Cosmological singularities in conformal Weyl gravity
Abstract
In this work, we study the issue of the past and future cosmological singularities in the context of the fourthorder conformal Weyl gravity. In particular, we investigate the emergent universe scenario proposed by Ellis et al., and find the stability conditions of the corresponding Einstein static state using the fixed point approach. We show that depending on the values of the parameters of the conformal Weyl gravity theory, there are possibilities for having initially stable emergent states for an FRW universe with both the positive and negative spatial curvatures. This represents that the conformal Weyl gravity can be free of the initial singularity problem. Then, following Barrow et al., we address the possible types of the finitetime future cosmological singularities. We discuss how these singularities also can be avoided in the context of this theory.
1 Introduction
The standard model of cosmology based on the Einstein general theory of relativity (GR) has a good agreement with the recent high resolution observations and provides an accurate overview of the cosmic history of our universe. However, it suffers from some fundamental problems such the initial big bang singularity problem, the flatness problem, the horizon problem, the magnetic monopoles and etc. Although the inflation scenario provides solutions for some of these problems, the initial big bang singularity problem still remains unsolved. Indeed, if the inflation scenario is realized by the dynamics of scalar fields coupled to Einstein gravity, then the HawkingPenrose singularity theorem [1, 2] proves that an inflationary universe is geodesically incomplete in its past. Therefore, there exists a singularity before the onset of inflation and consequently the inflationary scenario is not capable to yield the complete history of the very early universe. To deal with this shortcoming, a number of attempts have been made in construction of cosmological models which are initially nonsingular. There are two main proposals for this aim named as the “bouncing universe” and “emergent universe” scenarios. The bouncing universe scenario is based on a smooth transition of the universe at a finite radius from a contracting phase to an expanding phase. Then, by going backward in time, the universe collapses up to a finite value for the scale factor, before which it starts to expand again [3]. Although the bouncing scenario avoids the big bang singularity, the existence of the nonsingular bounce requires nonstandard matter fields violating energy conditions [4, 5]. In the emergent universe scenario, the universe emerges from a static state, namely as the “Einstein static universe (ESU)”, characterized by a nonzero spatial volume with a positive curvature [6, 7]. This scenario avoids the initial big bang singularity while preserving the standard energy conditions. However, due to the existence of varieties of perturbations, such as the quantum fluctuations and the variations of the average energy density of the system [8], the emergent universe model is unstable and suffers from a finetuning problem [9]. This unstability problem can be amended by the modifications of the cosmological field equations of Einstein’s GR. Indeed, there are two conditions which any gravitational theory should satisfy for admitting an emergent universe paradigm: (1) the existence and stability of the ESU, and (2) the possibility of joining the standard cosmological history by a graceful exit mechanism from the ESU. By this motivation, the emergent universe scenario and the stability issue of its Einstein static state has been explored in the context of various modified theories of gravity such as the loop quantum gravity [10, 11], f(R) gravity [12, 13, 14], f(T) gravity [15, 16, 17], EinsteinCartan theory [18, 19], Braneworlds [20, 21, 22] and Massive gravity [23, 24] as well as in the presence of vacuum energy corresponding to conformally invariant fields [25] among the others. Most of the mentioned references just focused on the first condition without addressing the later one.
On the other hand, after the discovery of the accelerating expansion of the universe, deeper studies of the features of the cosmic fluid responsible for this accelerating expansion, the so called “dark energy”, showed the plethora of new types of singularities different than the initial big bang singularity. These new types of singularities are future singularities for a universe and they are characterized by the violation of some of the energy conditions. This violation in turn results in the divergencies in some of the physical quantities such as the scale factor, energy density and pressure profiles. As instances for these type of singularities, one may refer to Big Rip singularity [26, 27, 28], Sudden singularity [29, 30, 31] and Big Freeze singularity [32]. There are also other classifications for the possible future singularities, see for example [33, 34, 35, 36, 37, 38, 39].
