Noether symmetry approach in \(f(\mathcal {G},T)\) gravity
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Abstract
We explore the recently introduced modified Gauss–Bonnet gravity (Sharif and Ikram in Eur Phys J C 76:640, 2016), \(f(\mathcal {G},T)\) pragmatic with \(\mathcal {G}\), the Gauss–Bonnet term, and T, the trace of the energymomentum tensor. Noether symmetry approach has been used to develop some cosmologically viable \(f(\mathcal {G},T)\) gravity models. The Noether equations of modified gravity are reported for flat FRW universe. Two specific models have been studied to determine the conserved quantities and exact solutions. In particular, the well known deSitter solution is reconstructed for some specific choice of \(f(\mathcal {G},T)\) gravity model.
Keywords
Dark Energy Gravity Model Cosmological Solution Bonnet Term Stringy Charged Black Hole1 Introduction
The cosmological aspect of research has proven to be the main resource of modern diverse theories of gravity. No doubt, the onset of General relativity (GR) being a physical theory has been a great success of the last century. However GR alone does not provide us with the sufficient fundamental platform to solve the problems like initial singularity, flatness issues, dark energy and dark matter problems. It also fails when one is interested to express this wide universe as a whole, particularly at the extreme conditions for the ultraviolet scales and for the expression of the quantum structure of spacetime. Few years back, the significant outcomes of the researchers have confirmed that the universe is expanding itself [2, 3, 4, 5, 6, 7, 8, 9]. A gigantic portion of this accelerating universe consists of mysterious substance, known as the dark energy and is believed to be the main cause of the acceleration of this expansion. Recent developments in cosmology have revealed new ideas to introduce the critical and observational innovations for this accelerated expanding universe.
Modified theories of gravity have provided the researchers with different aspects and directions to unveil the hidden realities behind the expansion of the universe. After being motivated and having started with the original theory, a number of theories have been structured by building intricate Lagrangians. For example, theories of gravity like f(R), f(R, T), \(f(\mathcal {G})\) and \(f(R,\mathcal {G})\) have been developed by combining curvature scalars, topological invariants and their derivatives. These modified theories address the purpose of solving the complexities related to the quantum gravity and provide with the conditions through which the accelerated expansion of universe is argued. Nojiri and Odintsov [10] were the pioneers to present the idea of implicit and explicit coupling of the curvature with the matter in f(R) gravity. Some reviews [11, 12, 13] were established about the f(R) gravity by different researchers and the consistency of its different cosmological models were also studied [14]. The equivalence between metric and Palatini formalisms in f(R) gravity is shown to be achieved using divergence free current [15]. Harko et al. [16] introduced a gravitational theory by including both the matter and curvature terms and is well known today as the f(R, T) gravity, where R is the scalar curvature and T is the trace of energymomentum tensor. The evolution of the universe through energy conditions along with the criteria of stability were discussed by Sharif and Zubair [17]. They also reestablished a variety of dark energy models, investigated the thermodynamical aspect and exactly solved the anisotropic universe in f(R, T) gravity [18].
The symmetry methods of approximations have played a pivotal role to work out the exact solutions of differential equations. These approximations smartly reduce the complexity involved in a system of nonlinear equations by finding the unknown parameters of equations. In particular, the Noether symmetries are not just a tool to deal with the solution of the dynamics, but also their existence provides favorable conditions so that we can choose physically and analytically the universe models according to our calculated observations. Sharif and Waheed [36] rescaled the energy of stringy charged black hole solutions using approximate symmetries. Kucukakca [37] determined the exact solutions of Bianchi typeI model, using Noether symmetries. Jamil et al. [38] used the Noether symmetry approach to find out \(f(\mathcal {T})\) explicitly for the phantom and quintessence models, where \(\mathcal {T}\) is the torsion scalar. Sharif and Shafique [39] discussed Noether symmetries in a modified scalartensor gravity. The exact solutions in f(R) gravity were also explored using Noether symmetries methods for FRW spacetime [42]. Similarly many authors have used Noether symmetries to investigate the cosmology in different contexts [43, 44, 45, 46, 47, 48, 49, 50, 51].
