This paper is devoted to explore modified $f(\mathcal{R})$ theories of gravity using Noether symmetry approach. For this purpose, Friedmann-Robertson-Walker spacetime is chosen to investigate the cosmic evolution. The study is mainly divided into two parts: Firstly Noether symmetries of metric $f(\mathcal{R})$ gravity are revisited and some new class of solutions with the help of conserved quantities are reported. It is shown that different scenarios of cosmic evolution can be discussed using Noether symmetries and one of the case indicates the chances for the existence of Big Rip singularity. Secondly, $f(\mathcal{R})$ theory coupled with scalar field has been discussed in detail. The Noether equations of modified gravity are reported with three subcases for flat Friedmann-Robertson-Walker universe. It is concluded that conserved quantities are quite helpful to find some important exact solutions in the cosmological contexts. Moreover, the scalar field involved in the modified gravity plays a vital role in the cosmic evolution and an accelerated expansion phase can be observed for some suitable choices of $f(\mathcal{R},φ,χ)$ gravity models.
In the framework of f(R) scalar-tensor cosmology, we use the Noether symmetry approach to find the cosmological models consistent with the Noether symmetry. We obtain the functions f(R) and H(a), or the corresponding differential equations, according to specific choices for the scalar field potential V(ϕ), the Brans-Dicke function ω(ϕ), some cosmological parameters, and the constants of motion.
This work deals with chameleon field cosmology (a scalar field nonminimally coupled to cold dark matter) in the background of flat Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time. Both classical and quantum cosmology have been investigated using Noether symmetry analysis of the underlying physical system. The Wheeler-DeWitt (WD) equation has been constructed on the minisuperspace and solutions have been obtained using conserved charge.
We discuss scalar-tensor cosmology with an extra $R^{-1}$ correction by the Noether Symmetry Approach. The existence of such a symmetry selects the forms of the coupling $ω(ϕ)$, of the potential $V(ϕ)$ and allows to obtain physically interesting exact cosmological solutions.
In this work, we consider F(R) alternative theories of gravity with an eye to Noether symmetry through the gauge theorem. For non-vacuum models, one finds Λ like gravity with energy density of Chaplygin Gas. We also obtain the effective equation of state parameter for corresponding cosmology and scale factor behavior with respect to cosmic time which show that the model provides viable EoS and scale factor with respect to observational data.
We discuss an extended Teleparallel gravity models comprising functions of scalar invariants constructed by torsion, torsion Gauss-Bonnet and boundary terms. We adopt the Noether Symmetry Approach to select the functional forms, the first integrals and, eventually, the exact solutions of the dynamics in the context of flat Friedman-Robertson-Walker cosmology. Standard perfect fluid matter, minimally coupled to geometry, is considered. Several physically consistent models are derived with related exact solutions.
A general approach to find out exact cosmological solutions in f(R)-gravity is discussed. Instead of taking into account phenomenological models, we assume, as a physical criterium, the existence of Noether symmetries in the cosmological f(R) Lagrangian. As a result, the presence of such symmetries selects viable models and allow to solve the equations of motion. We discuss also the case in which no Noether charge is present but general criteria can be used to achieve solutions.
This is a short communication in reference to the comments in (arXiv: 1605.05147) of our paper in Int. J. Mod. Phys. D 25, 1650051 (2016). We have tried to answer to the comments or criticism in the paper.
In the framework of $f(T)$-gravity theory, classical and quantum cosmology has been studied in the present work for FLRW space-time model. The Noether symmetry, a point-like symmetry of the Lagrangian is used to the physical system and a specific functional form of $f(T)$ is determined. A point transformation in the 2D augmented space restricts one of the variable to be cyclic so that the Lagrangian as well as the field equations are simplified so that they are solvable. Lastly for quantum cosmology, the WD equation is constructed and possible solution has been evaluated.
