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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.