# Andronikos Paliathanasis

**Andronikos Paliathanasis**?

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NameAndronikos Paliathanasis |
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## Pub CategoriesGeneral Relativity and Quantum Cosmology (34) Mathematics - Mathematical Physics (28) Mathematical Physics (28) Cosmology and Nongalactic Astrophysics (19) High Energy Physics - Theory (16) Mathematics - Analysis of PDEs (8) High Energy Physics - Phenomenology (2) Mathematics - Classical Analysis and ODEs (2) Mathematics - Differential Geometry (1) Physics - Classical Physics (1) |

## Publications Authored By Andronikos Paliathanasis

Recently a cubic Galileon cosmological model was derived by the assumption that the field equations are invariant under the action of point transformations. The cubic Galileon model admits a second conservation law which means that the field equations form an integrable system. The analysis of the critical points for this integrable model is the main subject of this work. Read More

Time-dependent analytic solutions of the Einstein-Skyrme system --gravitating Skyrmions--, with topological charge one are analyzed in detail. In particular, the question of whether these Skyrmions reach a spherically symmetric configuration for $t\rightarrow+\infty$ is discussed. It is shown that there is a static, spherically symmetric solution described by the Ermakov-Pinney system, which is fully integrable by algebraic methods. Read More

We perform the complete symmetry classification of the Klein-Gordon equation in maximal symmetric spacetimes. The central idea is to find all possible potential functions $V(t,x,y)$ that admit Lie and Noether symmetries. This is done by using the relation between the symmetry vectors of the differential equations and the elements of the conformal algebra of the underlying geometry. Read More

The Szekeres system is studied with two methods for the determination of conservation laws. Specifically we apply the theory of group invariant transformations and the method of singularity analysis. We show that the Szekeres system admits a Lagrangian and the conservation laws that we find can be derived by the application of Noether's theorem. Read More

A class of generalized Galileon cosmological models, which can be described by a point-like Lagrangian, is considered in order to utilize Noether's Theorem to determine conservation laws for the field equations. In the Friedmann-Lema\^itre-Robertson-Walker universe, the existence of a nontrivial conservation law indicates the integrability of the field equations. Due to the complexity of the latter, we apply the differential invariants approach in order to construct special power-law solutions and study their stability. Read More

For a family of Horndeski theories, formulated in terms of a generalized Galileon model, we study the integrability of the field equations in a Friedmann-Lema\^itre-Robertson-Walker spacetime. We are interested in point transformations which leave invariant the field equations. Noether's theorem is applied to determine the conservation laws for a family of models that belong to the same general class. Read More

A fourth-order theory of gravity is considered which in terms of dynamics has the same degrees of freedom and number of constraints as those of scalar-tensor theories. In addition it admits a canonical point-like Lagrangian description. We study the critical points of the theory and we show that it can describe the matter epoch of the universe and that two accelerated phases can be recovered one of which describes a de Sitter universe. Read More

We consider a Skyrme fluid with a constant radial profile in locally rotational Kantowski-Sachs spacetime. The Skyrme fluid is an anisotropic fluid with zero heat flux and with an equation of state parameter $w_{S}$ that $\left\vert w_{s}\right\vert \leq\frac{1}{3}$. From the Einstein field equations we define the Wheeler-DeWitt equation. Read More

We use a new mathematical approach to reconstruct the equation of state and the inflationary potential for the inflaton field from the spectral indices for the density perturbations $n_{s}$ and the tensor to scalar ratio $r$. According to the astronomical data, 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) $. Read More

In a homogeneous and isotropic universe with non-zero spatial curvature we consider the effects of gravitational particle production in the dynamics of the universe. We show that the dynamics of the universe in such a background are characterized by a single nonlinear differential equation which is significantly dependent on the rate of particle creation and whose solutions can be dominated by curvature effects at early times. For different particle creation rates we apply the singularity test in order to find the analytic solutions of the background dynamics. Read More

An algorithm is used to generate new solutions of the scalar field equations in homogeneous and isotropic universes. Solutions can be found for pure scalar fields with various potentials in the absence and presence of spatial curvature and other perfect fluids. A series of generalisations of the Chaplygin gas and bulk viscous cosmological solutions for inflationary universes are found. Read More

