Symmetry in Cosmology

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: closed (31 December 2018) | Viewed by 12382

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Guest Editor
Faculty of Physics, Department of Astronomy, Astrophysics and Mechanics, University of Athens, Panepistemiopolis, 157 83 Athens, Greece
Interests: mathematical physics; cosmology; general relativity; conservation laws; lie algebra; fundamental symmetry
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Special Issue Information

Dear Colleagues,

Symmetries is perhaps the strongest tool of physics and applied mathematics available for the manipulation of differential equations and the determination of actual solutions. They provide two major elements of significant importance:

a.    The invariant quantities which constitute the best variables for the study of a specific dynamical system and can be used to reduce the order of a differential equation or reduce the number of variables.
b.    Conserved currents which define surfaces in phase space, on which the evolution of the differential equation is constrained and can be used for the determination of analytical solutions.

The recent observational data on the cosmological scale—the main one being the observed acceleration of the Universe—require the development of new generalized models beyond the classical Friedman Robertson Walker (FRW) model of General Relativity. Today the FRW model is considered as the fundamental reference model on which all generalized models must reduce under specific assumptions.

In general, the field equations of a generalized cosmological model are nonlinear while new degrees of freedom and/or higher-order derivatives are introduced. The determination of analytical solutions as well as the study of integrability of those models is a subject of special interest.

The purpose of the present Special Issue entitled “Symmetry in Cosmology” is the presentation of new methods, techniques and results concerning the determination of new integrable systems in various significant generalized cosmological models.

Prof. Dr. Michael Tsamparlis
Guest Editor

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Keywords

  • Symmetries
  • Integrability
  • Dynamical systems
  • Dark energy
  • Modified theories of gravity

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Published Papers (3 papers)

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Research

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13 pages, 427 KiB  
Article
Analysis of the Angular Dependence of Time Delay in Gravitational Lensing
by Nicola Alchera, Marco Bonici, Roberta Cardinale, Alba Domi, Nicola Maggiore, Chiara Righi and Silvano Tosi
Symmetry 2018, 10(7), 246; https://doi.org/10.3390/sym10070246 - 1 Jul 2018
Cited by 2 | Viewed by 3075
Abstract
We consider an alternative formula for time delay in gravitational lensing. Imposing a smoothness condition on the gravitationally deformed paths followed by the photons from the source to the observer, we show that our formula displays the same degrees of freedom as the [...] Read more.
We consider an alternative formula for time delay in gravitational lensing. Imposing a smoothness condition on the gravitationally deformed paths followed by the photons from the source to the observer, we show that our formula displays the same degrees of freedom as the standard one. In addition to this, it is shown that the standard expression for time delay is recovered when small angles are involved. These two features strongly support the claim that the formula for time delay studied in this paper is the generalization to the arbitrary angles of the standard one, which is valid at small angles. This could therefore result in a useful tool in Astrophysics and Cosmology which may be applied to investigate the discrepancy between the various estimates of the Hubble constant. As an aside, two interesting consequences of our proposal for time delay are discussed: the existence of a constraint on the gravitational potential generated by the lens and a formula for the mass of the lens in the case of central potential. Full article
(This article belongs to the Special Issue Symmetry in Cosmology)
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42 pages, 431 KiB  
Article
Symmetries of Differential Equations in Cosmology
by Michael Tsamparlis and Andronikos Paliathanasis
Symmetry 2018, 10(7), 233; https://doi.org/10.3390/sym10070233 - 21 Jun 2018
Cited by 78 | Viewed by 5173
Abstract
The purpose of the current article is to present a brief albeit accurate presentation of the main tools used in the study of symmetries of Lagrange equations for holonomic systems and subsequently to show how these tools are applied in the major models [...] Read more.
The purpose of the current article is to present a brief albeit accurate presentation of the main tools used in the study of symmetries of Lagrange equations for holonomic systems and subsequently to show how these tools are applied in the major models of modern cosmology in order to derive exact solutions and deal with the problem of dark matter/energy. The key role in this approach are the first integrals of the field equations. We start with the Lie point symmetries and the first integrals defined by them, that is, the Hojman integrals. Subsequently, we discuss the Noether point symmetries and the well-known method for deriving the Noether integrals. By means of the Inverse Noether Theorem, we show that, to every Hojman quadratic first integral, it is possible to associate a Noether symmetry whose Noether integral is the original Hojman integral. It is emphasized that the point transformation generating this Noether symmetry need not coincide with the point transformation defining the Lie symmetry which produces the Hojman integral. We discuss the close connection between the Lie point and the Noether point symmetries with the collineations of the metric defined by the kinetic energy of the Lagrangian. In particular, the generators of Noether point symmetries are elements of the homothetic algebra of that metric. The key point in the current study of cosmological models is the introduction of the mini superspace, which is the space that is defined by the physical variables of the model, which is not the spacetime where the model evolves. The metric in the mini superspace is found from the kinematic part of the Lagrangian and we call it the kinetic metric. The rest part of the Lagrangian is the effective potential. We consider coordinate transformations of the original mini superspace metric in order to bring it to a form where we know its collineations, that is, the Killing vectors, the homothetic vector, etc. Then, we write the field equations of the cosmological model and we use the connection of these equations with the collineations of the mini superspace metric to compute the first integrals and subsequently to obtain analytic solutions for various allowable potentials and finally draw conclusions about the problem of dark energy. We consider the ΛCDM cosmological model, the scalar field cosmology, the Brans–Dicke cosmology, the f(R) gravity, the two scalar fields cosmology with interacting scalar fields and the Galilean cosmology. In each case, we present the relevant results in the form of tables for easy reference. Finally, we discuss briefly the higher order symmetries (the contact symmetries) and show how they are applied in the cases of scalar field cosmology and in the f(R) gravity. Full article
(This article belongs to the Special Issue Symmetry in Cosmology)

Review

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17 pages, 273 KiB  
Review
Symmetries in Classical and Quantum Treatment of Einstein’s Cosmological Equations and Mini-Superspace Actions
by Theodosios Christodoulakis, Alexandros Karagiorgos and Adamantia Zampeli
Symmetry 2018, 10(3), 70; https://doi.org/10.3390/sym10030070 - 16 Mar 2018
Cited by 13 | Viewed by 2988
Abstract
The use of automorphisms of the various Bianchi-type Lie algebras as Lie-point symmetries of the corresponding Einstein field equations entails a reduction of their order and ultimately leads to the entire solution space. When a valid reduced action principle exists, the symmetries of [...] Read more.
The use of automorphisms of the various Bianchi-type Lie algebras as Lie-point symmetries of the corresponding Einstein field equations entails a reduction of their order and ultimately leads to the entire solution space. When a valid reduced action principle exists, the symmetries of the configuration mini-supermetric space can also be used, in conjunction with the constraints, to provide local or non-local constants of motion. At the classical level, depending on their number, these integrals can even secure the acquisition of the entire solution space without any further solving of the dynamical equations. At the quantum level, their operator analogues can be used, along with the Wheeler–DeWitt equation, to define unique wave functions that exhibit singularity-free behavior at a semi-classical level. Full article
(This article belongs to the Special Issue Symmetry in Cosmology)
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