PERTURBATIVE LAGRANGIAN APPROACH TO GRAVITATIONAL-INSTABILITY

This paper deals with the time evolution in the matter era of perturbations in Friedman-Lema (i) over cap tre models with arbitrary density parameter Omega, with either a zero cosmological constant, Lambda = 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 in the Lagrangian perturbative series is the famous Zeldovich approximate solution (1970). The idea to consider the higher order terms was introduced by Moutarde et al. (1991), generalized by Bouchet et al. (1992), and further developed by many others. 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 (which is necessary to compute, for instance, the four point spatial correlation function or the corrections to the linear evolution of the two-point correlation function, as well as the secondary temperature anisotropies of the Cosmic Microwave Background). 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 approximations in the case of spherically symmetric perturbations. We also corn, pare this second order approximation and Zeldovich solution to N-body simulations with scale-free (n = -2) Gaussian initial conditions. We find that second order theory is both simple and powerful.