The ALPINE-CRISTAL-JWST survey: Spatially resolved star formation relations at z∼5

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,̊m μ m line has recently emerged as a key probe of the cold ISM and star formation in the early Universe. 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. 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. 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_ ̊m CII yields depletion times of $0.5–1$,Gyr, while a surface brightness-dependent W_ ̊m CII accounting for local ISM conditions, places some galaxies with high gas density in the starburst regime ($<0.1$,Gyr). 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.