Abstract:
In this paper we explore the relationship between the existence of eternal inflation and the initial conditions leading to inflation. We demonstrate that past and future completion of inflation is related, in that past-incomplete inflation can not be future eternal. Bubble universes nucleating close to the initial conditions hypersurface have the largest Lorentz boosts and experience the highest anisotropy. Consequently, their probability to collide upon formation is one. Thus instead of continuing eternally inflation ends soon after it starts. The difficulty in actualizing eternal inflation originates from the breaking of two underlying symmetries: Lorentz invariance and the general covariance of the theory which lead to an inconsistency of Einstein equations. Eternal inflation may not be eternal.

Abstract:
The basic workings of inflationary models are summarized, along with the arguments that strongly suggest that our universe is the product of inflation. It is argued that essentially all inflationary models lead to (future-)eternal inflation, which implies that an infinite number of pocket universes are produced. Although the other pocket universes are unobservable, their existence nonetheless has consequences for the way that we evaluate theories and extract consequences from them. The question of whether the universe had a beginning is discussed but not definitively answered. It appears likely, however, that eternally inflating universes do require a beginning.

Abstract:
``Eternal inflation'' is often confused with ``chaotic inflation''. Moreover, the term ``chaotic inflation'' is being used in three different meanings (all of them unrelated to eternal inflation). I make some suggestions in an effort to untangle this terminological mess. I also give a brief review of the origins of eternal inflation.

Abstract:
We study various probability measures for eternal inflation by applying their regularization prescriptions to models where inflation is not eternal. For simplicity we work with a toy model describing inflation that can interpolate between eternal and non-eternal inflation by continuous variation of a parameter. We investigate whether the predictions of four different measures (proper time, scale factor cutoff, stationary and causal {diamond}) change continuously with the change of this parameter. We will show that {only} for the stationary measure the predictions change continuously. For the proper-time and the scale factor cutoff, the predictions are strongly discontinuous. For the causal diamond measure, the predictions are continuous only if the stage of the slow-roll inflation is sufficiently long.

Abstract:
We investigate the condition for eternal inflation to take place in the noncommutative spacetime. We find that the possibility for eternal inflation's happening is greatly suppressed in this case. If eternal inflation cannot happen in the low energy region where the noncommutativity is very weak (the UV region), it will never happen during the whole inflationary history. Based on these conclusions, we argue that an initial condition for eternal inflation is available from the property of spacetime noncommutativity.

Abstract:
We discuss cosmological models for an eternal universe. Physical observables show no singularity from the infinite past to the infinite future. While the universe is evolving, there is no beginning and no end - the universe exists forever. The early state of inflation is described in two different, but equivalent pictures. In the freeze frame the universe emerges from an almost static state with flat geometry. After entropy production it shrinks and "thaws" slowly from a "freeze state" with extremely low temperature. The field transformation to the second "big bang picture" (Einstein frame) is singular. This "field singularity" is responsible for an apparent singularity of the big bang. Furthermore, we argue that past-incomplete geodesics do not necessarily indicate a singularity or beginning of the universe. Proper time ceases to be a useful concept for physical time if particles become massless. We propose to define physical time by counting the number of zeros of a component of the wave function. This counting is independent of the choice of coordinates and frames, and applies to massive and massless particles alike.

Abstract:
Models of eternal inflation predict a stochastic self-similar geometry of the universe at very large scales and allow existence of points that never thermalize. I explore the fractal geometry of the resulting spacetime, using coordinate-independent quantities. The formalism of stochastic inflation can be used to obtain the fractal dimension of the set of eternally inflating points (the ``eternal fractal''). I also derive a nonlinear branching diffusion equation describing global properties of the eternal set and the probability to realize eternal inflation. I show gauge invariance of the condition for presence of eternal inflation. Finally, I consider the question of whether all thermalized regions merge into one connected domain. Fractal dimension of the eternal set provides a (weak) sufficient condition for merging.

Abstract:
In this work we have carried out an approach between the nonsingular scientific cosmologies (without the initial singularity, the big bang), specially the cyclic models, and the Nietzsche's thought of the eternal recurrence. Moreover, we have pointed out reasons for the Nietzsche's search for scientific proofs about the eternal recurrence in the decade of 1880's.

Abstract:
As a result of discussions with Bousso and Vilenkin I want to return to the question of whether the multiverse is past-eternal or if there was a beginning. Not surprisingly, given three people, there were three answers. However, the discussions have led to some common ground. The multiverse being past-eternal, or at least extremely old has content and potential phenomenological implications. I will discuss how the oldness of the multiverse is connected with recent speculations of Douglas.

Abstract:
Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique covering numbers of graph are explored in this paper. Among other results, we characterize bipartite and triangle-free graphs with domination and eternal domination numbers equal to two, trees with equal m-eternal domination and clique covering numbers, and two classes of graphs with equal domination, eternal domination and clique covering numbers.