In this paper we will analyse the third quantization of a model of super-group field cosmology with gauge symmetry. The effect of creation and annihilation of bosonic and fermionic universes in the multiverse will also be analysed. We will also construct a third quantized Noether's charge which will be conserved even when the number of universes is not conserved. Finally, we will construct both the third quantized BRST and the third quantized anti-BRST charges for this theory and use them to show that the creation and annihilation of universes is a unitarity process.
The Noether gauge symmetry approach is revisited to study various quintom scenarios (those that arise by the presence of two dynamical scalar fields) to comprehend the role of dark energy in our universe. For such models, we obtain smooth parameterizations of the equation of state of dark energy across the boundary of cosmological constant $w_Λ=-1$. This study gives rise to two new cases of the potential $V(ϕ, σ)$, due to a quintom field in which nonlinear coupling of the scalar fields arise. Besides we report that a few cases of Noether gauge symmetries and their invariants in [Adnan Aslam, e…
This paper studies the cosmological equations for a scalar field Phi in the framework of a quantum gravity modified Einstein--Hilbert Lagrangian where G and Lambda are dynamical variables. It is possible to show that there exists a Noether symmetry for the point Lagrangian describing this scheme in a FRW universe. Our main result is that the Noether Symmetry Approach fixes both Lambda = Lambda(G) and the potential V = V(Phi) of the scalar field. The method does not lead, however, to easily solvable equations, by virtue of the higher dimensionality of the reduced configuration space involved, t…
In this work a generic set of boundary conditions for $\mathcal{N}=1$ SUGRA is proposed. This conditions defines that Hamiltonian charges equals Noether ones, including supercharge.
We consider a cosmological model in a Friedmann--Lema\^ıtre--Robertson--Walker background space with an ideal gas defined in Weyl Integrable gravity. In the Einstein-Weyl theory a scalar field is introduced in a geometric way. Furthermore, the scalar field and the ideal gas interact in the gravitational Action Integral. Furthermore, we introduce a potential term for the scalar field potential and we show that the field equations admit a minisuperspace description. Noether's theorem is applied for the constraint of the potential function and the corresponding conservation laws are constructed. …
The paper deals with phantom scalar field cosmology in Einstein gravity. At first using Lie symmetry, the coupling function to the kinetic term and the potential function of the scalar field and the equation of state parameter of the matter field are determined and a simple solution is obtained. Subsequently, Noether symmetry is imposed on the Lagrangian of the system. The symmetry vector is obtained and the potential takes a very general form from which potential using Lie Symmetry can be obtained as a particular case. Then we choose a point transformation $(a,ϕ)\rightarrow(u,v)$ such that on…
Generalized Noether's theory is a useful method for researching the modified gravity theories about the conserved quantities and symmetries. A generally Gauss-Bonnet gravity $f(R,\mathcal{G})$ theory was proposed as an alternative gravity model. Through the generalized Noether symmetry, polynomial and product forms of the $f(R,\mathcal{G})$ theory with corresponding conserved quantities and symmetries are researched. Then suitable general forms of the polynomial form $f(R,\mathcal{G}) \!=\! k_1 R^n + (6)^{\frac{n}{2}}(-1)^{n+1} k_1 \mathcal{G}^{\frac{n}{2}}$ and the product form $f(R,\mathcal{…
A new symmetry for Newtonian Dynamics is analyzed, this corresponds to going to an accelerated frame, which introduces a constant gravitational field into the system and subsequently. We consider the addition of a linear contribution to the gravitational potential \(ϕ\) which can be used to cancel the gravitational field induced by going to the accelerated from, the combination of these two operations produces then a symmetry. This symmetry leads then to a Noether current which is conserved. The conserved charges are analyzed in special cases. The charges may not be conserved if the Noether cu…
We apply the Noether Symmetry Approach to point-like teleparallel Lagrangians in view to derive minisuperspaces suitable for Quantum Cosmology. Adopting the Arnowitt-Deser-Misner formalism, we find out related Wave Functions of the Universe. Specifically, by means of appropriate changes of variables suggested by the existence of Noether symmetries, it is possible to obtain the cosmological Hamiltonians whose solutions are classical trajectories interpretable as observable universes.