The theory of conformal Weyl gravity has been proposed as an alternative theory to Einstein’s GR for the high energy limit and is based on the assumption of the local conformal invariance in the geometry of spacetime [40, 41]. There are many attempts for obtaining the Einstein’s GR as the low energy limit of the conformal Weyl theory, by dynamically breaking the conformal invariance, with varying degrees of success and difficulty. In this direction, the main problem is that by breaking the conformal symmetry, the quantum fluctuations bring back the EinsteinHilbert action with a repulsive rather than attractive gravity. However, one may refer to the recent work of Maldacena [42] where it is shown that in the four dimension, the conformal gravity with a Neumann boundary condition is classically equivalent to the Einstein gravity with a cosmological constant. It has been also suggested by Kazanas and Mannheim that the conformal Weyl gravity can be considered an independent theory on its own right instead of looking for a low energy GR limit. By this motivation, the exact vacuum solutions as well as the cosmological solutions of this theory have been found [43, 44, 45, 46]. It is shown that in the cosmological setup, the conformal Weyl gravity can solve naturally both the cosmological constant and the flatness problems in GR . Beside these successes of the conformal Weyl gravity, it is argued that this theory fails to fulfill simultaneously the observational constraints on the present cosmological parameters and on the primordial light element abundances [47]. For the static setup of the theory, it is shown that the de Sitter space can be found as the vacuum solution of this theory and there is no need to an ad hoc introduction of the cosmological constant to the gravitational action and its corresponding field equations [43]. Moreover, in the Newtonian limit of the theory, there are also linear modification terms to the exterior Schwarzschildde Sitter solutions that could explain the rotation curves of galaxies without any need to the mysterious dark matter [43, 48, 49]. Another argument about this theory is given in [50]. It was argued that the conformal Weyl gravity does not agree with the predictions of general relativity in the limit of weak fields and nonrelativistic velocities. Nevertheless, it was counterargued in [51], that in the presence of macroscopic long range scalar fields, the standard Schwarzschild phenomenology of GR is still recovered. In order to check the viability of the conformal Weyl gravity, the deflection of light and time delay in the exterior of a static spherically symmetric source were investigated in [52, 53], and it was found that the parameters of the theory fit the experimental constraints. Also, it is shown in [54] that one can find a class of wormhole geometries satisfying the energy conditions in the neighborhood of the throat of wormhole. This is in a clear contrast to the solutions in GR where exotic matter fields are needed to support a wormhole geometry.
In this work, regarding the above mentioned interesting features of the conformal Weyl gravity, we are motivated to study the issue of the past and future cosmological singularities in the context of this theory. Such a study can be also well motivated even one does not consider the Weyl gravity theory as an independent theory on its own right, but as the high energy limit alternative to GR where the being of singularities are important and inevitable issues. The organization of this paper is as follows. In Sect. 2, we review the conformal Weyl gravity theory and its cosmological field equations. In Sect. 3, we explore the existence and stability conditions for an ESU within the emergent universe scenario. In Sect. 4, we address the possible future finitetime cosmological singularities. Finally, in Sect. 5, we give our concluding remarks.
2 Conformal Weyl gravity theory
In the next section, using the cosmological field Eqs. (8), (9) and (10), we study the problem of initial big bang singularity which exists in the standard model of cosmology based on Einstein’s GR. We show that the conformal Weyl gravity admits the emergent universe scenario and then it can be free of the initial singularity problem.
3 Emergent universe paradigm: the initially nonsingular state and its stability analysis
As mentioned in the introduction, the emergent universe paradigm in GR is unstable against prevailing perturbations in very early universe and is expected to decay rapidly [9]. Therefore this scenario was developed in modified gravity theories with the hope to improve the stability at the high energy regimes. Thus, the conformal Weyl gravity as a successful theory in solving some of the fundamental problems in standard model of cosmology also deserves for studying the singularity problems in detail, which is the aim of this and the next sections.
The existence conditions for an ESU in the context of conformal Weyl gravity
\(\omega \) values  \(\lambda \) values  k values 

\(\omega >1/3\)  \(\lambda >0\)  \(k<0\) 
\(\lambda <0\)  \(k>0\)  
\(1<\omega <1/3\)  \(\lambda >0\)  \(k>0\) 
\(\lambda <0\)  \(k<0\)  
\(\omega <1\)  \(\lambda >0\)  \(k<0\) 
\(\lambda <0\)  \(k>0\) 
One notes that since the scale factor of the initial ESU \(a_{ES}\) has very small size, the corresponding density in (12) will be extremely high unless the scalar field S is very small too. In turn, this represents that the matter density \(\rho _{ES}\) can be arbitrary small too as the consequence of the smallness of S. This is one of the unique and interesting properties of the fourth order conformal Weyl gravity. The interpretation is that the Universe can born from nothing and any cosmological inhomogeneities such as those due to the large scale structures represent a perturbation around the vacuum state \(T_{\mu \nu }=0\), which has mentioned also by Mannheim [45]. Indeed, this is a direct result of the vanishing rank four Weyl tensor \(C_{\mu \nu \alpha \beta }\) and consequently the rank two gravitational tensor \(W_{\mu \nu }\) in the field equation of conformal Weyl gravity (4) for an FRW metric. There is no such an interesting property for an FRW universe in the context of the second order Einstein’s GR theory.
