High-redshift quasars found in sloan digital sky survey commissioning data. IV. Luminosity function from the fall equatorial stripe sample

This is the fourth paper in a series aimed at finding high-redshift quasars from five-color (u′g′r′i′z′) imaging data taken along the celestial equator by the Sloan Digital Sky Survey during its commissioning phase. In this paper, we use the color-selected sample of 39 luminous high-redshift quasars presented in Paper III to derive the evolution of the quasar luminosity function over the range 3.6 < z < 5.0 and -27.5 < M₁₄₅₀ < -25.5 (Ω = 1, H₀ = 50 km s^-1 Mpc^-1). We use the selection function derived in Paper III to correct for sample incompleteness. The luminosity function is estimated using three different methods: (1) the 1/V_a estimator; (2) a maximum likelihood solution, assuming that the density of quasars depends exponentially on redshift and as a power law in luminosity; and (3) Lynden-Bell's nonparametric C^- estimator. All three methods yield consistent results. The luminous quasar density decreases by a factor of ∼6 from z = 3.5 to z = 5.0, consistent with the decline seen from several previous optical surveys at z < 4.5. The luminosity function follows ψ(L) ∝ L^-2.5 for z ∼ 4 at the bright end, significantly flatter than the bright-end luminosity function ψ(L) ∝ L^-3.5 found in previous studies for z < 3, suggesting that the shape of the quasar luminosity function evolves with redshift as well, and that the quasar evolution from z = 2 to z = 5 cannot be described as pure luminosity evolution. Possible selection biases and the effect of dust extinction on the redshift evolution of the quasar density are also discussed.