Models of exponential and power-law acceleration of the Universe in Horndeski theory without ghosts and Laplace instabilities

We present a designer method for flat Friedman-Robertson-Walker space-times within the framework of Horndeski’s scalar-tensor theory. As a result, one can find subclasses of Horndeski’s scalar-tensor theory within which cosmological solutions exist without ghosts and Laplace instabilities. As an example, we found subclasses of Horndeski’s theory for exponential and power-law inflation models of the Universe without pathologies.