In this work we study a modified version of f ( R ) gravity in which higher order kinetic terms of a scalar field are added in the action of vacuum f ( R ) gravity .
This type of theory is a type of k -essence f ( R ) gravity , and it belongs to the general class of f ( R, \phi,X ) theories of gravity , where \phi is a scalar field and X = \frac { 1 } { 2 } \partial ^ { \mu } \phi \partial _ { \mu } \phi .
We focus on the inflationary phenomenology of the model , in the slow-roll approximation , and we investigate whether viable inflationary evolutions can be realized in the context of this theory .
We use two approaches , firstly by imposing the slow-roll conditions and by using a non-viable vacuum f ( R ) gravity .
As we demonstrate , the spectral index of the primordial scalar perturbations and the tensor-to-scalar ratio of the resulting theory can be compatible with the latest observational data .
In the second approach , we fix the functional form of the Hubble rate as a function of the e -foldings number , and we modify well-known vacuum f ( R ) gravity reconstruction techniques , in order to find the k -essence f ( R ) gravity which can realize the given Hubble rate .
Accordingly , we calculate the slow-roll indices and the corresponding observational indices , and we also provide general formulas of these quantities in the slow-roll approximation .
As we demonstrate , viability can be obtained in this case too , however the result is strongly model dependent .
In addition , we discuss when ghosts can occur in the theory , and we investigate under which conditions ghosts can be avoided by using a particular class of models .
Finally , we qualitatively discuss the existence of inflationary attractors for the non-slow-roll theory , and we provide hints towards finding general de Sitter attractors for the theory at hand .