  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.

Astrophysics of Galaxies