A certain vector-tensor (VT) theory is revisited. It was proposed and analyzed as a theory of electromagnetism without the standard gauge invariance. Our attention is first focused on a detailed variational formulation of the theory, which leads to both a modified Lorentz force and the true energy momentum tensor of the vector field. The theory is then applied to cosmology.A complete gauge invariant treatment of the scalar perturbations is presented. For appropriate gauge invariant variables describing the scalar modes of the vector field (A-modes), it is proved that the evolution equations of these modes do not involve the scalar modes appearing in General Relativity (GR-modes), which are associated to the metric and the energy momentum tensor of the cosmological fluids. However, the A-modes modify the standard gauge invariant equations describing the GR-modes. By using the new formalism, the evolution equations of the A-perturbations are derived and separately solved and, then, the correction terms -due to the A-perturbationsappearing in the evolution equations of the GR-modes are estimated. The evolution of these correction terms is studied for an appropriate scale. The relevance of these terms depends on both the spectra and the values of the normalization constants involved in extended electromagnetism.Further applications of the new formalism will be presented elsewhere.
We study relevant cosmological topics in the framework of a certain vector-tensor theory of gravitation (hereafter VT). This theory is first compared with the so-called extended electromagnetism (EE). These theories have a notable resemblance and both explain the existence of a cosmological constant. It is shown that, in EE, a positive dark energy density requires a Lagrangian leading to quantum ghosts, whereas VT is free from these ghosts. On account of this fact, the remainder of the paper is devoted to study cosmology in the framework of VT. Initial conditions, at high redshift, are used to solve the evolution equations of all the VT scalar modes. In particular, a certain scalar mode characteristic of VT -which does not appear in general relativity (GR)-is chosen in such a way that it evolves separately. In other words, the scalar modes of the standard model based on GR do not affect the evolution of the VT characteristic mode; however, this scalar mode influences the evolution of the standard GR ones. Some well known suitable codes (CMBFAST and COSMOMC) have been modified to include our VT initial conditions and evolution equations, which are fully general. One of the resulting codes -based on standard statistical methods-has been used to fit VT predictions and observational evidences about both Ia supernovae and cosmic microwave background anisotropy. Seven free parameters are used in this fit. Six of them are often used in GR cosmology and the seventh one is characteristic of VT. From the statistical analysis it follows that VT seems to be advantageous against GR in order to explain cosmological observational evidences.
Permutation entropy measures the complexity of a deterministic time series via a data symbolic quantization consisting of rank vectors called ordinal patterns or simply permutations. Reasons for the increasing popularity of this entropy in time series analysis include that (i) it converges to the Kolmogorov–Sinai entropy of the underlying dynamics in the limit of ever longer permutations and (ii) its computation dispenses with generating and ad hoc partitions. However, permutation entropy diverges when the number of allowed permutations grows super-exponentially with their length, as happens when time series are output by dynamical systems with observational or dynamical noise or purely random processes. In this paper, we propose a generalized permutation entropy, belonging to the class of group entropies, that is finite in that situation, which is actually the one found in practice. The theoretical results are illustrated numerically by random processes with short- and long-term dependencies, as well as by noisy deterministic signals.
A certain vector-tensor (VT) theory of gravitation was tested in previous papers. In the background universe, the vector field of the theory has a certain energy density, which is appropriate to play the role of vacuum energy (cosmological constant). Moreover, this background and its perturbations may explain the temperature angular power spectrum of the cosmic microwave background (CMB) obtained with WMAP (Wilkinson Map Anisotropy Probe), and other observations, as e.g., the Ia supernova luminosities. The parametrized post-Newtonian limit of the VT theory has been proved to be identical to that of general relativity (GR), and there are no quantum ghosts and classical instabilities. Here, the stationary spherically symmetric solution, in the absence of any matter content, is derived and studied. The metric of this solution is formally identical to that of the Reissner-Nordström-de Sitter solution of GR, but the role of the electrical charge is played by a certain quantity Γ depending on both the vector field and the parameters of the VT theory. The black hole and cosmological horizons are discussed. The radius of the VT black hole horizon deviates with respect to that of the Kottler-Schwarzschildde Sitter radius. Realistic relative deviations depend on Γ and reach maximum values close to 30 per cent. For large enough Γ values, there is no any black hole horizon, but only a cosmological horizon. The radius of this last horizon is almost independent of the mass source, the vector field components, and the VT parameters. It essentially depends on the cosmological constant value, which has been fixed by using cosmological observational data (CMB anisotropy, galaxy correlations and so on).
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