  Following the success of the Plateau de Bure high-z Blue Sequence Survey
(PHIBSS), we present the PHIBSS2 legacy program, a survey of the molecular gas
properties of star-forming galaxies on and around the star formation main
sequence (MS) at different redshifts using NOEMA. This survey significantly
extends the existing sample of star-forming galaxies with CO molecular gas
measurements, probing the peak epoch of star formation (z=1-1.6) as well as its
building-up (z=2-3) and winding-down (z=0.5-0.8) phases. The targets are drawn
from the GOODS, COSMOS, and AEGIS deep fields and uniformly sample the MS in
the stellar mass (M*) - star formation rate (SFR) plane with log(M*/Msun) =
10-11.8. We describe the survey strategy and sample selection before focusing
on the results obtained at z=0.5-0.8, where we report 60 CO(2-1) detections out
of 61 targets. We determine their molecular gas masses and separately obtain
disc sizes and bulge-to-total (B/T) luminosity ratios from HST I-band images.
The median molecular gas-to-stellar mass ratio, gas fraction, and depletion
time as well as their dependence with M* and offset from the MS follow
published scaling relations for a much larger sample of galaxies spanning a
wider range of redshifts. The galaxy-averaged Kennicutt-Schmidt (KS) relation
between molecular gas and SFR surface densities is strikingly linear, pointing
towards similar star formation timescales within galaxies at any given epoch.
In terms of morphology, the molecular gas content, the SFR, the disc stellar
mass, and the disc molecular gas fraction do not seem to correlate with B/T and
the stellar surface density, which suggests an ongoing supply of fresh
molecular gas to compensate for the build-up of the bulge. Our measurements do
not yield any significant variation of the depletion time with B/T and hence no
strong evidence for morphological quenching within the scatter of the MS.