We consider the Noetherian symmetries of second-order ODEs subjected to forces with nonzero curl. Both position and velocity dependent forces are considered. In the former case the first integrals are shown to follow from the symmetries of the celebrated Emden-Fowler equation. Read More

**Affiliations:**

^{1}Dalhousie U., Math. Dept.,

^{2}Valparaiso U., Catolica,

^{3}Chile Austral U., Valdivia

We investigate Kantowski-Sachs models in Einstein-{\ae}ther theory with a perfect fluid source using the singularity analysis to prove the integrability of the field equations and dynamical system tools to study the evolution. We find an inflationary source at early times, and an inflationary sink at late times, for a wide region in the parameter space. The results by A. Read More

In the cosmological scenario in $f\left( T\right) $ gravity, we find analytical solutions for an isotropic and homogeneous universe containing a dust fluid and radiation and for an empty anisotropic Bianchi I universe. The method that we apply is that of movable singularities of differential equations. For the isotropic universe, the solutions are expressed in terms of a Laurent expansion, while for the anisotropic universe we find a family of exact Kasner-like solutions in vacuum. Read More

We show that the recent results of \ [Int. J. Mod. Read More

Two essential methods, the symmetry analysis and of the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations are discussed. The main similarities and differences of these two different methods are given. Read More

The integrability of higher-order theories of gravity is of importance in the determining the properties of these models and so their viability as models of reality. An important tool in the establishment of integrability is the singularity analysis. We apply this analysis to the case of fourth-order theory of gravity $f(R) = R + qR^{n}$ to establish those values of the free parameters $q$ and $n$ for which integrability in this sense exists. Read More

We study the Lie and Noether point symmetries of a class of systems of second-order differential equations with $n$ independent and $m$ dependent variables ($n\times m$ systems). We solve the symmetry conditions in a geometric way and determine the general form of the symmetry vector and of the Noetherian conservation laws. We prove that the point symmetries are generated by the collineations of two (pseudo)metrics, which are defined in the spaces of independent and dependent variables. Read More

We present the solution space for the case of a minimally coupled scalar field with arbitrary potential in a FLRW metric. This is made possible due to the existence of a nonlocal integral of motion corresponding to the conformal Killing field of the two-dimensional minisuperspace metric. The case for both spatially flat and non flat are studied first in the presence of only the scalar field and subsequently with the addition of non interacting perfect fluids. Read More

We discuss the relation between Noether (point) symmetries and discrete symmetries for a class of minisuperspace cosmological models. We show that when a Noether symmetry exists for the gravitational Lagrangian then there exists a coordinate system in which a reversal symmetry exists. Moreover as far as concerns the scale-factor duality symmetry of the dilaton field, we show that it is related to the existence of a Noether symmetry for the field equations, and the reversal symmetry in the normal coordinates of the symmetry vector becomes scale-factor duality symmetry in the original coordinates. Read More

In a flat Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) geometry, we consider the expansion of the universe powered by the gravitationally induced `adiabatic' matter creation. To demonstrate how matter creation works well with the expanding universe, we have considered a general creation rate and analyzed this rate in the framework of dynamical analysis. The dynamical analysis hints the presence of a non-singular universe (without the big bang singularity) with two successive accelerated phases, one at the very early phase of the universe (i. Read More

We apply as selection rule to determine the unknown functions of a cosmological model the existence of Lie point symmetries for the Wheeler-DeWitt equation of quantum gravity. Our cosmological setting consists of a flat Friedmann-Robertson-Walker metric having the scale factor $a(t)$, a scalar field with potential function $V(\phi)$ minimally coupled to gravity and a vector field of its kinetic energy is coupled with the scalar field by a coupling function $f(\phi)$. Then, the Lie symmetries of this dynamical system are investigated by utilizing the behavior of the corresponding minisuperspace under the infinitesimal generator of the desired symmetries. Read More

