# \(f(\mathcal {R},\varphi ,\chi )\) cosmology with Noether symmetry

## Abstract

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 \(f(\mathcal {R},\varphi ,\chi )\) gravity models.

## 1 Introduction

The accelerated expansion of universe and modified theories of gravity have been two heavily debated topic of discussions in the last two decades. It has been argued that the mysteries like dark energy and dark matter, initial singularity problem and flatness issues can be well addressed in the context of modified or alternative theories of gravity. Based upon original theory of general relativity (GR), a number of modifications have been proposed by constructing intricate Lagrangians. The most discussed and viable theories include \(f(\mathcal {R})\) theory of gravity. Nojiri and Odintsov [1] are among the pioneers who discussed the possible coupling of matter with curvature. Some review papers can be really helpful to understand the viable features of \(f(\mathcal {R})\) gravity [2, 3, 4]. Further modifications like \(f(\mathcal {R},T)\) gravity [5] and a recently proposed \(f(\mathcal {R},\varphi ,\chi )\) theory of gravity [6] are also among successful theories of gravity. It is expected that these modified theories of gravity may well address the issues of late-time cosmic acceleration using some specific choices of cosmological models.

Noether symmetry approach is the most elegant and systematic way to compute conserved quantities [7]. These symmetries smartly minimize the complexities involved in a system of non-linear partial differential equations (PDEs) and many new solutions can be constructed using conserved quantities. In fact, the conservation laws play an important role in studying different physical phenomenon. The integrability of PDEs depends on the number of conservation laws. According to Noether’s theorem, any differentiable symmetry of the action for a physical system corresponds to some conservation law. This theorem is very important as it provides the information about the conservation laws in physical theories including GR. According to Noether theorem [8], the translational and rotational symmetries of any object are the consequence of the conservation laws of linear and angular momentum. Many authors have used this theorem in recent years to discuss some important issues in different cosmological contexts.

Sharif and Waheed [9] computed the energy contents of stringy charged black hole solutions with the help of approximate symmetries. Kucukakca [10] used Noether symmetries to investigate the exact solutions of Bianchi type-*I* spacetime. The exact solutions in \(f(\mathcal {R})\) gravity have been explored using Noether symmetries methods for Friedmann–Robertson–Walker (FRW) spacetime [11]. Fazlollahi [12] used Noether gauge symmetries to obtain an effective equation of state parameter for corresponding cosmology and it was concluded that the model provided viable cosmic scale factor with respect to observational data. In a recent paper, Bahamonde et al. [13] provided a class of new exact spherically symmetric solutions in the context of \(f(\mathcal {R},\varphi ,\chi )\) theory using Noether’s symmetry approach. In another paper [14], teleparallel gravity models have been studied by adopting the Noether symmetry approach and some exact solutions are derived in the context of flat FRW cosmology. Shamir and Ahmad [15, 16] used Noether symmetries to investigate the exact solutions of the field equations in \(f(\mathcal {G},T)\) theory of gravity and discussed some cosmologically important \(f(\mathcal {G},T)\) gravity models with both isotropic and anisotropic backgrounds. It was reported that specific models of modified Gauss-Bonnet gravity may be used to reconstruct \(\Lambda \)CDM cosmology without involving any cosmological constant. Thus it seems interesting to use Noether symmetries to further explore the universe with a hope of some fruitful results.

In this paper, we are focused to investigate \(f(\mathcal {R},\varphi ,\chi )\) gravity using Noether symmetry approach. We choose the flat FRW spacetime for this purpose. The paper is organized in the following way: Some basics of \(f(\mathcal {R},\varphi ,\chi )\) gravity are given in Sect. 2. Section 3 provides a detailed discussion about symmetry reduced Lagrangian and Noether equations for FRW universe model in \(f(\mathcal {R},\varphi ,\chi )\) gravity. Cosmological solutions based upon conserved quantities are presented in Sect. 4. Last section gives a brief summary of the results.