In the context of Weyl gravity there exists the possibility of having ESU for the ordinary matter fields possessing the equation of state \(\omega \ge 0\), and there is no need for exotic matter fields.
The case of \(\omega =1\), representing the cosmological constant field, does not correspond to an small size initial emergent ESU for the conformal Weyl gravity theory. Indeed, as the equation of state parameter \(\omega \) approaches to \(1\), we have \(a_{ES}\rightarrow \infty \). This fact has two meanings: (1) there is an extremely large static de Sitter limit for the theory, and (2) by approaching the equation of state parameter \(\omega \) to \(1\), possibly due to the perturbation in \(T_{\mu \nu }\) in very early universe [45], the initial small size nonsingular Einstein static state enlarges and enters to an inflationary phase which makes hint to a natural graceful exit mechanism. At the end of this section, we will discuss this case again.
There is no nonsingular spatially flat ESU for the conformal Weyl theory, i.e., for \(k=0\).
The case of \(\omega>1/3,~\lambda >0\) and \(k<0\) has a particular importance in the sense that it can provide a solution for the flatness problem in the standard model of cosmology, see [45].
The case \(\omega =1/3\) represents a singular state, i.e., \(a_{ES}=0\). Then, any perturbation around \(T_{\mu \nu }=0\) toward the equation of state \(\omega =1/3\) for matter fields yields an initially singular universe for the conformal Weyl gravity.
The conditions guaranteing both of the existence and stability of an ESU in the conformal Weyl gravity theory
\(\omega \) values  \(\lambda \) values  k values 

\(\omega >1/3\)  \(\lambda >0\)  \(k<0\) 
\(1<\omega <1/3\)  \(\lambda >0\)  \(k>0\) 
\(\omega <1\)  \(\lambda <0\)  \(k>0\) 
Until here, we have proved that the conformal Weyl gravity admits an stable ESU for the emergent universe scenario. As we addressed before, the matter density supporting this ESU can be arbitrary small, representing the possibility for being born from nothing [45]. Beside the possibility of being born from nothing for the universe, the ESU scenario in the conformal Weyl gravity has one another important theoretical advantage in comparison to the ESU in Einstein gravity as well as to some of the other modified gravity theories. This advantage is related to the fact that the de Sitter solution is a vacuum solution of the conformal Weyl theory. This vacuum solution is capable of triggering the inflation such that under the prevailing perturbations the universe can transit from the oscillations around its small size ESU to its subsequent inflationary phase and then produces whole of the cosmic history [43]. This advantage is supported by the fact that for both the FRW solution and the de Sitter solution (represented by \(R_{\mu \nu }=\Lambda g_{\mu \nu }\)), both of the Weyl tensor \(C_{\mu \nu \alpha \beta }\) and the gravitational tensor \(W_{\mu \nu }\) vanish. Then, as mentioned by Mannheim [45], the FRW cosmology is also the case that the energy momentum tensor is everywhere zero, and then it is another vacuum solution of the theory. Thus, any cosmological inhomogeneities such as those in the largescale structure of the universe is a consequence of a perturbation around \(T_{\mu \nu }=0\). In the presence of normal matter, such a property can be achieved nontrivially by an interplay between the various fields of the theory. In the conformal Weyl gravity, the perturbations around \(T_{\mu \nu }=0\) are responsible for both the generation the matter fields (\(\rho (t)\) and p(t)) associated to the inhomogeneities in the large scale structures and the transition from one nonsingular vacuum state to its subsequent another vacuum state deriving the inflation. This unique feature of having these two vacuum solutions in the conformal Weyl gravity may provides a natural mechanism for the graceful exit from the ESU to its subsequent inflationary phase and then to the radiation and matter dominated eras. We aim to study and elaborate on this important advantage of the conformal Weyl theory and report it elsewhere [55].
 (1)
An ever oscillating radiation dominated universe model with the finite minimum and maximum sizes.
