Hans-Jürgen Schmidt

New exact solutions for power‐law inflation Friedmann models

We consider the spatially flat Friedmann model For a ≈ tp, especially, if p ≥ 1, this is called power-law inflation. For the Lagrangian L = Rm with p = − (m − 1) (2m − 1)/(m − 2) power-law inflation is an exact solution, as it is for Einstein gravity with a minimally coupled scalar field ϕ in an exponential potential V(ϕ) = exp (μϕ) and also for the higher-dimensional Einstein equation with a special Kaluza-Klein ansatz. The synchronized coordinates are not adapted to allow a closed-form solution, so we write The general solutions reads Q(a) = (ab + C)f/b with free integration constant C (C = 0 gives exact power-law inflation) and m-dependent values b and f: f = −2 + 1/p, b = (4m − 5)/(m − 1). Finally, special solutions for the closed and open Friedmann model are found.

Inhomogeneous Cosmological Models Containing Homogeneous Inner Hypersurface Geometry. Changes of the BIANCHI Type.

There are investigated such cosmological models which instead of the usual spatial homogeneity property only fulfil the condition that in a certain synchronized system of reference all spacelike sections t = const. are homogeneous manifolds. This allows time-dependent changes of the BIANCHI type. Discussing differential-geometrical theorems it is shown which of them are permitted. Besides the trivial case of changing into type I there exist some possible changes between other types. However, physical reasons like energy inequalities partially exclude them. Es werden kosmologische Modelle betrachtet, die anstelle der ublichen raumlichen Homogenitatseigenschaft nur die Bedingung erfullen, das in einem gewissen synchronisierten Bezugssystem alle Raumschnitte t = const. homogene Mannigfaltigkeiten sind. Neben dem trivialen Fall des Wechsels zu Typ I gibt es einige mogliche Anderungen zwischen anderen Typen. Physikalische Grunde wie Energieungleichungen schliesen diese jedoch teilweise aus.

The massive scalar field in a closed Friedmann universe model - new rigorous results

For the minimally coupled scalar field in Einsteins theory of gravitation we look for the space of solutions within the class of closed Friedmann universe models. We prove D ≥ 1, where D ≥ is the dimension of the set of solutions which can be integrated up to t ∞ (D > 0 was conjectured by PAGE (1984)). We discuss concepts like “the probability of the appearance of a sufficiently long inflationary phase” and argue that it is primarily a probability measure μ in the space V of solutions (and not in the space of initial conditions) which has to be applied. μ is naturally defined for Bianchi-type I cosmological models because V is a compact cube. The problems with the closed Friedmann model (which led to controversial claims in the literature) will be shown to originate from the fact that V has a complicated non-compact non-Hausdorff Geroch topology: no natural definition of μ can be given. We conclude: the present state of our universe can be explained by models of the type discussed, but thereby the anthropic principle cannot be fully circumvented. Fur das minimal gekoppelte Skalarfeld in Einsteinscher Gravitationstheorie betrachten wir den Losungsraum in der Klasse der geschlossenen Friedmannmodelle des Universums. Wir beweisen, das D ≥ 1 gilt, wobei D die Dimension der Menge derjenigen Losungen ist, die sich bis t ∞ integrieren lassen. (D > 0 war von PAGE (1984) vermutet worden.) Wir diskutieren Begriffe wie, ”die Wahrscheinlichkeit des Auftretens einer hinreichend langen inflationaren Phase„ und argumentieren, das man primar ein Wahrscheinlichkeitsmas μ im Raum der Losungen V (und nicht im Raum der Anfangswerte) anwenden mus. Fur kosmologische Modelle vom Bianchi-Typ I gibt es eine naturliche Definition fur μ, da dort V ein kompakter Wurfel ist. Die Probleme mit dem geschlossenen Friedmannmodell (die zu Kontroversen in der Literatur fuhrten) ergeben sich aus der Tatsache, das V dort eine komplizierte Geroch-Topologie hat, die weder kompakt noch hausdorffsch ist, so das sich keine naturliche Definition fur μ angeben last. Wir schliesen: Zwar last sich der heutige Zustand unseres Universums durch Modelle des hier diskutierten Typs erklaren, jedoch last sich dabei das anthropische Prinzip nicht vollig umgehen.