Constant-roll $k$-Inflation Dynamics
  In this work we shall investigate the phenomenological implications of the
constant-roll condition on a $k$-Inflation theory of gravity. The latter
theories are particularly promising, since these remained robust to the results
of GW170817, since these have a gravitational wave speed $c_T=1$ in natural
units. We shall mainly focus on the phenomenology of the $k$-Inflation models,
with the only assumption being the slow-roll condition imposed on the first and
fourth slow-roll parameters, and the constant-roll condition for the evolution
of the scalar field. We present in detail the final form of the gravitational
equations of motion that control the dynamics of the cosmological system, with
the constant-roll condition imposed, and by using a conveniently, from the
perspective of analytical manipulations, chosen potential, we express the
slow-roll indices and the resulting observational indices of the theory as
functions of the $e$-foldings number. The results of our analysis indicate that
the constant-roll $k$-Inflation theory can be compatible with the Planck 2018
data, for a wide range of the free parameters. Also we examine in a
quantitative way the effects of the constant-roll condition on the parameter
$f_{NL}^{equil}$ on which the bispectrum is proportional, in the equilateral
momentum approximation, and we demonstrate that the effect of the constant-roll
condition is non-trivial. In effect, non-Gaussianities in the theory may be
enhanced, a phenomenon which is known to be produced by constant-roll scalar
theories of gravity in general.
