  In this article, we propose different background models of extended theories
of gravity, which are minimally coupled to the SM fields, to explain the
possibility of genesis of dark matter without affecting the SM particle sector.
We modify the gravity sector by allowing quantum corrections motivated from (1)
local $f(R)$ gravity and (2) non-minimally coupled gravity with SM sector and
dilaton field. Next we apply conformal transformation on the metric to
transform the action back to the Einstein frame. We also show that an effective
theory constructed from these extended theories of gravity and SM sector looks
exactly the same. Using the relic constraint observed by Planck 2015, we
constrain the scale of the effective field theory ($\Lambda_{UV}$) as well as
the dark matter mass ($M$). We consider two cases- (1) light dark matter (LDM)
and (2) heavy dark matter (HDM), and deduce upper bounds on thermally averaged
cross section of dark matter annihilating to SM particles. Further we show that
our model naturally incorporates self interactions of dark matter. Using these
self interactions, we derive the constraints on the parameters of the (1) local
$f(R)$ gravity and (2) non-minimally coupled gravity from dark matter self
interaction. Finally, we propose some different UV complete models from a
particle physics point of view, which can give rise to the same effective
theory that we have deduced from extended theories of gravity.
