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 twenty radio galaxies reaching out to z \approx 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 where 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 the traditional cosmological tests , where 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 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 transitions 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 \Omega _ { m } = 0.3 , and \Omega _ { \Lambda } = 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 to the traditional cosmological tests .