arXiv subjects

Andronikos Paliathanasis

Publications and source records attributed to Andronikos Paliathanasis.

At least 217 records · Page 12Linked to original sources

arXiv2019

Singularity Analysis of a Variant of the Painlev{é}--Ince Equation

We examine by singularity analysis an equation derived by reduction using Lie point symmetries from the Euler--Bernoulli Beam equation which is the Painlevé--Ince Equation with additional terms. The equation possesses the same leading-order behaviour and resonances as the Painlevé--Ince Equation and has a Right Painlevé Series. However, it has no Left Painlev% é Series. A conjecture for the existence of Left Painlevé Series for ordinary differential equations is given.

nlin.SI↗

arXiv2019

Lie symmetries and similarity solutions for rotating shallow water

We study a nonlinear system of partial differential equations which describe rotating shallow water with an arbitrary constant polytropic index $γ$ for the fluid. In our analysis we apply the theory of symmetries for differential equations and we determine that the system of our study is invariant under a five dimensional Lie algebra. The admitted Lie symmetries form the $\left\{ 2A_{1}\oplus _{s}2A_{1}\right\} \oplus _{s}A_{1}$ Lie algebra for $γ\neq 1$ and $2A_{1}\oplus _{s}3A_{1}$ for $γ=1$. The application of the Lie symmetries is performed with the derivation of the corresponding zero-order Lie invariants which applied to reduce the system of partial differential equations into integrable systems of ordinary differential equations. For all the possible reductions the algebraic or closed-form solutions are presented. Travel-wave and scaling solutions are also determined.

math-ph↗

arXiv2019

Cyclic Szekeres Universes

We consider the Szekeres universe with an inhomogeneous dust fluid and a homogeneous and isotropic ghost matter source with equation of state $p_{g}=\left( γ-1\right) ρ_{g},$ where $γ$ is a constant. The field equations determine two families of spacetimes which describe homogeneous Kantowski-Sachs universes and inhomogeneous Friedmann universe. The ghost field Einstein permits static and cyclic solutions to exist. The stability of the Einstein static and cyclic solutions are studied with a critical point analysis.

gr-qc↗

arXiv2019

Dynamics of nonlinear interacting dark energy models

We investigate the cosmological dynamics of interacting dark energy models in which the interaction function is a nonlinear in terms of the energy densities. Considering explicitly the interaction between a pressureless dark matter and a scalar field, minimally coupled to Einstein gravity, we explore the dynamics of the spatially flat FLRW universe for the exponential potential of the scalar field.. We perform the stability analysis for the three nonlinear interaction models of our consideration through the analysis of critical points and we investigate the cosmological parameters and we discuss the physical behaviour at the critical points. From the analysis of the critical points we find a number of possibilities that include the stable late time accelerated solution, $w$CDM-like solution, radiation-like solution and moreover the unstable inflationary solution as well.

gr-qc↗

arXiv2019

Bianchi IX cosmologies in the Einstein-Skyrme system in a sector with non-trivial topological charge

The dynamics of the most general Bianchi IX cosmology with three time dependent scale factors for the Einstein-Skyrme system is analyzed. For the Skyrmion, a generalized hedgehog ansatz with unit baryon charge is introduced. The most remarkable feature of this ansatz is that, in the above topologically non-trivial sector with unit topological charge, the Skyrme field equations are identically satisfied on any Bianchi IX metric. We will show that due to this feature the complete set of coupled Einstein-Skyrme field equations can be deduced from a suitable minisuperspace Lagrangian. The latter allows to perform a systematic study of the integrability properties of the Einstein-Skyrme system for the Bianchi IX cosmology. Moreover, some analytic and algebraic solutions for the Einstein-Skyrme model are derived. Another remarkable consequence of the present formalism is that it is possible to derive the Wheeler de-Witt equation for the Bianchi IX metric in the Einstein-Skyrme cosmology in which all the effects of the Skyrmion are encoded in an effective potential of the minisuperspace Lagrangian.

