The Two-Point Correlation Function of Gamma-ray Bursts
  In this paper, we examine the spacial distribution of gamma-ray bursts (GRBs)
using a sample of 373 objects. We subdivide the GRB data into two redshift
intervals over the redshift range $0<z< 6.7$. We measure the two-point
correlation function (2PCF), $\xi(r)$ of the GRBs. In determining the
separation distance of the GRB pairs, we consider two representative
cosmological models: a cold dark matter universe plus a cosmological constant
$\Lambda$, with $(\Omega_{{\rm m}}, \Omega_{{\rm \Lambda}})=(0.28,0.72)$ and an
Einstein-de Sitter (EdS) universe, with $(\Omega_{{\rm m}}, \Omega_{{\rm
\Lambda}})=(1,0)$. We find a $z$-decreasing correlation of the GRB
distribution, which is in agreement with the predictions of the current
structure formation theory. We fit a power-law model $\xi(r)=(r/r_0)^{-\gamma}$
to the measured $\xi(r)$ and obtain an amplitude and slope of $r_0= 1235.2 \pm
342.6~h^{-1}$ Mpc and $\gamma = 0.80\pm 0.19 $ ($1\sigma$ confidence level)
over the scales $r=200$ to $10^4~h^{-1}$ Mpc. Our result provide a supplement
to the measurement of matter correlation on large scales, while the matter
distribution below $200~h^{-1}$ Mpc is usually described by the correlation
function of galaxies.
