As deeper galaxy catalogues are soon to come, it becomes even more importantto measure large-scale fluctuations in the catalogues with robust statistics that cover all moments of the galaxy distribution.In this paper we reinforce a direct analysis of
galaxy data by employing the Germ-Grain method to calculate thefamily of Minkowski Functionals. We introduce a new code, suitable for the analysis of large data sets without smoothingand without the construction of excursion sets. We provide new tools to measure correlation properties, putting emphasis onexplicitly isolating non-Gaussian correlations with the help of integral-geometric relations. As a first application we present the analysis of large-scale fluctuations in the luminous red galaxy sample of Sloan Digital Sky Survey data release 7 data. We findsignificant deviations from the $Lambda$ cold dark matter mock catalogues on samples as large as $500h^{-1}$Mpc (more than $3sigma$)and slight deviations of around $2sigma$ on $700h^{-1}$Mpc, and we investigate possible sources of these deviations.
In this first paper we present a Lagrangian framework for the description of structure formation in general relativity, restricting attention to irrotational dust matter. As an application we present a self-contained derivation of a general-relativis
tic analogue of Zeldovichs approximation for the description of structure formation in cosmology, and compare it with previous suggestions in the literature. This approximation is then investigated: paraphrasing the derivation in the Newtonian framework we provide general-relativistic analogues of the basic system of equations for a single dynamical field variable and recall the first-order perturbation solution of these equations. We then define a general-relativistic analogue of Zeldovichs approximation and investigate its implications by functionally evaluating relevant variables, and we address the singularity problem. We so obtain a possibly powerful model that, although constructed through extrapolation of a perturbative solution, can be used to put into practice nonperturbatively, e.g. problems of structure formation, backreaction problems, nonlinear properties of gravitational radiation, and light-propagation in realistic inhomogeneous universe models. With this model we also provide the key-building blocks for initializing a fully relativistic numerical simulation.
