# Inflation and acceleration of the universe by nonlinear magnetic monopole fields

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## Abstract

Despite impressive phenomenological success, cosmological models are incomplete without an understanding of what happened at the big bang singularity. Maxwell electrodynamics, considered as a source of the classical Einstein field equations, leads to the singular isotropic Friedmann solutions. In the context of Friedmann–Robertson–Walker (FRW) spacetime, we show that singular behavior does not occur for a class of nonlinear generalizations of the electromagnetic theory for strong fields. A new mathematical model is proposed for which the analytical nonsingular extension of FRW solutions is obtained by using the nonlinear magnetic monopole fields.

## 1 Introduction

Cosmology has experienced remarkable advances in recent decades as a consequence of tandem observations of type-Ia supernovae and the cosmic microwave background. These observations suggest that cosmological expansion is accelerating [1]. The last two decades have witnessed enormous progress in our understanding of the source of this accelerated expansion. Furthermore, standard cosmology assumes that at the beginning, there must have been an initial singularity – a breakdown in the geometric structure of space and time – from which spacetime suddenly started evolving [2]. The standard cosmological model, with the source of Maxwell electrodynamics based on the Friedmann–Robertson–Walker (FRW) geometry, leads to a cosmological singularity at a finite time in the past. In order to solve this puzzle, researchers have proposed many different mechanisms in the literature, such as nonminimal couplings, a cosmological constant, nonlinear Lagrangians with quadratic terms in the curvature, scalar inflation fields, modified gravity theories, quantum gravity effects, and nonlinear electrodynamics without modification of general relativity [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].

One possible solution is to explore the evolution while avoiding the cosmic initial singularity contained in a given nonlinear effect of electromagnetic theory [10, 13, 16]. 1934, the nonlinear electrodynamics Lagrangian known as the Born–Infeld Lagrangian was published by the physicists Max Born and Leopold Infeld [28]. This Lagrangian has the amusing feature of turning into Maxwell theory for low electromagnetic fields; moreover, the nonlinear Lagrangian is invariant under the duality transformation.

To solve the initial singularity problem, the early stages of the universe are assumed to be dominated by the radiation of nonlinear modifications of Maxwell’s equations, which include a large amount of electromagnetic and gravitational fields. This is true inasmuch as strong magnetic fields in the early universe can cause deviations from linear electrodynamics to nonlinear electrodynamics [6, 7]. By following recently published procedures [16, 17], in this paper the nonlinear magnetic monopole (NMM) fields are used to show the source of the acceleration of the universe without an initial singularity.

In this paper, we investigate a cosmological model of the universe with NMM fields coupled to gravity. The structure of the paper is as follows: In Sect. 1, we briefly introduce NMM fields and consider the universe to be filled by pure nonlinear magnetic fields. In Sect. 2, we show that the universe accelerates without an initial singularity until it reaches the critical value of the scale factor. In Sect. 3, we check the classical stability of the universe under the deceleration phase. In Sect. 4, we report our conclusions.

## 2 Nonlinear magnetic monopole fields and a nonsingular FRW universe

*R*is the Ricci scalar, and \(\alpha \) is the fine-tuning parameter of \({\mathcal {L}}_{EM}\) Maxwell fields. \({\mathcal {L}}_{NMM}\) is the Lagrangian of the NMM fields. From a conceptual point of view, this action has the advantage that it does not invoke any unobserved entities such as scalar fields, higher dimensions, or brane worlds. Furthermore, we can ignore the Maxwell fields (\(\alpha =0\)), because they are weak compared to the dominant NMM fields in the very early epochs and inflation. However, in the literature there are many proposals of cosmological solutions based on the Maxwell fields plus corrections [11, 12, 16, 17, 24, 25, 26]. Herein, our main aim is to use this method to show that it yields an accelerated expansion phase for the evolution of the universe in the NMM field regime. The new ingredient we add is a modification of the electrodynamics, which has no Maxwell limit. The Einstein field equation and the NMM field equation are derived from the action

*P*is the magnetic monopole charge. Furthermore, it is noted that in the weak field limit the NMM Lagrangian does not yield the linear Maxwell Lagrangian [31]. In this work, following a standard procedure, we consider the pure magnetic field under the following NMM field Lagrangian suggested in Ref. [31]:

*l*are the positive constants. The constant parameter \(\beta \) will be fixed according to other parameters. The NMM field Lagrangian is folded into the homogeneous and isotropic FRW spacetime

*a*is a scale factor, to investigate the effects on the acceleration of the universe.