gr-qc↗

arXiv2019

Two-fluid solutions of particle-creation cosmologies

Cosmological evolution driven incorporating continuous particle creation by the time-varying gravitational field is investigated. We consider a spatially flat, homogeneous and isotropic universe with two matter fluids in the context of general relativity. One fluid is endowed with gravitationally induced `adiabatic' particle creation, while the second fluid simply satisfies the conservation of energy. We show that the dynamics of the two fluids is entirely controlled by a single nonlinear differential equation involving the particle creation rate, $Γ(t)$. We consider a very general particle creation rate, $Γ(t)$, that reduces to several special cases of cosmological interest, including $Γ=$ constant, $% Γ\propto 1/H^{n}$ ($n\in \mathbb{N}$), $Γ\propto \exp (1/H)$. Finally, we present singular algebraic solutions of the gravitational field equations for the two-fluid particle creation models and discuss their stability.

gr-qc↗

arXiv2018

Cosmological constraints on an exponential interaction in the dark sector

Cosmological models where dark matter (DM) and dark energy (DE) interact with each other are the general scenarios in compared to the non-interacting models. The interaction is usually motivated from the phenomenological ground and thus there is no such rule to prefer a particular interaction between DM and DE. Being motivated, in this work, allowing an exponential interaction between DM and DE in a spatially flat homogeneous and isotropic universe, we explore the dynamics of the universe through the constraints of the free parameters where the strength of the interaction is characterized by the dimensionless coupling parameter $ξ$ and the equation of state (EoS) for DE, $w_x$, is supposed to be a constant. The interaction scenario is fitted using the latest available observational data. Our analyses report that the observational data permit a non-zero value of $ξ$ but it is very small and consistent with $ξ=0$. From the constraints on $w_x$, we find that both phantom ($w_x< -1$) and quintessence ($w_x> -1$) regimes are equally allowed but $w_x$ is very close to `$-1$'. The overall results indicate that at the background level, the interaction model cannot be distinguished from the base $Λ$-cold dark matter model while from the perturbative analyses, the interaction model mildly deviates from the base model. We highlight that, even if we allow DM and DE to interact in an exponential manner, but according to the observational data, the evidence for a non-zero coupling is very small.

gr-qc↗

arXiv2018

Noether's Theorem and Symmetry

In Noether's original presentation of her celebrated theorm of 1918 allowance was made for the dependence of the coefficient functions of the differential operator which generated the infinitesimal transformation of the Action Integral upon the derivatives of the depenent variable(s), the so-called generalised, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades this dimunition of the power of Noether's Theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this special issue we emphasise the generality of Noether's Theorem in its original form and explore the applicability of even more general coefficient functions by alowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence upon the independent variables

math-ph↗

arXiv2018

Symmetries of spacetimes embedded with an Electromagnetic String Fluid

The electromagnetic string fluid (EMSF) is an anisotropic charged string fluid interacting with a strong magnetic field. In this fluid we consider the double congruence defined by the 4-velocity of the fluid $u^a$ and the unit vector $n^a$ along the magnetic field. Using the standard 1+3 decomposition defined by the vector $u^a$ and the 1+1+2 decomposition defined by the double congruence ${u^a,n^a}$ we determine the kinematic and the dynamic quantities of an EM string fluid in both decompositions. In order to solve the resulting field equations we consider simplifying assumptions in the form of collineations. We decompose the generic quantity $L_{X}g_{ab}$ in a trace $ψ$ and and a traceless part $H_{ab}$. Because all collineations are expressible in terms of the quantity $L_{X}g_{ab}$ it is possible to compute the Lie derivative of all tensors defined by the metric i.e. the Ricci tensor, the Weyl tensor etc. This makes possible the effects of any assumed collineation on the gravitational field equations. This is done as follows. Using relevant identities of Differential Geometry we express the quantity $L_{X}R_{ab}$ where $R_{ab}$ is the Ricci tensor in terms of the two irreducible parts $ψ,H_{ab}$. Subsequently using the gravitational field equations we compute the same quantity $L_{X}R_{ab}$ in terms of the Lie derivative of the dynamic variables. We equate the two results and find the field equations in the form $L_{X}{\text{Dynamic variable}}=F(ψ,H_{ab},{\text{ dynamic variables}})$. This result is general and holds for all gravitational systems and in particular for the EMSF. Subsequently we specialize our study at two levels. We consider the case of a Conformal Killing Vector (CKV) parallel to $u^a$ and a CKV parallel to $n^a$.

