Clustering of high-redshift (z ≥ 2.9) quasars from the sloan digital sky survey

We study the two-point correlation function of a uniformly selected sample of 4426 luminous optical quasars with redshift 2.9 ≤ z ≤ 5.4 selected over 4041 deg² from the Fifth Data Release of the Sloan Digital Sky Survey. We fit a power-law to the projected correlation function w _p(r_p) to marginalize over redshift-space distortions and redshift errors. For a real-space correlation function of the form ξ(r) = (r/r₀)^-γ, the fitted parameters in comoving coordinates are r₀ = 15.2 ± 2.7 h^-1 Mpc and γ = 2.0 ± 0.3, over a scale range 4 h^-1 Mpc ≤ r_p ≤ 150 h^-1 Mpc. Thus high-redshift quasars are appreciably more strongly clustered than their z ≈ 1.5 counterparts, which have a comoving clustering length r₀ ≈ 6.5 h^-1 Mpc. Dividing our sample into two redshift bins, 2.9 ≤ z ≤ 3.5 and z ≥ 3.5, and assuming a power-law index γ = 2.0, we find a correlation length of r₀ = 16.9 ± 1.7 h^-1 Mpc for the former and r₀ = 24.3 ± 2.4 h^-1 Mpc for the latter. Strong clustering at high redshift indicates that quasars are found in very massive, and therefore highly biased, halos. Following Martini & Weinberg, we relate the clustering strength and quasar number density to the quasar lifetimes and duty cycle. Using the Sheth & Tonnen halo mass function, the quasar lifetime is estimated to lie in the range ∼4-50Myr for quasars with 2.9 ≤ z ≤ 3.5, and ∼30-600 Myr for quasars with z ≥ 3.5. The corresponding duty cycles are ∼0.004-0.05 for the lower redshift bin and ∼0.03-0.6 for the higher redshift bin. The minimum mass of halos in which these quasars reside is (2-3) × 10¹² h^-1 M_⊙ for quasars with 2.9 ≤ z ≤ 3.5 and (4-6) × 10¹² h^-1 M _⊙ for quasars with z ≥ 3.5; the effective bias factor b _eff increases with redshift, e.g., b_eff ∼ 8 at z = 3.0 and b_eff ∼ 16 at z = 4.5.