Sub-millimetre compactness as a critical dimension to understand the Main Sequence of star-forming galaxies

We study the interstellar medium (ISM) properties as a function of the molecular gas size of 82 infrared-selected galaxies on and above the main sequence at $z \sim 1.3$. Molecular gas sizes are measured on ALMA images that combine CO(2-1), CO(5-4) and underlying continuum observations. We include CO(4-3),CO(7-6)+[CI]($^3 P_2-^3P_1$), [CI]($^3 P_1-^3P_0$) observations for a subset of the sample. The $\geqslant 46 \%$ of our galaxies have a compact molecular gas reservoir as these lie $\geqslant 1 \sigma$ below the optical mass-size relation of disks. Compact galaxies on and above the main sequence have higher CO excitation and star formation efficiency than galaxies with extended molecular gas reservoirs, as traced by CO(5-4)/CO(2-1) and CO(2-1)/$L_{\rm IR, SF}$ ratios. Average CO+[CI] spectral line energy distributions indicate higher excitation in compacts relative to extended sources. Using multiple molecular gas mass tracers, and conversion factors tailored to their ISM conditions, we measure lower gas fractions in compact main-sequence galaxies compared to extended sources. These results are consistent with a picture in which mergers have driven the gas in the nuclear regions, enhancing the CO excitation and star formation efficiency. We suggest that the sub-millimetre compactness, defined as the ratio between the molecular gas and stellar size, is an unavoidable information to be used with the main sequence offset to describe the ISM properties of galaxies, at least above $M_{\star} \geqslant 10^{10.6}$ M$_{\odot}$, where our observations fully probe the main sequence scatter. Compact main-sequence galaxies are consistent with being an early post-starburst population following a merger-driven starburst episode, stressing the important role of mergers in the evolution of massive galaxies.