Lagrangian Perturbation Approach to the Formation of Large–scale Structure

The present lecture notes address three columns on which the Lagrangian perturbation approach to cosmological dynamics is based: 1. the formulation of a Lagrangian theory of self–gravitating flows in which the dynamics is described in terms of a single field variable; 2. the procedure, how to obtain the dynamics of Eulerian fields from the Lagrangian picture, and 3. a precise definition of a Newtonian cosmology framework in which Lagrangian perturbation solutions can be studied. While the first is a discussion of the basic equations obtained by transforming the Eulerian evolution and field equations to the Lagrangian picture, the second exemplifies how the Lagrangian theory determines the evolution of Eulerian fields including kinematical variables like expansion, vorticity, as well as the shear and tidal tensors. The third column is based on a specification of initial and boundary conditions, and in particular on the identification of the average flow of an inhomogeneous cosmology with a “Hubble–flow”. Here, we also look at the limits of the Lagrangian perturbation approach as inferred from comparisons with N–body simulations and illustrate some striking properties of the solutions. ∗ to appear in: Proc. Int. School of Physics Enrico Fermi, Course CXXXII, Varenna 1995. 1. Lagrangian Theory of Self–gravitating Flows The description of fluid motions in cosmology has been largely studied in an Eulerian coordinate system ~x, i.e., a rectangular non–rotating frame in Euclidean space. Quite recently, it has become popular to study fluid motions in a Lagrangian coordinate system ~ X, i.e., a curvilinear, possibly rotating frame in Euclidean space which is defined such as to move with the fluid. Since the Lagrangian description has a number of advantages over the Eulerian one, and since this description enjoys many applications in the recent cosmology literature, it is important to elucidate in proper language the Lagrangian formalism. Since the lectures by François Bouchet and Peter Coles (this volume) explore the field of recent applications of the Lagrangian perturbation theory, I here concentrate on the basic architecture of a Lagrangian theory of structure formation. I do this in Newtonian cosmology, the lecture by Sabino Matarrese (this volume) gives an extension 2 to the framework of General Relativity. Accordingly, I kept my reference list short, since more references may be found in the other lectures. That the Lagrangian approach is experiencing a revival in cosmology is good news; I consider it the natural frame to describe fluid motions, since this description is formally close to the mechanics of point particles. If you consult old textbooks on hydrodynamics, you will find that the Lagrangian picture was considered too complicated for practical purposes beyond problems with high symmetry, and therefore has not been pursued further. I hope that, after this and the related lectures, you will be convinced of the opposite. Let us start with the basic system of equations in Newtonian cosmology describing the motion of a pressureless fluid in the gravitational field which is generated by its own density. We think at applications for a (dominating) collisionless component in the Universe; the gravitational dynamics we describe is thought to act as an attractor for the baryonic matter component which is “lighted up” by physics not described by these equations. With this assumption the fluid motion in Eulerian space is completely characterized by its velocity field ~v(~x, t) and its density field ̺(~x, t) > 0. The fluid has to obey the familiar evolution equations for these fields, ∂t~v = −(~v · ∇)~v + ~g , (1a) ∂t̺ = −∇ · (̺~v) , (1b) where the gravitational field ~g(~x, t) is constrained by the (Newtonian) field equations ∇× ~g = ~0 , (1c) ∇ · ~g = Λ− 4πGρ ; (1d) Λ denotes the cosmological constant. (Strictly speaking, ~g in eq. (1a) is a force per unit inertial mass, whereas in eqs. (1c,d) ~g is the field strength associated with gravitational mass. That we set both equal is the content of Einstein’s equivalence principle of inertial and gravitational mass.) Alternatively, we can write the eqs. (1c) and (1d) in terms of a single Poisson equation for the gravitational potential, which we do not need in the following. Hereafter, we call the system (1) the Euler–Newton system. One important issue to learn about the Lagrangian treatment is the fact that both evolution equations (1a) and (1b) can be integrated exactly in Lagrangian space, velocity and density will therefore not appear as dynamical variables later. To see this, we first look at the basic Lagrangian field variable which is the trajectory field of fluid elements, or the deformation field of the medium, respectively (Fig.1a): ~x = ~ f( ~ X, t) ; ~ X := ~ f( ~ X, t0) , (2a) 3 where ~ X denote the Lagrangian coordinates which label fluid elements, ~x are the positions of these elements in Eulerian space at the time t, and ~ f is the trajectory of fluid elements for constant ~ X. (Notice: the Eulerian positions ~x are here viewed not as independent variables (i.e., coordinates), but as dependent fields of Lagrangian coordinates; therefore, we employ the letter ~ f for the sake of clarity. Independent variables are now ( ~ X, t) instead of (~x, t).)