Slowly expanding stable dust spacetimes

Fajman, David; Ofner, Maximilian; Wyatt, Zoe

doi:10.48550/arxiv.2107.00457

Cited by 5 publications

(10 citation statements)

References 28 publications

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“…On the other hand, in the same problem if the cosmological constant is turned off (i.e., the vacuum energy is absent), then the perturbed solutions do come back to the background solution asymptotically [27,28]. Similar phenomena of asymptotic stability are observed in the study of [38], where the background solutions once again are simply the Milne universe that does not contain a non-zero energy density. In order to reach a definite conclusion about forming large-scale structures by non-linear perturbations of a background with non-vanishing energy density would require further analysis including radiation sources and possibly more observational inputs.…”

Section: Motivation and Overviewmentioning

confidence: 79%

“…However, we do not claim that turning off the cosmological constant leads to instability but simply we are unable to reach a definite conclusion with the currently available method. It is worth pointing out that [38] proved the non-linear asymptotic stability of the Milne universe (devoid of background matter density) including a dust source (slowly expanding spacetime). In such a case, the individual dust particles travel along their respective geodesics and the sound speed vanishes.…”

Section: ∇[γ]mentioning

confidence: 98%

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The nonlinear stability of n+1 dimensional FLRW spacetimes

Mondal¹

2022

Preprint

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We prove nonlinear Lyapunov stability of a family of 'n + 1'-dimensional cosmological models of general relativity locally isometric to the Friedman Lemaître Robertson Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein-Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant n = 3 case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein-Euler field equations in constant mean extrinsic curvature spatial harmonic gauge (CMCSH). In order to handle Euler's equations, we construct energy from a current that is similar to the one derived by Christodoulou [1] (and which coincides with Christodoulou's current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein-Euler field equations into a coupled elliptichyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant Λ is included, which suggests that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the non-linearities in the case of small data. A few physical consequences of this stability result are discussed.

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Section: Motivation and Overviewmentioning

confidence: 79%

Section: ∇[γ]mentioning

confidence: 98%

The nonlinear stability of n+1 dimensional FLRW spacetimes

Mondal¹

2022

Preprint

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“…In the case of dust K = 0, [34] showed that small perturbations of the homogeneous fluid solutions are globally regular for all α > 1/2. If K ∈ (0, 1/3), work by some of the present authors showed that homogeneous fluid solutions are globally regular under irrotational perturbations for α = 1 [14]. The case α > 1 without the irrotational restriction was shown in [34].…”

mentioning

confidence: 72%

The Stability of Relativistic Fluids in Linearly Expanding Cosmologies

Fajman¹,

Maximilian²,

Oliynyk³

et al. 2023

Preprint

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In this paper we study cosmological solutions to the Einstein-Euler equations. We first establish the future stability of nonlinear perturbations of a class of homogeneous solutions to the relativistic Euler equations on fixed linearly expanding cosmological spacetimes with a linear equation of state p = Kρ for the parameter values K ∈ (0, 1/3). This removes the restriction to irrotational perturbations in earlier work [15], and relies on a novel transformation of the fluid variables that is well-adapted to Fuchsian methods. We then apply this new transformation to show the global regularity and stability of the Milne spacetime under the coupled Einstein-Euler equations, again with a linear equation of state p = Kρ, K ∈ (0, 1/3). Our proof requires a correction mechanism to account for the spatially curved geometry. In total, this is indicative that structure formation in cosmological fluid-filled spacetimes requires an epoch of decelerated expansion.

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“…Following these earlier studies several additional ones have been performed addressing the stability issue on manifolds admitting negative spatial Einstein metrics (hyperbolic manifolds for the 3 + 1 dimensional case) including source terms in the case of small data. These include the Λ−vacuum (Λ > 0) [38,37], Klein-Gordon field [40,39] (with Λ = 0), dust [30], and Vlasov matter [44] cases, in particular. The stability of these special solutions incorporating several matter sources that satisfy the energy condition we prescribe for the analysis to hold true provides some 'weak' support for our conclusion.…”

Section: Stability Resultsmentioning

confidence: 99%

Einstein flow with matter sources: stability and convergence

Moncrief,

Mondal

2021

Preprint

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Two recent articles [1,2] suggested an interesting dynamical mechanism within the framework of the vacuum Einstein flow (or Einstein-Λ flow if a positive cosmological constant Λ is included) which suggests that many closed (compact without boundary) manifolds that do not support homogeneous and isotropic metrics at all will nevertheless evolve to be asymptotically compatible with the observed approximate homogeneity and isotropy of the physical universe. These studies however did not include matter sources. Therefore the aim of the present study is to include suitable matter sources and investigate whether one is able to draw a similar conclusion.

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Slowly expanding stable dust spacetimes

Cited by 5 publications

References 28 publications

The nonlinear stability of n+1 dimensional FLRW spacetimes

The nonlinear stability of n+1 dimensional FLRW spacetimes

The Stability of Relativistic Fluids in Linearly Expanding Cosmologies

Einstein flow with matter sources: stability and convergence

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