Does the lockstep growth between black holes and bulges create their mass relation?

Recent studies have revealed a strong relation between sample-averaged black-hole (BH) accretion rate (BHAR) and star formation rate (SFR) among bulge-dominated galaxies, i.e., "lockstep" BH-bulge growth, in the distant universe. This relation might be closely related to the BH-bulge mass correlation observed in the local universe. To understand further BH-bulge coevolution, we present ALMA CO(2-1) or CO(3-2) observations of 7 star-forming bulge-dominated galaxies at z=0.5-2.5. Using the ALMA data, we detect significant ($>3\sigma$) CO emission from 4 objects. For our sample of 7 galaxies, we measure (or constrain with upper limits) their CO line fluxes and estimate molecular gas masses ($M_{gas}$). We also estimate their stellar masses ($M_{star}$) and SFRs by modelling their spectral energy distributions (SEDs). Using these physical properties, we derive the gas-depletion timescales ($t_{dep} = M_{gas}/SFR$) and compare them with the bulge/BH growth timescales ($t_{grow} = M_{star}/SFR \sim M_{BH}/BHAR$). Our sample generally has $t_{dep}$ shorter than $t_{grow}$ by a median factor of $\gtrsim 4$, indicating that the cold gas will be depleted before significant bulge/BH growth takes place. This result suggests that the BH-bulge lockstep growth is mainly responsible for maintaining their mass relation, not creating it. We note that our sample is small and limited to $z<2.5$; JWST and ALMA will be able to probe to higher redshifts in the near future.