Alpha, Betti and the Megaparsec Universe: on the Topology of the Cosmic Web

We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them. For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, $\alpha$. As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of $\alpha$, and their relation to the morphological patterns in the Cosmic Web, we first study them within the context of simple heuristic Voronoi clustering models. Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy content. The evolution and scale-dependence of the Betti numbers is shown to reflect the hierarchical evolution of the Cosmic Web and yields a promising measure of cosmological parameters. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field.Comment: 42 pages, 14 figure

Course of subthreshold manic symptoms and related risk factors in the general population: A three-year follow-up study

Objectives: Subthreshold manic symptoms (subM) are a risk factor for the onset and recurrence of bipolar disorder (BD). Individuals with subM may benefit from preventive interventions, however, their development is hampered by a lack of knowledge on subM prevalence and subsequent course. This study examines subM characteristics, course, and risk factors for an unfavourable course. Methods: In a Dutch representative, population-based sample aged 18–64 (N = 4618), we assessed subM, defined as the occurrence of manic core symptoms (elation/irritability), without meeting full DSM-IV criteria for BD I or II in the past 3 years. Comparison groups had either no manic symptoms (noM) or hypomania/mania in the context of BD (mBD) in the past 3 years. Furthermore, we differentiated a mild and moderate type of subM, based on the number of manic symptoms. A subsequent three-year course was assessed prospectively. Results: SubM had a three-year prevalence of 4.9%. Its prevalence, characteristics, and course were in between noM and mBD, and there were few differences between mild and moderate subM. Over the 3-year follow-up, 25.0% of individuals with subM had persistent subM and another 6.1% transitioned to mBD. Eleven significant risk factors for this unfavourable course were found. The most important were a history of depression/dysthymia (OR 3.75, p ≤ 0.001), living alone (OR 2.61, p ≤ 0.01) and elevated neuroticism score (OR 1.21, p ≤ 0.001). Conclusions: This study supports the validity and clinical relevance of subM as a BD prodrome. It demonstrates that subM symptoms often persist or increase during follow-up and identifies 11 risk factors that are associated with an unfavourable course

Precision Electroweak Measurements on the Z resonance.

We report on the final electroweak measurements performed with data taken at the Z resonance by the experiments operating at the electron–positron colliders SLC and LEP. The data consist of 17 million Z decays accumulated by the ALEPH, DELPHI, L3 and OPAL experiments at LEP, and 600 thousand Z decays by the SLD experiment using a polarised beam at SLC. The measurements include cross-sections, forward–backward asymmetries and polarised asymmetries. The mass and width of the Z boson, mZ and ΓZ, and its couplings to fermions, for example the ρ parameter and the effective electroweak mixing angle for leptons, are precisely measured: The number of light neutrino species is determined to be 2.9840±0.0082, in agreement with the three observed generations of fundamental fermions. The results are compared to the predictions of the Standard Model (SM). At the Z-pole, electroweak radiative corrections beyond the running of the QED and QCD coupling constants are observed with a significance of five standard deviations, and in agreement with the Standard Model. Of the many Z-pole measurements, the forward–backward asymmetry in b-quark production shows the largest difference with respect to its SM expectation, at the level of 2.8 standard deviations. Through radiative corrections evaluated in the framework of the Standard Model, the Z-pole data are also used to predict the mass of the top quark, , and the mass of the W boson, . These indirect constraints are compared to the direct measurements, providing a stringent test of the SM. Using in addition the direct measurements of mt and mW, the mass of the as yet unobserved SM Higgs boson is predicted with a relative uncertainty of about 50% and found to be less than at 95% confidence level