Non-minimally Coupled Einstein-Gauss-Bonnet Gravity with Massless
  Gravitons: The Constant-roll Case
  In this letter we study the behavior of non-minimally coupled
Einstein-Gauss-Bonnet gravity theories with the constant-roll condition.
Recalling the results of the striking GW170817 event, we demand that the
velocity of the gravitational waves is equated to unity in natural units,
meaning that $c_T^2=1$. This is a powerful restriction since it leads to a
decrease in the degrees of freedom and subsequently reveals a connection
between the scalar functions of the theory which presumably have different
origins. In this framework, we shall assume that a scalar potential is present
and can be extracted easily from the equations of motion by simply designating
the scalar coupling functions. Obviously, a different approach is feasible but
such choice will prove to be extremely convenient. Afterwards, we impose
certain approximations in order to facilitate our study. Each assumption is
capable of producing different phenomenology so a summary of all the possible
configurations for Hubble's parameter, its derivative and the scalar potential
along with the corresponding assumptions are present at the end of the paper.
We show that compatibility under the constant-roll assumption can be achieved
for a variety of model functions and different approaches, although one must
always be aware of the imposed approximations since the possibility of a model
producing viable results while simultaneously violating even a single
approximation exists. Finally, in the end, a new formalism which leads to a
plethora of convenient coupling functions according to the readers choice is
presented. Utilizing such formalism may lead to new cases for coupling
functions which have not been used yet, but are in fact able of producing
viable phenomenology for the inflationary era.