We consider $f\left(R\right) $-gravity in a Friedmann-Lema\^itre-Robertson-Walker spacetime with zero spatial curvature. We apply the Killing tensors of the minisuperspace in order to specify the functional form of $f\left(R\right) $ and the field equations to be invariant under Lie-B\"acklund transformations which are linear in the momentum (contact symmetries). Consequently, the field equations to admit quadratic conservation laws given by Noether's Theorem. Read More

We consider the application of group invariant transformations in order to constrain a flat isotropic and homogeneous cosmological model, containing of a Brans-Dicke scalar field and a perfect fluid with a constant equation of state parameter $w$, where the latter is not interacting with the scalar field in the gravitational action integral. The requirement that the Wheeler-DeWitt equation be invariant under one-parameter point transformations provides us with two families of power-law potentials for the Brans-Dicke field, in which the powers are functions of the Brans-Dicke parameter $\omega_{BD}$ and the parameter $w$. The existence of the Lie symmetry in the Wheeler-DeWitt equation is equivalent to the existence of a conserved quantity in field equations and with oscillatory terms in the wavefunction of the universe. Read More

We consider quintessence scalar field cosmology in which the Lagrangian of the scalar field is modified by the Generalized Uncertainty Principle. We show that the perturbation terms which arise from the deformed algebra are equivalent with the existence of a second scalar field, where the two fields interact in the kinetic part. Moreover, we consider a spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker spacetime (FLRW), and we derive the gravitational field equations. Read More

We give a general method to find exact cosmological solutions for scalar-field dark energy in the presence of perfect fluids. We use the existence of invariant transformations for the Wheeler De Witt (WdW) equation. We show that the existence of a point transformation under which the WdW equation is invariant is equivalent to the existence of conservation laws for the field equations, which indicates the existence of analytical solutions. Read More

We study the Lie point symmetries of a general class of partial differential equations (PDE) of second order. An equation from this class naturally defines a second-order symmetric tensor (metric). In the case the PDE is linear on the first derivatives we show that the Lie point symmetries are given by the conformal algebra of the metric modulo a constraint involving the linear part of the PDE. Read More

**Authors:**Andronikos Paliathanasis

In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new geometric method which relates the point symmetries of the differential equations with the collineations of the underlying manifold where the motion occurs. Read More

We develop a new method in order to classify the Bianchi I spacetimes which admit conformal Killing vectors (CKV). The method is based on two propositions which relate the CKVs of 1+(n-1) decomposable Riemannian spaces with the CKVs of the (n-1) subspace and show that if 1+(n-1) space is conformally flat then the (n-1) spacetime is maximally symmetric. The method is used to study the conformal algebra of the Kasner spacetime and other less known Bianchi type I matter solutions of General Relativity. Read More

We propose to use dynamical symmetries of the field equations, in order to classify the dark energy models in the context of scalar field (quintessence or phantom) FLRW cosmologies. Practically, symmetries provide a useful mathematical tool in physical problems since they can be used to simplify a given system of differential equations as well as to determine the integrability of the physical system. The requirement that the field equations admit dynamical symmetries results in two potentials one of which is the well known Unified Dark Matter (UDM) potential and another new potential. Read More

We consider the two scalar field cosmology in a FRW spatially flat spacetime where the scalar fields interact both in the kinetic part and the potential. We apply the Noether point symmetries in order to define the interaction of the scalar fields. We use the point symmetries in order to write the field equations in the normal coordinates and we find that the Lagrangian of the field equations which admits at least three Noether point symmetries describes linear Newtonian systems. Read More

We prove a theorem concerning the Noether symmetries for the area minimizing Lagrangian under the constraint of a constant volume in an n-dimensional Riemannian space. We illustrate the application of the theorem by a number of examples. Read More

Symmetries play a crucial role in physics and, in particular, the Noether symmetries are a useful tool both to select models motivated at a fundamental level, and to find exact solutions for specific Lagrangians. In this work, we consider the application of point symmetries in the recently proposed metric-Palatini Hybrid Gravity in order to select the $f({\cal R})$ functional form and to find analytical solutions for the field equations and for the related Wheeler-DeWitt (WDW) equation. We show that, in order to find out integrable $f({\cal R})$ models, conformal transformations in the Lagrangians are extremely useful. Read More

