  The high-redshift star formation rate (SFR) is difficult to measure directly
even by modern approaches. Long-duration gamma-ray bursts (GRBs) can be
detected to the edge of the visible universe because of their high
luminorsities. The collapsar model of long gamma-ray bursts indicates that they
may trace the star formation history. So long gamma-ray bursts may be a useful
tool of measuring the high-redshift SFR. Observations show that long gamma-ray
bursts prefer to form in a low-metallicity environment. We study the
high-redshift SFR up to z~8.3 considering the Swift GRBs tracing the star
formation history and the cosmic metallicity evolution in different background
cosmological models including $\Lambda$CDM, quintessence, quintessence with a
time-varying equation of state, and brane-world model. We use latest Swift GRBs
including two highest-$z$ GRBs, GRB 080913 at $z=6.7$ and GRB 090423 at
$z=8.3$. We find that the SFR at $z>4$ shows a steep decay with a slope of
$\sim -5.0$ in $\Lambda$CDM. In the other three models, the high-redshift SFR
is slightly different from $\Lambda$CDM model, and also shows a steep decay.
