A model-independent determination of the expansion and acceleration rates of the universe as a function of redshift and constraints on dark energy

Determination of the expansion and acceleration history of the universe is one of the fundamental goals of cosmology. Detailed measurements of these rates as a function of redshift can provide new physical insights into the nature and evolution of the dark energy, which apparently dominates the global dynamics of the universe at the present epoch. We present here dimensionless coordinate distances y(z) to 20 radio galaxies reaching out to z ≈ 1.8, the redshift range currently not covered by supernova standard-candle observations. There is very good agreement between coordinate distances to radio galaxies and supernovae for the redshift range in which these measurements overlap, suggesting that neither is plagued at this level by unknown systematic errors. We develop a simple numerical method for a direct determination of the expansion and acceleration rates, E(z) and q(z), from the data, which makes no assumptions about the underlying cosmological model or the equation of state parameter w. This differential method is in contrast to the traditional cosmological tests, in which particular model equations are integrated and then compared with the observations. The new approach is model independent, but at a cost of being noisier and highly sensitive to the amount and quality of the available data. We illustrate the method by applying it to the currently available supernova data and the data on radio galaxies presented here. We derive the expansion rate of the universe as a function of redshift, E(z), and for the first time we obtain a direct estimate of the acceleration rate of the universe as a function of redshift, q(z), in a way that is independent of assumptions regarding the dark energy and its redshift evolution. The current observations indicate that the universe makes a transition from acceleration to deceleration at a redshift greater than 0.3, with a best-fit estimate of about 0.45; this transition redshift and our determinations of E(z) are broadly in agreement with the currently popular Friedmann-Lemaitre cosmology with Ω_m = 0.3 and Ω_Λ = 0.7, even though no model assumptions are made in deriving the fits for E(z) and q(z). With the advent of much better and richer data sets in the future, our direct method can provide a useful complementarity and an independent check on the traditional cosmological tests.