At the Frontier of Spacetime pp 133-157 | Cite as

# The Lambda-CDM Model Is Not an Universal Attractor of the Brans–Dicke Cosmology

## Abstract

By means of the tools of the dynamical systems theory it is shown that the general relativity de Sitter solution is an attractor of the Jordan frame (dilatonic) Brans–Dicke theory only for the exponential potential \(U(\varphi )\propto \exp \varphi \), which corresponds to the quadratic potential \(V(\phi )\propto \phi ^2\) in terms of the original Brans–Dicke field \(\phi =\exp \varphi \), or for potentials which approach to \(\exp \varphi \) at the stable point. I find bounds on the Brans–Dicke coupling constant \(\omega _\textsc {bd}\), which are consistent with well-known results.

## Keywords

Cold Dark Matter Jordan Frame Sitter Solution Exponential Potential Cold Dark Matter Model## Notes

### Acknowledgments

I want to thank T Asselmeyer-Maluga for inviting me to join this initiative to celebrate the 80th birthday of one of the greatest minds of the XX century, our dear fiend Carl H. Brans, to whom I am profoundly indebted. I want to acknowledge, also, J D Barrow, S D Odintsov and A Alho, for pointing to me several indispensable bibliographic references. Thanks are due to CONACyT of México for support of this research. I am grateful to SNI-CONACyT for continuous support of my research activity.

## References

- 1.C. Brans, R.H. Dicke, Phys. Rev.
**124**, 925–935 (1961)ADSMathSciNetCrossRefGoogle Scholar - 2.
- 3.Y. Fujii, K.-I. Maeda,
*The Scalar-Tensor Theory of Gravitation*(Cambridge University Press, UK, 2003)zbMATHGoogle Scholar - 4.V. Faraoni,
*Cosmology in Scalar-Tensor Gravity*(Kluwer Academic Publishers, The Netherlands, 2004)CrossRefzbMATHGoogle Scholar - 5.J. Khoury, A. Weltman, Phys. Rev. Lett.
**93**, 171104 (2004), arXiv:astro-ph/0309300; Phys. Rev. D**69**, 044026 (2004), arXiv:astro-ph/0309411 - 6.S. Sen, T.R. Seshadri, Int. J. Mod. Phys. D
**12**, 445–460 (2003), arXiv:gr-qc/0007079; D.F. Torres, Phys. Rev. D**66**, 043522 (2002), arXiv:astro-ph/0204504; N. Banerjee, D. Pavon, Class. Quant. Grav.**18**, 593 (2001), arXiv:gr-qc/0012098 - 7.P.A.R. Ade et al., Planck Collaboration. Astron. Astrophys.
**571**, A23 (2014), arXiv:1303.5083 - 8.P.J.E. Peebles, B. Ratra, Rev. Mod. Phys.
**75**, 559–606 (2003), arXiv:astro-ph/0207347; V. Sahni, A.A. Starobinsky. Int. J. Mod. Phys. D**9**, 373–444 (2000), arXiv:astro-ph/9904398 Google Scholar - 9.O. Hrycyna, M. Szydlowski, Phys. Rev. D
**88**(6), 064018 (2013), arXiv:1304.3300 - 10.O. Hrycyna, M. Kamionka, M. Szydlowski, Phys. Rev. D
**90**(12), 124040 (2014), arXiv:1404.7112 - 11.O. Hrycyna, M. Szydlowski, JCAP
**1312**, 016 (2013), arXiv:1310.1961 - 12.J.D. Barrow, K. Maeda, Nucl. Phys. B
**341**, 294–308 (1990)ADSCrossRefGoogle Scholar - 13.J.D. Barrow, Phys. Rev. D
**51**, 2729–2732 (1995)ADSCrossRefGoogle Scholar - 14.E. Elizalde, S. Nojiri, S.D. Odintsov, D. Saez-Gomez, V. Faraoni, Phys. Rev. D
**77**, 106005 (2008), arXiv:0803.1311 - 15.C.M. Will, Living Rev. Rel.
**9**, 3 (2006), arXiv:gr-qc/0510072; B. Bertotti, L. Iess, P. Tortora, Nature**425**, 374 (2003) - 16.I. Quiros, R. García-Salcedo, T. Gonzalez, F.A. Horta-Rangel, Phys. Rev. D
**92**(4), 044055 (2015), arXiv:1506.05420 - 17.V. Acquaviva, C. Baccigalupi, S.M. Leach, A.R. Liddle, F. Perrotta, Phys. Rev. D
**71**, 104025 (2005), arXiv:astro-ph/0412052 - 18.R. Nagata, T. Chiba, N. Sugiyama, Phys. Rev. D
**69**, 083512 (2004), arXiv:astro-ph/0311274 - 19.J.E. Lidsey, D. Wands, E.J. Copeland, Phys. Rept.
**337**, 343–492 (2000), arXiv:hep-th/9909061 - 20.E.J. Copeland, A.R. Liddle, D. Wands, Phys. Rev. D
**57**, 4686 (1998), arXiv:gr-qc/9711068 Google Scholar - 21.A.A. Coley, arXiv:gr-qc/9910074
- 22.E.J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D
**15**, 1753–1936 (2006), arXiv:hep-th/0603057 Google Scholar - 23.L.A. Urena-Lopez, JCAP
**1203**, 035 (2012), arXiv:1108.4712 - 24.C.G. Boehmer, N. Chan, arXiv:1409.5585
- 25.R. García-Salcedo, T. Gonzalez, F.A. Horta-Rangel, I. Quiros, D. Sanchez-Guzmán, Eur. J. Phys.
**36**(2), 025008 (2015), arXiv:1501.04851 - 26.V.I. Arnold, Ordinary Differential Equations, translated from Russian and edited by R.A. Silverman (The MIT Press, Cambridge, Massachusets and London, England, 1973); D.K. Arrowsmith, C.M. Place, Introduction to Dynamical Systems (Cambridge University Press, UK, 1990); L. Perko, Differential Equations and Dynamical Systems (Springer, USA, 2001)Google Scholar
- 27.A. Cid, G. Leon, Y. Leyva, arXiv:1506.00186
- 28.J.-P. Uzan, Rev. Mod. Phys.
**75**, 403 (2003), arXiv:hep-ph/0205340 - 29.F. Wu, X. Chen, Phys. Rev. D
**82**, 083003 (2010), arXiv:0903.0385 - 30.F.S. Accetta, L.M. Krauss, P. Romanelli, Phys. Lett. B
**248**, 146 (1990)ADSCrossRefGoogle Scholar - 31.X. Chen, M. Kamionkowski, Phys. Rev. D
**60**, 104036 (1999), arXiv:astro-ph/9905368 - 32.V. Faraoni, Class. Quant. Grav.
**26**, 145014 (2009), arXiv:0906.1901 - 33.J. Wainwright, G.F.R. Ellis, Dynamical Systems in Cosmology (Cambridge University Press, Cambridge, 1997); A.A. Coley, Dynamical Systems and Cosmology (Kluwer Academic Publishers, Dordrecht Boston London, 2003)Google Scholar