*p*by varying the action as follows:

*a*denotes the derivatives with respect to the cosmic time. The most important condition for the accelerated universe is \(\rho +3p<0\). Here, the NMM field is used as the main source of gravity. Using Eqs. (9) and (10), it is found that

*p*from Eqs. (12) and (13), and integrating, the evolution of the magnetic field under the change of the scale factor is obtained as follows:

*p*, but there is no singularity point at \(a(t)\rightarrow 0\) and \(a(t)\rightarrow \infty \). Hence, one finds that, as shown in Fig. 1,

*p*(\(\rho =-p)\) at the beginning of the universe (\(a=0\)), similarly to a model of the \( \Lambda \)CDM. The absence of singularities is also shown in the literature [16, 17] by using a different model of nonlinear electrodynamics.

## 3 Evolution of the universe

### 3.1 A test of causality with speed of the sound

## 4 Conclusion

In this paper, we used the model of NMM fields with parameters \(\beta \) and *l* for the sources of the gravitational field. This model is not scale-invariant because of the free parameters \(\beta \) and *l*, so the energy-momentum tensor is not zero. We consider the universe to be magnetic and to accelerate with the help of NMM field sources. After the inflation period, it was shown that the universe is homogeneous and isotropic. The acceleration of the universe is bounded at \(a(t)<a_{c}(t)= \frac{1}{2^{7/12}}\frac{\sqrt{B_{0}}}{\root 4 \of {\beta }}\).

We also showed that, at the time of the Big Bang, there was no singularity in the energy density, pressure, or curvature terms. After some time, the universe approaches flat spacetime. We checked causality and found that it satisfies the classical stability where the speed of sound should be less than the local light speed. Hence, nonlinear sources, such as NMM fields at the early regime of the universe, allow accelerated expansion with inflation and without dark energy. This model of NMM fields can also be used to describe the evolution of the universe. We noted that at the weak NMM field, there is no Maxwell’s limit, so the inflation and the acceleration of the universe can be analyzed by using different types of fields. We manage to smooth the singularity of the magnetic universe by using only the NMM fields that were strong in the accelerated phase of the universe. In our model, in the early regime of the universe, NMM fields are very strong, making the effects of the usual electromagnetic fields negligible. We leave for a future publication the use of NMM fields with the usual Maxwell fields, and scalar fields, to investigate this problem more deeply. Another future project is to find the relationship between the different types of NMM fields and the possible existence of wormholes in the universe and their effect of Hawking radiation in relation to our previous work [35, 36, 37, 38].

## Notes

### Acknowledgements

The author would like to thank Prof. Dr. Mustafa Halilsoy for reading the manuscript and giving valuable suggestions. The author is grateful to the editor and anonymous referees for their valuable and constructive suggestions.

