Abstract
We consider the Einstein-dust equations with positive cosmological constant \({\lambda}\) on manifolds with time slices diffeomorphic to an orientable, compact 3-manifold \({S}\). It is shown that the set of standard Cauchy data for the Einstein-\({\lambda}\)-dust equations on \({S}\) contains an open (in terms of suitable Sobolev norms) subset of data which develop into solutions that admit at future time-like infinity a space-like conformal boundary \({{\mathcal J}^+}\) that is \({C^{\infty}}\) if the data are of class \({C^{\infty}}\) and of correspondingly lower smoothness otherwise. The class of solutions considered here comprises non-linear perturbations of FLRW solutions as very special cases. It can conveniently be characterized in terms of asymptotic end data induced on \({{\mathcal J}^+}\). These data must only satisfy a linear differential equation. If the energy density is everywhere positive they can be constructed without solving differential equations at all.
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Bartnik, R., Isenberg, J.: The constraint equations. In: Chruściel, P.T., Friedrich, H. (Eds.) The Einstein equations and the large scale behaviour of gravitational fields. Birkhäuser, Basel (2004)
Beyer F.: Investigations of solutions of Einstein’s field equations close to lambda-Taub-NUT. Class. Quantum Gravity 25, 235005 (2008)
Friedrich H.: On the hyperbolicity of Einstein’s and other gauge field equations. Commun. Math. Phys. 100, 525–543 (1985)
Friedrich H.: Existence and structure of past asymptotically simple solution of Einstein’s field equations with positive cosmological constant. J. Geom. Phys. 3, 101–117 (1986)
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)
Friedrich H.: On the global existence and the asymptotic behaviour of solutions to the Einstein-Maxwell-Yang-Mills equations. J. Differ. Geometry 34, 275–345 (1991)
Friedrich H.: Einstein equations and conformal structure: existence of anti-de Sitter-type space-times. J. Geom. Phys. 17, 125–184 (1995)
Friedrich H.: Evolution equations for gravitating ideal fluid bodies in general relativity. Phys. Rev. D 57, 2317–2322 (1998)
Friedrich H.: Conformal geodesics on vacuum space-times. Commun. Math. Phys. 235, 513–543 (2003)
Friedrich H.: Smooth non-zero rest-mass evolution across time-like infinity. Ann. Henri Poincaré 16, 2215–2238 (2015)
Friedrich, H.: Geometric asymptotics and beyond. In: Bieri, L., Yau, S.T. (Eds.) Surveys in differential geometry, vol.20. International Press, Boston (2015). arXiv:1411.3854
Friedrich, H., Rendall, A.: The Cauchy Problem for the Einstein Equations. In: Schmidt, B. (Ed.) Einstein’s field equations and their physical implications. Springer, Lecture Notes in Physics, Berlin (2000)
Hadžić M., Speck J.: The global future stability of the FLRW solutions to the Dust–Einstein system with a positive cosmological constant. J. Hyperb. Differ. Equations 12, 87 (2015)
Hawking S., Ellis G.: The large scale structure of space-time. Cambridge University Press, Cambridge (1973)
Kato T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181–205 (1975)
Lübbe C., Valiente Kroon J.A.: A conformal approach to the analysis of the non-linear stability of radiation cosmologies. Ann. Phys. 328, 1 (2013)
Penrose R.: Zero rest-mass fields including gravitation: asymptotic behaviour. Proc. R. Soc. Lond. A 284, 159–203 (1965)
Penrose R.: Cycles of time. Vintage, London (2011)
Reula O.: Exponential decay for small non-linear perturbations of expanding flat homogeneous cosmologies. Phys. Rev. D 60, 083507 (1999)
Ringström H.: Future stability of the Einstein-non-linear scalar field system. Invent. math. 173, 123–208 (2008)
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Communicated by P. T. Chruściel
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Friedrich, H. Sharp Asymptotics for Einstein-\({\lambda}\)-Dust Flows. Commun. Math. Phys. 350, 803–844 (2017). https://doi.org/10.1007/s00220-016-2716-6
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DOI: https://doi.org/10.1007/s00220-016-2716-6