F(R,G) Cosmology through Noether Symmetry Approach

Recent observational data indicate that the current expansion of the universe is accelerating [1,2,3,4,5,6,7,8], not only expanding. Then this acceleration is explained by the existence of a dark energy, which could result from a cosmological constant Λ as the simplest candidate with the equation of state parameter $w_{\mathsf{\Lambda }}=-1$, or may also be explained in the context of modified gravity models. The nature and origin of the dark energy has not been persuasively explained yet. In addition to the cosmological constant, there are different kinds of candidates for dark energy such as quintessence or phantom in the literature, and it is not even clear what type of candidates to the dark energy occur in the present universe. Therefore, there have been a number of attempts [9,10,11,12,13,14,15] to modify gravity to explain the origin of dark energy.

A possible modification of the standard general relativistic gravitational Lagrangian includes a wider number of curvature invariants $R,R_{ij}{R}^{ij}$ and $R_{ijkl}{R}^{ijkl}$ among others. In the so-called Gauss-Bonnet (GB) gravity theories the gravitational Lagrangian consists of a $F(R,G)$ function, where the GB invariant G is defined as $G=R^2-4R_{ij}{R}^{ij}+R_{ijkl}{R}^{ijkl}$. Considering the GB invariant G in dynamical equations one can recover all the curvature budget coming from the Riemann tensor. Due to of the fact that the GB invariant comes out from defining quantum fields in curved spacetimes, it should be important to take it in the context of the extended theories of gravity. It is shown in [13] that the quintessence paradigm can be recovered in the framework of $F(R,G)$ theories of gravity. The $F(R,G)$ gravity theories are generalizations $f(R)$ and $f(G)$ theory of gravities which are offered by higher order gravities, and use combinations of higher order curvature invariants constructed from the Ricci and Gauss-Bonnet scalars. In [14], some classes of $F(R,G)$ gravity have been studied with respect to the successful realization of the dark energy and of the inflationary era. We refer to readers the latest review [15] on developments of modified gravity in cosmology, emphasizing on inflation, bouncing cosmology and late-time acceleration era.

If a Lagrangian $\mathcal{L}$ for a given dynamical system admits any symmetry, this property should strongly be related with Noether symmetries that describe physical features of differential equations possessing a Lagrangian $\mathcal{L}$ in terms of first integrals admitted by them [16,17]. This can actually be seen in two ways. Firstly, one can consider a strict Noether symmetry approach [18,19,20,21] which yields ${\pounds }_{\mathbf{X}}\mathcal{L}=0$, where ${\pounds }_{\mathbf{X}}$ is the Lie derivative operator along $\mathbf{X}$. On the other side, one could use the classical Noether symmetry approach with a functional term [22,23,24,25] which is a generalization of the strict Noether symmetry approach in the sense that the Noether symmetry equation includes a divergence of a functional boundary term. The classical Noether symmetry approach was originally established by Emmy Noether [26] and it gives a connection between a Noether symmetry and the existence of a first integral expressed in a simple form. Not only the classical Noether symmetries but also the strict ones are useful in a variety of problems arising from physics and applied mathematics. Both types of symmetries lead to the first integrals. Which type of symmetry works, i.e., gives any conserved quantity, in the first instance this is what is important. The classical Noether symmetries are directly related with the conserved quantities (first integrals) or conservation laws [17]. The strict Noether symmetry approach represents how Noether’s theorem and cyclic variables are related. It is known that the conserved quantities are also related to the existence of cyclic variables into the dynamics by the strict Noether symmetry. However, it is usually required a clever choice of cyclic variables because of that the equations for the change of coordinates have not a unique solution which is also not well defined on the whole space, and thus it is not unique to find those of the cyclic variables (see References [27] for details). Furthermore, we refer to the interested readers the recent review on symmetries in differential equations [28].

