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

6. Conclusions

In this paper we have presented a relation between cosmic density and velocity fields following a purely Lagrangian derivation (Eq.(22)) that is self-consistent in the second-order model. The advantage of working in the Lagrangian picture is that the over-density field obtained from the continuity equation is self-consistent and automatically satisfies conservation of momentum. For simplicity, one can adopt the condition (19) of parallelism of the fields and , satisfied at first order and by a large class of irrotational solutions in second-order Lagrangian perturbation theory, and the result (22) holds accurately at both orders. However, as pointed out in Sect. 3, we note that (19) is not a sine qua non condition but a weaker condition of parallelism of the gradients of the fields would yield the same result. It is shown that this condition is well satisfied for generic initial conditions. In each case, the function is given explicitly in Appendix B.1 and Appendix B.2 respectively. Its range of validity is limited by the epoch when the inversion map becomes singular, i.e. up to orbit-crossing time, while its principal range of validity should strictly be estimated by , where is the displacement field with respect to the Hubble-flow. This condition is probably very conservative in view of the success of these approximations if followed up to shell-crossing.

By specifying the initial data at and keeping only the leading terms in the solutions, we are able to recover the leading terms of previous reconstruction models as shown in Sect. 5. Furthermore, the time dependence of the orbits in our model is established by solving the Lagrange-Newton system at each order. This leads to a unique and consistent result for the Eulerian density constrast in terms of the peculiar-velocities.

We finally wish to show some skepticism about the usefulness of the reconstruction formulae given. We want them to be applicable to present day observational data. However, we know confidently that shell-crossing has occurred, i.e. the interesting regime is no longer covered. Instead one would have to investigate a relationship between smoothed cosmic over-density and peculiar-velocity fields, where the smoothing window scale has to ensure the absence of vorticity and multi-stream systems in the average flow. Given the present work we should ask whether there is a formally proper way to incorporate smoothing into a consistent Lagrangian reconstruction formalism. With regard to this point, we think that solutions of the Lagrange-Newton system may possibly provide reasonable tools for reconstructing average fields, since averages in Eulerian space of any vector function are straightforward to calculate: . However, we still encounter the fundamental problem of averaging over multi-stream flows, which may not be properly handled in the framework of the Lagrange-Newton system. Rather, such a description has to be based on an approximation of the Vlasov-Poisson system. This major improvement of the methods lies well beyond the scope of present reconstruction techniques and requires the construction of Vlasov-Poisson type approximations.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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