Molecular gas in the Herschel-selected strongly lensed submillimeter
  galaxies at z~2-4 as probed by multi-J CO lines
  (abridged) We present the IRAM-30m observations of multiple-J CO and CI line
emission in a sample of redshift ~2-4 Herschel-ATLAS SMGs. A non-negligible
effect of differential lensing is found for the CO emission lines, which could
have caused significant underestimations of the linewidths, hence of the
dynamical masses. The CO SLEDs are found to be similar to those of the local
starburst-dominated ULIRGs and of the previously studied SMGs. After correcting
for lensing amplification, we derived the global properties of the bulk of
molecular gas in the SMGs using non-LTE radiative transfer modelling. The gas
thermal pressure is found to be correlated with star formation efficiency.
Further decomposing the CO SLEDs into two excitation components, we find a
low-excitation component, which is less correlated with star formation, and a
high-excitation one which is tightly related to the on-going star-forming
activity. Additionally, tight linear correlations between the FIR and CO line
luminosities have been confirmed for the $J \ge 5$ CO lines, implying that
these CO lines are good tracers of star formation. The [CI](2-1) lines follow
the tight linear correlation between the luminosities of the [CI](2-1) and the
CO(1-0) line found in local starbursts, indicating that CI lines could serve as
good total molecular gas mass tracers for high-redshift SMGs. The total mass of
the molecular gas reservoir, $(1-30) \times 10^{10} M_\odot$, suggests a
typical molecular gas depletion time ~20-100 Myr and a gas to dust mass ratio
${\delta}_{\rm GDR}$~30-100. The ratio between CO line luminosity and the dust
mass appears to be slowly increasing with redshift for the SMGs, which need to
be further confirmed. Finally, through comparing the linewidth of CO and H2O
lines, we find that they agree well in almost all our SMGs, confirming that the
emitting regions are co-spatially located.
