# Spherical Skyrmion black holes as gravitational lenses

## Abstract

In this article, we extend the strong deflection limit to calculate the deflection angle for a class of geometries which are asymptotically locally flat. In particular, we study the deflection of light in the surroundings of spherical black holes in Einstein–Skyrme theory. We find the deflection angle in this limit, from which we obtain the positions and the magnifications of the relativistic images. We compare our results with those corresponding to the Schwarzschild and the global monopole (Barriola–Vilenkin) spacetimes.

## 1 Introduction

The presence of supermassive black holes at the center of most galaxies, in particular the Milky Way [1] and the closest one M87 [2], has led to a growing interest in the optical effects in their neighborhood. It is believed that the observation of some of these effects, including direct imaging, will be possible in the near future [3, 4, 5]. Regarding gravitational lensing by compact objects possessing a photon sphere – like black holes –, besides the primary and secondary images, there exist two infinite sets of the denominated relativistic images [6], produced by light rays passing close to the photon sphere, then having large deflection angles. In this case, the deflection angle admits a logarithmic approximation dubbed the *strong deflection limit*, which allows for obtaining analytically the positions, the magnifications, and the time delays of the relativistic images. This approximate procedure was firstly introduced for the Schwarzschild black hole [7, 8, 9, 10, 11], extended to the Reissner–Nordström spacetime [12], and then generalized to any spherically symmetric and asymptotically flat geometries [13]. This method was recently simplified and improved [14]. The continued advances in gravitational lensing observations has lead to a growing interest in the analysis of lensing effects as a possible test of gravitational theories in the strong regime. Many analysis of strong deflection lenses have been considered in recent years [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] both within general relativity as well as in alternative theories of gravity. The lensing effects of rotating black holes have also been considered in the literature [48, 49, 50]. In this case, the deformation of the shadow is another related topic of great interest [51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63].

Within this context, it is relevant the study of strong deflection gravitational lensing when General Relativity is coupled with the action describing the strong interactions of baryons and mesons. Such an action corresponds to Skyrme’s theory [64, 65, 66] (detailed reviews are [67, 68, 69, 70]) which describes the low energy limit of QCD [71, 72, 73]. The dynamical variable of the Skyrme action is a scalar field *U* taking value in *SU*(*N*) (here we will consider the *SU*(2) case). The agreement of the theoretical predictions of Skyrme theory with experiments is very good (a partial list of relevant references is [67, 68, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82] and references therein). Due to these reasons, the Einstein–Skyrme system has attracted a lot of attention. In a series of seminal papers Droz, Heusler, and Straumann [83] (following the findings of Luckock and Moss [84]) constructed black hole solutions with a non-trivial Skyrme hair with a spherically symmetric ansatz. The issue of linear stability has also been analyzed in [85]. On the other hand, until very recently, there were basically no analytic solution in the Einstein–Skyrme system. Here we want to remark that the search for analytic configurations in the Skyrme and Einstein–Skyrme theories is not just of academic interest.^{1} Using some recent results on the generalization of the hedgehog ansatz [88, 89, 90, 91, 92, 93, 94, 95, 96] an analytic spherically symmetric black hole in the *SU*(2) Einstein–Skyrme theory has been constructed [97]. In this black hole, the effects of the Skyrme are manifest and so it offers the intriguing possibility to analyze a gravitational lens which includes the effects of strong interactions. Such a possibility will be explored in the present paper.

The article is organized as follows. In Sect. 2, we introduce the *U*(2) Einstein–Skyrme system and we present the spherically symmetric black hole solution. In Sect. 3, we extend the strong deflection limit for the deflection angle to a class of spherically symmetric spacetimes which are not necessarily asymptotically Minkowski. In Sect. 4, we obtain the angular positions and the magnifications of the relativistic images. In Sect. 5, we apply the method to the Skyrmion black hole. Finally, in Sect. 6, we analyze the results obtained. We adopt Planck units, so that \(G=c=\hslash =1\).

