| 1. |
- Oude Groeniger, Hans, 1993-
(författare)
-
Quiescent regimes in cosmology
- 2023
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is about cosmological solutions to Einstein’s equations of general relativity, in particular spacetimes whose mean curvature diverges. Moreover, we consider anisotropic spacetimes with big bang singularities. In this setting the singularity is expected to generically be oscillatory if no matter is present. However, complementary to an oscillatory singularity is the notion of quiescence, i.e. the convergence of the eigenvalues of the expansion-normalized Weingarten map . This thesis contains results related to two regimes in which quiescence is expected to occur, namely the presence of certain geometrical features or the satisfaction of an algebraic condition on the eigenvalues of .Paper A is concerned with Bianchi type spacetimes with an orthogonal perfect fluid, and we show that generically their initial singularity is anisotropic and quiescent. The quiescence that occurs may be understood as a consequence of the Abelian subgroup of the isometry group acting orthogonally-transitively. These results are then used to obtain asymptotics for solutions to the Klein-Gordon equation on backgrounds of this type.Paper B is about Bianchi type spacetimes with an orthogonal stiff fluid. Bianchi type is known as exceptional, for the fact that the dynamics of vacuum and orthogonal perfect fluid cosmologies of this type have the same degrees of freedom as those of Bianchi type or . This is due to the not necessarily acting orthogonally-transitively for type . The main result is that, generically, the initial singularity of such solutions is anisotropic and quiescent, and the eigenvalues of converge to strictly positive values. Here quiescence is a result of the stiff fluid matter, which allows for the algebraic condition on the eigenvalues of to be satisfied. Complementary to this generic behaviour are the spacetimes with special geometrical features, in particular those in which the does act orthogonally-transitively, and those that (asymptotically) satisfy a polarization condition. In these cases it occurs that the smallest limit of the eigenvalues of is negative. This is in contrast with type or cosmologies with an orthogonal stiff fluid, for which the eigenvalues of always converge to strictly positive limits. As a secondary result we obtain a concise way to represent the dynamics.In paper C, which is joint work with Oliver Petersen and Hans Ringström, we consider CMC initial data to the Einstein-nonlinear scalar field equations for a certain class of potentials. The main result is that if a certain bound on expansion-normalized quantities holds, if an algebraic condition on the eigenvalues of is satisfied, and if the eigenvalues of remain separated over the manifold, then there exists a threshold for the initial mean curvature, which, if surpassed, guarantees that the development has a quiescent big bang singularity. By this we mean past global existence of the development until the blowup of the Kretschmann scalar, and convergence of the eigenvalues of . We also obtain asymptotics for the eigenvalues of and expansion-normalized quantities relating to the scalar field. Combining the main result with results by Ringström concerning Bianchi class A solutions leads to a proof of the future and past global non-linear stability of a large class of spatially locally homogeneous solutions.
|
|
| 2. |
- Andreasson, Håkan, 1966, et al.
(författare)
-
Proof of the cosmic no-hair conjecture in the T-3-Gowdy symmetric Einstein-Vlasov setting
- 2016
-
Ingår i: Journal of the European Mathematical Society. - : EMS Publishing House. - 1435-9855 .- 1435-9863. ; 18:7, s. 1565-1650
-
Tidskriftsartikel (refereegranskat)abstract
- The currently preferred models of the universe undergo accelerated expansion induced by dark energy. One model for dark energy is a positive cosmological constant. It is consequently of interest to study Einstein's equations with a positive cosmological constant coupled to matter satisfying the ordinary energy conditions: the dominant energy condition etc. Due to the difficulty of analysing the behaviour of solutions to Einstein's equations in general, it is common to either study situations with symmetry, or to prove stability results. In the present paper, we do both. In fact, we analyse, in detail, the future asymptotic behaviour of T-3-Gowdy symmetric solutions to the Einstein-Vlasov equations with a positive cosmological constant. In particular, we prove the cosmic no-hair conjecture in this setting. However, we also prove that the solutions are future stable (in the class of all solutions). Some of the results hold in a more general setting. In fact, we obtain conclusions concerning the causal structure of T-2-symmetric solutions, assuming only the presence of a positive cosmological constant, matter satisfying various energy conditions and future global existence. Adding the assumption of T-3-Gowdy symmetry to this list of requirements, we obtain C-0-estimates for all but one of the metric components. There is consequently reason to expect that many of the results presented in this paper can be generalised to other types of matter.
