The recursion relation in Lagrangian perturbation theory

Abstract

We derive a recursion relation in the framework of Lagrangian perturbation\ntheory, appropriate for studying the inhomogeneities of the large scale\nstructure of the universe. We use the fact that the perturbative expansion of\nthe matter density contrast is in one-to-one correspondence with standard\nperturbation theory (SPT) at any order. This correspondence has been recently\nshown to be valid up to fourth order for a non-relativistic, irrotational and\ndust-like component. Assuming it to be valid at arbitrary (higher) order, we\nexpress the Lagrangian displacement field in terms of the perturbative kernels\nof SPT, which are itself given by their own and well-known recursion relation.\nWe argue that the Lagrangian solution always contains more non-linear\ninformation in comparison with the SPT solution, (mainly) if the\nnon-perturbative density contrast is restored after the displacement field is\nobtained.\n