Five-dimensional metric $f(R)$ gravity and the accelerated universe
  The metric $f(R)$ theories of gravity are generalized to five-dimensional
spacetimes. By assuming a hypersurface-orthogonal Killing vector field
representing the compact fifth dimension, the five-dimensional theories are
reduced to their four-dimensional formalism. Then we study the cosmology of a
special class of $f(R)=\alpha R^m$ models in a spatially flat FRW spacetime. It
is shown that the parameter $m$ can be constrained to a certain range by the
current observed deceleration parameter, and its lower bound corresponds to the
Kaluza-Klein theory. It turns out that both expansion and contraction of the
extra dimension may prescribe the smooth transition from the deceleration era
to the acceleration era in the recent past as well as an accelerated scenario
for the present universe. Hence five-dimensional $f(R)$ gravity can naturally
account for the present accelerated expansion of the universe. Moreover, the
models predict a transition from acceleration to deceleration in the future,
followed by a cosmic recollapse within finite time. This differs from the
prediction of the five-dimensional Brans-Dicke theory but is in consistent with
a recent prediction based on loop quantum cosmology.