 (2)
An eternally oscillating small size emergent universe standing as the seed of a large scale universe. In this case, the large scale universe can born and exit from this finite size state and then experience an inflationary phase by the prevailing perturbations around \(T_{\mu \nu }=0\).
Believing in the history of the universe having subsequent phase transitions from the radiation and dust to the dark energy dominated phases seems to be more in the favor of the above second interpretation. This is also supported by the current observations about the accelerating expansion of the universe in the sense that the repulsive dark energy profile makes unlikely the being of a contracting phase. Here, regarding the square roots in (18), one sees that to have the oscillatory modes around the scale factor \(a_0=\sqrt{\frac{k}{4\lambda S^2}}\), the conditions \(k<0\) and \(\lambda >0\) are required. These conditions are in agreement with our results for an emergent ESU given in Table 2. Also it is seen that \(\rho \) should satisfy the condition \(\rho _0<k^2/16\lambda \) which represents that the initial density of ESU state is finite for a finite and nonzero values of k and \(\lambda \) parameters, respectively.
4 The possible future singularities
In this section, following Barrow et al. [56, 57], we study the possible types of finitetime singularities may happen in the future of an FRW universe in the context of the conformal Weyl gravity. To do this, based on our study in the past section for the possibility of having an initially nonsingular cosmology in the context of this theory, we construct initially nonsingular scale factors and then investigate the behavior of \(H(t),~{\ddot{a}}(t),~\rho (t)\) and p(t) versus the cosmic time.
The possible finite time future singularities for an FRW cosmology
Singularity type  t  a(t)  \(\rho (t)\)  p(t) 

I  \(t\rightarrow t_s\)  \(a\rightarrow \infty \)  \(\rho \rightarrow \infty \)  \(\mid p\mid \rightarrow \infty \) 
II  \(t\rightarrow t_s\)  \(a\rightarrow a_s\)  \(\rho \rightarrow \rho _s\)  \(\mid p\mid \rightarrow \infty \) 
III  \(t\rightarrow t_s\)  \(a\rightarrow a_s\)  \(\rho \rightarrow \infty \)  \(\mid p\mid \rightarrow \infty \) 
IV  \(t\rightarrow t_s\)  \(a\rightarrow a_s\)  \(\rho \rightarrow 0\)  \(\mid p\mid \rightarrow 0\) 
The case of \(t \rightarrow t_s\) with \(0<n<1\) and \(0<q\le 1\).
For this case we have$$\begin{aligned}&a(t_s)\rightarrow a_s,~~\dot{a}(t_s)\rightarrow +\infty ,~~ H(t_s)\rightarrow \nonumber \\&\quad +\infty ,~~{\ddot{a}}(t_s)\rightarrow +\infty ,\end{aligned}$$(26)This case represents the type III singularity as in the table (3), or Big Freeze singularity. For this case, the evolution of the scale factor a(t), density \(\rho (t)\) and pressure p(t) are plotted in Fig. 3 for two typical values of q.$$\begin{aligned}&\rho (t_s) \rightarrow \infty ,~~ p(t_s)\rightarrow \infty . \end{aligned}$$(27)The case of \(t \rightarrow t_s\) with \(n=1\) and \(0<q\le 1\).
For this case we find$$\begin{aligned}&a(t_s)\rightarrow a_s, ~~\dot{a}(t_s)\rightarrow \dot{a}_s>0,~~ \nonumber \\&\quad H(t_s)\rightarrow H_s>0,~~{\ddot{a}}(t_s)\rightarrow {\ddot{a}}_s\le 0 \end{aligned}$$(28)where all the \(a_s,~\dot{a}_s,~H_s,~{\ddot{a}}_s,~\rho _s\) and \(p_s\) are finite. Then, there is no finitetime future singularity for this case. Here, the zero acceleration case happens for \(q=1\). Similarly, the evolution of the scale factor a(t), density \(\rho (t)\) and pressure p(t) are plotted in Fig. 4 for two typical values of q.$$\begin{aligned}&\rho (t_s) \rightarrow \rho _s<0,~~ p(t_s)\rightarrow p_s \end{aligned}$$(29)The case of \(t \rightarrow t_s\) with \(1<n<2\) and \(0<q\le 1\).
