  Recently we showed that in FLRW cosmology, the contribution from higher
curvature terms in any generic metric gravity theory to the energy-momentum
tensor is of the perfect fluid form. Such a geometric perfect fluid can be
interpreted as a fluid remaining from the beginning of the universe where the
string theory is thought to be effective. Just a short time after the beginning
of the Universe, it is known that the Einstein-Hilbert action is assumed to be
modified by adding all possible curvature invariants. We propose that the
observed late-time accelerating expansion of the Universe can be solely driven
by this geometric fluid. To support our claim, we specifically study the
quadratic gravity field equations in $D$-dimensions. We show that the field
equations of this theory for the FLRW metric possess a geometric perfect fluid
source containing two critical parameters $\sigma_1$ and $\sigma_2$. To analyze
this theory concerning its parameter space $(\sigma_1, \sigma_2)$, we obtain
the general second-order nonlinear differential equation governing the
late-time dynamics of the deceleration parameter $q$. Hence using some
present-day cosmological data as our initial conditions, our findings for the
$\sigma_2=0$ case are as follows: $ (i)$ In order to have a positive energy
density for the geometric fluid $\rho_g$, the parameter $\sigma_1$ must be
negative for all dimensions up to $D = 11$, $(ii)$ For a suitable choice of
$\sigma_1$, the deceleration parameter experiences signature changes in the
past and future, and in the meantime it lies within a negative range which
means that the current observed accelerated expansion phase of the Universe can
be driven solely by the curvature of the spacetime, $(iii)$ $q$ experiences a
signature change and as the dimension $D$ of spacetime increases, this
signature change happens at earlier and later times, in the past and future,
respectively.
