  In the present work the primordial Big-Bang Nucleosynthesis (BBN) and weakly
interacting massive particle (WIMP) dark matter are discussed in a certain
class of modified gravitational theories, namely $f(R) \sim R^n$ gravity. The
new gravitational model is characterized by a single parameter $n$. First we
determine the conditions under which the theoretical predictions for the
$^{4}$He abundance are in agreement with the observations. More precisely,
during BBN the physics is known and all the parameters are known. The only free
parameter to be constrained is the power $n$ related to the new gravitational
model. After that, for cold dark matter we use the value of $n$ determined from
the BBN considerations and determine how the mass of the dark matter particle
is related to the annihilation cross section in order for the cold dark matter
constraint to be satisfied.
