General background conditions for K-bounce and adiabaticity

We study the background conditions for a bounce uniquely driven by a single scalar field model with a generalized kinetic term $K(X)$, without any additional matter field. At the background level we impose the existence of two turning points where the derivative of the Hubble parameter $H$ changes sign and of a bounce point where the Hubble parameter vanishes. We find the conditions for $K(X)$ and the potential which ensure the above requirements. We then give the examples of two models constructed according to these conditions. One is based on a quadratic $K(X)$, and the other a $K(X)$ which is avoiding divergences of the second time derivative of the scalar field, which may otherwise occur. An appropriate choice of the initial conditions can lead to a sequence of consecutive bounces, or oscillations of $H$. In the region where these models have a constant potential they are adiabatic on any scale and because of this they may not conserve curvature perturbations on super-horizon scales. While at the perturbation level one class of models is free from ghosts and singularities of the classical equations of motion, in general gradient instabilities are present around the bounce time, because the sign of the squared speed of sound is opposite to the sign of the time derivative of $H$. We discuss how this kind of instabilities could be avoided by modifying the Lagrangian by introducing Galileion terms in order to prevent a negative squared speed of sound around the bounce.