Figure 13 - uploaded by Claes Uggla
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Examples of (parts of) spike chains. In (a) the continuous lines represent the spike timeline (z = 0 trajectory) of T Hi-T R3-T R1-T Hi-T R3-T R1-T R3 ; the dashed lines are T R1-T R3-T R1. In (b) we have T Hi-T R3-T Jo (where T Jo consists of three pieces: S Lo-T R1-T Hi ).
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According to Belinskii, Khalatnikov and Lifshitz (BKL), a generic spacelike
singularity is characterized by asymptotic locality: Asymptotically, toward the
singularity, each spatial point evolves independently from its neighbors, in an
oscillatory manner that is represented by a sequence of Bianchi type I and II
vacuum models. Recent investigations...
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Citations
... Relaxing the orthogonal transitivity condition allows multiple Kasner eras and a non-terminating sequence of transitions. The permanent spike is replaced by the so-called joint spike transition [8], which is a more elaborate spike transition that straddles two Kasner eras. The results of [7] is therefore incomplete until a new exact solution describing the joint spike transition is found and numerically matched. ...
We produce numerical evidence that the joint spike transitions between Kasner eras of $G_2$ cosmologies are described by the non-orthogonally transitive $G_2$ spike solution. A new matching procedure is developed for this purpose.
... Due to this power law dependence of the physical variables in the scalar t, the evolution of these models back in time towards this initial singularity is rather straightforwards, although that does not mean that there could not be some interesting structure near the initial singularity (see, for example, [30]). However, there cannot be any unusual behaviour arising from evolution alone such as oscillatory temporal behaviour (see, for example, [9]) or spikes (see, for example, [31]). ...
The Einstein field equations for a class of irrotational non-orthogonally transitive G 2 cosmologies are written down as a system of partial differential equations. The equilibrium points are self-similar and can be written as a one-parameter, five-dimensional, ordinary differential equation (ODE). The corresponding cosmological models both evolve and have one-dimension of inhomogeneity. The major mathematical features of this ODE are derived, and a cosmological interpretation is given. The relationship to the exceptional Bianchi models is explained and exploited to provide a conjecture about future generalizations. It is assumed throughout that the source is a perfect fluid with a linear equation of state.
... The non-local nature of spikes brings the the local nature of the conjecture into doubt. See [5] for a comprehensive introduction. Since 2012 the focus has shifted to the possible role of spikes in the formation of large-scale structures as the Universe expands [6][7][8]. ...
... Are there simple enough seed solutions that generate spiky solutions with such isometries? Lastly, we are working on applying the new technique to the exact vacuum non-OT G 2 spike solution from [9] and the stiff spike solution from [10], to help improve the numerical simulation and matching in an upcoming paper that is an extension of [5,24]. ...
The Geroch/Stephani transformation is a solution-generating transformation, and may generate spiky solutions. The spikes in solutions generated so far are either early-time permanent spikes or transient spikes. We want to generate a solution with a late-time permanent spike. We achieve this by applying Stephani’s transformation with the rotational Killing vector field (KVF) of the locally rotationally symmetric Jacobs solution. The late-time permanent spike occurs along the cylindrical axis. Using a mixed KVF, the generated solution also features a rich variety of transient structures. We introduce a new technique to analyse these structures. Our findings lead us to discover a transient behaviour, which we call the overshoot transition. These discoveries compel us to revise the description of transient spikes.
... The structures turn out to be permanent spikes, which essentially arise from a discontinuous limit among Kasner solutions towards the initial big bang singularity (at the bang time for brevity). Further numerical studies reveal recurring transient structures which appear more spiky towards the bang time (so-called high-velocity spikes [3] and joint low/high velocity spike transitions [4,5]). Spiky structures present a possible classical (non-quantum mechanical) way to seed large scale structure formation in the Universe [6][7][8]. ...
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In this paper, we expand our previous work [1], using the entire family of Bianchi type V solutions as seed solutions of the Stephani transformation. Among the generated solutions, we observe a number of interesting phenomena. The most interesting phenomenon is the intersecting spikes. Other interesting phenomena are saddle states and close-to-FL epoch.
