  We explain the effect of dark matter (flat rotation curve) using modified
gravitational dynamics. We investigate in this context a low energy limit of
generalized general relativity with a nonlinear Lagrangian ${\cal L}\propto
R^n$, where $R$ is the (generalized) Ricci scalar and $n$ is parameter
estimated from SNIa data. We estimate parameter $\beta$ in modified
gravitational potential $V(r) \propto -\frac{1}{r}(1+(\frac{r}{r_c})^{\beta})$.
Then we compare value of $\beta$ obtained from SNIa data with $\beta$ parameter
evaluated from the best fitted rotation curve. We find $\beta \simeq 0.7$ which
becomes in good agreement with an observation of spiral galaxies rotation
curve. We also find preferred value of $\Omega_{m,0}$ from the combined
analysis of supernovae data and baryon oscillation peak. We argue that although
amount of "dark energy" (of non-substantial origin) is consistent with SNIa
data and flat curves of spiral galaxies are reproduces in the framework of
modified Einstein's equation we still need substantial dark matter. For
comparison predictions of the model with predictions of the $\Lambda$CDM
concordance model we apply the Akaike and Bayesian information criteria of
model selection.
