Resolving the Excess of Long GRBs at low redshift in the Swift Era
  Utilizing more than 100 long gamma-ray bursts (LGRBs) in the Swift-Ryan-2012
sample that include the observed redshifts and jet angles, Le & Mehta performed
a timely study of the rate-density of LGRBs with an assumed broken power law
GRB spectrum and obtained a GRB-burst-rate functional form that gives
acceptable fits to the preSwift and Swift redshift and jet angle distributions.
The results indicated an excess of LGRBs at redshift below $z \sim 2$ in the
Swift sample. In this work, we are investigating if the excess is caused by the
cosmological Hubble constant $H_0$, the $\gamma$-ray energy released
$E_\gamma$, the low- and high-energy indices ($\alpha, \beta$) of the Band
function, the minimum and maximum jet angles $\theta_{\rm j,min}$ and
$\theta_{\rm j, max}$, or that the excess is due to a biased in the
Swift-Ryan-2012 sample. Our analyses indicate that none of the above physical
parameters resolved the excess problem, but suggesting that the
Swift-Ryan-2012~sample is biased with possible afterglow selection effect. The
following model physical parameter values provide the best fit to the
Swift-Ryan-2012 and preSwift~samples: the Hubble constant $H_0 = 72 \, {\rm km
s^{-1} Mpc^{-1}}$, the energy released $E_\gamma \sim 4.47 \times 10^{51}$ erg,
the energy indices $\alpha \sim 0.9$ and $\beta \sim -2.13$, the jet angles of
$\theta_{\rm j,max} \sim 0.8$ rad, and $\theta_{\rm j, min} \sim 0.065$ and
$\sim 0.04$ rad for preSwift and Swift, respectively, $s \sim -1.55$ the jet
angle power-law index, and a GRB formation rate that is similar to the Hopkins
& Beacom observed star formation history and as extended by Li. Using the Swift
Gamma-Ray Burst Host Galaxy Legacy Survey (SHOALS) Swift-Perley LGRB sample and
applying the same physical parameter values as above, however, our model
provides consistent results with this data set and indicating no excess of
LGRBs at any redshift.