## 2 The *SU*(2) Einstein–Skyrme system

*SU*(2) Skyrme field is a

*SU*(2)-valued scalar field so that the Einstein–Skyrme action is described by

*e*are fixed by comparison with experimental data. The Skyrme fields satisfy the dominant energy condition [98].

*e*. Thus, all the values of the parameter

*e*in the above window can be considered as reasonable.

It is interesting to note that, when \(\Lambda =0\), the above spherical black hole in Einstein–Skyrme theory can be interpreted as the black hole of Barriola–Vilenkin type [99] (since \(f(r) \rightarrow 1-8\pi K<1\) when \(r\rightarrow \infty \)) but in which the Skyrme coupling \(\lambda \) gives an explicit contribution to *f*(*r*) of order \(1/r^{2}\). The role of this term will be apparent in the following analysis.

## 3 Deflection angle in the strong deflection limit

*asymptotically locally flat*. The most famous example of an asymptotically locally flat is the Barriola–Vilenkin metric [99] which describes the space-time of a global monopole. This means that we are interested in extending the strong deflection limit to the asymptotically locally flat scenario, i.e. in which the functions

*A*(

*r*) and \(B(r)^{-1}\) approach a positive constant when \(r\rightarrow \infty \), but this constant is not necessarily the number 1 as in the usual case of Minkowski asymptotics.

^{2}The radius of the event horizon \(r_{h}\) is given by the largest root of the function

*A*(

*r*). We assume in what follows that all the metric functions are positive and finite for \(r>r_{h}\). The photon sphere corresponds to an unstable circular orbit for massless particles. We define

*r*. We assume that the equation \(D(r)=0\) has at least a positive solution, being the radius of the photon sphere \(r_{m}\) the largest one of them.

*E*(energy) and

*L*(angular momentum), and the movement is confined to a plane, which can be taken with constant \(\vartheta = \pi /2\), without losing generality. By parameterizing the trajectory with an affine parameter, it is straightforward to verify that

*m*denotes evaluation in \(r=r_m\) in the corresponding functions. The integral \(I_R\) is defined by

*u*, the deflection angle in the strong deflection limit is given by

## 4 Relativistic images

*n*th image, which for the first set of relativistic images results

*n*th relativistic image is defined by the quotient of the solid angles subtended by the image and the source

*n*. All images are very faint because their magnifications are proportional to \((u_{m}/D_{ol})^2\).

*s*corresponds to the angular separation between the position of the first relativistic image and the limiting value of the others \(\theta _{\infty }\), and

*r*is the quotient between the flux of the first image and the flux coming from all the other images. For high alignment, these observables take the simple form [13]:

## 5 Application to the Skyrmion black hole

*M*, and the parameters of the model

*K*and \(\lambda \). The deflection angle in the strong deflection limit is univocally determined for the Skyrmion black hole lens by replacing Eqs. (76), (77), and (78) in Eq. (51). Once the black hole and light source positions are determined, the angular positions of the relativistic images and their magnifications can be found by using Eqs. (61), (62), and (65), while the observables defined in the previous section by Eqs. (66), (69), and (70). In the case that \(K=0\), the geometry (71) becomes Schwarzschild and the corresponding values \(u_m^{\mathrm {Schw}} = 3 \sqrt{3} M\), \(a_1^{\mathrm {Schw}}=1\), and \(a_2^{\mathrm {Schw}}=\ln [216(7-4\sqrt{3})]-\pi \) are recovered.

In Planck units, using Eq. (6), we have that \(K=3.33 \times 10^{-41}\) and \(0.0241\le K\lambda \le 0.0400\). The solar mass in these units is \(M_{\odot } = 9.14 \times 10^{37}\); then, for a black hole with \(M=10\ M_{\odot }\) we obtain \(2.87 \times 10^{-80} \le K \lambda /M^{2} \le 4.78\times 10^{-80}\), while for the supermassive Galactic black hole with \(M = 4 \times 10^{6}\ M_{\odot }\) we have that \(1.80\times 10^{-91}\le K\lambda /M^{2}\le 2.99\times 10^{-91}\). So, in these cases, our first order Taylor expansion above is justified. We see that for the Skyrmion black hole, the deviations of the strong deflection limit coefficients and observables from those corresponding to a Schwarzschild spacetime with the same mass in a possible astrophysical scenario are extremely small.

