Wave function of the Universe in the early stage of its evolution

In quantum cosmological models, constructed in the framework of Friedmann-Robertson-Walker metrics, a nucleation of the Universe with its further expansion is described as a tunneling transition through an effective barrier between regions with small and large values of the scale factor $a$ at non-zero (or zero) energy. The approach for describing this tunneling consists of constructing a wave function satisfying an appropriate boundary condition. There are various ways for defining the boundary condition that lead to different estimates of the barrier penetrability and the tunneling time. In order to describe the escape from the tunneling region as accurately as possible and to construct the total wave function on the basis of its two partial solutions unambiguously, we use the tunneling boundary condition that the total wave function must represent only the outgoing wave at the point of escape from the barrier, where the following definition for the wave is introduced: the wave is represented by the wave function whose modulus changes minimally under a variation of the scale factor $a$. We construct a new method for a direct non-semiclassical calculation of the total stationary wave function of the Universe, analyze the behavior of this wave function in the tunneling region, near the escape point and in the asymptotic region, and estimate the barrier penetrability. We observe oscillations of modulus of wave function in the external region starting from the turning point which decrease with increasing of $a$ and which are not shown in semiclassical calculations. The period of such an oscillation decreases uniformly with increasing $a$ and can be used as a fully quantum dynamical characteristic of the expansion of the Universe.Comment: 19 pages, 21 files for 10 EPS figures, LaTeX svjour style. The Sec.2 (formalism of Wheeler-De Witt equation) is reduced. In Sec.3.1 definition of the outgoing wave from barrier is defined more accurately. In Sec.4.1 semiclassical calculations of wavew function and penetrability are performed and comparison with results in fully quantum approach is adde

Rotational Surfaces in L3\mathbb{L}^3 and Solutions in the Nonlinear Sigma Model

The Gauss map of non-degenerate surfaces in the three-dimensional Minkowski space are viewed as dynamical fields of the two-dimensional O(2,1) Nonlinear Sigma Model. In this setting, the moduli space of solutions with rotational symmetry is completely determined. Essentially, the solutions are warped products of orbits of the 1-dimensional groups of isometries and elastic curves in either a de Sitter plane, a hyperbolic plane or an anti de Sitter plane. The main tools are the equivalence of the two-dimensional O(2,1) Nonlinear Sigma Model and the Willmore problem, and the description of the surfaces with rotational symmetry. A complete classification of such surfaces is obtained in this paper. Indeed, a huge new family of Lorentzian rotational surfaces with a space-like axis is presented. The description of this new class of surfaces is based on a technique of surgery and a gluing process, which is illustrated by an algorithm.Comment: PACS: 11.10.Lm; 11.10.Ef; 11.15.-q; 11.30.-j; 02.30.-f; 02.40.-k. 45 pages, 11 figure

Transitions of cardio-metabolic risk factors in the Americas between 1980 and 2014

Describing the prevalence and trends of cardiometabolic risk factors that are associated with non-communicable diseases (NCDs) is crucial for monitoring progress, planning prevention, and providing evidence to support policy efforts. We aimed to analyse the transition in body-mass index (BMI), obesity, blood pressure, raised blood pressure, and diabetes in the Americas, between 1980 and 2014