Astron. Astrophys. 323, 295-304 (1997)

1. Introduction

An important aim of cosmology is to determine a relationship between the cosmic density contrast and the peculiar-velocity field under the assumption that these fields have evolved under the action of gravity. In the Eulerian theory of gravitational instability, retaining only the linear growing-mode solution, this relationship is simply given by ¹

where is the usual linear growth factor, such that in an Einstein-de Sitter universe. Beyond linear theory, in the weakly non-linear regime, (1) has been used to good effect as an estimate of the density contrast field from observed peculiar-velocities (e.g. Bertschinger & Dekel 1989; Dekel et al. 1990; Bertschinger et al. 1990). On the other hand, the POTENT method and other quasi-linear reconstruction procedures involve the Zel'dovich approximation (ZA)(Nusser et al. 1991; Nusser & Dekel 1992). The ZA does not however, strictly speaking, provide an exact algebraic relation between the fields and . It can however yield (as shown in Nusser et al. 1991) a relation between the fields that is self-consistent within the order of the approximation, i.e. within . In the ZA, from the Lagrangian integral of the continuity equation we have

and the fields are mapped back to Eulerian space via the ZA map for the orbits of the particles. Initial conditions are specified at an arbitrary early time . Also, it is customary to make the assumption of smoothness at early times, , by virtue of the amplitude of the fluctuations in the microwave background. From (2) we obtain the linear relation (1) by truncating the expansion of the determinant (2) to the lowest order in the perturbation field and replacing Lagrangian by Eulerian coordinates ().

These schemes have been generalized further (Gramann 1993a, 1993b; for their Eulerian analogue, see Bernardeau 1992) by starting out with a parametrization of the particle orbits of the type:

In the case of a flat Universe, the leading second-order term in perturbation theory (Bouchet et al. 1992) is indeed . Therefore, the coordinate map (3) is in principle well-motivated (for a numerical comparison of the different models, see e.g. Dekel 1994). However, as shown in Gramann (1993b), orbits of the type (3) do not yield a self-consistent relation for the cosmic density and velocity fields; the density fields obtained via (a) the continuity equation and (b) the Euler equation differ within the order of the approximation. The reason for this is that, as we shall see, (3) is only an approximate second-order solution of the Lagrange-Newton system. Thus, we wish to call the attention of the reader to the fact that the reconstruction models derived from (3) and, more generally, from an ansatz including higher orders in ,

(for arbitrary N), do not follow a rigorous line of analysis (in the case of an arbitrary ) for the following reasons:

I In these models, the parametrization of the orbits is given ad hoc and it is not derived as a solution of the Lagrangian evolution equations for the flow field. In particular, the absence of lower-order growing modes, that are present in the general perturbation theory solution, implies that the derivation of through the continuity equation neglects couplings of these modes with leading-order terms.

II The density contrast obtained from the continuity equation in perturbation theory and satisfying Euler's equation are mutually incompatible.

III There is no reason to justify that is a good perturbative parameter (in terms of the convergence of the solutions) and that a meaningful generalization of the family of solutions where the ZA belongs can be realized as a polynomial in .

It is easy to see that (II) follows as a consequence of (I). Regarding (I) and (III), it has been shown that the correct coordinate map between Eulerian and Lagrangian coordinates in Lagrangian perturbation theory is obtained by calculating the perturbative solution of the Lagrange-Newton system for the trajectories . These evolution equations are obtained by transforming the Euler-Newton system to Lagrangian coordinates and eliminating all Eulerian fields by using exact Lagrangian integrals for the Eulerian acceleration and density (Buchert & Götz 1987; Buchert 1989). Within this scheme, the ZA is recovered as a subclass of solutions (Buchert 1992), and hence, one can obtain a self-consistent solution for the over-density field in terms of the velocities as in Nusser et al. (1991). Furthermore, the second- and third-order solutions obtained in the Lagrangian framework (Buchert & Ehlers 1993, Buchert 1994; and in a slightly different approach Bouchet et al. 1992, 1995; Catelan 1995) are unique and well-defined at all times from a fiducial time , where initial data are specified, ² until shell-crossing, provided we impose periodic boundary conditions and fix some global gauge conditions (Ehlers & Buchert 1997) as opposed to the orbits described by polynomials in , which are expected to have poor convergence (for a discussion in the context of the variational approach to the solutions see Susperregi & Binney 1994). Although the leading terms of the longitudinal parts of the perturbation solutions seem to confirm this polynomial approach (e.g. Bouchet et al. 1992, for the second-order solutions), this is not a mathematically consistent motivation for using as an expansion parameter by going to higher orders in perturbation theory. On the contrary, the structure of the Lagrangian perturbation scheme is such that, starting at third order, the presence of interaction terms among perturbations of different orders are an obstacle to convergence, and in fact for all terms are interaction terms (see Buchert 1994). Therefore, for an ansatz of the form (4) cannot reproduce gravitational dynamics, as stated in (III). This is an especially critical point to bear in mind in the context of reconstruction models in which higher-order couplings of flow field gradients show up in the density-velocity relation.

In this paper, we approach the reconstruction problem from a purely Lagrangian perspective using the solutions of the Lagrange-Newton system obtained by Buchert & Ehlers (1993). The goal of the paper is to obtain a self-consistent solution for the over-density field that is valid to second order, thus extending the results of Nusser et al. 1991 in the context of the ZA. Our results are purely Lagrangian, but in obtaining them we rely on the fact that there is a one-to-one mapping between Lagrangian and Eulerian coordinates and hence, the reconstruction formulae presented are only valid up to orbit-crossing time. The basic formalism is laid out in Sect. 2, and in Sect. 3 we discuss its application to two different classes of initial conditions. In Sect. 4 we discuss the self-consistency of solutions in perturbation theory models. Sect. 5 compares the model presented with previous reconstruction procedures and Sect. 6 sums up.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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