This paper is devoted to investigate the recently proposed modified Gauss-Bonnet $f(\mathcal{G},T)$ gravity, with $\mathcal{G}$, the Gauss-Bonnet term, coupled with ${T}$, the trace of energy-momentum tensor. We have used the Noether symmetry methodology to discuss some cosmologically important $f(\mathcal{G},T)$ gravity models with anisotropic background. In particular, the Noether symmetry equations for modified $f(\mathcal{G},T)$ gravity are reported for locally rotationally symmetric Bianchi type $I$ universe. Explicitly, two models have been proposed to explore the exact solutions and the…
This work aims to investigate some possible emergence of relativistic compact stellar objects in modified $f(\mathcal{G})$ gravity using Noether symmetry approach. For this purpose, we assume static spherically symmetric spacetime in the presence of isotropic matter distribution. We construct Noether symmetry generators along with associated conserved quantities by considering the standard choice of viable $f(\mathcal{G})$ gravity model i.e. $f(\mathcal{G})= α\mathcal{G}^{n}$, where $α$ is the model parameter. In particular, we use conservation relation acquired from the classical Noether appr…
In this paper, we study anisotropic universe using Noether symmetries in modified gravity. In particular, we choose locally rotationally symmetric Bianchi type-$I$ universe for the analysis in $f(R,\mathcal{G})$ gravity, where $R$ is the Ricci scalar and $\mathcal{G}$ is the Gauss-Bonnet invariant. Firstly, a model $f(R,\mathcal{G})=f_0R^l+f_1\mathcal{G}^n$ is proposed and the corresponding Noether symmetries are investigated. Further, we have also recovered the Noether symmetries for $f(R)$ and $f(\mathcal{G})$ theories of gravity. Secondly, some important cosmological solutions are reconstru…
The Noether Symmetry Approach is applied to interacting quintessence cosmology with the aim to search for exact solutions and select scalar-field self-interaction potentials. It turns out that the solutions found are compatible with the accelerated expansion of the Universe and with observational dataset, as the SNeIa Pantheon data.
We consider extensions of General Relativity based on the non-local function $f(R, \Box^{-1} R)$, where $R$ is the Ricci curvature scalar and the non-locality is due to the term $\Box^{-1} R$. We focus on cosmological minisuperspaces and select viable models by the Noether Symmetry Approach. Then we find viable exact solutions pointing out the role of non-locality in cosmology.
The probability representation, in which cosmological quantum states are described by a standard positive probability distribution, is constructed for minisuperspace models selected by Noether symmetries. In such a case, the tomographic probability distribution provides the classical evolution for the models and can be considered an approach to select "observable" universes. Some specific examples, derived from Extended Theories of Gravity, are worked out. We discuss also how to connect tomograms, symmetries and cosmological parameters.
We consider the Noether Symmetry Approach for a cosmological model derived from a tachyon scalar field $T$ with a Dirac-Born-Infeld Lagrangian and a potential $V(T)$. Furthermore, we assume a coupled canonical scalar field $ϕ$ with an arbitrary interaction potential $B(T,ϕ)$. Exact solutions are derived consistent with the accelerated behavior of cosmic fluid.
We consider curvature-teleparallel $F(R,T)$ gravity, where the gravitational Lagrangian density is given by an arbitrary function of the Ricci scalar $R$ and the torsion scalar $T$. Using the Noether Symmetry Approach, we show that the functional form of the $F(R, T)$ function, can be determined by the presence of symmetries . Furthermore, we obtain exact solutions through to the presence of conserved quantities and the reduction of cosmological dynamical system. Example of particular cosmological models are considered.
