  We investigate the consequences of non-minimal gravitational coupling to
matter and study how it differs from the case of minimal coupling by choosing
certain simple forms for the nature of coupling, The values of the parameters
are specified at $z=0$ (present epoch) and the equations are evolved backwards
to calculate the evolution of cosmological parameters. We find that the Hubble
parameter evolves more slowly in non-minimal coupling case as compared to the
minimal coupling case. In both the cases, the universe accelerates around
present time, and enters the decelerating regime in the past. Using the latest
Union2 dataset for supernova Type Ia observations as well as the data for
baryon acoustic oscillation (BAO) from SDSS observations, we constraint the
parameters of Linder exponential model in the two different approaches. We find
that there is a upper bound on model parameter in minimal coupling. But for
non-minimal coupling case, there is range of allowed values for the model
parameter.
