The origin of the high star formation rates ( SFR ) observed in high-redshift dusty star-forming galaxies is still unknown .
Large fractions of dense molecular gas might provide part of the explanation , but there are few observational constraints on the amount of dense gas in high-redshift systems dominated by star formation .
In this paper , we present the results of our Atacama large millimeter array ( ALMA ) program targeting dense-gas tracers ( HCN ( 5-4 ) , HCO ^ { + } ( 5-4 ) , and HNC ( 5-4 ) ) in five strongly lensed galaxies from the South Pole Telescope ( SPT ) submillimeter galaxy sample .
We detected two of these lines ( S/N > 5 ) in SPT-125-47 at z=2.51 and tentatively detected all three ( S/N \sim 3 ) in SPT0551-50 at z=3.16 .
Since a significant fraction of our target lines is not detected , we developed a statistical method to derive unbiased mean properties of our sample taking into account both detections and non-detections .
On average , the HCN ( 5-4 ) and HCO ^ { + } ( 5-4 ) luminosities of our sources are a factor of \sim 1.7 fainter than expected , based on the local L ’ _ { HCN ( 5 - 4 ) } -L _ { IR } relation , but this offset corresponds to only \sim 2 \sigma if we consider sample variance .
We find that both the HCO ^ { + } /HCN and HNC/HCN flux ratios are compatible with unity .
The first ratio is expected for photo-dominated regions ( PDRs ) while the second is consistent with PDRs or X-ray dominated regions ( XDRs ) and/or mid-infrared ( IR ) pumping of HNC .
Our sources are at the high end of the local relation between the star formation efficiency , determined using the L _ { IR } / [ CI ] and L _ { IR } /CO ratios , and the dense-gas fraction , estimated using the HCN/ [ CI ] and HCN/CO ratios .
Finally , in SPT0125-47 , which has the highest signal-to-noise ratio , we found that the velocity profiles of the lines tracing dense ( HCN , HCO ^ { + } ) and lower-density ( CO , [ CI ] ) molecular gas are similar .
In addition to these lines , we obtained one robust and one tentative detection of ^ { 13 } CO ( 4-3 ) and found an average I _ { { } ^ { 12 } CO ( 4 - 3 ) } /I _ { { } ^ { 13 } CO ( 4 - 3 ) } flux ratio of 26.1 _ { -3.5 } ^ { +4.5 } , indicating a young but not pristine interstellar medium .
We argue that the combination of large and slightly enriched gas reservoirs and high dense-gas fractions could explain the prodigious star formation in these systems .