On the other hand, being the present spherical black hole asymptotically locally flat, it is reasonable to compare it with another asymptotically locally flat black hole, the obvious candidate being the Barriola–Vilenkin black hole [99]. This geometry can be recovered from Eq. (71) if we take \(\lambda =0\) and we identify *K* with the usual parameter \(\eta ^2\). With these replacements, the equations above provide the strong deflection limit for the Barriola–Vilenkin spacetime, which was previously studied in Ref. [45]. To give a precise estimate of the mass of the Barriola–Vilenkin black hole is not easy (and, indeed, there is no common agreement in the literature on this issue). However, a natural order of magnitude for the mass of a black hole whose “source” is a topological defect is around 10–100 TeV (which is the order of magnitude for the gravitating topological defects appearing in the standard model, see, for instance Ref. [101] and the references therein). In this case, the effects of the Skyrme term could become quite relevant compared with the Barriola–Vilenkin black hole. But the lensing distances for these small mass black holes should be very short, because the observables \(\theta _{\infty }\) and *s* are proportional to \(M/D_{ol}\).

## 6 Discussion

We have extended the strong deflection limit to a class of spherically symmetric spacetimes which are asymptotically locally flat. From this logarithmic approximation for the deflection angle, we have presented the analytical expressions for the positions and the magnifications of the relativistic images, and for three useful observables. Although some asymptotically locally flat geometries were analyzed by using the strong deflection limit (e.g. [45]), a systematic approach was missing in the literature.

We have applied the formalism to a spherically symmetric black hole solution of the *SU*(2) Einstein–Skyrme system. This model is of interest due to its close relation with the low energy limit of QCD. The metric possesses a solid angle deficit as the Barriola–Vilenkin black hole and a Reissner–Norsdröm like term with a fixed positive constant replacing the square of the charge. To the best of authors knowledge, this is the first analytic derivation of the strong deflection limit in Einstein–Skyrme theory. We have analytically obtained the strong deflection limit coefficients, the positions and the magnifications of the relativistic images in terms of them, as well as the standard observables.

We have also compared the strong deflection limit of the present spherical black hole in Einstein–Skyrme theory with those corresponding to the Schwarzschild and the Barriola–Vilenkin geometries. We have found that the deviations from the results corresponding to the Schwarzschild spacetime are extremely small (tens orders of magnitude) for a mass range from a few solar masses to supermassive objects, like the astrophysical black holes of interest. Consequently, the observation of these deviations is not expected with current or foreseeable future astronomical facilities. On the other hand, the deviations from the Barriola–Vilenkin black hole could be relevant when the black hole mass is of the typical order of magnitude of the masses of gravitating topological defects of the standard model [101]. But gravitational lensing in this case will require very short lensing distances, i.e. the presence of these very small size black holes under controlled conditions in a terrestrial laboratory. This is an intriguing possibility in view of the recent proposals on the production of black holes in particle accelerators [102].

## Footnotes

## Notes

### Acknowledgements

This work has been supported by Universidad de Buenos Aires and by CONICET (E.F.E. and C.M.S.), and by the Fondecyt Grant 1160137 (F.C.). The Centro de Estudios Científicos (CECS) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt (F.C.).

