Backreaction in Cosmological Models

Most cosmological models studied today are based on the assumption of homogeneity and isotropy. Observationally one can find evidence that supports these assumptions on very large scales, the strongest being the almost isotropy of the Cosmic Microwave Background radiation after assigning the whole dipole to our proper motion relative to this background. However, on small and on intermediate scales up to several hundreds of Mpcs, there are strong deviations from homogeneity and isotropy. Here the problem arises how to relate the observations with the homogeneous and isotropic models. The usual proposal for solving this problem is to assume that Friedmann-Lemaitre models describe the mean observables. Such mean values may be identified with spatial averages. For Newtonian fluid dynamics the averaging procedure has been discussed in detail in Buchert and Ehlers (1997), leading to an additional backreaction term in the Friedmann equation. We use the Eulerian linear approximation and the `Zel'dovich approximation' to estimate the effect of the backreaction term on the expansion. Our results indicate that even for domains matching the background density in the mean, the evolution of the scale factor strongly deviates from the Friedmann solution, critically depending on the velocity field inside.Comment: 4 pages LaTeX, 2 figures, a4wide.sty include

Improving the accuracy of estimators for the two-point correlation function

We show how to increase the accuracy of estimates of the two-point correlation function without sacrificing efficiency. We quantify the error of the pair-counts and of the Landy-Szalay estimator by comparing with exact reference values. The standard method, using random point sets, is compared to geometrically motivated estimators and estimators using quasi Monte-Carlo integration. In the standard method the error scales proportional to $1/\sqrt{N_r}$, with $N_r$ the number of random points. In our improved methods the error is scaling almost proportional to $1/N_q$, where $N_q$ is the number of points from a low discrepancy sequence. In an example we achieve a speedup by a factor of $10^4$ over the standard method, still keeping the same level of accuracy. We also discuss how to apply these improved estimators to incompletely sampled galaxy catalogues.Comment: 11 pages, 6 figures, submitted to A&

Dimensionality and morphology of particle and bubble clusters in turbulent flow

We conduct numerical experiments to investigate the spatial clustering of particles and bubbles in simulations of homogeneous and isotropic turbulence. Varying the Stokes parameter and the densities, striking differences in the clustering of the particles can be observed. To quantify these visual findings we use the Kaplan--Yorke dimension. This local scaling analysis shows a dimension of approximately 1.4 for the light bubble distribution, whereas the distribution of very heavy particles shows a dimension of approximately 2.4. However, clearly separate parameter combinations yield the same dimensions. To overcome this degeneracy and to further develop the understanding of clustering, we perform a morphological (geometrical and topological) analysis of the particle distribution. For such an analysis, Minkowski functionals have been successfully employed in cosmology, in order to quantify the global geometry and topology of the large-scale distribution of galaxies. In the context of dispersed multiphase flow, these Minkowski functionals -- being morphological order parameters -- allow us to discern the filamentary structure of the light particle distribution from the wall-like distribution of heavy particles around empty interconnected tunnels.Comment: 12 pages, 8 figure

A comparison of estimators for the two-point correlation function

Nine of the most important estimators known for the two-point correlation function are compared using a predetermined, rigorous criterion. The indicators were extracted from over 500 subsamples of the Virgo Hubble Volume simulation cluster catalog. The ``real'' correlation function was determined from the full survey in a 3000Mpc/h periodic cube. The estimators were ranked by the cumulative probability of returning a value within a certain tolerance of the real correlation function. This criterion takes into account bias and variance, and it is independent of the possibly non-Gaussian nature of the error statistics. As a result for astrophysical applications a clear recommendation has emerged: the Landy & Szalay (1993) estimator, in its original or grid version Szapudi & Szalay (1998), are preferred in comparison to the other indicators examined, with a performance almost indistinguishable from the Hamilton (1993) estimator.Comment: aastex, 10 pages, 1 table, 1 figure, revised version, accepted in ApJ