The collapsar model of long gamma-ray bursts ( GRBs ) indicates that they may trace the star formation history .
So long GRBs may be a useful tool of measuring the high-redshift star formation rate ( SFR ) .
The collapsar model explains GRB formation via the collapse of a rapidly rotating massive star with M > 30 M _ { \odot } into a black hole , which may imply a decrease of SFR at high redshift .
However , we find that the Swift GRBs during 2005-2012 are biased tracing the SFR , including a factor about ( 1 + z ) ^ { 0.5 } , which is in agreement with recent results .
After taking this factor , the SFR derived from GRBs does not show steep drop up to z \sim 9.4 .
We consider the GRBs produced by rapidly rotating metal-poor stars with low masses to explain the high-redshift GRB rate excess .
The chemically homogeneous evolution scenario ( CHES ) of rapidly rotating stars with mass larger than 12 M _ { \odot } is recognized as a promising path towards collapsars in connection with long GRBs .
Our results indicate that the stars in the mass range 12 M _ { \odot } < M < 30 M _ { \odot } for low enough metallicity Z \leq 0.004 with the GRB efficiency factor 10 ^ { -5 } can fit the derived SFR with good accuracy .
Combining these two factors , we find that the conversion efficiency from massive stars to GRBs is enhanced by a factor of 10 , which may be able to explain the excess of the high-redshift GRB rate .
We also investigate the cosmic reionization history using the derived SFR .
The GRB-inferred SFR would be sufficient to maintain cosmic reionization over 6 < z < 10 and reproduce the observed optical depth of Thomson scattering to the cosmic microwave background .