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.

Den här avhandlingen handlar om kosmologiska lösningar till Einsteins ekvationer i allmän relativitetsteori, särskilt om rumtider där medelkrökning divergerar. Specifikt studerar vi anisotropa rumtider med en Big Bang-singularitet. Under dessa omständigheter förväntar man sig i allmänhet en oscillerande singularitet i frånvaro av materia. Men, som komplement till oscillerande singulariteter finns begreppet ”quiescence”, det vill säga, konvergens av egenvärdena av den expansionsnormaliserade Weingartenavbildningen . Denna avhandling innehåller resultat beträffande två regimer där quiescence sker, nämligen då vissa geometriska villkor är uppfyllda, eller då ett algebraiskt villkor på egenvärdena av är uppfyllt.

Artikel A behandlar Bianchi typ -rumtider med en ortogonal ideal vätska, och vi visar att deras ursprungssingulariteter i allmänhet är anisotropa och quiescenta. Den quiescence som sker kan förstås som en konsekvens av att den Abelska undergruppen av isometrigruppen agerar ortogonalt transitivt. Resultaten används sedan för att erhålla asymptotik för lösningar av Klein–Gordon-ekvationen på bakgrunder av denna typ.

Artikel B handlar om Bianchi typ -rumtider med en ortogonal stel vätska. Bianchi typ kallas för exceptionell, på grund av att vakuums eller ortogonala ideala vätskekosmologier av denna typ har samma antal frihetsgrader som Bianchi typ och . Det är på grund av att inte nödvandigtvis agerar ortogonalt transitivt för typ . Huvudresultatet är att singulariteten är, generellt sett, anisotrop och quiescent, och egenvärdena av konvergerar till strikt positiva gränsvärden. Här är quiescence en följd av den stela västkan, vilken gör att det algebraiska villkoret på egenvärdena av uppfylls. Likväl finns det ett komplement till det generiska beteendet, nämligen rumtider med vissa geometriska egenskaper, särskilt de i vilket agerar ortogonalt transitivt, och de som (asymptotiskt) uppfyller ett polariseringsvillkor. I dessa faller händer det att det minsta av gränsvärdena av egenvärdena av är negativt. Detta står i kontrast med kosmologier av typ eller med en ortogonal stel vätska, för vilka egenvärdena av alltid konvergerar till strikt positiva gränsvärden. Som ett sekundärt resultat får vi ett får vi ett koncist sätt för att representera dynamiken.

I papper C betraktar vi CMC begynnelse data till Einsteins ekvationer kopplat till ett icke-linjärt skälarfält för en viss klass av potentialer. Huvudresultatet är att, om en uppskattning för vissa expansionsnormaliserade storheter gäller, om ett algebraiska villkor på egenvärdena av är uppfyllt, och om egenvärdena av förblir distinkta över mång- falden, finns det en undre gräns för den initiala medelkrökning som, om den överskrids, garanterar att utvecklingen har en quiescent Big Bang singularitet. I synnerhet menar vi global existens bakåt i tiden av utvecklingen fram till att Kretschmann-skalären exploderar, och konvergensen av egenvärdena av . Vi erhåller också asymptotik för egenvärdena av och expansionsnormaliserade storheter relaterade till skalärfältet. Kombinationen av detta med resultater av Ringström beträffende Bianchi klass A lösningar leder till ett bevis av global icke-linjär stabilitet för en stor klass av rumsligt lokalt homogena lösningar.

We study the asymptotic behaviour of Bianchi type VI₀ spacetimes with orthogonal perfect fluid matter satisfying Einstein’s equations. In particular, we prove a conjecture due to Wainwright about the initial singularity on such backgrounds. Using the expansion-normalized variables of Wainwright–Hsu, we demonstrate that for a generic solution the initial singularity is vacuum dominated, anisotropic and silent. In addition, by employing known results on Bianchi backgrounds, we obtain convergence results on the asymptotics of solutions to the Klein–Gordon equation on all backgrounds of this type, except for one specific case.

