Statefinder diagnostic and constraints on the Palatini f(R) gravity
  theories
  We focus on a series of $f(R)$ gravity theories in Palatini formalism to
investigate the probabilities of producing the late-time acceleration for the
flat Friedmann-Robertson-Walker (FRW) universe. We apply statefinder diagnostic
to these cosmological models for chosen series of parameters to see if they
distinguish from one another. The diagnostic involves the statefinder pair
$\{r,s\}$, where $r$ is derived from the scale factor $a$ and its higher
derivatives with respect to the cosmic time $t$, and $s$ is expressed by $r$
and the deceleration parameter $q$. In conclusion, we find that although two
types of $f(R)$ theories: (i) $f(R) = R + \alpha R^m - \beta R^{-n}$ and (ii)
$f(R) = R + \alpha \ln R - \beta$ can lead to late-time acceleration, their
evolutionary trajectories in the $r-s$ and $r-q$ planes reveal different
evolutionary properties, which certainly justify the merits of statefinder
diagnostic. Additionally, we utilize the observational Hubble parameter data
(OHD) to constrain these models of $f(R)$ gravity. As a result, except for
$m=n=1/2$ of (i) case, $\alpha=0$ of (i) case and (ii) case allow $\Lambda$CDM
model to exist in 1$\sigma$ confidence region. After adopting statefinder
diagnostic to the best-fit models, we find that all the best-fit models are
capable of going through deceleration/acceleration transition stage with
late-time acceleration epoch, and all these models turn to de-Sitter point
($\{r,s\}=\{1,0\}$) in the future. Also, the evolutionary differences between
these models are distinct, especially in $r-s$ plane, which makes the
statefinder diagnostic more reliable in discriminating cosmological models.
