Abstract
We study the Einstein-scalar field system with positive cosmological constant and spherically symmetric characteristic initial data given on a truncated null cone. We prove well-posedness, global existence and exponential decay in (Bondi) time, for small data. From this, it follows that initial data close enough to de Sitter data evolves to a causally geodesically complete spacetime (with boundary), which approaches a region of de Sitter asymptotically at an exponential rate; this is a non-linear stability result for de Sitter within the class under consideration, as well as a realization of the cosmic no-hair conjecture.
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Anderson M.T.: Existence and stability of even dimensional asymptotically de Sitter spaces. Annales Henri Poincaré 6, 801–820 (2005) arXiv:gr-qc/0408072
Beyer F.: Non-genericity of the Nariai solutions: I. Asymptotics and spatially homogeneous perturbations. Class. Quantum Grav. 26, 235015 (2009) arXiv:0902.2531
Beyer F.: Non-genericity of the Nariai solutions: II. Investigations within the Gowdy class. Class. Quantum Grav. 26, 235016 (2009) arXiv:0902.2532
Beyer, F.: The cosmic no-hair conjecture: a study of the Nariai solutions. In: Ruffini, R., Damour, T., Jantzen, R.T. (eds.) Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity (2009). arXiv:1012.0056
Brady P., Chambers C., Krivan W., Laguna P.: Telling tails in the presence of a cosmological constant. Phys. Rev. D 55, 7538–7545 (1997) arXiv:gr-qc/9611056
Costa J.L., Alho A., Natário J.: Spherical linear waves in de Sitter spacetime. J. Math. Phys. 53, 024001 (2012) arXiv:1107.0802
Choptuik M.: Universality and scaling in gravitational collapse of a massless scalar field. Phys. Rev. Lett. 70, 9–12 (1993)
Christodoulou D.: The problem of a self-gravitating scalar field. Commun. Math. Phys. 105, 337–361 (1986)
Christodoulou D.: The formation of black holes and singularities in spherically symmetric gravitational collapse. Commun. Pure Appl. Math. 44, 339–373 (1991)
Christodoulou, D.: The formation of black holes in general relativity. EMS Monographs in Mathematics, European Mathematical Society, Zürich (2009). arXiv:0805.3880
Friedrich H.: On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure. Commun. Math. Phys. 107, 587–609 (1986)
Gundlach, C., Martín-García, J.: Critical phenomena in gravitational collapse. Living Rev. Relativ. 10 (2007). http://www.livingreviews.org/lrr-2007-5
Lübbe, C., Valiente Kroon, J.: A conformal approach for the analysis of the non-linear stability of pure radiation cosmologies. arXiv:1111.4691
Maeda H., Nozawa M.: Generalized Misner-Sharp quasi-local mass in Einstein–Gauss–Bonnet gravity. Phys. Rev. D77, 064031 (2008) arXiv:0709.1199
Nakao, K.: On a quasilocal energy outside the cosmological horizon. arXiv:gr-qc/9507022
Rendall A.D.: Asymptotics of solutions of the Einstein equations with positive cosmological constant. Annales Henri Poincare 5, 1041–1064 (2004) arXiv:gr-qc/0312020
Ringstrom H.: Future stability of the Einstein-non-linear scalar field system. Invent. Math. 173, 123–208 (2008)
Rodnianski, I., Speck, J.: The stability of the irrotational Euler-Einstein system with a positive cosmological constant. arXiv:0911.5501
Speck, J.: The nonlinear future-stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant. Accepted for publication in Selecta Mathematica. arXiv:1102.1501
Tchapnda N., Sophonie B., Noutchegueme N.: The surface-symmetric Einstein-Vlasov system with cosmological constant. Math. Proc. Camb. Phil. Soc. 138, 541–553 (2005) arXiv:gr-qc/0304098
Tchapnda N., Sophonie B., Rendall A.D.: Global existence and asymptotic behavior in the future for the Einstein-Vlasov system with positive cosmological constant. Class. Quant. Grav. 20, 3037–3049 (2003) arXiv:gr-qc/0305059
Wald Robert M.: Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant. Phys. Rev. D28, 2118–2120 (1983)
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Communicated by Piotr T. Chrusciel.
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Costa, J.L., Alho, A. & Natário, J. The Problem of a Self-Gravitating Scalar Field with Positive Cosmological Constant. Ann. Henri Poincaré 14, 1077–1107 (2013). https://doi.org/10.1007/s00023-012-0215-7
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DOI: https://doi.org/10.1007/s00023-012-0215-7