Skip to main content
SpringerLink
Log in

Propagation of Massive Scalar Fields in Pre-Big Bang Cosmologies

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We investigate the linear and semilinear massive Klein–Gordon equations in geometrical frameworks of type “Conformal Cyclic Cosmology” of R. Penrose, or “Singular Bouncing Scenario” as well. We give sufficient conditions on the decay of the mass to the fields be able to propagate across the Big-Bang.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. However when \({\hat{g}}\) is a solution of the Vacuum Einstein Equations \(R_{\mu \mu }-\frac{1}{2}Rg_{\mu \mu }+\Lambda g_{\mu \mu }=0\), we have \(R_{{\hat{g}}}=2\frac{n+1}{n-1}\Lambda \), and the case \((\xi =0,m^2=\frac{n+1}{2n}\Lambda )\) is equivalent to the conformal invariant massless case. In particular Friedrich in [13] studied a non-linear version of (1.9) as well as the coupling to the background, which in the present article is prescribed.

References

  1. Bachelot, A.: Wave asymptotics at a time cosmological singularity: classical and quantum scalar field. Commun. Math. Phys. 369, 973–1020 (2019)

    MATH  Google Scholar 

  2. Bachelot, A., Bachelot-Motet, A.: Waves on accelerating dodecahedral universes. Class. Quantum Grav. 34(5), 055010–39 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Battefeld, D., Peter, P.: A critical review of classical bouncing cosmologies. Phys. Rep. 571, 1–66 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  4. Beyer, F., LeFloch, P.G.: Second-order hyperbolic Fuchsian systems and applications. Class. Quantum Grav. 27, 245012 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  5. Beyer, F., Oliynyk, T. A., Olvera-Santamarìa, J. A.: The Fuchsian approach to global existence for hyperbolic equations (2019). arXiv:1907.04071

  6. Brandenberger, R., Peter, P.: Bouncing cosmologies: progress and problems. Found. Phys. 47(6), 797–850 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  7. Cagnac, F., Choquet-Bruhat, Y.: Solution globale d’une équation non linéaire sur une variété hyperbolique. J. Math. Pures Appl. (9) 63, 377–390 (1984)

    MathSciNet  MATH  Google Scholar 

  8. Camara da Silva, U., Alves Lima, A.L., Sotkov, G.M.: Scale factor duality for conformal cyclic cosmologies. JHEP 11, 090 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  9. Del Santo, D., Kinoshita, T., Reissig, M.: Klein-Gordon Type Equations with a Singular Time-dependent Potential. Rend. Istit. Mat. Univ. Trieste 39(11–12), 141–175 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Ebert, M.R., Nascimento, W.N.: A classification for wave models with time-dependent mass and speed of propagation. Adv. Differ. Equ. 23, 847–888 (2018)

    MATH  Google Scholar 

  11. Ebert, M.R., Reissig, M.: Regularity theory and global existence of small data solutions to semi-linear de Sitter models with power non-linearity. Nonlinear Anal.: Real World Appl. 40, 14–54 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Fournodavlos, G., Luk, J.: Asymptotically Kasner-like singularities. arXiv:2003.13591

  13. Friedrich, H.: Smooth non-zero rest-mass evolution across time-like infinity. Ann. Henri Poincaré 16, 2215–2238 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  14. Friedrich, H.: Sharp asymptotics for Einstein-\(\lambda \)-dust flows. Commun. Math. Phys. 350, 803–844 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  15. Galstian, A., Yagdjian, K.: Global solutions for semilinear Klein–Gordon equations in FLRW spacetimes. Nonlinear Anal. 113, 339–356 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Galstian, A., Yagdjian, K.: Global in time existence of self-interacting scalar field in de Sitter spacetimes. Nonlinear Anal.: Real World Appl. 34, 110–139 (2017)

    Article  MathSciNet  Google Scholar 

  17. Gasperini, M., Veneziano, G.: The Pre-Big Bang scenario in string cosmology. Phys. Rep. 373, 1–212 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  18. Joudioux, J.: Conformal scattering for a nonlinear wave equation. J. Hyperbolic Differ. Equ. 9(1), 1–65 (2012)

