COALAS III: The ATCA CO(1-0) look at the growth and death of H$α$ emitters in the Spiderweb protocluster at z=2.16

We obtain CO(1-0) molecular gas measurements with ATCA on a sample of 43 spectroscopically confirmed H$α$ emitters in the Spiderweb protocluster at $z=2.16$ and investigate the relation between their star formation and cold gas reservoirs as a function of environment. We achieve a CO(1-0) detection rate of $\sim23\pm12\%$ with 10 dual CO(1-0) and H$α$ detections at $10<\log M_{*}/M_\odot<11.5$. In addition, we obtain upper limits for the remaining sources. In terms of total gas fractions ($F_{gas}$), our sample is divided into two different regimes with a steep transition at $\log M_{*}/M_\odot\approx10.5$. Galaxies below that threshold have gas fractions that in some cases are close to unity, indicating that their gas reservoir has been replenished by inflows from the cosmic web. However, objects at $\log M_{*}/M_\odot>10.5$ display significantly lower gas fractions and are dominated by AGN (12 out of 20). Stacking results yield $F_{gas}\approx0.55$ for massive emitters excluding AGN, and $F_{gas}\approx0.35$ when examining only AGN candidates. Furthermore, depletion times show that most H$α$ emitters may become passive by $1<z<1.6$, concurrently with the surge and dominance of the red sequence in the most massive clusters. Our analyses suggest that galaxies in the outskirts of the protocluster have larger molecular-to-stellar mass ratios and lower star formation efficiencies than in the core. However, star formation across the protocluster remains consistent with the main sequence, indicating that evolution is primarily driven by the depletion of the gas reservoir towards the inner regions. We discuss the relative importance of in-/outflow processes in regulating star formation during the early phases of cluster assembly and conclude that a combination of feedback and overconsumption may be responsible for the rapid cold gas depletion these objects endure.