Third-order perturbative solutions in Lagrangian perturbation theory with pressure

The structure formation scenario based on gravitational instability has been studied for a long time. The Lagrangian perturbative method for the cosmological fluid describes the nonlinear evolution of density fluctuation rather well. Zel’dovich [1] proposed a linear Lagrangian approximation for dust fluid. This approximation is called the Zel’dovich approximation (ZA) [1–6]. ZA describes the evolution of density fluctuation better than the Eulerian approximation [7–9]. After that, the secondand the thirdorder perturbative solutions for dust fluid were derived [10–17]. Recently the effect of the pressure in the cosmological fluid has been considered. At first, the effect of the pressure is originated from velocity dispersion using the collisionless Boltzmann equation [18,19]. Buchert and Domı́nguez [18] showed that when the velocity dispersion is regarded as small and isotropic it produces effective ‘‘pressure’’ or viscosity terms. Furthermore, they posited the relation between mass density and pressure P, i.e., an ‘‘equation of state’’. Adler and Buchert [20] have formulated the Lagrangian perturbation theory for a barotropic fluid. Morita and Tatekawa [21] and Tatekawa et al. [22] solved the Lagrangian perturbation equations for a polytropic fluid up to the second-order. Hereafter, we call this model the ‘‘pressure model’’. Although the higher-order perturbative solution is expected to improve the approximation, there is a counterargument. Let us consider the evolution of the spherical void for dust fluid. Because the exact solution has already been derived, we can discuss the accuracy of the perturbative solutions. In this case, when we increase the order of Lagrangian approximation, contrary to expectation, the description becomes worse [7,8,23]. Especially, when we stop the order of the perturbation until even order (the second-order), the perturbative solution describes the contraction of a void at a later. For the pressure model, although we do not know the exact solution for the evolution of the spherical void, the same problem may arise. In fact, according to a comparison between the first-order solution and the full-order