The GRB Redshift Distribution: Implications for Abundance Evolution, Star Formation, and Cosmology

It has been claimed that the \emph{Swift} long gamma-ray bursts (LGRBs) do not trace the star formation history (SFH) in $\Lambda$CDM. In this paper, we confirm that the latest \emph{Swift} sample of GRBs reveals an increasing evolution in the GRB rate relative to the star formation rate (SFR) at high redshifts. One may eliminate the observed discrepancy between the GRB rate and the SFR by assuming a modest evolution, parameterized as $(1+z)^{0.5}$---an effect that perhaps implies a cosmic evolution in metallicity. However, we find a relatively higher metallicity cut of $Z=0.68Z_{\odot}$ than was seen in previous studies, which suggested that LGRBs occur preferentially in metal poor environments, i.e., $Z\sim0.1-0.3Z_{\odot}$. Here, we use a simple power-law approximation to the high-\emph{z} ($\ga 3.8$) SFH, i.e., $R_{\rm SF}\propto[(1+z)/4.8]^{\alpha}$, to examine how the high-\emph{z} SFR may be impacted by a possible abundance evolution in the \emph{Swift} GRB sample. For an expansion history consistent with $\Lambda$CDM, we obtain $\alpha=-3.7_{-2.3}^{+0.5}$, and using this updated SFH, we show that the observed redshift distribution of \emph{Swift} GRBs can be reproduced with reasonable accuracy. We also carry out a comparative analysis of the GRB redshift distribution for the $R_{\rm h}=ct$ Universe. We show that the GRB rate for this cosmology is slightly different from that in $\Lambda$CDM and also requires an extra evolutionary effect, which can be explained by a metallicity cut of $Z=0.59Z_{\odot}$, also higher than previous studies. Assuming that the GRB rate is related to the SFR and an evolving metallicity, we find that the GRB data constrain the slope of the high-\emph{z} SFR to be $\alpha\sim-4.6$. With this updated SFH, we find that in the $R_{\rm h}=ct$ Universe, as was the case for $\Lambda$CDM, the observed redshift distribution can also be fitted very well.