The cosmological principle assume that the universe is homogeneous and isotropic in large scale structure and the geometrical model that satisfies these properties is Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. In [19], it has been discussed the strict Noether symmetry approach for spatially flat FLRW spacetime in GB cosmology, where it was pointed out that the existence of Noether symmetries is capable of selecting suitable $F(R,G)$ models to integrate dynamics by the identification of suitable cyclic variables. After this work, the classical Noether symmetries of flat FLRW spacetime have been computed by [25], where the authors were used Noether symmetries as a geometric criterion to select the form of $F(R,G)$ function. Due to the richness of the classical Noether symmetry approach, we deduced throughout this study that it is better to use the classical Noether symmetry approach to find Noether symmetries in $F(R,G)$ gravity as in [25], rather that the approach used in [19]. If there exists any Noether symmetry with a selection of physically interesting forms of $F(R,G)$ function, then this allows us to write out the constants of motion which reduce dynamics. Furthermore, the reduced dynamics results exactly solvable cosmological model by a straightforward way. In fact, choosing an appropriate $F(R,G)$ Lagrangian, it is possible to find out conserved Noether currents which will be useful to solve dynamics. This approach is very powerful due to the fact that it allows us to find a closed system of equations, where we do not need to impose the particular form of $F(R,G$) which is selected by the classical Noether symmetry itself. To this aim, it is possible to consider flat FLRW background metric and demonstrate that it is possible to find exact solutions via the Noether Symmetry Approach. In this study we again underline the generality of Noether’s Theorem in its original form by considering the standard cosmological model.

This paper is organized as follows. In the following section, we will present an analysis of the classical Noether symmetry approach including a boundary function for the point-like $F(R,G)$ Lagrangian according to the spatially flat FLRW background. In Section 3, we will apply the classical Noether theorem to the $F(R,G)$ Lagrangian obtained in Section 2 for the flat FLRW model. In Section 4, we classify the Noether symmetries with respect to some specific forms of $F(R,G)$, and search the cosmological solutions of $F(R,G)$ gravity by considering both the vacuum and the non-vacuum cases. Finally, in Section 5, we will provide a summary of the main results obtained in the paper.

In this section we briefly present the general formalism of $F(R,G)$ gravity. The action for $F(R,G)$ gravity is given by where ${\kappa }^2=8\pi G_N$, $G_N$ is the Newton constant and ${\mathcal{L}}_m$ represents the matter Lagrangian. Variation of the action (1) with respect to the metric tensor $g_{ij}$ we obtain the modified field equations where we have defined the following expressions In the above field equations, ${\nabla }_i$ is the covariant derivative operator associate with $g_{ij}$, $\square \equiv g^{ij}{\nabla }_i{\nabla }_j$ is the covariant d’Alembertian operator, and $T_{ij}^m$ describes the ordinary matter. It is clear from the field Equation (2) that the form of $F(R,G)$ determine the dynamical behaviour of the theory.

In this study, we consider the spatially flat FLRW metric where $a(t)$ is the scale factor of the Universe. Then, the Hubble parameter H is usually defined by $H\equiv \dot{a}/a$, and R and G become where the overdot denotes a derivative with respect to the time coordinate, t. For a perfect fluid matter with comoving observer $u_i={\delta }_i^0$, the energy momentum tensor is $T_{ij}=(\rho +p)u_i{u}_j+p{g}_{ij}$, where $\rho$ is the energy density and p is the isotropic pressure measured by the observer $u_i$. Let us assume that the matter fluid will be given under the form of a perfect fluid with the equation of state $p=w\rho$ satisfying the standard continuity equation $\dot{\rho }+3(1+w)\rho \dot{a}/a=0$ which yields a solution $\rho ={\rho }_{m0}{a}^{-3(1+w)}$, where ${\rho }_{m0}$ is the energy density of the present universe, and w is a constant parameter. Thus, in the flat FLRW background with a perfect fluid matter, the field Equation (2) for the $F(R,G)$ gravity are given by

In terms of the Hubble parameter H, the gravitational field Equations (6) and (7) for $F(R,G)$ gravity have the following form where ${\rho }_{eff}$ and $p_{eff}$ are respectively the effective energy density and pressure of the universe, which are defined as Here we observe from (8) that ${\rho }_{eff}+p_{eff}=-\frac{2}{{\kappa }^2}\dot{H}$.

Recently the strict Noether symmetries of GB cosmology for the flat FLRW spacetime have been calculated, and choosing some functional form of the $F(R,G)$, the Noether symmetries related to these functional forms have been achieved [19]. Afterwards, the classical Noether symmetries have also been calculated by [25]. Both of these studies were performed in the vacuum case. In this work, after reviewing the vacuum case, we aim to generalize these studies to the non-vacuum case using the classical Noether symmetry approach described below.