## 2 Some basics of \(f(\mathcal {R},\varphi ,\chi )\) gravity

\(\mathcal {R}\) is the Ricci Scalar and \(\varphi \) is the scalar field,

\(\chi =-\frac{\epsilon }{2}{\partial }^u\varphi {\partial }_u\varphi \), \(\epsilon \) being a parameter such that when equal to 1 represents canonical scalar field and when equal to \(-1\) represents a phantom scalar field.

Since \(f(\mathcal {R},\varphi , \chi )\) is a multivariate analytic function and its partial derivatives will be involved in many equations in this paper, so for simplicity we use the notations \(f(\mathcal {R},\varphi , \chi )\equiv f\), \(f_{\mathcal {R}}\equiv \frac{\partial f}{\partial {\mathcal {R}}}\), \(f_{\varphi }\equiv \frac{\partial f}{\partial {\varphi }}\) and \(f_\chi \equiv \frac{\partial f}{\partial {\chi }}\).

*S*in Eq. (1) with respect to the metric tensor

## 3 Symmetry reduced Lagrangian and Noether equations

Noether symmetries have become quite essential practice to investigate the solutions of non-linear differential equations. In this section, we develop the point-like Lagrangian for FRW spacetime in the context of \(f(\mathcal {R},\varphi ,\chi )\) gravity and apply Noether symmetry approach to find the corresponding determining equations. The existence of this approach implies the uniqueness of the vector field in the associated tangent space. Thus, the vector field acts like symmetry generator which further provides the conserved quantities helpful in exploring the exact solutions of modified field equations.

*d*-dimensional configuration space \(Q=\{ q^i,~i=1,...,d \}\). Also it would be worthwhile to mention here that components \(\zeta \) and \({\beta }^{i}\) of the above Noether symmetry generator are the multivariable functions of

*t*and \(q^i\). For example, in our case \(\zeta \equiv \zeta (t,a,\mathcal {R},\varphi ,\chi )\) and \(q^i\equiv q^i(t,a,\mathcal {R},\varphi ,\chi )\), \(i=1,...4\). Also for the existence of Noether symmetry, the Lagrangian (15) must satisfy

\(\Psi \equiv \Psi (t,q^i)\) is known as the Noether gauge function,

\(D_t\) is the total derivative defined as \(D_t\equiv \frac{\partial }{\partial t}+\dot{q}^i \frac{\partial }{\partial q^i}\) and \(\dot{{\beta }}^i\equiv D_t{\beta }^i-\dot{q}^iD_t\zeta \).

*Y*is given by

## 4 Conserved quantities

In this section, we solve the system of PDEs (25)–(32) to get the Noether symmetries \(Y=\zeta \partial _t +{\beta }^i\partial _i\). Since the above system depends on the function \(f(\mathcal {R},\varphi ,\chi )\) along with some other unknowns, so it is difficult to find a simultaneous solution without assigning some appropriate values to \(f(\mathcal {R},\varphi ,\chi )\). However, if we see first equation of the the system (32), we come across with at least a trivial symmetry \(Y=\partial _t\) independent of the choice of any specific \(f(\mathcal {R},\varphi ,\chi )\) gravity model. Thus, we start our analysis by choosing different cosmological models.

### 4.1 \(f(\mathcal {R})\) gravity

*t*only and Eq. (32) simplifies to

*a*is obtained as

### 4.2 \(f(\mathcal {R},\varphi ,\chi )\) gravity

In this section, we investigate the conserved quantities for a broader class. For this purpose, we consider the following cases:

\(\mathbf {Case(i)}\): \(f(\mathcal {R},\varphi ,\chi )=\varphi \mathcal {R}^n,~~~n\ne 0,1.\)

\(\mathbf {Case(ii)}\): \(f(\mathcal {R},\varphi ,\chi )=\varphi ^m \mathcal {R}^{\frac{19}{14}},~~~m\ne 0,1.\)

*m*can play an important role in the evolution of universe. For example, when we assume scalar field as a quadratic function of cosmic time and \(\mathcal {R}=\mathcal {R}_0t^{-\frac{63m}{5}}\), the evolution of scale factor becomes interesting. It can be seen from Fig. 2 that for \(m>1\), the scale factor remains positive and shows an increasing behavior. In fact, an accelerated expansion phase is observed as

*m*increases. Also, when \(m<1\), the scale factor shows a negative and decreasing behavior. Thus with the considered parametric values involved in the analysis, the scalar field becomes very important when \(m>1\).

