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- Oude Groeniger, Hans, 1993-
(författare)
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Quiescent regimes in cosmology
- 2023
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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.
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| 2. |
- Ringström, Hans, 1972-
(författare)
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Linear Systems Of Wave Equations On Cosmological Backgrounds With Convergent Asymptotics
- 2020
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Ingår i: Astérisque. - : SOC MATHEMATIQUE FRANCE. - 0303-1179 .- 2492-5926. ; :420, s. V-
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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.
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