$k$-essence non-minimally coupled with Gauss-Bonnet invariant for   inflation
  In this paper, we investigated inflationary solutions for a subclass of Horndeski models where a scalar field is non-minimally coupled with the Gauss-Bonnet invariant. Examples of canonical scalar field and $k$-essence to support the early-time acceleration are considered. The formalism to calculate the perturbations in FRW universe and to derive the spectral index and the tensor-to-scalar ratio is furnished. 
Ratbay Myrzakulov and Lorenzo Sebastiani
gr-qc