For this case we have$$\begin{aligned}&a(t_s)\rightarrow a_s,~~\dot{a}(t_s)\rightarrow \dot{a}_s>0,~~ H(t_s) \nonumber \\&\quad \rightarrow H_s>0,~~ {\ddot{a}}(t_s)\rightarrow \infty \end{aligned}$$(30)This case represents a type II singularity, i.e. a cosmological sudden singularity. For this case, the evolution of the scale factor a(t), density \(\rho (t)\) and pressure p(t) are plotted in Fig. 5 for typical values of q.$$\begin{aligned}&\rho (t_s) \rightarrow \rho _s <0,~~ p(t_s)\rightarrow \infty . \end{aligned}$$(31)The case of \(t \rightarrow t_s\) with \(n\ge 2\) and \(0<q\le 1\).
For this case we have$$\begin{aligned}&a(t_s)\rightarrow a_s,~~\dot{a}(t_s)\rightarrow \dot{a}_s>0,~~ H(t_s)\nonumber \\&\quad \rightarrow H_s>0,~~ {\ddot{a}}(t_s)\rightarrow {\ddot{a}}_s\le 0 \end{aligned}$$(32)where all \(a_s,~\dot{a}_s,~H_s,~{\ddot{a}}_s,~\rho _s\) and \(p_s\) are finite. Then, for this case all the physical quantities remain finite. The evolution of the scale factor a(t), density \(\rho (t)\) and pressure p(t) are plotted in Fig. 6 for two typical values of q.$$\begin{aligned}&\rho (t_s) \rightarrow \rho _s<0,~~ p(t_s)\rightarrow p_s. \end{aligned}$$(33)Then, we see that for an initially nonsingular universe which evolves according to (21), in general there are type II and III cosmological singularities in the context of conformal Weyl gravity. The type IV singularity, i.e \(t\rightarrow t_s,~a\rightarrow a_s,~\rho \rightarrow 0\) and \(p\rightarrow 0\), in the context of this theory with the scale factor (21) can be achieved for the very small values of the scalar field S values (\(S\rightarrow 0\)) such that \(S \dot{a}_s\rightarrow 0\) and \(S^2 {\ddot{a}}_s\rightarrow 0\). Even in the cases of divergent \(\dot{a}(t)\) and \({\ddot{a}}(t)\), the limiting values of \(S \dot{a}_s\rightarrow 0\) and \(S^2 {\ddot{a}}_s\rightarrow 0\) can be obtained as the result of \(0\times \infty \) type limits which require \(\frac{{\ddot{a}}(t)}{\dot{a}^2(t)}\) and \(\frac{\dddot{a}(t)}{{\ddot{a}}^2(t)}\) to be finite, respectively. Finally, for the scale factor (21), for \(0\le t\le t_s\) we have \(a_{ES}\le a(t)\le a_s\), then the big rip singularity defined as \(t\rightarrow t_s,~a\rightarrow \infty ,~\rho \rightarrow \infty \) and \(p\rightarrow \infty \) can not be achieved for a finite time. However, one may consider the scale factorfor a universe evolving from an initial nonsingular state \(a_{ES}\) where \(0\le t \le t_s\) and \(n\ge 1\). Then, through the field Eqs. (24) and (25), for \(t\rightarrow t_s\), we find \(a\rightarrow \infty ,~\rho \rightarrow \infty \) and \(p\rightarrow \infty \) representing a Big Rip type singularity. We conclude our analysis for the possible future cosmological singularities in the context of the conformal Weyl gravity in the following two points.$$\begin{aligned} a(t)=a_{ES}\left( 1+\frac{t}{t_s t} \right) ^n, \end{aligned}$$(34)
 (1)

In general, there are possibilities for having the types II, III and I cosmological singularities for an initially nonsingular universe evolving according to the scale factors (21) and (34), respectively. The type IV singularity requires that the scalar field S takes very small values and \(\frac{{\ddot{a}}(t)}{\dot{a}^2(t)}\) and \(\frac{\dddot{a}(t)}{{\ddot{a}}^2(t)}\) to be finite. Then, the final fate of the cosmos in the conformal Weyl gravity can be a sudden, big freeze, big rip or big brake singularity depending on the evolution of the scale factor a(t) and scalar field S value.