### References

- 1.S. Capoziello, V. Faraoni,
*Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics*(Springer Science+Business Media B.V., New York, 2011)MATHGoogle Scholar - 2.B. Craps, arXiv:1001.4367 [hep-th]
- 3.F.V. Mukhanov et al., Phys. Rev. Lett.
**68**, 1969–1972 (1992)ADSMathSciNetCrossRefGoogle Scholar - 4.E.N. Saridakis, M. Tsoukalas, Phys. Rev. D
**93**, 124032 (2016)Google Scholar - 5.H. Sheikhahmadi, E.N. Saridakis, A. Aghamohammadi, K. Saaidi, JCAP
**1610**(10), 021 (2016)Google Scholar - 6.Garcia-Salcedo, Ricardo et al., Int. J. Mod. Phys. A
**15**, 4341–4354 (2000)Google Scholar - 7.C.S. Camara et al., Phys. Rev. D
**69**, 123504 (2004)ADSMathSciNetCrossRefGoogle Scholar - 8.E. Elizalde et al., Phys. Lett. B
**574**, 1–7 (2003)ADSCrossRefGoogle Scholar - 9.C. Quercellini, M. Bruni, A. Balbi, D. Pietrobon, Phys. Rev. D
**78**, 063527 (2008)ADSCrossRefGoogle Scholar - 10.V.A. De Lorenci, R. Klippert, M. Novello, J.M. Salim, Phys. Rev. D
**65**, 063501 (2002)ADSCrossRefGoogle Scholar - 11.M. Novello et al., Phys. Rev. D
**69**, 127301 (2004)ADSCrossRefGoogle Scholar - 12.M. Novello et al., Class. Quantum Gravity
**24**, 3021–3036 (2007)ADSCrossRefGoogle Scholar - 13.D.N. Vollick, Phys. Rev. D
**78**, 063524 (2008)ADSMathSciNetCrossRefGoogle Scholar - 14.O. Akarsu, T. Dereli, Int. J. Theor. Phys.
**51**, 612–621 (2012)CrossRefGoogle Scholar - 15.A.W. Beckwith, J. Phys. Conf. Ser.
**626**(1), 012058 (2015)CrossRefGoogle Scholar - 16.S.I. Kruglov, Phys. Rev. D
**92**(12), 123523 (2015)ADSMathSciNetCrossRefGoogle Scholar - 17.S.I. Kruglov, Int. J. Mod. Phys. D
**25**(4), 1640002 (2016)ADSMathSciNetCrossRefGoogle Scholar - 18.G. Gecim, Y. Sucu, Adv. High Energy Phys.
**9**, 2056131 (2017)Google Scholar - 19.Y. Cai, E.N. Saridakis, M.R. Setare, J. Xia, Phys. Rep.
**493**, 1–60 (2010)ADSMathSciNetCrossRefGoogle Scholar - 20.E.N. Saridakis, S.V. Sushkov, Phys. Rev. D
**81**, 083510 (2010)ADSCrossRefGoogle Scholar - 21.E.N. Saridakis, Eur. Phys. J. C
**67**, 229–235 (2010)ADSCrossRefGoogle Scholar - 22.O. Akarsu, T. Dereli, N. Oflaz, Class. Quantum Gravity
**32**(21), 215009 (2015)ADSCrossRefGoogle Scholar - 23.N. Breton, R. Lazkoz, A. Montiel, JCAP
**1210**, 013 (2012)ADSCrossRefGoogle Scholar - 24.M. Novello, S.E.P. Bergliaffa, Phys. Rep.
**463**, 127–213 (2008)ADSMathSciNetCrossRefGoogle Scholar - 25.V.F. Antunes, M. Novello, Gravit. Cosmol.
**22**(1), 1–9 (2016)ADSMathSciNetCrossRefGoogle Scholar - 26.E. Bittencourt, U. Moschella, M. Novello, J.D. Toniato, Phys. Rev. D
**90**(12), 123540 (2014)ADSCrossRefGoogle Scholar - 27.O. Akarsu, T. Dereli, N. Katirci, M.B. Sheftel, Gen. Relativ. Gravit.
**47**(5), 61 (2015)ADSCrossRefGoogle Scholar - 28.M. Born, L. Infeld, Proc. R. Soc. Lond. A
**144**, 425–451 (1934)ADSCrossRefGoogle Scholar - 29.R. Durrer, A. Neronov, Astron. Astrophys. Rev.
**21**, 62 (2013)ADSCrossRefGoogle Scholar - 30.K.E. Kunze, Plasma Phys. Control. Fusion
**55**, 124026 (2013)ADSCrossRefGoogle Scholar - 31.M. Halilsoy, A. Ovgun, S. Habib Mazharimousavi, Eur. Phys. J. C
**74**, 2796 (2014)Google Scholar - 32.R. Tolman, P. Ehrenfest, Phys. Rev.
**36**(12), 1791 (1930)Google Scholar - 33.A. Balbi, EPJ Web Conf.
**58**, 02004 (2013)CrossRefGoogle Scholar - 34.R. Garcia-Salcedo, T. Gonzalez, I. Quiros, Phys. Rev. D
**89**(8), 084047 (2014)ADSCrossRefGoogle Scholar - 35.A. Ovgun, M. Halilsoy, Astrophys Space Sci.
**361**, 214 (2016)Google Scholar - 36.I. Sakalli, A. Ovgun, Eur. Phys. J. Plus
**131**, 184 (2016)CrossRefGoogle Scholar - 37.A. Ovgun, I. Sakalli, I. Theor. Math. Phys.
**190**, 120 (2017)Google Scholar - 38.A. Ovgun, Int. J. Theor. Phys.
**55**, 2919 (2016)CrossRefGoogle Scholar

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