    Article  MathSciNet  Google Scholar 

  19. Klein, D., Reschke, J.: Pre-big Bang geometric extensions of inflationary cosmologies. Ann. Henri Poincaré 19(2), 565–606 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  20. Lübbe, C.: Conformal scalar fields, isotropic singularities and conformal cyclic cosmologies (2013). arXiv:1312.2059 [gr-qc]

  21. Meissner, K.A., Nurowski, P.: Conformal transformations and the beginning of the Universe. Phys. Rev. D 95(8), 084016 (2017)

    ADS  MathSciNet  Google Scholar 

  22. Nakamura, M.: The Cauchy problem for semi-linear Klein–Gordon equations in de Sitter spacetime. J. Math. Anal. Appl. 410, 445–454 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Newman, E.: A fundamental solution to the CCC equations. Gen. Rel. Grav. 46, 1717 (2014)

    ADS  MATH  Google Scholar 

  24. Nicolas, J.-P.: Non linear Klein–Gordon equation on Schwarzschild-like metrics. J. Math. Pures Appl. 74, 35–58 (1995)

    ADS  MathSciNet  MATH  Google Scholar 

  25. Nicolas, J.-P.: A nonlinear Klein–Gordon equation on Kerr metrics. J. Math. Pures Appl. 81(9), 885–914 (2002)

    MathSciNet  MATH  Google Scholar 

  26. Penrose, R.: Cycles of Time: An Extraordinary New View of the Universe. Bodley Head, London (2010)

    MATH  Google Scholar 

  27. Penrose, R.: The Big Bang and its dark-matter content: whence, whither, and wherefore. Found. Phys. 48, 1177–1190 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  28. Rendall, A.D.: Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity. Class. Quant. Grav. 17(16), 3305–3316 (2000)

    ADS  MathSciNet  MATH  Google Scholar 

  29. Ringström, H.: Future stability of the Einstein non-linear scalar field system. Invent. Math. 173, 123–208 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  30. Ringström, H.: On the Topology and Future Stability of the Universe. Oxford Mathematical Monographs. Oxford University Press, Oxford (2013)

    MATH  Google Scholar 

  31. Ringström, H.: Linear systems of wave equations on cosmological backgrounds with convergent asymptotics, preprint (2017). arXiv:1707.02803

  32. Ringström, H.: A unified approach to the Klein–Gordon equation on Bianchi backgrounds. Commun. Math. Phys. 372(2), 599–656 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  33. Rodnianski, I., Speck, J.: A regime of linear stability for the Einstein-scalar field system with applications to nonlinear big bang formation. Ann. Math. (2) 187(1), 65–156 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Rodnianski, I., Speck, J.: Stable big bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Selecta Math. (N.S.) 24(5), 4293–4459 (2018)

    MathSciNet  MATH  Google Scholar 

  35. Strauss, W.A.: On continuity of functions with values in various Banach spaces. Pac. J. Math. 19(3), 543–551 (1966)

    MathSciNet  MATH  Google Scholar 

  36. Teschl, G.: Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics, vol. 140. Amer. Math. Soc., Providence (2012)

    MATH  Google Scholar 

  37. Tod, P.: The equations of conformal cyclic cosmology, Gen. Relativ. Gravit. 47, no. 3, Art. 17 (2015)

  38. Vasy, A.: The wave equation on asymptotically de Sitter-like spaces. Adv. Math. 223, 49–97 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author wants to thank the anonymous referees for their valuable comments for improving the manuscript.

Author information

Authors and Affiliations

  1. Université de Bordeaux, Institut de Mathématiques, UMR CNRS 5251, 33405, Talence Cedex, France

    Alain Bachelot

Authors
  1. Alain Bachelot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alain Bachelot.

Additional information

Communicated by P. Chrusciel

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bachelot, A. Propagation of Massive Scalar Fields in Pre-Big Bang Cosmologies. Commun. Math. Phys. 380, 973–1001 (2020). https://doi.org/10.1007/s00220-020-03880-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-020-03880-4

Search

Navigation