Cosmological Tests Using GRBs, the Star Formation Rate and Possible
  Abundance Evolution
  The principal goal of this paper is to use attempts at reconciling the Swift
long gamma-ray bursts (LGRBs) with the star formation history (SFH) to compare
the predictions of $\Lambda$CDM with those in the $R_{\rm h}=ct$ Universe. In
the context of the former, we confirm that the latest Swift sample of GRBs
reveals an increasing evolution in the GRB rate relative to the star formation
rate (SFR) at high redshifts. The observed discrepancy between the GRB rate and
the 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 previous
studies, which suggested that LGRBs occur preferentially in metal poor
environments, i.e., $Z\sim0.1-0.3Z_{\odot}$. 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 Swift GRB sample. For an
expansion history consistent with $\Lambda$CDM, we find that the Swift redshift
and 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 rate
is slightly different from that in $\Lambda$CDM, but also requires an extra
evolutionary effect, with a metallicity cut of $Z=0.44Z_{\odot}$. Assuming that
the SFR and GRB rate are related via an evolving metallicity, we find that the
GRB 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 rather
well.
