The multi-measure cosmological model and its peculiar effective potential

The multi-measures model applied to cosmology has been recently shown to reproduce qualitatively the expected stages of the Universe evolution, along with some unexpected features. In this article, we continue its exploration with a detailed study of the effective potential of the model. An analysis of the limits of applicability of the effective potential show that during most of the Universe evolution, the effective potential is a very good approximation of the actual potential term. There is, however, a deviation between the two occurring in the earliest moments of the evolution, which has an important role on the behavior of the inflaton scalar field. In the studied cases, during this initial time, the inflaton is increasing in absolute value, which seems consistent with "climbing up" the effective potential. To investigate this behavior, we use numerical integration to find the numerical potential and we show that during the early stage of the evolution, the shape of the numerical potential is very different from that of the effective one and instead of a left plateau followed by steep slope, one observes only a slope with additional local maximum and minimum. This result demonstrates that for complicated equations of motion of the inflaton, one should not rely only on the notion of kinetic and effective potential terms to describe the problem as they may not be accurate in the entire numerical domain.Comment: 9 pages, 5 figure

Special cases of the Multi-Measure Model -- understanding the prolonged inflation

The multi-measure model (MMM), in which one modifies the action to include both the Riemannian measure and a non-Riemannian one, has proven to be able to produce viable Universe evolution scenarios. In this article we consider two special cases of the multi-measure model, in which we remove some of the non-linearity of the Lagrangian or we decouple the two kinetic terms entirely. We show numerically that one can still get the needed evolutionary stages, but furthermore, one can obtain a sufficient number of e-folds of the early inflation in both cases. Furthermore, we connect the model with hyperinflationary models and investigate how the different epochs are born from the interplay between the two scalar fields. We also demonstrate that there is a dynamically induces slow-roll epoch, which is prolonged by the complicated movement of the two scalars in the field space. Finally, we show that while the adiabatic speed of sound can become imaginary, the phase speed of sound is equal to one.Comment: 9 pages, 4 figure

Solving systems of transcendental equations involving the Heun functions

The Heun functions have wide application in modern physics and are expected to succeed the hypergeometrical functions in the physical problems of the 21st century. The numerical work with those functions, however, is complicated and requires filling the gaps in the theory of the Heun functions and also, creating new algorithms able to work with them efficiently. We propose a new algorithm for solving a system of two nonlinear transcendental equations with two complex variables based on the M\"uller algorithm. The new algorithm is particularly useful in systems featuring the Heun functions and for them, the new algorithm gives distinctly better results than Newton's and Broyden's methods. As an example for its application in physics, the new algorithm was used to find the quasi-normal modes (QNM) of Schwarzschild black hole described by the Regge-Wheeler equation. The numerical results obtained by our method are compared with the already published QNM frequencies and are found to coincide to a great extent with them. Also discussed are the QNM of the Kerr black hole, described by the Teukolsky Master equation.Comment: 17 pages, 4 figures. Typos corrected, one figure added, some sections revised. The article is a rework of the internal report arXiv:1005.537

Cosmological solutions from models with unified dark energy and dark matter and with inflaton field

Recently, few cosmological models with additional non-Riemannian volume form(s) have been proposed. In this article we use Supernovae type Ia experimental data to test one of these models which provides a unified description of both dark energy via dynamically generated cosmological constant and dark matter as a "dust" fluid due to a hidden nonlinear Noether symmetry. It turns out that the model allows various scenarios of the future Universe evolution and in the same time perfectly fits contemporary observational data. Further, we investigate the influence of an additional inflaton field with a step like potential. With its help we can reproduce the Universe inflation epoch, matter dominated epoch and present accelerating expansion in a seamless way. Interesting feature is that inflaton undergoes a finite change during its evolution. It can be speculated that the inflaton asymptotic value is connected to the vacuum expectation value of the Higgs field.Comment: 10 pages, 4 figures, prepared for the Proceedings of the XII. International Workshop Lie Theory And Its Applications In Physics (2017