In this paper, we present a complete Noether Symmetry analysis in the framework of scalar-tensor cosmology. Specifically, we consider a non-minimally coupled scalar field action embedded in the FLRW spacetime and provide a full set of Noether symmetries for related minisuperspaces. The presence of symmetries implies that the dynamical system becomes integrable and then we can compute cosmological analytical solutions for specific functional forms of coupling and potential functions selected by the Noether Approach.
We consider a general theory of all possible quadratic, first-order derivative terms of the non-metricity tensor in the framework of Symmetric Teleparallel Geometry. We apply the Noether Symmetry Approach to classify those models that are invariant under point transformations in a cosmological background and we use the symmetries of these models to reduce the dynamics of the system in order to find analytical solutions.
We show that the recent results of \ [Int. J. Mod. Phys. D 25 (2016) 1650051] on the application of Lie/Noether symmetries in scalar field cosmology are well-known in the literature while the problem could have been solved easily under a coordinate transformation. That follows from the property, that the admitted group of invariant transformations of dynamical system is independent on the coordinate system.
We consider the cosmology derived from $f(T,B)$ gravity where $T$ is the torsion scalar and $B=\frac{2}{e}\partial_μ(e T^μ)$ a boundary term. In particular we discuss how it is possible to recover, under the same standard, the teleparallel $f(T)$ gravity, the curvature $f(R)$ gravity and the teleparallel-curvature $f(R,T)$ gravity, which are particular cases of $f(T,B)$. We adopt the Noether Symmetry Approach to study the related dynamical systems and to find out cosmological solutions.
The standard $Λ$CDM model of cosmology is formulated as a simple modified gravity coupled to a single scalar field ("darkon") possessing a non-trivial hidden nonlinear Noether symmetry. The main ingredient in the construction is the use of the formalism of non-Riemannian spacetime volume-elements. The associated Noether conserved current produces stress-energy tensor consisting of two additive parts -- dynamically generated dark energy and dark matter components non-interacting among themselves. Noether symmetry breaking via an additional scalar "darkon" potential introduces naturally an inter…
We quantize a flat FRW cosmology in the context of the $f(R)$ gravity by Noether symmetry approach. We explicitly calculate the form of $f(R)$ for which such symmetries exist. It is shown that the existence of a Noether symmetry yields a general solution of the Wheeler-DeWitt equation where can be expressed as a superposition of states of the form $e^{iS}$. In terms of Hartle criterion, this type of wave function exhibits classical correlations, i.e. the emergent of classical universe is expected due to the oscillating behavior of the solutions of Wheeler-DeWitt equation. According to this int…
We discuss the Noether Symmetry Approach in the framework of Gauss-Bonnet cosmology showing that the functional form of the $F(R, {\cal G})$ function, where $R$ is the Ricci scalar and ${\cal G}$ is the Gauss-Bonnet topological invariant, can be determined by the presence of symmetries. Besides, the method allows to find out exact solutions due to the reduction of cosmological dynamical system and the presence of conserved quantities. Some specific cosmological models are worked out
The Noether symmetry of a generic $f(R)$ cosmological model is investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generators of the desired symmetry. We explicitly calculate the form of $f(R)$ for which such symmetries exist. It is shown that the resulting form of $f(R)$ yields a power law expansion for the cosmological scale factor. We also obtain the effective equation of state parameter for the corresponding cosmology and show that our model can provide a gravitational alternative to the quintessence.
