## 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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