Perturbative Lagrangian Approach to Gravitational Instability

Abstract

This paper deals with the time evolution in the matter era of perturbations in Friedman-Lemaitre models with arbitrary density parameter $Ω$, with either a zero cosmological constant, $Λ= 0$, or with a non-zero cosmological constant in a spatially flat Universe. Unlike the classical Eulerian approach where the density contrast is expanded in a perturbative series, this analysis relies instead on a perturbative expansion of particles trajectories in Lagrangian coordinates. This brings a number of advantages over the classical analysis. In particular, it enables the description of stronger density contrasts. Indeed the linear term is the famous Zel'dovich approximate solution (1970). We present here a systematic and detailed account of this approach. We give analytical results (or fits to numerical results) up to the third order. We then proceed to explore the link between the lagrangian description and statistical measures. We show in particular that Lagrangian perturbation theory provides a natural framework to compute the effect of redshift distortions, using the skewness of the density distribution function as an example. Finally, we show how well the second order theory does as compared to other approximat- ions in the case of spherically symmetric perturbations. We also compare this second order approximation and Zel'dovich solution to N-body simulations in the description of large-scale structure formation starting from a power law (n=-2) power spectrum of Gaussian perturbation. We find that second order theory is both simple and powerful.