Topology and Geometry of the CfA2 Redshift Survey

We analyse the redshift space topology and geometry of the nearby Universe by computing the Minkowski functionals of the Updated Zwicky Catalogue (UZC). The UZC contains the redshifts of almost 20,000 galaxies, is 96% complete to the limiting magnitude m_Zw=15.5 and includes the Center for Astrophysics (CfA) Redshift Survey (CfA2). From the UZC we can extract volume limited samples reaching a depth of 70 hMpc before sparse sampling dominates. We quantify the shape of the large-scale galaxy distribution by deriving measures of planarity and filamentarity from the Minkowski functionals. The nearby Universe shows a large degree of planarity and a small degree of filamentarity. This quantifies the sheet-like structure of the Great Wall which dominates the northern region (CfA2N) of the UZC. We compare these results with redshift space mock catalogues constructed from high resolution N-body simulations of two Cold Dark Matter models with either a decaying massive neutrino (tauCDM) or a non-zero cosmological constant (LambdaCDM). We use semi-analytic modelling to form and evolve galaxies in these dark matter-only simulations. We are thus able, for the first time, to compile redshift space mock catalogues which contain galaxies, along with their observable properties, rather than dark matter particles alone. In both models the large scale galaxy distribution is less coherent than the observed distribution, especially with regard to the large degree of planarity of the real survey. However, given the small volume of the region studied, this disagreement can still be a result of cosmic variance.Comment: 14 pages including 10 figures. Accepted for publication in Monthly Notice

Tensor Minkowski Functionals for random fields on the sphere

We generalize the translation invariant tensor-valued Minkowski Functionals which are defined on two-dimensional flat space to the unit sphere. We apply them to level sets of random fields. The contours enclosing boundaries of level sets of random fields give a spatial distribution of random smooth closed curves. We obtain analytic expressions for the ensemble expectation values for the matrix elements of the tensor-valued Minkowski Functionals for isotropic Gaussian and Rayleigh fields. We elucidate the way in which the elements of the tensor Minkowski Functionals encode information about the nature and statistical isotropy (or departure from isotropy) of the field. We then implement our method to compute the tensor-valued Minkowski Functionals numerically and demonstrate how they encode statistical anisotropy and departure from Gaussianity by applying the method to maps of the Galactic foreground emissions from the PLANCK data.Comment: 1+23 pages, 5 figures, Significantly expanded from version 1. To appear in JCA

Causality theory for closed cone structures with applications

We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular differentiability assumptions. We prove the validity of most results of the regular Lorentzian causality theory including causal ladder, Fermat's principle, notable singularity theorems in their causal formulation, Avez-Seifert theorem, characterizations of stable causality and global hyperbolicity by means of (smooth) time functions. For instance, we give the first proof for these structures of the equivalence between stable causality, $K$-causality and existence of a time function. The result implies that closed cone structures that admit continuous increasing functions also admit smooth ones. We also study proper cone structures, the fiber bundle analog of proper cones. For them we obtain most results on domains of dependence. Moreover, we prove that horismos and Cauchy horizons are generated by lightlike geodesics, the latter being defined through the achronality property. Causal geodesics and steep temporal functions are obtained with a powerful product trick. The paper also contains a study of Lorentz-Minkowski spaces under very weak regularity conditions. Finally, we introduce the concepts of stable distance and stable spacetime solving two well known problems (a) the characterization of Lorentzian manifolds embeddable in Minkowski spacetime, they turn out to be the stable spacetimes, (b) the proof that topology, order and distance (with a formula a la Connes) can be represented by the smooth steep temporal functions. The paper is self-contained, in fact we do not use any advanced result from mathematical relativity.Comment: Latex2e, 138 pages. Work presented at the meetings "Non-regular spacetime geometry", Firenze, June 20-22, 2017, and "Advances in General Relativity", ESI Vienna, August 28 - September 1, 2017. v3: added distance formula for stably causal (rather than just stable) spacetimes. v4: Added a few regularity results, final versio

Morphological Image Analysis of Quantum Motion in Billiards

Morphological image analysis is applied to the time evolution of the probability distribution of a quantum particle moving in two and three-dimensional billiards. It is shown that the time-averaged Euler characteristic of the probability density provides a well defined quantity to distinguish between classically integrable and non-integrable billiards. In three dimensions the time-averaged mean breadth of the probability density may also be used for this purpose.Comment: Major revision. Changes include a more detailed discussion of the theory and results for 3 dimensions. Now: 10 pages, 9 figures (some are colored), 3 table