We study the metric $f(R)$ cosmology using Noether symmetry approach by utilizing the behavior of the corresponding Lagrangian under infinitesimal generators of the desired symmetry. The existence of Noether symmetry of the cosmological $f(R)$ minisuperspace helps us to find out the form of $f(R)$ function for which such symmetry exist. It is shown that the resulting form for $f(R)$ yields a power law expansion for the cosmic scale factor. We also show that in the corresponding Noether symmetric quantum model, the solutions to the Wheeler-DeWitt equation can be expressed as a superposition of …
We investigate the main features of the flat Friedmann-Lemaıtre-Robertson-Walker cosmological models in the f(T) teleparallel gravity. In particular, a general approach to find out exact cosmological solutions in f (T) gravity is discussed. Instead of taking into account phenomenological models, we consider as a selection criterion, the existence of Noether symmetries in the cosmological f(T) point-like Lagrangian. We find that only power-law models admit extra Noether symmetries. A complete analysis of such cosmological models is developed.
We derive the Lie and the Noether conditions for the equations of motion of a dynamical system in a $n-$dimensional Riemannian space. We solve these conditions in the sense that we express the symmetry generating vectors in terms of the special projective and the homothetic vectors of the space. Therefore the Lie and the Noether symmetries for these equations are geometric symmetries or, equivalently, the geometry of the space is modulating the motion of dynamical systems in that space. We give two theorems which contain all the necessary conditions which allow one to determine the Lie and the…
We summarize the use of Noether symmetries in Minisuperspace Quantum Cosmology. In particular, we consider minisuperspace models, showing that the existence of conserved quantities gives selection rules that allow to recover classical behaviors in cosmic evolution according to the so called Hartle criterion. Such a criterion selects correlated regions in the configuration space of dynamical variables whose meaning is related to the emergence of classical observable universes. Some minisuperspace models are worked out starting from Extended Gravity, in particular coming from scalar tensor, f(R)…
We discuss the Hamiltonian dynamics for cosmologies coming from Extended Theories of Gravity. In particular, minisuperspace models are taken into account searching for Noether symmetries. The existence of conserved quantities gives selection rule to recover classical behaviors in cosmic evolution according to the so called Hartle criterion, that allows to select correlated regions in the configuration space of dynamical variables. We show that such a statement works for general classes of Extended Theories of Gravity and is conformally preserved. Furthermore, the presence of Noether symmetries…
We develop the $n$-dimensional cosmology for $f(\mathcal{G})$ gravity, where $\mathcal{G}$ is the \emph{Gauss-Bonnet} topological invariant. Specifically, by the so-called Noether Symmetry Approach, we select $f(\mathcal{G})\simeq \mathcal{G}^k$ power-law models where $k$ is a real number. In particular, the case $k = 1/2$ for $n=4$ results equivalent to General Relativity showing that we do not need to impose the action $R+f(\mathcal{G})$ to reproduce the Einstein theory. As a further result, de Sitter solutions are recovered in the case where $f(\mathcal{G})$ is non-minimally coupled to a sc…
In this paper, we analyze static traversable wormholes via Noether symmetry technique in modified Gauss-Bonnet $f(\mathcal{G})$ theory of gravity (where $\mathcal{G}$ represents Gauss-Bonnet term). We assume isotropic matter configuration and spherically symmetric metric. We construct three $f(\mathcal{G})$ models, i.e, linear, quadratic and exponential forms and examine the consistency of these models. The traversable nature of wormhole solutions is discussed via null energy bound of the effective stress-energy tensor while physical behavior is studied through standard energy bounds of isotro…
A generalized teleparallel cosmological model, $f(T_\mathcal{G},T)$, containing the torsion scalar $T$ and the teleparallel counterpart of the Gauss-Bonnet topological invariant $T_{\mathcal{G}}$, is studied in the framework of the Noether Symmetry Approach. As $f(\mathcal{G}, R)$ gravity, where $\mathcal{G}$ is the Gauss-Bonnet topological invariant and $R$ is the Ricci curvature scalar, exhausts all the curvature information that one can construct from the Riemann tensor, in the same way, $f(T_\mathcal{G},T)$ contains all the possible information directly related to the torsion tensor. In th…
