Bursting at the seams: the star-forming main sequence and its scatter at z=3-9 using NIRCam photometry from JADES

We present a comprehensive study of the star-forming main sequence (SFMS) and its scatter at redshifts $3 \leq z \leq 9$, using NIRCam photometry from the JADES survey in the GOODS-S and GOODS-N fields. Our analysis is based on a sample of galaxies that is stellar mass complete down to $\log \left(M_{\star}/M_{\odot}\right) \approx 8.1$. The redshift evolution of the SFMS at an averaging timescale of 10 Myr follows a relation, quantified by the specific star-formation rates (sSFR$_{10}$), of $\mathrm{sSFR}\propto(1+z)^μ$ with $μ= 2.30^{+0.03}_{-0.01}$, in good agreement with theoretical predictions and the specific mass accretion rate of dark matter halos. We find that the SFMS normalisation varies in a complex way with the SFR averaging timescale, reflecting the combined effects of bursty star formation and rising star formation histories (SFHs). We quantify the scatter of the SFMS, revealing that it decreases with longer SFR averaging timescales, from $σ_{\rm{int}} \approx 0.4-0.5~\mathrm{dex}$ at 10 Myr to $σ_{\rm{int}} \approx 0.2~\mathrm{dex}$ at 100 Myr, indicating that shorter-term fluctuations dominate the scatter, although long-term variations in star formation activity are also present. Our findings suggest that bursty SFHs are more pronounced at lower stellar masses. Furthermore, we explore the implications of our results for the observed over-abundance of UV-bright galaxies at $z > 10$, concluding that additional mechanisms, such as top-heavy initial mass functions, increased star-formation efficiencies, or increased burstiness in star formation are needed to explain these observations. Finally, we emphasize the importance of accurate stellar mass completeness limits when fitting the SFMS, especially for galaxies with bursty SFHs.