  We propose a cosmological scenario in which the universe undergoes through a
non-singular bounce, and after the bounce, it decelerates having a matter-like
dominated evolution during some regime of the deceleration era, and finally at
the present epoch it evolves through an accelerating stage. Our aim is to study
such evolution in the context of Chern-Simons corrected F(R) gravity theory and
confront the model with various observational data. Using the reconstruction
technique, and in addition by employing suitable boundary conditions, we
determine the form of F(R) for the entire possible range of the cosmic time.
The form of F(R) seems to unify a non-singular bounce with a dark energy epoch,
in particular, from a non-singular bounce to a deceleration epoch and from a
deceleration epoch to a late time acceleration era. It is important to mention
that the bouncing scenario in the present context is an asymmetric bounce, in
particular, the Hubble radius monotonically increases and asymptotically
diverges at the late contracting era, while it seems to decrease with time at
the present epoch. Such evolution of the Hubble radius leads to the primordial
perturbation modes generate at the deep contracting era when all the
perturbation modes lie within the horizon. We calculate the scalar and tensor
power spectra, and as a result, the primordial observables are found to be in
agreement with the latest Planck 2018 constraints. In this regard, the
Chern-Simons term seems to have considerable effects on the tensor perturbation
evolution, however keeping intact the scalar part of the perturbation with that
of in the case of a vacuum F(R) model, and as a result, the Chern-Simons term
proves to play an important role in making the observable quantities consistent
with the Planck results. Furthermore the theoretical expectation of the dark
energy observables are confronted with the Planck+SNe+BAO data.
