×

zbMATH — the first resource for mathematics

Cosmological Newtonian limits on large spacetime scales. (English) Zbl 1402.83116
Summary: We establish the existence of 1-parameter families of \(\epsilon\)-dependent solutions to the Einstein-Euler equations with a positive cosmological constant \(\Lambda > 0\) and a linear equation of state \(p =\epsilon^{2}{K}\rho\), \(0 < K \leq 1/3\), for the parameter values \(0 < \epsilon < \epsilon_{0}\). These solutions exist globally on the manifold \(M = (0, 1] \times \mathbb{R}^{3}\), are future complete, and converge as \(\epsilon \searrow 0\) to solutions of the cosmological Poisson-Euler equations. They represent inhomogeneous, nonlinear perturbations of a FLRW fluid solution where the inhomogeneities are driven by localized matter fluctuations that evolve to good approximation according to Newtonian gravity.

MSC:
83F05 Relativistic cosmology
53Z05 Applications of differential geometry to physics
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
Software:
GADGET ; RAMSES
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adams R.A., Fournier J.J.F.: Sobolev Spaces, second ed. Academic Press Inc., Cambridge (2003)
[2] Arfken G.B.: Mathematical Methods for Physicists, third ed.. Academic Press Inc, Cambridge (1985) · Zbl 0135.42304
[3] Brauer, U.; Rendall, A.; Reula, O., The cosmic no-hair theorem and the non-linear stability of homogeneous Newtonian cosmological models, Classical Quantum Gravity, 11, 2283, (1994) · Zbl 0815.53092
[4] Browning, G.; Kreiss, H.-O., Problems with different time scales for nonlinear partial differential equations, SIAM J. Appl. Math., 42, 704-718, (1982) · Zbl 0506.35006
[5] Crocce, M.; Fosalba, P.; Castander, F. J.; Gaztañaga, E., Simulating the universe with MICE: the abundance of massive clusters, Mon. Not. R. Astron. Soc., 403, 1353-1367, (2010)
[6] Evrard, A. E.; MacFarland, T. J.; Couchman, H. M.P.; Colberg, J. M.; Yoshida, N.; White, S. D.M.; Jenkins, A.; Frenk, C. S.; Pearce, F. R.; Peacock, J. A.; etal., Galaxy Clusters in hubble volume simulations: cosmological constraints from sky survey populations, Astrophys. J., 573, 7, (2002)
[7] Helmut, F., 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) · Zbl 0659.53056
[8] Helmut, F., On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations, J. Differ. Geom., 34, 275-345, (1991) · Zbl 0737.53070
[9] Helmut, F., Sharp asymptotics for Einstein-\({\lambda}\)-dust flows, Commun. Math. Phys., 350, 803-844, (2017) · Zbl 1360.83008
[10] Grafakos L.: Classical Fourier Analysis, third ed. Graduate Texts in Mathematics. Springer, New York (2014) · Zbl 1304.42001
[11] Grafakos L.: Modern Fourier analysis, third ed., Graduate Texts in Mathematics. Springer, New York (2014) · Zbl 1304.42002
[12] Green, S. R.; Wald, R. M., Newtonian and relativistic cosmologies, Phys. Rev. D, 85, 063512, (2012)
[13] Hadi, M.; Speck, J., The global future stability of the FLRW solutions to the dust-Einstein system with a positive cosmological constant, J. Hyperb. Differ. Equ., 12, 87-188, (2015) · Zbl 1333.35281
[14] Hahn, O.; Angulo, R. E., An adaptively refined phase-space element method for cosmological simulations and collisionless dynamics, Mon. Not. R. Astron. Soc., 455, 1115-1133, (2016)
[15] Klainerman, S.; Majda, A., Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pure Appl. Math., 34, 481-524, (1981) · Zbl 0476.76068
[16] Klainerman, S.; Majda, A., Compressible and incompressible fluids, Commun. Pure.Appl.Math., 35, 629-651, (1982) · Zbl 0478.76091
[17] Koch, H.: Hyperbolic equations of second order. Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg (1990) · Zbl 0717.35058