... The first alternative is to form a 'permanent' spike, followed by an R 1 transition, and lastly to resolve the spike. This was described in [6] as the joint spike transition. This alternative is more commonly encountered (assuming that permanent spikes are more commonly encountered than no-spike at the end of a Kasner era). ...
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Principles in the form of heuristic guidelines or generally accepted dogma
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... Presumably asymptotic silence is a necessary requirement for asymptotic locality, but it is not sufficient. As we will see, generic singularities are not only associated with asymptotic locality, but also with recurring spikes, non-local evolution along certain timelines that is described by PDEs [15,16,17], and it seems that this type of behaviour is also associated with asymptotic silence (i.e., ultra-strong gravity does not always lead to asymptotic locality, nor to asymptotic silence as illustrated by null singularities); for various examples of asymptotic silence and asymptotic silence-breaking, see [14]. ...
... It follows that the singularity occurs at τ → −∞ and that t −t =Ĥ −1 exp(3τ )/3. By choosing a gauge in which the temporal constantt is set to zero, so that the singularity occurs at t = 0, and by scalingẽ α i with the temporal constantĤ −1 appropriately, we obtain the previous result (17). ...
... As in the case of SH class A models, transitions can be concatenated to yield heteroclinic chains [13,17]. In the present case this means that T N1 , T R1 and T R3 transitions form (BKL) 'Iwasawa chains,' which describe the oscillatory evolution along individual timelines. ...
This is a review of recent progress concerning generic spacelike
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... In a recent paper [9] it was shown that recurring spike formation could be analytically described by means of combining the exact inhomogeneous solutions found by Lim [7] into concatenation blocks that connect families of timelines affected by spike formation at common initial and final Kasner states. It was also shown that the concatenation blocks come in two versions, so-called high velocity transitions, T Hi , and joint low/high velocity spike transitions, T Jo , where the word transition, as in the BKL case, signifies a change from one Kasner state to another. ...
... Let us now consider the Kasner map that is induced by the spike transitions T Hi and T Jo . It is shown in [9] that, remarkably, the two types of transitions give rise to the same Kasner map which we refer to as the spike map. This map is obtained by applying the BKL map twice, see [9], which results in the following relations: ...
... It is shown in [9] that, remarkably, the two types of transitions give rise to the same Kasner map which we refer to as the spike map. This map is obtained by applying the BKL map twice, see [9], which results in the following relations: ...
In this paper we explore stochastical and statistical properties of so-called
recurring spike induced Kasner sequences. Such sequences arise in recurring
spike formation, which is needed together with the more familiar BKL scenario
to yield a complete description of generic spacelike singularities. In
particular we derive a probability distribution for recurring spike induced
Kasner sequences, complementing similar available BKL results, which makes
comparisons possible. As examples of applications, we derive results for
so-called large and small curvature phases and the Hubble-normalized Weyl
scalar.
A quantum analysis of the vacuum Bianchi IX model is performed, focusing in particular on the chaotic nature of the system. The framework constructed here is general enough for the results to apply in the context of any theory of quantum gravity, since it includes only minimal approximations that make it possible to encode the information of all quantum degrees of freedom in the fluctuations of the usual anisotropy parameters. These fluctuations are described as canonical variables that extend the classical phase space. In this way, standard methods for dynamical systems can be applied to study the chaos of the model. Two specific methods are applied that are suitable for time-reparametrization invariant systems. First, a generalized version of the Misner-Chitre variables is constructed, which provides an isomorphism between the quantum Bianchi IX dynamics and the geodesic flow on a suitable Riemannian manifold, extending, in this way, the usual billiard picture. Second, the fractal dimension of the boundary between points with different outcomes in the space of initial data is numerically analyzed. While the quantum system remains chaotic, the main conclusion is that its strength is considerably diminished by quantum effects as compared to its classical counterpart.