 (2)

The conformal Weyl gravity possesses one interesting property: the unique type of coupling the scalar field S to the dynamical quantities a(t), H(t), and \({\ddot{a}}(t)\). Indeed, these dynamical quantities are the origins of various type of future cosmological singularities. For the singularities that the divergencies in \(\rho (t)\) and p(t) are due to the divergencies in H and \({\ddot{a}}(t)\), one may consider the following finitetime finetunings: \(S\,H(t_s)\rightarrow finite\) and \(S\,{\ddot{a}}(t_s)\rightarrow finite\). Then, by these finetunings, the scalar field S can control and regularize the corresponding singularities and makes free the conformal theory from any singularity problem. The smallness of S as a result of being the FRW solution as the vacuum solution for the conformal Weyl theory guarantees the avoidance of finite time future singularities. There is no such a possibility in Einstein’s GR for a scalar field as the matter source of the field equations. Then, the conformal Weyl gravity can provide a cosmological model which is free of both the past and future singularities.
5 Conclusion
In the present work, we studied the issue of the past and future cosmological singularities in the context of the fourthorder conformal Weyl gravity. For the past singularity problem, we investigated the emergent universe scenario proposed by Ellis et al. We obtained the stability conditions for the corresponding ESU using the fixed point approach. We showed that depending on the values of the parameters of the conformal Weyl gravity theory, there are possibilities for having initially stable ESUs for an FRW universe with both the positive and negative spatial curvatures. In particular, it is found that there exists the possibility of having ESU for the ordinary matter fields possessing the equation of state \(\omega \ge 0\), but the conformal Weyl theory is not capable to have a nonsingular spatially flat ESU. In contrast to GR, here we found that the matter density of the initial ESU, i.e \(\rho _{ES}\), can be arbitrary small which represents the possibility of being born form nothing. Moreover, the fact that both the FRW solution and de Sitter solution are vacuum solutions of the conformal Weyl theory provides a unique feature for this theory in the sense that the initially nonsingular emergent universe can gracefully exit from one vacuum to the other vacuum solution. We discussed that by approaching the equation of state parameter \(\omega \) to \(1\), possibly due to the perturbation in \(T_{\mu \nu }=0\) in the early universe [45], the initial small size nonsingular ESU enlarges and enters to an inflationary phase. In the conformal Weyl theory, the perturbations around \(T_{\mu \nu }=0\) are responsible for both the generation of the matter fields (\(\rho (t)\) and p(t)) associated to the inhomogeneities in the large scale structures and for the transition from one nonsingular vacuum state to its subsequent another vacuum state deriving the inflation. This unique feature of having these two vacuum solutions in the conformal Weyl gravity may provides a natural mechanism for a graceful exit from the ESU to inflationary phase and then to its subsequent radiation and matter dominated eras. We aim to study and elaborate on this point and report such a possibility elsewhere [55].
Then, following Barrow et al., we addressed the possible types of the finitetime future cosmological singularities such as Big Rip, Sudden, Big Freeze and Big Brake singularities. We discussed that these future cosmological singularities can be the fate of the universe in the context of this theory in a general setup. However, the conformal Weyl gravity possesses one interesting property: the unique type of coupling of the scalar field S to the dynamical quantities a(t), H(t), and \({\ddot{a}}(t)\). Indeed, these dynamical quantities are the origins of various type of future cosmological singularities. For the singularities that the divergencies in \(\rho (t)\) and p(t) are due to the divergencies in H(t) and \({\ddot{a}}(t)\), by considering the finitetime finetunings \(S\,H(t_s)\rightarrow finite\) and \(S\,{\ddot{a}}(t_s)\rightarrow finite\), the scalar field S can control and regularize the corresponding singularities and make free the conformal Weyl theory from any singularity problem. The smallness of S as a result of being the FRW solution as the vacuum solution for the conformal Weyl theory guarantees the avoidance of finite time future singularities. There is no such a possibility in Einstein’s GR for a scalar field as the matter source of the field equations. Then, the conformal Weyl gravity can provide a cosmological model which is free of both the past and future singularities.
Footnotes
 1.
The differential Eq. (14) possesses the Jacobi elliptic function \(x_1=A\,cn(\omega t+ \delta )\) as its solution which satisfies the equations \(x_1^\prime =(1x_1^2)(1k_1^2 +k_1^2x_1^2)\) and \(x_1^{\prime \prime }=(12k_1^2)x_12k_1^2x_1^3\) where \(k_1\) is an arbitrary constant. In the case of \(k_1=0\), the solution reduces to the form \(x_1=A\cos (\omega t +\delta )\).
Notes
Acknowledgements
The author deeply thanks Prof. Metin Gürses for useful comments.
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