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 spacetimes within the framework of Horndeski’s scalar–tensor theory. In this method, as ansatzes, we use the following assumptions. First, we set the propagation speeds [Formula: see text], [Formula: see text] in the tensor and scalar sectors of perturbations, the tensor-to-scalar ratio r and other perturbation parameters to be constant. Following the observational data, we believe that [Formula: see text], [Formula: see text] (LIGO Scientific Collab., Virgo Collab. and Fermi Gamma-Ray Burst Monitor Team, Astrophys. J. Lett. 848, L13 (2017)). This ansatz allows us to exclude ghosts and Laplace instabilities at the initial stage. Second, we choose the dependence of the Hubble parameter on the derivative of the scalar field [Formula: see text]. Since the characteristics of perturbations [Formula: see text], [Formula: see text] and others are initially expressed through H, [Formula: see text] and the Horndeski potentials, the ansatzes with the gravity equations give a system of differential equations for the scalar field and the unknown Horndeski potentials. Knowledge of [Formula: see text] and [Formula: see text] gives the law of the Universe expansion [Formula: see text]. 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.