\(\mathbf {Case(iii)}\): \(f(\mathcal {R},\varphi ,\chi )=\varphi \mathcal {R}+h(\chi )\)

## 5 Outlook

The exact solutions of Noether equations have been divided mainly in two parts. The first case is focussed on revisiting usual metric \(f(\mathcal {R})\) gravity. It is worthwhile to mention here that the \(f(\mathcal {R})\) gravity is constrained with the condition that \(f_{\mathcal {R}\mathcal {R}}\ne 0\). Thus for the sake of simplicity, we have considered \(f(\mathcal {R},\varphi ,\chi )=f_0\mathcal {R}^{\frac{3}{2}}\), where \(f_0\ne 0\) is a real arbitrary parameter. The obtained Noether symmetries turn out to be same as already available in literature [17, 20]. However, we use conserved quantities to investigate both numerical and analytical solutions to study the evolution of universe. After using some appropriate initial condition and suitable values of parameters involved, a numerical solution is obtained. It is evident from the behavior of scale factor as shown in Fig. 1 that universe expanded at an early time with deceleration phase and then later on accelerated cosmic expansion could be observed. Analytical approach provides an exact solution that have already been suggested to exist in the context of \(f(\mathcal {R})\) gravity [18, 21]. We also report a new exact solution (45) in the context of \(f(\mathcal {R})\) gravity which for a special case indicates the existence of Big Rip singularity.

The second part deals with the more general form of \(f(\mathcal {R},\varphi , \chi )\) gravity model. Three subcases have been discussed under this category. The first case provides the exact solutions of FRW universe when \(f(\mathcal {R},\varphi ,\chi )=\varphi \mathcal {R}^n,~~~n\ne 0,1\). Five conserved quantities have been obtained in this case. Three of them have been used to provide a new class of exact solutions in the context of \(f(\mathcal {R},\varphi , \chi )\) gravity.

We further consider \(f(\mathcal {R},\varphi ,\chi )=\varphi ^m \mathcal {R}^{\frac{19}{14}}\) for finding the Noether symmetry generators and corresponding first integrals. Here we also get five conserved quantities as in the previous case. It is due to the fact that both cases are similar in nature, the only difference is that in first case Ricci power law model is used while in this case scalar field power law form is used for a better comparative analysis. Many solutions are possible using conserved quantities in this case, however, one solution has been reported for the discussion. It is worth mentioning that the scalar field parameter

*m*plays an important role in the evolution of universe. For example, when scalar field is assumed as a quadratic function of cosmic time and \(\mathcal {R}=\mathcal {R}_0t^{-\frac{63m}{5}}\), the evolution of scale factor becomes interesting. It can be seen from Fig. 2 that for \(m>1\), the scale factor remains positive and shows an increasing behavior. In fact, an accelerated expansion phase is observed as*m*increases. Also, when \(m<1\), the scale factor shows a negative and decreasing behavior. Thus with the considered parametric values involved in this analysis, the scalar field becomes very important when \(m>1\).The last cases provides the Noether symmetries for \(f(\mathcal {R},\varphi ,\chi )=\varphi \mathcal {R}+h(\chi )\). Here we obtain two Noether symmetry generators after solving the determining equations simultaneously. Many cosmic solutions can be constructed here for different choices of the function \(h(\chi )\). However for the present analysis, we consider the simplest case \(h(\chi )=k_1\chi +k_2\), where \(k_1\ne 0\) and \(k_2\) are arbitrary real coefficients. A conserved quantity in this case suggests that the growth of scale factor depends on the scalar field and varies with inverse proportion. A non-linear differential equation has been formed in terms of scalar field \(\varphi \). We can try for the possible solution for the scalar field with some suitable initial conditions. A numerical solution is reported for canonical scalar field \(\epsilon =1\). The evolution of scale factor with blue colored curve as shown in Fig. 3 seems more physical due to increasing behavior.

