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On the Cosmic No-Hair Conjecture in T2-symmetric non-linear scalar field spacetimes
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider spacetimes solving the Einstein non-linear scalar field equations with T2-symmetry and show that they admit an areal time foliation in the expanding direction. In particular, we prove global existence and uniqueness of solutions to the corresponding system of evolution equations for all future times. The only assumption we have to make is that the potential is a non-negative smooth function.

In the special case of a constant potential, a setting which is equivalent to a linear scalar field on a background with a positive cosmological constant, we achieve detailed asymptotic estimates for the different components of the spacetime metric. This result holds for all T3-Gowdy symmetric metrics and extends to certain T2-symmetric ones satisfying an a priori decay property. Building upon these asymptotic estimates, we show future causal geodesic completeness and prove the Cosmic No-Hair conjecture.

Keywords [en]
No-Hair, no hair, scalar field, T2, T3, Gowdy, symmetry, geodesic, complete, areal foliation
National Category
Geometry Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-218325OAI: oai:DiVA.org:kth-218325DiVA, id: diva2:1164365
Note

QC 20171218

Available from: 2017-12-11 Created: 2017-12-11 Last updated: 2017-12-19Bibliographically approved
In thesis
1. Strong Cosmic Censorship and Cosmic No-Hair in spacetimes with symmetries
Open this publication in new window or tab >>Strong Cosmic Censorship and Cosmic No-Hair in spacetimes with symmetries
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of three articles investigating the asymptotic behaviour of cosmological spacetimes with symmetries arising in Mathematical General Relativity.

In Paper A and B, we consider spacetimes with Bianchi symmetry and where the matter model is that of a perfect fluid. We investigate the behaviour of such spacetimes close to the initial singularity ('Big Bang'). In Paper A, we prove that the Strong Cosmic Censorship conjecture holds in non-exceptional Bianchi class B spacetimes. Using expansion-normalised variables, we further show detailed asymptotic estimates. In Paper B, we prove similar estimates in the case of stiff fluids.

In Paper C, we consider T2-symmetric spacetimes satisfying the Einstein equations for a non-linear scalar field. To given initial data, we show global existence and uniqueness of solutions to the corresponding differential equations for all future times. In the special case of a constant potential, a setting which is equivalent to a linear scalar field on a background with a positive cosmological constant, we investigate in detail the asymptotic behaviour towards the future. We prove that the Cosmic No-Hair conjecture holds for solutions satisfying an additional a priori estimate, an estimate which we show to hold in T3-Gowdy symmetry.

Abstract [sv]

Denna avhandling består av tre artiklar som undersöker det asymptotiska beteendet hos kosmologiska rumstider med symmetrier som uppstår i Matematisk Allmän Relativitetsteori.

I Artikel A och B studerar vi rumstider med Bianchi symmetri och där materiemodellen är en ideal fluid. Vi undersöker beteendet av sådana rumstider nära ursprungssingulariteten ('Big Bang'). I Artikel A bevisar vi att den Starka Kosmiska Censur-förmodan håller för icke-exceptionella Bianchi klass B-rumstider. Med hjälp av expansions-normaliserade variabler visar vi detaljerade asymptotiska uppskattningar. I Artikel B visar vi liknande uppskattningar för stela fluider.

I Artikel C betraktar vi T2-symmetriska rumstider som uppfyller Einsteins ekvationer för ett icke-linjärt skalärfält. För givna begynnelsedata visar vi global existens och entydighet av lösningar till motsvarande differentialekvationer för all framtid. I det speciella fallet med en konstant potential, en situation som motsvarar ett linjärt skalärfält på en bakgrund med en positiv kosmologisk konstant, undersöker vi i detalj det asymptotiska beteendet mot framtiden. Vi visar att den Kosmiska Inget-Hår-förmodan håller för lösningar som uppfyller en ytterligare a priori uppskattning, en uppskattning som vi visar gäller i T3-Gowdy-symmetri.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2017. p. 39
Series
TRITA-MAT-A ; 2017:06
Keywords
Cosmic Censorship, Cosmic No-Hair, Big Bang, symmetry, Bianchi, T2-symmetry, Gowdy, general relativity, cosmology, late time, initial time, asymptotic behaviour, perfect fluid, scalar field, Einstein equations
National Category
Geometry Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-220400 (URN)978-91-7729-632-4 (ISBN)
Public defence
2018-02-19, F3, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20171220

Available from: 2017-12-20 Created: 2017-12-19 Last updated: 2017-12-20Bibliographically approved

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