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The principal goal of this paper is to use attempts at reconciling the Swiftlong gamma-ray bursts (LGRBs) with the star formation history (SFH) to comparethe predictions of $\Lambda$CDM with those in the $R_{\rm h}=ct$ Universe. Inthe context of the former, we confirm that the latest Swift sample of GRBsreveals an increasing evolution in the GRB rate relative to the star formationrate (SFR) at high redshifts. The observed discrepancy between the GRB rate andthe SFR may be eliminated by assuming a modest evolution parameterized as$(1+z)^{0.8}$---perhaps indicating a cosmic evolution in metallicity. However,we find a higher metallicity cut of $Z=0.52Z_{\odot}$ than was seen in previousstudies, which suggested that LGRBs occur preferentially in metal poorenvironments, i.e., $Z\sim0.1-0.3Z_{\odot}$. We use a simple power-lawapproximation to the high-\emph{z} ($\ga 3.8$) SFH, i.e., $R_{\rmSF}\propto[(1+z)/4.8]^{\alpha}$, to examine how the high-\emph{z} SFR may beimpacted by a possible abundance evolution in the Swift GRB sample. For anexpansion history consistent with $\Lambda$CDM, we find that the Swift redshiftand luminosity distributions can be reproduced with reasonable accuracy if$\alpha=-2.41_{-2.09}^{+1.87}$. For the $R_{\rm h}=ct$ Universe, the GRB rateis slightly different from that in $\Lambda$CDM, but also requires an extraevolutionary effect, with a metallicity cut of $Z=0.44Z_{\odot}$. Assuming thatthe SFR and GRB rate are related via an evolving metallicity, we find that theGRB data constrain the slope of the high-\emph{z} SFR in $R_{\rm h}=ct$ to be$\alpha=-3.60_{-2.45}^{+2.45}$. Both cosmologies fit the GRB/SFR data ratherwell.

Cosmological Tests Using GRBs, the Star Formation Rate and Possible Abundance Evolution