Clustering of High Redshift ($z\ge 2.9$) Quasars from the Sloan Digital Sky Survey

(Abridged) We study the two-point correlation function of a uniformly selected sample of 4,426 luminous optical quasars with redshift $2.9 \le z\le 5.4$ selected over 4041 deg$^2$ from the Fifth Data Release of the Sloan Digital Sky Survey. For a real-space correlation function of the form $\xi(r)=(r/r_0)^{-\gamma}$, the fitted parameters in comoving coordinates are $r_0 = 15.2 \pm 2.7 h^{-1}$ Mpc and $\gamma = 2.0 \pm 0.3$, over a scale range $4\le r_p\le 150 h^{-1}$ Mpc. Thus high-redshift quasars are appreciably more strongly clustered than their $z \approx 1.5$ counterparts, which have a comoving clustering length $r_0 \approx 6.5 h^{-1}$ Mpc. Dividing our sample into two redshift bins: $2.9\le z\le 3.5$ and $z\ge 3.5$, and assuming a power-law index $\gamma=2.0$, we find a correlation length of $r_0 = 16.9 \pm 1.7 h^{-1}$ Mpc for the former, and $r_0 = 24.3 \pm 2.4 h^{-1}$ Mpc for the latter. Following Martini & Weinberg, we relate the clustering strength and quasar number density to the quasar lifetimes and duty cycle. Using the Sheth & Tormen halo mass function, the quasar lifetime is estimated to lie in the range $4\sim 50$ Myr for quasars with $2.9\le z\le 3.5$; and $30\sim 600$ Myr for quasars with $z\ge 3.5$. The corresponding duty cycles are $0.004\sim 0.05$ for the lower redshift bin and $0.03\sim 0.6$ for the higher redshift bin. The minimum mass of halos in which these quasars reside is $2-3\times 10^{12}\ h^{-1}M_\odot$ for quasars with $2.9\le z\le 3.5$ and $4-6\times 10^{12}\ h^{-1}M_\odot$ for quasars with $z\ge 3.5$.