The ALPINE-CRISTAL-JWST survey: Spatially resolved star formation relations at

Context. Star formation governs galaxy evolution, shaping stellar mass assembly and gas consumption across cosmic time. The Kennicutt-Schmidt (KS) relation, linking the star formation rate (SFR) and gas surface densities, is fundamental to understand star formation regulation, yet remains poorly constrained at z > 2 due to observational limitations and uncertainties in locally calibrated gas tracers. The [CII] 158 μm line has recently emerged as a key probe of the cold ISM and star formation in the early Universe. Aims. We investigate whether the resolved [CII]–SFR and KS relations established at low redshift remain valid at 4 < z < 6 by analysing 13 main-sequence galaxies from the ALPINE and CRISTAL surveys, using multi-wavelength data (HST, JWST, ALMA) at ∼2 kpc resolution. Methods. We performed pixel-by-pixel spectral energy distribution (SED) modelling with CIGALE on resolution-homogenised images. We developed a statistical framework to fit the [CII]–SFR relation that accounts for pixel covariance and compare our results to classical fitting methods. We tested two [CII]-to-gas conversion prescriptions to assess their impact on inferred gas surface densities and depletion times. Results. We find a resolved [CII]–SFR relation with a slope of 0.87 ± 0.15 and intrinsic scatter of 0.19 ± 0.03 dex, which is shallower and tighter than previous studies at z ∼ 5. The resolved KS relation is highly sensitive to the [CII]-to-gas conversion factor: using a fixed global α[CII] yields depletion times of 0.5–1 Gyr, while a surface brightness-dependent W[CII], accounting for local ISM conditions, places some galaxies with high gas density in the starburst regime (< 0.1 Gyr). Conclusions. Future inputs from both simulations and observations are required to better understand how the [CII]-to-gas conversion factor depends on local ISM properties. We need to break this fundamental limit to properly study the KS relation at z ≳ 4.