Eulerian Perturbative Approximations in Planar Symmetry

Abstract

In planar symmetry (one-dimensional peculiar motions), using the Eulerian perturbative framework and an arbitrary initial profile, we compute the analytical evolution of the peculiar velocity and the density contrast up to the fourth order (E1-E4 approximations). From these results and the exact (Lagrangian) solutions, the accuracy of the Ej (j ≤ 4) approaches in describing the quasi-linear evolution of some particular initial profiles (overdense and underdense planar halos) are studied. For the peculiar velocity, E2 works well. On the contrary, only E3 and E4 are good descriptors of the quasi-linear regime of the density contrast in underdensities, and the situation is worse for the density contrast in overdensities. We have also analyzed the true power of the Eulerian theory in general planar problems. In spite of the apparent weakness of the Eulerian scheme in planar problems (in relation to the Lagrangian formalism, which leads to the exact solution even in the first-order approximation), we showed that this formalism is capable of yielding the exact solution to some plane-symmetric gravitational instability problems. For the E1-E4 approximations, the local relation agrees with the exact one. Moreover, for the evolution of planar cores (a top-hat initial profile or the central shell of an arbitrary inhomogeneity), it is possible to make, in a relatively simple way, a superapproach E∞, which leads to the exact solution (peculiar velocity and density contrast). We observe finally that the Eulerian formalism may, however, be a poor tracer for the evolution of planar halos. Only when a superapproach is viable does the Eulerian theory rival the Lagrangian one.