Lagrangian Description for the Cosmic Fluid

Abstract

The Lagrangian description for the cosmological fluid can be usefully applied to the structure formation scenario. This description provides a relatively accurate model even in a quasi-linear regime. Zel’dovich [?] proposed a linear Lagrangian approximation for dust fluid. This approximation is called the Zel’dovich approximation (ZA) [?, ?, ?, ?]. In ZA and its extended models, pressure was ignored. Recently, Lagrangian approximation in which the effect of pressure was taken into consideration have been analyzed. Buchert and Dominguez [?] discussed the effect of velocity dispersion using the collisionless Boltzmann equation They argued that models of a large-scale structure should be constructed for a flow describing the average motion of a multi-stream system. Then they showed that when the velocity dispersion is regarded as small and isotropic it produces effective “pressure” or viscosity terms. Furthermore, they derived the relation between mass density ρ and pressure P , i.e., an “equation of state.” Hereafter, we call this model the Euler-Jeans-Newton (EJN) model. Actually, Adler and Buchert [?] have formulated the Lagrangian perturbation theory for a barotropic fluid. Morita and Tatekawa [?] derived the linear perturbative solutions for the polytropic fluid in Einstein-de Sitter Universe model. Then Tatekawa et al. [?] showed the solutions in generic Friedmann Universe models. In the Lagrangian approximation, the displacement of the fluid element from homogeneous distribution is regarded as a perturbative quantity. Using this formalism, the matter density is described with exact form. Furthermore we can obtain relatively good description for the density field, because the relation between the density and the displacement is nonlinear.