All degrees of freedom related to the torsion scalar can be explored by analysing, the $f(T,T_G)$ gravity formalism where, $T$ is a torsion scalar and $T_G$ is the teleparallel counterpart of the Gauss-Bonnet topological invariant term. The well-known Noether symmetry approach is a useful tool for selecting models that are motivated at a fundamental level and determining the exact solution to a given Lagrangian, hence we explore Noether symmetry approach in $f(T,T_G)$ gravity formalism with three different forms of $f(T,T_G)$ and study how to establish nontrivial Noether vector form for each o…
The existence of a Noether symmetry for a given minisuperspace cosmological model is a sort of selection rule to recover classical behaviours in cosmic evolution since oscillatory regimes for the wave function of the universe come out. The so called Hartle criterion to select correlated regions in the configuration space of dynamical variables can be directly connected to the presence of a Noether symmetry and we show that such a statement works for generic extended theories of gravity in the framework of minisuperspace approximation. Examples and exact cosmological solutions are given for non…
In the framework of phantom quintessence cosmology, we use the Noether Symmetry Approach to obtain general exact solutions for the cosmological equations. This result is achieved by the quintessential (phantom) potential determined by the existence of the symmetry itself. A comparison between the theoretical model and observations is worked out. In particular, we use type Ia supernovae and large scale structure parameters determined from the 2-degree Field Galaxy Redshift Survey (2dFGRS)and from the Wide part of the VIMOS-VLT Deep Survey (VVDS). It turns out that the model is compatible with t…
Teleparallel Horndeski theory offers an avenue through which to circumvent the speed constraint of gravitational waves in an efficient manner. However, this provides an even larger plethora of models due to the increase in action terms. In this work we explore these models in the context of cosmological systems. Using Noether point symmetries, we classify the dynamical systems that emerge from Teleparallel Horndeski cosmologies. This approach is very effective at selecting specific models in the general class of second-order Teleparallel scalar-tensor theories, as well as for deriving exact so…
We carried out the detailed group classification of the potential in Klein-Gordon equation in anisotropic Riemannian manifolds. Specifically, we consider the Klein-Gordon equations for the four-dimensional anisotropic and homogeneous spacetimes of Bianchi I, Bianchi\ III and Bianchi V. We derive all the closed-form expressions for the potential function where the equation admits Lie and Noether symmetries. We apply previous results which connect the Lie symmetries\ of the differential equation with the collineations of the Riemannian space which defines the Laplace operator, and we solve the c…
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-Lemaître-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.
We study a generalized nonlocal theory of gravity which, in specific limits, can become either the curvature non-local or teleparallel non-local theory. Using the Noether Symmetry Approach, we find that the coupling functions coming from the non-local terms are constrained to be either exponential or linear in form. It is well known that in some non-local theories, a certain kind of exponential non-local couplings are needed in order to achieve a renormalizable theory. In this paper, we explicitly show that this kind of coupling does not need to by introduced by hand, instead, it appears natur…
We analyze the cosmology of a general Scalar-Tensor theory which encompasses generalized Brans-Dicke theory, Gauss-Bonnet gravity, non-minimal derivative gravity, generalized Galileon gravity and also the general k-essence type models. Instead of taking into account phenomenological considerations we adopt a Noether symmetry approach, as a physical criterion, to single out the form of undetermined functions in the action. These specified functions symmetrize equations of motion in the simplest possible form which result in exact solutions. Demanding de Sitter, power-law and bouncing universe s…
We study Palatini f(R) cosmology using Noether symmetry approach for the matter dominated universe. In order to construct a point-like Lagrangian in the flat FRW space time, we use the dynamical equivalence between f(R) gravity and scalar-tensor theories. The existence of Noether symmetry of the cosmological f(R) Lagrangian helps us to find out the form of f(R) and the exact solutions for cosmic scale factor. We show that this symmetry always exist for f(R)~R^n and the Noether constant is a function of the Newton's gravitational constant and the current matter content of the universe.