We determine the Lie point symmetries of the Schr\"{o}dinger and the Klein Gordon equations in a general Riemannian space. It is shown that these symmetries are related with the homothetic and the conformal algebra of the metric of the space respectively. We consider the kinematic metric defined by the classical Lagrangian and show how the Lie point symmetries of the Schr\"{o}dinger equation and the Klein Gordon equation are related with the Noether point symmetries of this Lagrangian. Read More

We prove a general theorem which allows the determination of Lie symmetries of Laplace equation in a general Riemannian space using the conformal group of the space. Algebraic computing is not necessary. We apply the theorem in the study of the reduction of Laplace equation in certain classes of Riemannian spaces which admit a gradient Killing vector, a gradient Homothetic vector and a special Conformal Killing vector. Read More

Conformally related metrics and Lagrangians are considered in the context of scalar-tensor gravity cosmology. After the discussion of the problem, we pose a lemma in which we show that the field equations of two conformally related Lagrangians are also conformally related if and only if the corresponding Hamiltonian vanishes. Then we prove that to every non-minimally coupled scalar field, we may associate a unique minimally coupled scalar field in a conformally related space with an appropriate potential. Read More

We study the reduction of the heat equation in Riemannian spaces which admit a gradient Killing vector, a gradient homothetic vector and in Petrov Type D,N,II and Type III space-times. In each reduction we identify the source of the Type II hidden symmetries. More specifically we find that a) If we reduce the heat equation by the symmetries generated by the gradient KV the reduced equation is a linear heat equation in the nondecomposable space. Read More

A detailed study of the modified gravity, f(R) models is performed, using that the Noether point symmetries of these models are geometric symmetries of the mini superspace of the theory. It is shown that the requirement that the field equations admit Noether point symmetries selects definite models in a self-consistent way. As an application in Cosmology we consider the Friedman -Robertson-Walker spacetime and show that the only cosmological model which is integrable via Noether point symmetries is the $(R^{b}-2\Lambda) ^{c}$ model, which generalizes the Lambda Cosmology. Read More

We give two theorems which show that the Lie point and the Noether symmetries of a second-order ordinary differential equation of the form (D/(Ds))(((Dx^{i}(s))/(Ds)))=F(x^{i}(s),x^{j}(s)) are subalgebras of the special projective and the homothetic algebra of the space respectively. We examine the possible extension of this result to partial differential equations (PDE) of the form A^{ij}u_{ij}-F(x^{i},u,u_{i})=0 where u(x^{i}) and u_{ij} stands for the second partial derivative. We find that if the coefficients A^{ij} are independent of u(x^{i}) then the Lie point symmetries of the PDE form a subgroup of the conformal symmetries of the metric defined by the coefficients A^{ij}. Read More

We generalize the two dimensional autonomous Hamiltonian Kepler Ermakov dynamical system to three dimensions using the sl(2,R) invariance of Noether symmetries and determine all three dimensional autonomous Hamiltonian Kepler Ermakov dynamical systems which are Liouville integrable via Noether symmetries. Subsequently we generalize the autonomous Kepler Ermakov system in a Riemannian space which admits a gradient homothetic vector by the requirements (a) that it admits a first integral (the Riemannian Ermakov invariant) and (b) it has sl(2,R) invariance. We consider both the non-Hamiltonian and the Hamiltonian systems. Read More

We employ a three fluid model in order to construct a cosmological model in the Friedmann Robertson Walker flat spacetime, which contains three types of matter dark energy, dark matter and a perfect fluid with a linear equation of state. Dark matter is described by dust and dark energy with a scalar field with potential V({\phi}). In order to fix the scalar field potential we demand Lie symmetry invariance of the field equations, which is a model-independent assumption. Read More

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. Read More

We prove two theorems which relate the Lie point symmetries and the Noether symmetries of a dynamical system moving in a Riemannian space with the special projective group and the homothetic group of the space respectively. The theorems are applied to classify the two dimensional Newtonian dynamical systems, which admit a Lie point/Noether symmetry. Two cases are considered, the non-conservative and the conservative forces. Read More