## References

- 1.S. Gillessen, P.M. Plewa, F. Eisenhauer, R. Sari, I. Waisberg, M. Habibi, O. Pfuhl, E. George, J. Dexter, S. von Fellenberg, T. Ott, R. Genzel, Astrophys. J.
**837**, 30 (2017)ADSCrossRefGoogle Scholar - 2.A.E. Broderick, R. Narayan, J. Kormendy, E.S. Perlman, M.J. Rieke, S.S. Doeleman, Astrophys. J.
**805**, 179 (2015)ADSCrossRefGoogle Scholar - 3.H. Falcke, S.B. Markoff, Class. Quantum Gravity
**30**, 244003 (2013)ADSCrossRefGoogle Scholar - 4.T. Johannsen, Class. Quantum Gravity
**33**, 113001 (2016)ADSCrossRefGoogle Scholar - 5.C. Bambi, Rev. Mod. Phys.
**89**, 025001 (2017)ADSMathSciNetCrossRefGoogle Scholar - 6.K.S. Virbhadra, G.F.R. Ellis, Phys. Rev. D
**62**, 084003 (2000)ADSMathSciNetCrossRefGoogle Scholar - 7.C. Darwin, Proc. R. Soc. Lond. A
**249**, 180 (1959)ADSMathSciNetCrossRefGoogle Scholar - 8.J.-P. Luminet, Astron. Astrophys.
**75**, 228 (1979)ADSGoogle Scholar - 9.H.C. Ohanian, Am. J. Phys.
**55**, 428 (1987)ADSCrossRefGoogle Scholar - 10.R.J. Nemiroff, Am. J. Phys.
**61**, 619 (1993)ADSCrossRefGoogle Scholar - 11.V. Bozza, S. Capozziello, G. Iovane, G. Scarpetta, Gen. Relativ. Gravit.
**33**, 1535 (2001)ADSCrossRefGoogle Scholar - 12.E.F. Eiroa, G.E. Romero, D.F. Torres, Phys. Rev. D
**66**, 024010 (2002)ADSMathSciNetCrossRefGoogle Scholar - 13.V. Bozza, Phys. Rev. D
**66**, 103001 (2002)ADSCrossRefGoogle Scholar - 14.N. Tsukamoto, Phys. Rev. D
**95**, 064035 (2017)ADSMathSciNetCrossRefGoogle Scholar - 15.K.S. Virbhadra, D. Narasimha, S.M. Chitre, Astron. Astrophys.
**337**, 1 (1998)ADSGoogle Scholar - 16.K.S. Virbhadra, G.F.R. Ellis, Phys. Rev. D
**65**, 103004 (2002)ADSMathSciNetCrossRefGoogle Scholar - 17.E.F. Eiroa, D.F. Torres, Phys. Rev. D
**69**, 063004 (2004)ADSCrossRefGoogle Scholar - 18.N. Tsukamoto, Y. Gong, Phys. Rev. D
**95**, 064034 (2017)ADSMathSciNetCrossRefGoogle Scholar - 19.V. Bozza, Phys. Rev. D
**78**, 103005 (2008)ADSCrossRefGoogle Scholar - 20.K.S. Virbhadra, C.R. Keeton, Phys. Rev. D
**77**, 124014 (2008)ADSCrossRefGoogle Scholar - 21.K.S. Virbhadra, Phys. Rev. D
**79**, 083004 (2009)ADSCrossRefGoogle Scholar - 22.A. Bhadra, Phys. Rev. D
**67**, 103009 (2003)ADSMathSciNetCrossRefGoogle Scholar - 23.E.F. Eiroa, Phys. Rev. D
**73**, 043002 (2006)ADSMathSciNetCrossRefGoogle Scholar - 24.N. Mukherjee, A.S. Majumdar, Gen. Relativ. Gravit.