gr-qc↗

arXiv2018

Szekeres Universes with Homogeneous Scalar Fields

We consider the existence of an "inflaton" described by an homogeneous scalar field in the Szekeres cosmological metric. The gravitational field equations are reduced to two families of solutions which describe the homogeneous Kantowski-Sachs spacetime and an inhomogeneous FLRW(-like) spacetime with spatial curvature a constant. The main differences with the original Szekeres spacetimes containing only pressure-free matter are discussed. We investigate the stability of the two families of solution by studying the critical points of the field equations. We find that there exist stable solutions which describe accelerating spatially-flat FLRW geometries.

gr-qc↗

arXiv2018

The past and future dynamics of quintom dark energy models

We study the phase space of the quintom cosmologies for a class of exponential potentials. We combine normal forms expansions and the center manifold theory in order to describe the dynamics near equilibrium sets. Furthermore, we construct the unstable and center manifold of the massless scalar field cosmology motivated by the numerical results given in Lazkoz and Leon (Phys Lett B 638:303. arXiv:astro-ph/0602590, 2006). We study the role of the curvature on the dynamics. Several monotonic functions are defined on relevant invariant sets for the quintom cosmology. Finally, conservation laws of the cosmological field equations and algebraic solutions are determined by using the symmetry analysis and the singularity analysis.

gr-qc↗

arXiv2018

Dynamical analysis and cosmological viability of varying $G$ and $Λ$ cosmology

The cosmological viability of varying $G\left( t\right) $ and $Λ\left( t\right) $ cosmology is discussed by determining the cosmological eras provided by the theory. Such a study is performed with the determination of the critical points while stability analysis is performed. The application of Renormalization group in the ADM formalism of General Relativity provides a modified second-order theory of gravity where varying $G\left( t\right) $ plays the role of a minimally coupled field, different from that of Scalar-tensor theories, while $Λ\left( t\right) =Λ\left( G\left( t\right) \right) $ is a potential term. We find that the theory provides two de Sitter phases and a tracking solution. In the presence of matter source, two new critical points are introduced, where the matter source contributes to the universe. One of those points describes the $Λ$CDM cosmology and in order for the solution at the point to be cosmologically viable, it has to be unstable. Moreover, the second point, where matter exists, describes a universe where the dark energy parameter for the equation of state has a different value from that of the cosmological constant.

gr-qc↗

arXiv2018

Quadratic conservation laws and collineations: a discussion

Every second order system of autonomous differential equations can be described by an autonomous holonomic dynamical system with a Lagrangian part, an effective potential and a set of generalized forces. The kinematic part of the Lagrangian defines the kinetic metric which subsequently defines a Riemannian geometry in the configuration space. We consider the generic function $I=K_{ab}(t,q^{c})\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q^{c})\dot{q}% ^{a}+K(t,q^{c})$ and require the quadratic first integral condition $dI/dt=0$ without involving any type of symmetry Lie or Noether. Condition $dI/dt=0$ leads to a system of equations involving the coefficients $K_{ab}% (t,q^{c}),K_{a}(t,q^{c}),K(t,q^{c})$ whose solution will produce all possible quadratic first integrals of the original system of autonomous differential equations. We show that the new system of equations relates the quadratic first integrals of the holonomic system with the geometric collineations of the kinetic metric and in particular with the Killing tensors of order two. We consider briefly various results concerning the Killing tensors of second-order and prove a general formula which gives in the case of a flat kinetic metric the generic Killing tensor in terms of the vectors of the special projective algebra of the kinetic metric. This establishes the connection between the geometry defined by the kinetic metric and the quadratic first integrals of the original system of differential equations.

