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The global future stability of the FLRW solutions to the dust-Einstein system with a positive cosmological constant. (English) Zbl 1333.35281

The authors of this interesting paper study the problem connected with small perturbations of the Friedman-Lemaitre-Robertson-Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant. Here, they consider the case in which the space-like Cauchy hypersurfaces are diffeomorphic to \(\mathbb{T}^3\). It is assumed that the dust is a fluid under zero pressure. Unknowns are the space-time manifold \(\mathcal{M}\), the Lorentzian space-time metric \(g_{\mu\nu }\), the dust mass-energy density \(\rho \), and the dust four-velocity \(u^{\mu }\), which is a future-directed vector field. The dust-Einstein system has the form: \[ \mathrm{Ric}_{\mu\nu }-(1/2)Rg_{\mu\nu }+\Lambda g_{\mu\nu } = T_{\mu\nu } \quad (\mu ,\nu = 0,1,2,3), \]
\[ D_{\alpha }T^{\alpha\mu }=0 \quad (\nu = 0,1,2,3), \]
\[ g_{\alpha\beta }u^{\alpha }u^{\beta }=-1. \] Here, \(\mathrm{Ric}_{\mu\nu }\) is the Ricci tensor of a metric \(g_{\mu\nu }\), \(R\) is the scalar curvature of the same \(g_{\mu\nu }\), \(\Lambda >0\) is a fixed cosmological constant, \(T_{\mu\nu }\) is the energy-momentum tensor of the dust, \(T_{\mu\nu }=\rho u_{\mu }u_{\nu }\) (\(\mu ,\nu = 0,1,2,3\)), and \(D_{\mu }\) is the covariant derivative associated with \(g_{\mu\nu }\). It is known that the above stated system of nonlinear partial differential equations (PDE) admits a family of explicit solutions known as FLRW solutions. For the fixed \(\Lambda \) these solutions are quadruple \(((\-\infty ,\infty )\times \mathbb{T}^3,\tilde{g},\tilde{u},\tilde{\rho })\); here, \(\mathbb{T}^3=[-\pi ,\pi ]^3\) is the 3-dimensional torus and the metric \(\tilde{g}=-dt^2+a^2(t) \sum\limits_{i=1}^{3}(dx^i)^2\), \(\tilde{u}=(1,0,0,0)\), \(\tilde{\rho }=a^{-3}(t)\bar{\rho }\), \(a(t)\sim Ce^{Ht}\).
The main result proved in the present paper concerns the above stated solutions. The main statement is that the above mentioned solutions are globally future-stable. In other words, it means that under small perturbations of the FLRW data given on the torus \(\mathbb{T}^3\) maximal globally hyperbolic evolution changes are launched. The space-time with a boundary \(([0,\infty )\times \mathbb{T}^3, g_{\mu\nu })\) is future causally geodesically complete. In addition, \((g_{\mu\nu }, u^{\mu },\rho=e^{-3\Omega }\varrho )\) solves the unmodified dust-Einstein system.

MSC:

35Q76 Einstein equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35L99 Hyperbolic equations and hyperbolic systems
35Q31 Euler equations
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83F05 Relativistic cosmology
35B35 Stability in context of PDEs
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