**39**, 583 (2007)ADSCrossRefGoogle Scholar - 25.G.N. Gyulchev, S.S. Yazadjiev, Phys. Rev. D
**75**, 023006 (2007)ADSMathSciNetCrossRefGoogle Scholar - 26.S. Chen, J. Jing, Phys. Rev. D
**80**, 024036 (2009)ADSMathSciNetCrossRefGoogle Scholar - 27.Y. Liu, S. Chen, J. Jing, Phys. Rev. D
**81**, 124017 (2010)ADSCrossRefGoogle Scholar - 28.E.F. Eiroa, C.M. Sendra, Class. Quantum Gravity
**28**, 085008 (2011)ADSCrossRefGoogle Scholar - 29.E.F. Eiroa, Phys. Rev. D
**71**, 083010 (2005)ADSCrossRefGoogle Scholar - 30.R. Whisker, Phys. Rev. D
**71**, 064004 (2005)ADSCrossRefGoogle Scholar - 31.A.S. Majumdar, N. Mukherjee, Int. J. Mod. Phys. D
**14**, 1095 (2005)ADSCrossRefGoogle Scholar - 32.C.R. Keeton, A.O. Petters, Phys. Rev. D
**73**, 104032 (2006)ADSMathSciNetCrossRefGoogle Scholar - 33.S. Pal, S. Kar, Class. Quantum Gravity
**25**, 045003 (2008)ADSCrossRefGoogle Scholar - 34.A.Y. Bin-Nun, Phys. Rev. D
**81**, 123011 (2010)ADSCrossRefGoogle Scholar - 35.A.Y. Bin-Nun, Phys. Rev. D
**82**, 064009 (2010)ADSCrossRefGoogle Scholar - 36.E.F. Eiroa, C.M. Sendra, Phys. Rev. D
**86**, 083009 (2012)ADSCrossRefGoogle Scholar - 37.S. Chakraborty, S. SenGupta, J. Cosmol. Astropart. Phys.
**07**, 045 (2017)ADSCrossRefGoogle Scholar - 38.K. Sarkar, A. Bhadra, Class. Quantum Gravity
**23**, 6101 (2006)ADSCrossRefGoogle Scholar - 39.G.N. Gyulchev, I.Zh. Stefanov, Phys. Rev. D
**87**, 063005 (2013)ADSCrossRefGoogle Scholar - 40.C. Ding, C. Liu, Y. Xiao, L. Jiang, R.G. Cai, Phys. Rev. D
**88**, 104007 (2013)ADSCrossRefGoogle Scholar - 41.E.F. Eiroa, C.M. Sendra, Phys. Rev. D
**88**, 103007 (2013)ADSCrossRefGoogle Scholar - 42.M. Azreg-Aïnou, Phys. Rev. D
**87**, 024012 (2013)ADSCrossRefGoogle Scholar - 43.E.F. Eiroa, C.M. Sendra, Eur. Phys. J. C
**74**, 3171 (2014)ADSCrossRefGoogle Scholar - 44.J. Badía, E.F. Eiroa, Eur. Phys. J. C
**77**, 779 (2017)ADSCrossRefGoogle Scholar - 45.H. Cheng, J. Man, Class. Quantum Gravity
**28**, 015001 (2011)ADSCrossRefGoogle Scholar - 46.N. Tsukamoto, Phys. Rev. D
**94**, 124001 (2016)ADSMathSciNetCrossRefGoogle Scholar - 47.N. Tsukamoto, Phys. Rev. D
**95**, 084021 (2017)ADSMathSciNetCrossRefGoogle Scholar - 48.V. Bozza, F. De Luca, G. Scarpetta, Phys. Rev. D
**74**, 063001 (2006)ADSCrossRefGoogle Scholar - 49.V. Bozza, G. Scarpetta, Phys. Rev. D
**76**, 083008 (2007)ADSCrossRefGoogle Scholar - 50.G.V. Kraniotis, Class. Quantum Gravity
**28**, 085021 (2011)ADSCrossRefGoogle Scholar - 51.H. Falcke, F. Melia, E. Agol, Astrophys. J.
**528**, L13 (2000)ADSCrossRefGoogle Scholar - 52.K. Hioki, K.I. Maeda, Phys. Rev. D
**80**, 024042 (2009)ADSCrossRefGoogle Scholar - 53.C. Bambi, K. Freese, Phys. Rev. D
**79**, 043002 (2009)ADSMathSciNetCrossRefGoogle Scholar - 54.N. Tsukamoto, Z. Li, C. Bambi, J. Cosmol. Astropart. Phys.
