According to current cosmological theory, the rate of expansion of the universe depends on the average energy densities of matter, radiation, and a possible vacuum energy described by a cosmological constant, &Lambda;, in the Einstein equation.<p>Observations of galaxies and radiation, along with an assumption that we hold no special place in the universe, imply an isotropic and homogeneous energy distribution, for which the universal rate of expansion for most of the history of the universe may be constructed to depend only on present values of the dimensionless matter and vacuum energy density parameters, &Omega;_M and &Omega;_&Lambda;, respectively, and the present rate of expansion of the universe, H₀. Over the past decade, much progress has been made in determining the values of the three density parameters using a variety of independent methods. In particular, observations of type Ia supernovae in the late 1990s provided the first evidence that &Lambda; &ne; 0 and that universal expansion is accelerating.<p>This study has determined values for &Omega;_M and &Omega;_&Lambda; using the brightest quasars in the Sloan Digital Sky Survey Data Release 5, which are located at a range of distances - equivalently, a range of lookback times - that have not been accessible through any other observations. After fitting the apparent magnitudes of the brightest quasars at various redshifts to the distance modulus equation with a luminosity evolution term, values for the density parameters were determined to be &Omega;_M = 0.07 and &Omega;_&Lambda; = 1.13.

According to current cosmological theory, the rate of expansion of the universe depends on the average energy densities of matter, radiation, and a possible vacuum energy described by a cosmological constant, &Lambda;, in the Einstein equation.<p>Observations of galaxies and radiation, along with an assumption that we hold no special place in the universe, imply an isotropic and homogeneous energy distribution, for which the universal rate of expansion for most of the history of the universe may be constructed to depend only on present values of the dimensionless matter and vacuum energy density parameters, &Omega;_M and &Omega;_&Lambda;, respectively, and the present rate of expansion of the universe, H₀. Over the past decade, much progress has been made in determining the values of the three density parameters using a variety of independent methods. In particular, observations of type Ia supernovae in the late 1990s provided the first evidence that &Lambda; &ne; 0 and that universal expansion is accelerating.<p>This study has determined values for &Omega;_M and &Omega;_&Lambda; using the brightest quasars in the Sloan Digital Sky Survey Data Release 5, which are located at a range of distances - equivalently, a range of lookback times - that have not been accessible through any other observations. After fitting the apparent magnitudes of the brightest quasars at various redshifts to the distance modulus equation with a luminosity evolution term, values for the density parameters were determined to be &Omega;_M = 0.07 and &Omega;_&Lambda; = 1.13.