A cosmological model with a gravitational Lagrangian L _ { g } ( R ) \propto R + AR ^ { n } is set up to account for the presently observed re-acceleration of the universe .
The evolution equation for the scale factor a of the universe is analyzed in detail for the two parameters n = 2 and n = 4 / 3 , which were preferred by previous studies of the early universe .
The initial conditions are specified at a red-shift parameter z \approx 0 .
The fit to the observable data fixes the free parameter A .
The analysis shows that the model with n = 2 agrees better with present data .
Then , if we set w ( q ) = -1 at z = 0 , corresponding to the deceleration parameter q \approx - 1 / 2 , we find that at z \approx 0.5 , w ( q ) has evolved to w \approx - 0.72 , corresponding to q \approx 0 .
At z \approx 1 we find w \approx 0 corresponding to q \approx 1 / 2 .
These results are compared with the flat Friedmann model with cold matter and Lambda-term ( LCDM model ) for the same initial conditions at z \approx 0 .
The other choice of the model with n = 4 / 3 allows for big crunch .
However this possibility is eliminated by the fit of A to the present data .