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.