|
|
| 3. |
- Radermacher, Katharina Maria, 1987-
(författare)
-
Strong Cosmic Censorship and Cosmic No-Hair in spacetimes with symmetries
- 2017
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
|
|
| 4. |
- Ringström, Hans, 1972-
(författare)
-
A Unified Approach to the Klein-Gordon Equation on Bianchi Backgrounds
- 2019
-
Ingår i: Communications in Mathematical Physics. - : SPRINGER. - 0010-3616 .- 1432-0916. ; 372:2, s. 599-656
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we study solutions to the Klein-Gordon equation on Bianchi backgrounds. In particular, we are interested in the asymptotic behaviour of solutions in the direction of silent singularities. The main conclusion is that, for a given solution u to the Klein-Gordon equation, there are smooth functions u(i), i = 0, 1, on the Lie group under consideration, such that u(sigma) (. , sigma) - u(1) and u(. , sigma) - u(1)sigma - u(0) asymptotically converge to zero in the direction of the singularity (where s is a geometrically defined time coordinate such that the singularity corresponds to sigma -> -infinity). Here u(i), i = 0, 1, should be thought of as data on the singularity. Interestingly, it is possible to prove that the asymptotics are of this form for a large class of Bianchi spacetimes. Moreover, the conclusion applies for singularities that arematter dominated; singularities that are vacuum dominated; and even when the asymptotics of the underlying Bianchi spacetime are oscillatory. To summarise, there seems to be a universality as far as the asymptotics in the direction of silent singularities are concerned. In fact, it is tempting to conjecture that as long as the singularity of the underlying Bianchi spacetime is silent, then the asymptotics of solutions are as described above. In order to contrast the above asymptotics with the non-silent setting, we, by appealing to known results, provide a complete asymptotic characterisation of solutions to the Klein-Gordon equation on a flat Kasner background. In that setting, us does, generically, not converge.
|
|
| 5. |
- Ringström, Hans, 1972-
(författare)
-
Linear Systems Of Wave Equations On Cosmological Backgrounds With Convergent Asymptotics
- 2020
-
Ingår i: Astérisque. - : SOC MATHEMATIQUE FRANCE. - 0303-1179 .- 2492-5926. ; :420, s. V-
-
Tidskriftsartikel (refereegranskat)abstract
- The subject of the book is linear systems of wave equations on cosmological backgrounds with convergent asymptotics. The condition of convergence corresponds to the requirement that the second fundamental form, when suitably normalized, converges. The model examples are the Kasner solutions. The main result of the article is optimal energy estimates. However, we also derive asymptotics and demonstrate that the leading order asymptotics can be specified (also in situations where the asymptotics are not convergent). It is sometimes argued that if the factors multiplying the spatial derivatives decay exponentially (for a system of wave equations), then the spatial derivatives can be ignored. This line of reasoning is incorrect: we give examples of equations such that 1) the factors multiplying the spatial derivatives decay exponentially, 2) the factors multiplying the time derivatives are constants, 3) the energies of individual modes of solutions asymptotically decay exponentially, and 4) the energies of generic solutions grow as exp[exp(t)] as t -> infinity. When the factors multiplying the spatial derivatives grow exponentially, the Fourier modes of solutions oscillate with a frequency that grows exponentially. To obtain asymptotics, we fix a mode and consider the net evolution over one period. Moreover, we replace the evolution (over one period) with a matrix multiplication. We cannot calculate the matrices explicitly, but we approximate them. To obtain the asymptotics we need to calculate a matrix product where there is no bound on the number of factors, and where each factor can only be approximated. Nevertheless, we obtain detailed asymptotics. In fact, it is possible to isolate an overall behavior (growth/decay) from the (increasingly violent) oscillatory behavior. Moreover, we are also in a position to specify the leading order asymptotics.
|
|
| 6. |
|
|
| 7. |
- Ringström, Hans, 1972-
(författare)
-
On the future stability of cosmological solutions to Einstein's equations with accelerated expansion
- 2014
-
Ingår i: PROCEEDINGS OF THE INTERNATIONAL CONGRESS OF MATHEMATICIANS (ICM 2014), VOL II. - : KyungMoonSa Publishers. - 9788961058056 ; , s. 983-999
-
Konferensbidrag (refereegranskat)abstract
- The solutions of Einstein's equations used by physicists to model the universe have a high degree of symmetry. In order to verify that they are reasonable models, it is therefore necessary to demonstrate that they are future stable under small perturbations of the corresponding initial data. The purpose of this contribution is to describe mathematical results that have been obtained on this topic. A question which turns out to be related concerns the topology of the universe: what limitations do the observations impose? Using methods similar to ones arising in the proof of future stability, it is possible to construct solutions with arbitrary closed spatial topology. The existence of these solutions indicate that the observations might not impose any limitations at all.
|
|
| 8. |
- Ringström, Hans, 1972-
(författare)
-
On the Topology and Future Stability of the Universe
- 2013. - 1
-
Bok (refereegranskat)abstract
- The subject of the book is the topology and future stability of models of the universe. In standard cosmology, the universe is assumed to be spatially homogeneous and isotropic. However, it is of interest to know whether perturbations of the corresponding initial data lead to similar solutions or not. This is the question of stability. It is also of interest to know what the limitations on the global topology imposed by observational constraints are. These are the topics addressed in the book. The theory underlying the discussion is the general theory of relativity. Moreover, in the book, matter is modelled using kinetic theory. As background material, the general theory of the Cauchy problem for the Einstein–Vlasov equations is therefore developed.
|
|
| 9. |
|
|