Hawking’s singularity theorem says that cosmological solutions satisfying the strong energy condition and corresponding to initial data with positive mean curvature have a past singularity; any past timelike curve emanating from the initial hypersurface has length at most equal to the inverse of the mean curvature. However, the nature of the singularity remains unclear. We therefore ask the following question: If the initial hypersurface has sufficiently large mean curvature, does the curvature necessarily blow up towards the singularity?

In case the eigenvalues of the expansion-normalized Weingarten map are everywhere distinct and satisfy a certain algebraic condition (which in 3+1 dimensions is equivalent to them being positive), we prove that this is indeed the case in the CMC Einstein-non-linear scalar field setting. More specifically, we associate a set of geometric expansion-normalized quantities to any initial data set with positive mean curvature. These quantities are expected to converge, in the quiescent setting, in the direction of crushing big bang singularities; i.e. as the mean curvature diverges. Our main result says that if the mean curvature is large enough, relative to an appropriate Sobolev norm of these geometric quantities, and if the algebraic condition on the eigenvalues is satisfied, then a quiescent (as opposed to oscillatory) big bang singularity with curvature blow-up necessarily forms. This provides a stable regime of big bang formation without requiring proximity to any particular class of background solutions.

An important recent result by Fournodavlos, Rodnianski and Speck demonstrates stable big bang formation for all the spatially flat and spatially homogeneous solutions to the Einstein-scalar field equations satisfying the algebraic condition. As an application of our analysis, we obtain analogous stability results for any solution with induced data at a quiescent big bang singularity, in the sense introduced by the third author. In particular, we conclude stable big bang formation of large classes of spatially locally homogeneous solutions, of which the result by Fournodavlos, Rodnianski and Speck is a special case. Finally, since we here consider the Einstein-non-linear scalar field setting, we are also, combining the results of this article withan analysis of Bianchi class A solutions, able to prove both future and past global non-linear stability of a large class of spatially locally homogeneous solutions.

The exceptional Bianchi type VI_−1/9 cosmologies, with non-stiff fluid matter or in vacuum, are conjectured to generically undergo an infinite series of chaotic oscillations toward the initial singularity, as has been proven to occur for Bianchi type VIII and IX cosmologies. Cosmologies of the lower Bianchi types, i.e. except those of type VIII or IX, admit a two-dimensional Abelian subgroup of the isometry group, the G_2. In orthogonal perfect fluid cosmologies of all lower Bianchi types except for type VI_−1/9 the G₂ acts orthogonally-transitively, which is closely related to an eventual cessation of the oscillations and thus to a quiescent singularity. But due to a degeneracy in the momentum constraints, such cosmologies of type VI_−1/9 do not necessarily have this property. As a consequence, the dynamics of type VI_−1/9 orthogonal perfect fluid cosmologies have the same degrees of freedom as those of the higher types VIII and IX and their dynamics are expected to be markedly different compared to those of the other lower Bianchi types.

In this article we take a different approach to quiescence, namely the presence of an orthogonal stiff fluid. On the one hand, this completes the analysis of the initial singularity for all Bianchi orthogonal stiff fluid cosmologies. On the other hand, this allows us to get a grasp of the underlying dynamics of type VI_−1/9 perfect fluid cosmologies, in particular the effect of orthogonal transitivity as well as possible (asymptotic) polarization conditions. In particular, we show that a generic type VI_−1/9 cosmology with an orthogonal stiff fluid has similar asymptotics as a generic Bianchi type VIII or IX cosmology with an orthogonal stiff fluid. The only exceptionsto this genericity are solutions satisfying (asymptotic) polarization conditions, and solutions for which the G₂ acts orthogonally-transitively. Only in those cases may the limits of the eigenvalues of the expansion-normalized Weingarten map be negative. We also obtain a concise way to represent the dynamics which works more generally for the exceptional type VI_−1/9 orthogonal perfect fluid cosmologies, and obtain a monotonic function for the case of a orthogonal perfect fluid that is more stiff than a radiation fluid.