Analytic approaches to the structure formation in the accelerating universe

The exixtence of dark energy is a serious problem in modern cosmology. For the origin of the dark energy, many models including a cosmological constant have been proposed. Although these models can explain the present acceleration of the Universe, some of the models would not be able to explain the observed large-scale structure of the universe. Therefore, in order to constrain the models of the dark energy, we should consider the structure formation in the universe. From primordial density fluctuation, the large-scale structure is formed via its own self-gravitational instability. Even though numerical simulations are necessary to follow the full history of the structure formation, in order to understand the physics behind the structure formation, analytic approaches play important roles. In this review, we summarise various analytic approaches to the evolution of the density fluctuation in Newtonian cosmology and show they can be helpful to distinguish models when applied to the quasi-nonlinear region. We also mention several applications of the analytic approaches including the initial condition problems for cosmological N-body simulations, higher-order Lagrangian perturbation theory.