Extended matter bounce scenario in ghost free $f(R,\mathcal{G})$ gravity
  compatible with GW170817
  In the context of a ghost free $f(R,\mathcal{G})$ model, an extended matter
bounce scenario is studied where the form of the scale factor is given by $a(t)
= (a_0t^2 + 1)^n$. The ghost free character of the model is ensured by the
presence of a Lagrange multiplier, as developed in \cite{Nojiri:2018ouv}. The
conditions under which, in this model, the speed of gravitational waves becomes
equal to the speed of light (equal to one, in natural units), thus becoming
compatible with the striking event GW170817, is investigated. It is shown that
this happens for a class of Gauss-Bonnet (GB) coupling functions ($h(t)$) which
satisfies a constraint equation of the form $\ddot{h} = \dot{h}H$, with $H$ the
Hubble parameter. This constraint is then imposed on the ghost free
$f(R,\mathcal{G})$ gravity theory to be consistent with the GW170817 event,
subsequently, the corresponding non-singular bouncing cosmology with the
aforementioned scale factor is extensively studied. The forms of the coupling
function and Lagrange multiplier in the "low curvature limit" of the theory are
reconstructed, yielding a viable approximation for $n < 1/2$. Correspondingly,
by solving the cosmological perturbation equation, the main observable
quantities, namely the spectral index, tensor to scalar ratio, and the running
index are determined and confronted with the latest Planck 2018 data.
Consistency with the data is proven for those parametric regimes that which
correspond to $n < 1/2$. This makes the low curvature approximation a viable
one for calculating the scalar and tensor power spectra.