**06**, 043 (2014)ADSCrossRefGoogle Scholar - 55.O.Y. Tsupko, Phys. Rev. D
**95**, 104058 (2017)ADSMathSciNetCrossRefGoogle Scholar - 56.V. Perlick, O.Y. Tsupko, Phys. Rev. D
**95**, 104003 (2017)ADSMathSciNetCrossRefGoogle Scholar - 57.P.V.P. Cunha, C.A.R. Herdeiro, Gen. Relativ. Gravit.
**50**, 42 (2018)ADSCrossRefGoogle Scholar - 58.L. Amarilla, E.F. Eiroa, G. Giribet, Phys. Rev. D
**81**, 124045 (2010)ADSMathSciNetCrossRefGoogle Scholar - 59.L. Amarilla, E.F. Eiroa, Phys. Rev. D
**85**, 064019 (2012)ADSCrossRefGoogle Scholar - 60.L. Amarilla, E.F. Eiroa, Phys. Rev. D
**87**, 044057 (2013)ADSCrossRefGoogle Scholar - 61.A. Grenzebach, V. Perlick, C. Lämmerzahl, Phys. Rev. D
**89**, 124004 (2014)ADSCrossRefGoogle Scholar - 62.E.F. Eiroa, C.M. Sendra, Eur. Phys. J. C
**78**, 91 (2018)ADSCrossRefGoogle Scholar - 63.N. Tsukamoto, Phys. Rev. D
**97**, 064021 (2018)ADSCrossRefGoogle Scholar - 64.T. Skyrme, Proc. R. Soc. Lond. A
**260**, 127 (1961)ADSMathSciNetCrossRefGoogle Scholar - 65.T. Skyrme, Proc. R. Soc. Lond. A
**262**, 237 (1961)ADSMathSciNetCrossRefGoogle Scholar - 66.T. Skyrme, Nucl. Phys. B
**31**, 556 (1962)MathSciNetCrossRefGoogle Scholar - 67.D.I. Olive, P.C. West (eds.),
*Duality and Supersymmetric Theories*(Cambridge University Press, Cambridge, 1999)zbMATHGoogle Scholar - 68.N. Manton, P. Sutcliffe,
*Topological Solitons*(Cambridge University Press, Cambridge, 2007)zbMATHGoogle Scholar - 69.H. Weigel,
*Chiral soliton models for baryons*.*Springer Lecture Notes in Physics*, vol. 743 (Springer, Berlin, 2008)Google Scholar - 70.M. Shifman, A. Yung,
*Supersymmetric Solitons*(Cambridge University Press, Cambridge, 2009)CrossRefGoogle Scholar - 71.E. Witten, Nucl. Phys. B
**223**, 422 (1983)ADSCrossRefGoogle Scholar - 72.E. Witten, Nucl. Phys. B
**223**, 433 (1983)ADSCrossRefGoogle Scholar - 73.A.P. Balachandran, V.P. Nair, N. Panchapakesan, S.G. Rajeev, Phys. Rev. D
**28**, 2830 (1983)ADSMathSciNetCrossRefGoogle Scholar - 74.A.P. Balachandran, F. Lizzi, V.G.J. Rodgers, A. Stern, Nucl. Phys. B
**256**, 525 (1985)ADSCrossRefGoogle Scholar - 75.A.P. Balachandran, H. Gomm, R.D. Sorkin, Nucl. Phys. B
**281**, 573 (1987)ADSCrossRefGoogle Scholar - 76.J. Myrheim, Anyons, in
*Topological Aspect of Low-dimensional Systems*ed. by A. Comtet, T. Jolicoeur, S. Ouvry, F. David. Les Houches, Session LXIX (Springer, Berlin, 1999)Google Scholar - 77.G.S. Adkins, C.R. Nappi, E. Witten, Nucl. Phys. B