gr-qc↗

arXiv2018

The generic model of General Relativity

We develop a generic spacetime model in General Relativity which can be used to build any gravitational model within General Relativity. The generic model uses two types of assumptions: (a) Geometric assumptions additional to the inherent geometric identities of the Riemannian geometry of spacetime and (b) Assumptions defining a class of observers by means of their 4-velocity $u^{a}$ which is a unit timelike vector field. The geometric assumptions as a rule concern symmetry assumptions (the so called collineations). The latter introduces the 1+3 decomposition of tensor fields in spacetime. The 1+3 decomposition results in two major results. The 1+3 decomposition of $u_{a;b}$ defines the kinematic variables of the model (expansion, rotation, shear and 4-acceleration) and defines the kinematics of the gravitational model. The 1+3 decomposition of the energy momentum tensor representing all gravitating matter introduces the dynamic variables of the model (energy density, the isotropic pressure, the momentum transfer or heat flux vector and the traceless tensor of the anisotropic pressure) as measured by the defined observers and define the dynamics of he model. The symmetries assumed by the model act as constraints on both the kinematical and the dynamical variables of the model. As a second further development of the generic model we assume that in addition to the 4-velocity of the observers $u_{a}$ there exists a second universal vector field $n_{a}$ in spacetime so that one has a so called double congruence $(u_{a},n_{a})$ which can be used to define the 1+1+2 decomposition of tensor fields. The 1+1+2 decomposition leads to an extended kinematics concerning both fields building the double congruence and to a finer dynamics involving more physical variables.

gr-qc↗

arXiv2018

Black branes in four-dimensional conformal equivalent theories

The physical properties of static analytic solutions which describe black brane geometries are discussed. In particular we study the similarities and differences of analytic black brane/string solutions in the Einstein and Jordan frames. The comparison is made between vacuum power law $f(R)$ gravity solutions and their conformal equivalents in the Einstein frame. In our analysis we examine how the geometrical and physical properties of these analytic axisymmetric solutions - such as singularities, the temperature and the entropy - are affected as we pass from one frame to the other.

gr-qc↗

arXiv2018

Symmetries of Differential Equations in Cosmology

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 one 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, that is the space which 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.

gr-qc↗

arXiv2018

Reconstructions of the dark-energy equation of state and the inflationary potential

We use a mathematical approach based on the constraints systems in order to reconstruct the equation of state and the inflationary potential for the inflaton field from the observed spectral indices for the density perturbations $n_{s}$ and the tensor to scalar ratio $r$. From the astronomical data, we can observe that the measured values of these two indices lie on a two-dimensional surface. We express these indices in terms of the Hubble slow-roll parameters and we assume that $n_{s}-1=h\left( r\right) $. For the function $h\left( r\right) $, we consider three cases, where $h\left( r\right) $ is constant, linear and quadratic, respectively. From this, we derive second-order equations whose solutions provide us with the explicit forms for the expansion scale-factor, the scalar-field potential, and the effective equation of state for the scalar field. Finally, we show that for there exist mappings which transform one cosmological solution to another and allow new solutions to be generated from existing ones.

gr-qc↗

arXiv2018

Duality transformation and conformal equivalent scalar-tensor theories

We deal with the duality symmetry of the Dilaton field in cosmology and specifically with the so-called Gasperini-Veneziano duality transformation. In particular, we determine two conformal equivalent theories to the Dilaton field, and we show that under conformal transformations Gasperini-Veneziano duality symmetry does not survive. Moreover, we show that those theories share a common conservation law, of Noetherian kind, while the symmetry vector which generates the conservation law is an isometry only for the Dilaton field. Finally, we show that the Lagrangian of the Dilaton field is equivalent with that of the two-dimensional \textquotedblright hyperbolic oscillator\textquotedblright\ in a Lorentzian space whose $O(d,d)$ invariance is transformed to the Gasperini-Veneziano duality invariance in the original coordinates.

gr-qc↗