## Notes

### Acknowledgements

Many thanks to the anonymous reviewer for valuable suggestions to improve the paper. My thanks also go to Ms. Iffat Fayyaz and Ms. Tayyaba Naz for their valuable comments and discussions. The author acknowledges National University of Computer and Emerging Sciences, Islamabad, Pakistan for research reward program.

## References

- 1.S. Nojiri, S.D. Odintsov, Phy. Lett.
**599**, 137 (2004)ADSCrossRefGoogle Scholar - 2.A.D. Felice, S. Tsujikaswa, Living Rev. Rel.
**13**, 3 (2010)CrossRefGoogle Scholar - 3.S. Nojiri, S.D. Odintsov, Phys. Rept.
**505**, 59 (2011)ADSCrossRefGoogle Scholar - 4.K. Bamba, S. Capozziella, S. Nojiri, S.D. Odintsov, Astrophys. Space Sci.
**342**, 155 (2012)ADSCrossRefGoogle Scholar - 5.T. Harko, F.S.N. Lobo, S. Nojiri, S.D. Odinttsov, Phys. Rev. D
**84**, 024020 (2011)ADSCrossRefGoogle Scholar - 6.S. Bahamonde, C.G. Bohmer, F.S.N. Lobo, D. Saez-Gomez, Universe
**1**, 186 (2015)ADSCrossRefGoogle Scholar - 7.E. Noether, Invariante Variationsprobleme. Nachr. Ges. Wiss. Gottingen, Math. Phys. Kl.
**1918**, 235–257 (1918)zbMATHGoogle Scholar - 8.J. Hanc, S. Tuleja, M. Hancova, Am. J. Phys.
**72**, 428 (2004)ADSCrossRefGoogle Scholar - 9.M. Sharif, S. Waheed, Can. J. Phys.
**88**, 833 (2010)ADSCrossRefGoogle Scholar - 10.Y. Kucukakca, U. Camci, I. Semiz, Gen. Relat. Gravit.
**44**, 1893 (2012)ADSCrossRefGoogle Scholar - 11.S. Capozziello, A. Stabile, A. Troisi, Class. Quantum Grav.
**24**, 2153 (2007)ADSCrossRefGoogle Scholar - 12.H.R. Fazlollahi, Phy. Lett. B
**781**, 542 (2018)ADSCrossRefGoogle Scholar - 13.S. Bahamonde, K. Bamba, U. Camci, JCAP
**02**, 016 (2019)ADSCrossRefGoogle Scholar - 14.S. Bahamonde, U. Camci, S. Capozziello, Class. Quantum Grav.
**36**, 065013 (2019)ADSCrossRefGoogle Scholar - 15.M.F. Shamir, M. Ahmad, Eur. Phys. J. C
**77**, 55 (2017)ADSCrossRefGoogle Scholar - 16.M.F. Shamir, M. Ahmad, Mod. Phys. Lett. A
**32**, 1750086 (2017)ADSCrossRefGoogle Scholar - 17.I. Hussain, M. Jamil, F.M. Mahomed, Astrophys. Space Sci.
**337**, 373 (2011)ADSCrossRefGoogle Scholar - 18.S. Capozziello, Int. J. Mod. Phys. D
**11**, 483 (2002)ADSCrossRefGoogle Scholar - 19.P.J. Olver,
*Applications of Lie groups to differential equations*(Springer Science & Business Media, New York, 2000)zbMATHGoogle Scholar - 20.M.F. Shamir, A. Jhangeer, A.A. Bhatti, Chin. Phys. Lett.
**29**(8), 080402 (2012)CrossRefGoogle Scholar - 21.S. Capozziello, G. Lambiase, Gen. Relat. Grav.
**32**, 295 (2000)ADSCrossRefGoogle Scholar - 22.A.V. Astashenok, S. Nojiri, S.D. Odintsov, A.V. Yurov, Phy. Lett. B
**709**, 396 (2012)ADSCrossRefGoogle Scholar - 23.K. Bamba, R. Myrzakulov, S. Nojiri, S.D. Odintsov, Phys. Rev. D
**85**, 104036 (2012)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Funded by SCOAP^{3}