The cosmic no-hair theorem and the non-linear stability of homogeneous Newtonian cosmological models

Abstract

The validity of the cosmic no-hair theorem is investigated in the context of Newtonian cosmology with a perfect uid matter model and a positive cosmological constant. It is shown that if the initial data for an expanding cosmological model of this type is subjected to a small perturbation then the corresponding solution exists globally in the future and the perturbation decays in a way which can be described precisely. It is emphasized that no linearization of the equations or special symmetry assumptions are needed. The result can also be interpreted as a proof of the nonlinear stability of the homogeneous models. In order to prove the theorem we write the general solution as the sum of a homogeneous background and a perturbation. As a by-product of the analysis it is found that there is an invariant sense in which an inhomogeneous model can be regarded as a perturbation of a unique homogeneous model. A method is given for associating uniquely to each Newtonian cosmological model with compact spatial sections a spatially homogeneous model which incorporates its large-scale dynamics. This procedure appears very natural in the NewtonCartan theory which w e take as the starting point for Newtonian cosmology.] Member of CONICET 1

Critical Thresholds in 2D Restricted Euler-Poisson Equations

“…This hyperbolic-elliptic coupled system (1.1)-(1.3) describes the dynamic behavior of many important physical flows including the charge transport [22], plasma with collision [13], cosmological waves [2] and the expansion of the cold ions [12]. Let us mention that the Euler-Poisson equations could also be realized as the semi-classical limit of Schrödinger-Poisson equation and are found in the 'cross-section' of Vlasov-Poisson equations.…”

On the finite time blow-up of the Euler-Poisson equations in $\Bbb R^{2}$

“…The research of E. Tadmor was supported in part by NSF grants DMS07-07949 and FRG07-57227, and ONR grant N00014-91-J-1076. 1 In fact, div v remains uniformly upper-bounded for all non-vacuum paths, div v(·,t) ≤ sup {a:ρ 0 (a)>0}˘d iv v 0 (a), √ −nkc¯.…”

“…The hyperbolic-elliptic system (1.1) appears in a variety of different applications, from small scale models in charge transport and plasma collision, e.g., [18,8], to large scale dynamics of (clusters of) stars in cosmological waves, and expansion of the cold ions, e.g., [1,7].…”

Critical thresholds in multi-dimensional Euler-Poisson equations with radial symmetry

“…The parameter k is a scaled physical constant signifying the property of the underlying force; the force is repulsive if k > 0, and attractive if k < 0. This system describes dynamic behaviors of many important physical flows, from small scale models of charged transport [20,12], expansion of cold ions [9], and collisional plasma [10], to large scale models of cosmological waves [1,2]. For smooth solutions away from vacuum, (1.1) can be reduced to n t + ∇ · (nu) = 0, (1.2a)…”