**228**, 552 (1983)ADSCrossRefGoogle Scholar - 78.N.S. Manton, Phys. Lett. B
**110**, 54 (1982)ADSMathSciNetCrossRefGoogle Scholar - 79.E. Guadagnini, Nucl. Phys. B
**236**, 35 (1984)ADSCrossRefGoogle Scholar - 80.A.P. Balachandran, G. Marmo, B.S. Skagerstam, A. Stern,
*Classical Topology and Quantum States*(World Scientific, Singapore, 1991)CrossRefGoogle Scholar - 81.C.J. Houghton, N.S. Manton, P.M. Sutcliffe, Nucl. Phys. B
**510**, 507 (1998)ADSCrossRefGoogle Scholar - 82.Y.M. Cho, Phys. Rev. Lett.
**87**, 252001 (2001)ADSCrossRefGoogle Scholar - 83.S. Droz, M. Heusler, N. Straumann, Phys. Lett. B
**268**, 371 (1991)ADSMathSciNetCrossRefGoogle Scholar - 84.H. Luckock, I. Moss, Phys. Lett. B
**176**, 341 (1986)ADSCrossRefGoogle Scholar - 85.S. Droz, M. Heusler, N. Straumann, Phys. Lett. B
**271**, 61 (1991)ADSMathSciNetCrossRefGoogle Scholar - 86.P.D. Alvarez, F. Canfora, N. Dimakis, A. Paliathanasis, Phys. Lett. B
**773**, 401 (2017)ADSCrossRefGoogle Scholar - 87.L. Aviles, F. Canfora, N. Dimakis, D. Hidalgo, Phys. Rev. D
**96**, 125005 (2017)ADSCrossRefGoogle Scholar - 88.F. Canfora, Phys. Rev. D
**88**, 065028 (2013)ADSCrossRefGoogle Scholar - 89.F. Canfora, F. Correa, J. Zanelli, Phys. Rev. D
**90**, 085002 (2014)ADSCrossRefGoogle Scholar - 90.S. Chen, Y. Li, Y. Yang, Phys. Rev. D
**89**, 025007 (2014)ADSCrossRefGoogle Scholar - 91.S. Chen, Y. Yang, Nucl. Phys. B
**904**, 470 (2016)ADSCrossRefGoogle Scholar - 92.F. Canfora, M. Di Mauro, M.A. Kurkov, A. Naddeo, Eur. Phys. J. C
**75**, 443 (2015)ADSCrossRefGoogle Scholar - 93.E. Ayon-Beato, F. Canfora, J. Zanelli, Phys. Lett. B
**752**, 201 (2016)ADSMathSciNetCrossRefGoogle Scholar - 94.F. Canfora, N. Dimakis, A. Paliathanasis, Phys. Rev. D
**96**, 025021 (2017)ADSCrossRefGoogle Scholar - 95.A. Giacomini, M. Lagos, J. Oliva, A. Vera, Phys. Lett. B
**783**, 193 (2018)ADSCrossRefGoogle Scholar - 96.M. Astorino, F. Canfora, A. Giacomini, M. Ortaggio, Phys. Lett. B
**776**, 236 (2018)ADSMathSciNetCrossRefGoogle Scholar - 97.F. Canfora, H. Maeda, Phys. Rev. D
**87**, 084049 (2013)ADSCrossRefGoogle Scholar - 98.G.W. Gibbons, Phys. Lett. B
**566**, 171 (2003)ADSMathSciNetCrossRefGoogle Scholar - 99.M. Barriola, A. Vilenkin, Phys. Rev. Lett.
**63**, 341 (1989)ADSCrossRefGoogle Scholar - 100.S. Weinberg,
*Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity*(Wiley, New York, 1972)Google Scholar - 101.Y.M. Cho, K. Kimm, J.H. Yoon, Phys. Lett. B
**761**, 203 (2016)ADSCrossRefGoogle Scholar - 102.D.M. Eardley, S.B. Giddings, Phys. Rev. D
**66**, 044011 (2002)ADSMathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}