Oscillating solutions of the matter density contrast in Horndeski's theory

Evolution of the large scale structure of the Universe is described by galaxy bias and the growth rate function of the matter density contrast. The behavior of the growth rate function, in general, depends on the types of dark energy or modified gravity theories, because they modify Poisson's equation for gravity by changing the Newtonian constant in the quasi-static solutions. Whereas, another mode of the matter density contrast "oscillating mode" has not been studied well, although there is a possibility that it drastically changes the gravitational law. In this paper, we will construct the way to investigate oscillating solutions and show the concrete forms of the solutions by using the WKB approximation.