We demonstrate that the observationally inferred rapid onset of star formation after parental molecular clouds have assembled can be achieved by flow-driven cloud formation of atomic gas, using our previous three-dimensional numerical simulations. We post-process these simulations to approximate CO formation, which allows us to investigate the times at which CO becomes abundant relative to the onset of cloud collapse. We find that global gravity in a finite cloud has two crucial effects on cloud evolution. (a) Lateral collapse (perpendicular to the flows sweeping up the cloud) leads to rapidly increasing column densities above the accumulation from the one-dimensional flow. This in turn allows fast formation of CO, allowing the molecular cloud to ``appear rapidly. (b) Global gravity is required to drive the dense gas to the high pressures necessary to form solar-mass cores, in support of recent analytical models of cloud fragmentation. While the clouds still appear ``supersonically turbulent, this turbulence is relegated to playing a secondary role, in that it is to some extent a consequence of gravitational forces.
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