR data with C III] 1909 angstrom line observations and our physical models. We additionally investigated how ISM properties, such as burstiness, kappa(s), total gas density, n, and metallicity, Z, affect the deviation from the Sigma([C II]) - Sigma(SFR) relation in these sources. Methods. We present the VLT/X-shooter observations targeting the C III] lambda 1909 line emission in three galaxies at 5.5 < z < 7.0. We include archival X-shooter data of two other sources at 5.5 < z < 7.0 and the VLT/MUSE archival data of six galaxies at z similar to 2.We extend our sample of galaxies with eleven star-forming systems at 6 < z < 7.5, with either C III] or [C II] detection reported in the literature. Results. We detected C III] lambda lambda 1907, 1909 line emission in HZ10 and we derived the intrinsic, integrated flux of the C III] lambda 1909 line. We constrained the ISM properties for our sample of galaxies, kappa(s), n, and Z, by applying our physically motivated model based on the MCMC algorithm. For the most part, high-z star-forming galaxies show subsolar metallicities. The majority of the sources have log(kappa(s)) greater than or similar to 1, that is, they overshoot the Kennicutt-Schmidt (KS) relation by about one order of magnitude. Conclusions. Our findings suggest that the whole KS relation might be shifted upwards at early times. Furthermore, all the high-z galaxies of our sample lie below the Sigma([C II]) - Sigma(SFR) local relation. The total gas density, n, shows the strongest correlation with the deviation from the local Sigma([C II]) - Sigma(SFR) relation, namely, low-density high-z systems have lower [C II] surface brightness, in agreement with theoretical models.