The Noether symmetry generator for any point-like Lagrangian $\mathcal{L}$ is if there exists a function $K(t,a,R,G)$ and the Noether symmetry condition is satisfied, where $D_t=\frac{\partial }{\partial t}+{\dot{q}}^i\frac{\partial }{\partial q^i}$ is the total derivative operator and ${\mathbf{X}}^{\left[1\right]}$ is the first prolongation of Noether symmetry generator $\mathbf{X}$, i.e. where ${\dot{\eta }}^i(t,q^k,{\dot{q}}^k)=D_t{\eta }^i-{\dot{q}}^i{D}_t\xi$, $q^i=\{a,R,G\}$ are the generalized coordinates in the three-dimensional configuration space $Q\equiv \{q^i,i=1,2,3\}$ of the Lagrangian, whose tangent space is $TQ\equiv \{q^i,{\dot{q}}^i\}$. The energy functional $E_{\mathcal{L}}$ or the Hamiltonian of the Lagrangian $\mathcal{L}$ is defined by Using above definition of energy functional, the corresponding Noether flow I, which is a constant called the first integral of motion, has the expression which is a conserved quantity. The Noether flow (15) satisfies the conservation relation $D_{t}I=0$.

It is obviously seen from a general point of view that R and G are functions of $a,\dot{a}$ and $\ddot{a}$, which yields non-canonical dynamics. The Lagrange multipliers plays a main role so as to get a canonical point-like Lagrangian [29]. Using this key future in [19], it has been accomplished that the point-like Lagrangian for $F(R,G)$ gravity becomes canonical with suitable Lagrange multipliers, where both R and G behave like effective scalar fields. We left the details for finding a canonical point-like Lagrangian by Lagrange multipliers method to the Reference [19]. For the spatially flat FLRW spacetime (4), the Lagrangian for the action of $F(R,G)$ gravity (1) has the form where ${\dot{F}}_R=F_{RR}\dot{R}+F_{RG}\dot{G}$ and ${\dot{F}}_G=F_{GR}\dot{R}+F_{GG}\dot{G}$. By variation of the above Lagrangian with respect to the configuration space variables $a,R$ and G, we find respectively that in which the Equation (17) is equivalent to the field Equation (7). Then we note that R and G coincides with the definitions of the Ricci scalar and Gauss-Bonnet invariant given by (5), respectively. Now, we calculate the energy functional $E_{\mathcal{L}}$ for the Lagrangian density (16) which has the form It is explicitly seen that the energy function $E_{\mathcal{L}}$ vanishes due to the $(0,0)$-field Equation (6).

Let us consider the Noether symmetry conditions (12) for the point-like Lagrangian (16) to seek the dependent variables $\xi ,{\eta }^1,{\eta }^2,{\eta }^3$ which will be solved in order that the Lagrangian (16) would admit any Noether symmetry (11). For the flat FLRW spacetime (4), the Noether symmetry conditions (12) yield 27 partial differential equations as the following where V is defined as $V(a,R,G)=a^3(R{F}_R+G{F}_G-F)+2{\kappa }^2{\rho }_{m0}{a}^{-3w}$, which can be considered as an effective potential for the $F(R,G)$ gravity. Here, R and G act as two different scalar fields whose regimes can lead different phases of the cosmological evolution.

We note here that ${\eta }^1=0,{\eta }^2=0,{\eta }^3=0,K=const.$ and $\xi =const.$ are trivial solutions for the Noether symmetry Equation (21). This result implies that any form of $F(R,G)$ function admits the trivial Noether symmetry ${\mathbf{X}}_1=\partial /{\partial }_t$, i.e., energy conservation, whose Noether first integral or the Hamiltonian of the system vanishes, $I=-E_{\mathcal{L}}=0$. In the following section, we consider the form of $F(R,G)$ to find the corresponding Noether symmetries and solutions to the corresponding first integrals for each of the vacuum an the non-vacuum cases.

Using the symmetry condition (12) to the point-like Lagrangian (16), which will fix the form of $F(R,G)$, several different cases were classified in [25] according to whether the derivative $F_{RG}$ vanishes or not. If $F_{RG}=0$, it means $F(R,G)=f(R)+g(G)$, which is considered as the case (i) below, taking $f(R)=f_0{R}^n$ and $g(G)=g_0{G}^m$. Otherwise, if $F_{RG}\ne 0$, we will take the form of $F(R,G)$ function as the case (ii), i.e., $F(R,G)=f_0{R}^n{G}^m$, where n and m are real numbers.