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The exact relationship between the long gamma-ray burst (LGRB) rate and thecosmic star formation rate (CSFR) is essential for using LGRBs as cosmologicalprobes. In this work, we collect a large sample composed of 371 Swift LGRBswith known redshifts and prompt emission properties. We first compare therest-frame prompt properties of these bursts in different redshift bins,finding negligible redshift evolution of the luminosity of LGRBs with$L_{\mathrm{iso}}\gtrsim10^{51}\,\mathrm{erg\, s^{-1}}$ between $z\sim1$ and$z\sim4$. Then, by utilizing the CSFR obtained from the large-scalecosmological hydrodynamical simulation, the Illustris simulation, we calculatethe cumulative redshift distribution of LGRBs under different metallicitythresholds. After comparing with our sample, we find that the predictions witha moderate threshold between $0.3\,Z_{\odot}\leqslantZ_{\mathrm{th}}\leqslant1.0\,Z_{\odot}$ are consistent with the sample betweenredshift $0<z<3$, while at higher redshifts between $3<z<5$, all metallicitythresholds fit the data well. When changing to an empirical model based onobservations, the predictions show similar results as well. After comparingwith the metallicity distribution of the observed LGRB host galaxies between$0<z<1$, we confirm that the production of LGRBs in galaxies with super-solarmetallicity is suppressed. Nevertheless, considering that a significantfraction of stars are born in sub-solar metallicity environments at$z\gtrsim3$, we suggest that, as a first approximation, LGRBs can be used asdirect tracers of the CSFR in this redshift range.

Revisiting the Relationship between the Long GRB Rate and Cosmic Star Formation History Based on a Large Swift Sample