The molecular gas content of normal galaxies at z>4 is poorly constrained, because the commonly used molecular gas tracers become hard to detect. We use the [CII]158um luminosity, recently proposed as a molecular gas tracer, to estimate the molecular gas content in a large sample of main-sequence star-forming galaxies at z=4.4-5.9, with a median stellar mass of 10^9.7 Msun, drawn from the ALMA Large Program to INvestigate [CII] at Early times (ALPINE) survey. The agreement between molecular gas masses derived from [CII] luminosity, dynamical mass, and rest-frame 850um luminosity, extrapolated from the rest-frame 158um continuum, supports [CII] as a reliable tracer of molecular gas in our sample. We find a continuous decline of the molecular gas depletion timescale from z=0 to z=5.9, which reaches a mean value of (4.6+/-0.8)x10^8 yr at z~5.5, only a factor of 2-3 shorter than in present-day galaxies. This suggests a mild enhancement of star formation efficiency toward high redshifts, unless the molecular gas fraction significantly increases. Our estimates show that the rise in molecular gas fraction as reported previously, flattens off above z~3.7 to achieve a mean value of 63%+/-3 over z=4.4-5.9. This redshift evolution of the gas fraction is in line with the one of the specific star formation rate. We use multi-epoch abundance matching to follow the gas fraction evolution over cosmic time of progenitors of z=0 Milky Way-like galaxies in 10^13 Msun halos and of more massive z=0 galaxies in 10^14 Msun halos. Interestingly, the former progenitors show a monotonic decrease of the gas fraction with cosmic time, while the latter show a constant gas fraction from z=5.9 to z~2 and a decrease at z<2. We discuss three possible effects, namely outflows, halt of gas supplying, and over-efficient star formation, which may jointly contribute to the gas fraction plateau of the latter massive galaxies.
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