Towards a smooth unification from an ekpyrotic bounce to the dark energy
  era
  In the context of a ghost free $f(R,\mathcal{G})$ model, we present a
non-singular cosmological scenario in which the universe initially contracts
through an ekpyrotic phase having a bouncing like behaviour, and following the
bounce, it smoothly transits to a matter or radiation like deceleration era
which is further smoothly connected to the dark energy era at present epoch. We
consider the Gauss-Bonnet (GB) coupling function in such a way that it gets
compatible with the event GW170817. Using suitable reconstruction technique, we
obtain the non-trivial scalar field potential as well as the GB coupling
function. Such scalar potential and GB coupling function source a smooth
unified scenario from an ekpyrotic bounce to the dark energy era with an
intermediate deceleration era. The occurrence of ekpyrotic contraction phase
justifies the resolution of the anisotropic problem (also known as BKL
instability) in the background evolution. The Hubble radius shows an asymmetric
behaviour around the bounce, in particular, the evolution of the Hubble radius
leads to the generation era of the primordial perturbation modes far before the
bounce in the deep sub-Hubble regime. Accordingly we perform the scalar and
tensor perturbation evolution in the present context, and as a result, the
scalar power spectrum at large scale modes is found to be problematic. Thus an
extended scenario is proposed where we consider a pre-ekpyrotic phase having
the equation of state parameter is less than unity, and re-examine the scalar
and tensor power spectra, on large scales that cross the Hubble radius during
the pre-ekpyrotic stage. In this regard, the GB coupling function shows
considerable effects in reducing the tensor to scalar ratio compared to the
case where the GB coupling is absent. Furthermore the dark energy epoch is
consistent with the Planck+SNe+BAO data.