[18] Kreiss, H. O., Problems with different time scales for partial differential equations, Commun. Pure Appl. Math., 33, 399-439, (1980) · Zbl 0439.35043
[19] LeFloch, P.G., Wei, C.: The Global Nonlinear Stability of Self-gravitating Irrotational Chaplygin Fluids in a FRW Geometry. arXiv:1512.03754
[20] Liu, C.; Oliynyk, T. A., Newtonian limits of isolated cosmological systems on long time scales, Ann. Henri Poincaré, 19, 2157-2243, (2018) · Zbl 1394.83004
[21] Lottermoser, M., A convergent post-Newtonian approximation for the constraint equations in general relativity, Ann. l’ I.H.P. Phys. Thorique , 57, 279-317, (1992) · Zbl 0762.53053
[22] Lübbe, C.; AntonioValiente Kroon, J., A conformal approach for the analysis of the non-linear stability of radiation cosmologies, Ann. Phys., 328, 1-25, (2013) · Zbl 1263.83188
[23] Majda, A., Compressible fluid flow and systems of conservation laws in several space variables, Appl. Math. Sci., 53, 30-51, (2012)
[24] Oliynyk, T. A., An existence proof for the gravitating BPS monopole, Ann. Henri Poincaré, 7, 199-232, (2006) · Zbl 1097.81046
[25] Oliynyk, T. A., The Newtonian limit for perfect fluids, Commun. Math. Phys., 276, 131-188, (2007) · Zbl 1194.83018
[26] Oliynyk, T. A., Post-Newtonian expansions for perfect fluids, Commun. Math. Phys., 288, 847-886, (2009) · Zbl 1175.83027
[27] Oliynyk, T. A., Cosmological post-Newtonian expansions to arbitrary order, Commun. Math. Phys., 295, 431-463, (2010) · Zbl 1195.35286
[28] Oliynyk, T. A., A rigorous formulation of the cosmological Newtonian limit without averaging, J. Hyperb. Differ. Equ., 7, 405-431, (2010) · Zbl 1204.83030
[29] Oliynyk, T. A., The fast Newtonian limit for perfect fluids, Adv. Theor. Math. Phys., 16, 359-391, (2012) · Zbl 1269.83039
[30] Oliynyk, T. A., Cosmological Newtonian limit, Phys. Rev. D, 89, 124002, (2014)
[31] Oliynyk, T. A., The Newtonian limit on cosmological scales, Commun. Math. Phys., 339, 455-512, (2015) · Zbl 1352.35120
[32] Oliynyk, T. A., Future stability of the FLRW fluid solutions in the presence of a positive cosmological constant, Commun. Math. Phys., 346, 293-312, (2016) · Zbl 1346.83023
[33] Oliynyk, T.A., Robertson, C.: Linear cosmological perturbations on large scales via post-Newtonian expansions (in preparation)
[34] Ringström, H., Future stability of the Einstein-non-linear scalar field system, Invent. Math., 173, 123, (2008) · Zbl 1140.83314
[35] Rodnianski, I.; Speck, J., The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant, J. Eur. Math. Soc., 15, 2369-2462, (2013) · Zbl 1294.35164
[36] Speck, J., The nonlinear future stability of the FLRW family of solutions to the Euler-Einstein system with a positive cosmological constant, Sel. Math., 18, 633-715, (2012) · Zbl 1251.83071
[37] Springel, V., The Cosmological simulation code GADGET-2, Mon. Not. R. Astron. Soc., 364, 1105-1134, (2005)
[38] Springel, V.; White, S. D.M.; Jenkins, A.; Frenk, C. S.; Yoshida, N.; Gao, L.; Navarro, J.; Thacker, R.; Croton, D.; Helly, J.; Peacock, J. A.; Cole, S.; Thomas, P.; Couchman, H.; Evrard, A.; Colberg, J.; Pearce, F., Simulations of the formation, evolution and clustering of galaxies and quasars, Nature, 435, 629-636, (2005)
[39] Stein E.M.: Singular Integrals and Differentiability Properties of Functions, Monographs in Harmonic Analysis. Princeton University Press, Princeton (1970)
[40] Taylor M.E.: Partial Differential Equations III: Nonlinear Equations, Applied Mathematical Sciences. Springer, New York (2010)
[41] Teyssier, R., Cosmological hydrodynamics with adaptive mesh refinement A new high resolution code called RAMSES, Astron. Astrophys., 385, 337-364, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.