In this paper, we are interested to investigate \(f(\mathcal {G},T)\) gravity using Noether symmetries. For this purpose, we consider the flat FRW universe model. The arrangement for the paper is as follows. In Sect. 2, we provide the preliminary formalism for \(f( \mathcal {G},T)\) gravity. Section 3 gives the Noether equations of FRW universe model for \(f(\mathcal {G},T)\) gravity. Reconstruction of cosmological solutions is presented in Sect. 4. Last section provides a brief outlook of the paper.
2 \(f(\mathcal {G},T)\) gravity with field equations

Equation (5) are highly nonlinear differential equations and its not an easy task to solve them analytically. It is mentioned here that fiat FRW modified field equations are fourthorder differential equations involving unknowns like \(f(\mathcal {G}, {T})\). Here we are interested to find the Noether symmetries of \(f(\mathcal {G},T)\) gravity with fiat FRW background.

The advantage of exact solutions in modified gravity have gained much importance, particularly in the study of phase transitions and recent phenomenon of accelerated expansion of universe. The viable cosmological models can be found using Noether symmetries and hence some physically important solutions can be reconstructed.
3 Noether symmetries and \(f(\mathcal {G},T)\) gravity
Case(i):
Let us assume that \(f_{\mathcal {G}\mathcal {G}}=0\). Manipulating the Eqs. (26–35), we obtain \(\alpha =0\), \(\beta =0\), \(\gamma =0\), and \(\delta =0\), hence providing us with a trivial solution. Moreover, the conservation Eq. (36) is also satisfied in this case. Thus, we have to consider \(f_{\mathcal {G}\mathcal {G}}\ne 0\) to obtain a nontrivial solution.
Case(ii):
4 Reconstruction of cosmological solutions
5 Outlook
In this paper, we have discussed in detail about the Noether symmetries of the flat FRW universe model in \(f(\mathcal {G},\mathrm {T})\) gravity. Noether symmetries are not just a tool to deal with the solution of the dynamics, but also their existence provides favorable conditions so that we can choose physically and analytically the universe models according to our calculated observations. Lagrangian multipliers perform a big part to shape the Lagrangian into its canonical form and so as to reduce the dynamics to determine the exact solutions. We have worked out the Lagrangian for FRW universe model in \(f(\mathcal {G},\mathrm {T})\) theory. The existence of Noether charges is considered important in the literature and equation for conservation of charge plays an important role to investigate the Noether symmetries. The conservation equation for Noether charge has been developed. The exact solutions of Noether equations have been discussed for two cases of \(f(\mathcal {G},\mathrm {T})\) gravity models. The first case when \(f_{\mathcal {G}\mathcal {G}}=0\) yields trivial symmettries while we obtain nontrivial symmetries for the second case when \(f_{\mathcal {G}\mathrm {\textit{T}}}=0\) and \(f_{\mathcal {G}\mathcal {G}}\ne 0\). Thus we have also worked out the corresponding \(f(\mathcal {G},\mathrm {T})\) gravity model and the solution metric. It is concluded that the second case provides \(f(\mathcal {G},\textit{T})=a_{0}\mathcal {G}^{2}+b_{0}{} \textit{T}^{2}\) gravity model, where \(a_{0}\) and \(b_{0}\) are arbitrary constants. Furthermore, solutions in both cases satisfy the conservation equation for Noether charge.
We have also reconstructed an important cosmological solution by considering \(f(\mathcal {G},\textit{T})=\mathcal {G}^{k}{} \textit{T}^{1k}\), where k is an arbitrary real number. This model yields the wellknown deSitter solution already available in GR. It is mentioned here that many other cosmologically physical solutions may be reconstructed for some other choice of \(f(\mathcal {G},\textit{T})\) gravity models. We have discussed the exact solutions with only three cases. Many other solutions can be explored by assuming some other forms of \(f(\mathcal {G},\mathrm {T})\).
Notes
Acknowledgements
Many thanks to the anonymous reviewer for valuable comments and suggestions to improve the paper. This work was supported by National University of Computer and Emerging Sciences (NUCES).
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