An unexpectedly low-redshift excess of Swift gamma-ray burst rate
  Gamma-ray bursts (GRBs) are the most violent explosions in the Universe and
can be used to explore the properties of high-redshift universe. It is believed
that the long GRBs are associated with the deaths of massive stars. So it is
possible to use GRBs to investigate the star formation rate (SFR). In this
paper, we use Lynden-Bell's $c^-$ method to study the luminosity function and
rate of \emph{Swift} long GRBs without any assumptions. We find that the
luminosity of GRBs evolves with redshift as $L(z)\propto g(z)=(1+z)^k$ with
$k=2.43_{-0.38}^{+0.41}$. After correcting the redshift evolution through
$L_0(z)=L(z)/g(z)$, the luminosity function can be expressed as
$\psi(L_0)\propto L_0^{-0.14\pm0.02}$ for dim GRBs and $\psi(L_0)\propto
L_0^{-0.70\pm0.03}$ for bright GRBs, with the break point
$L_{0}^{b}=1.43\times10^{51}~{\rm erg~s^{-1}}$. We also find that the formation
rate of GRBs is almost constant at $z<1.0$ for the first time, which is
remarkably different from the SFR. At $z>1.0$, the formation rate of GRB is
consistent with the SFR. Our results are dramatically different from previous
studies. Some possible reasons for this low-redshift excess are discussed. We
also test the robustness of our results with Monte Carlo simulations. The
distributions of mock data (i.e., luminosity-redshift distribution, luminosity
function, cumulative distribution and $\log N-\log S$ distribution) are in good
agreement with the observations. Besides, we also find that there are
remarkable difference between the mock data and the observations if long GRB
are unbiased tracers of SFR at $z<1.0$.
