Spike oscillations

According to Belinskiǐ, 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 support this conjecture but with a modification: Apart from local BKL behavior there also exists formation of spatial structures (“spikes”) at, and in the neighborhood of, certain spatial surfaces that break asymptotic locality; the complete description of a generic spacelike singularity involves spike oscillations, which are described by sequences of Bianchi type I and certain inhomogeneous vacuum models. In this paper we describe how BKL and spike oscillations arise from concatenations of exact solutions in a Hubble-normalized state space setting, suggesting the existence of hidden symmetries and showing that the results of BKL are part of a greater picture.

Figure 1

The division of the Kasner circle ${\mathrm{K}}^{◯}$ of fixed points into six equivalent sectors and six LRS fixed points ${\mathrm{T}}_{\alpha }$ and ${\mathrm{Q}}_{\alpha }$, $\alpha =1$, 2, 3. Sector ($\alpha \beta \gamma$) is defined by ${\Sigma }_{\alpha }<{\Sigma }_{\beta }<{\Sigma }_{\gamma }$. The values of the extended Kasner parameter $\check{u}$, cf. (17), along ${\mathrm{K}}^{◯}$ are indicated by the values of $\check{u}$ at the LRS fixed points; in addition, $k=2\check{u}+1$ is given.

Figure 2

Projections of the two types of single frame transitions, ${\mathcal{T}}_{R_1}$ and ${\mathcal{T}}_{R_3}$, that exist in the $G_2$ case onto $({\Sigma }_1,{\Sigma }_2,{\Sigma }_3)$ space. Throughout this paper, the direction of time, as indicated by the arrows, is toward the past.

Figure 3

Projections onto $({\Sigma }_1,{\Sigma }_2,{\Sigma }_3)$ space of the single curvature transitions ${\mathcal{T}}_{N_1}$.

Figure 4

High velocity spike transitions ${\mathcal{T}}_{\mathsf{H}\mathsf{i}}$ and low velocity spike solutions ${\mathcal{S}}_{\mathsf{L}\mathsf{o}}$ projected onto $({\Sigma }_1,{\Sigma }_2,{\Sigma }_3)$ space. Each ${\mathcal{T}}_{\mathsf{H}\mathsf{i}}$ or ${\mathcal{S}}_{\mathsf{L}\mathsf{o}}$ corresponds to a family of curves parametrized by the spatial coordinate $z$. The thick (straight) lines represent the trajectories of the time lines of the spike surface $z=0$, i.e., spike surface trajectories ${\mathcal{T}}_S$. The thin curves represent the trajectories of time lines $0\ne |z|\gg ⃒1$. The thin straight lines represent $|z|\gg 1$ time lines; these are short BKL chains, i.e., sequences of curvature-frame transitions. The dashed lines represent constant time slices of a solution.

Figure 5

The “skeleton” of a ${\mathcal{S}}_{\mathsf{L}\mathsf{o}}$ spike solution described by the spike surface trajectory ${\mathcal{T}}_S$ (thick line) and the short BKL chain representing the trajectories associated with $|z|\gg 1$, cf. Fig. 4; in addition the dashed line $S_A$ corresponds to a “missing” curvature transition ${\mathcal{T}}_{N_1}$ that would reunite the spike surface $z=0$ with the $z\ne 0$ trajectories.

Figure 6

Projections of spike surface trajectories ${\mathcal{T}}_S$ onto $({\Sigma }_1,{\Sigma }_2,{\Sigma }_3)$ space.

Figure 7

In the OT case each sector is associated with a “trigger” ($N_1$ or $R_3$) except for sector (312) at which these variables are stable. As a consequence, OT BKL chains (i.e., concatenations of ${\mathcal{T}}_{N_1}$ and ${\mathcal{T}}_{R_3}$ transitions) are finite and terminate at sector (312).

Figure 8

OT spike chains in the projection onto $({\Sigma }_1,{\Sigma }_2,{\Sigma }_3)$ space; depicted are chains of the spike surface trajectories ($z=0$) and trajectories associated with $|z|\gg 1$ (which are OT BKL chains). An OT spike chain is a high velocity chain (i.e., a concatenation of ${\mathcal{T}}_{\mathsf{H}\mathsf{i}}$ transitions and ${\mathcal{T}}_{R_3}$ frame transitions) ending with a ${\mathcal{T}}_{R_3}$ transition (a) or a ${\mathcal{T}}_{\mathsf{H}\mathsf{i}}$ transition (b); or there is one additional (final) transition: a low velocity solution ${\mathcal{S}}_{\mathsf{L}\mathsf{o}}$ (c) or a $\mathsf{B}\mathsf{i}\mathsf{l}\mathsf{l}$ spiky feature (d). The latter is depicted by a dashed line.

Figure 9

Triggers for the evolution of the spike surface $z=0$ in the OT case. In sectors (213), (231), and (321), $R_3$ triggers frame transitions ${\mathcal{T}}_{R_3}$. Since $N_1=0$ at $z=0$, $N_1$ does not exist as a trigger; instead spatial gradients ($S$) take the role as triggers and lead to a spike surface trajectory ${\mathcal{T}}_S$; see Fig. 6. These triggers cover sectors (312) and (132) with $u>2$. Sector (312) and sector (132) with $u<2$ are stable in this context.

Figure 10

Triggers for general $G_2$ models. Figures (a) and (b) generalize Figs. 7a, 9, respectively.

Figure 11

A BKL chain is an (in general) infinite heteroclinic chain consisting of ${\mathcal{T}}_{N_1}$, ${\mathcal{T}}_{R_1}$ and ${\mathcal{T}}_{R_3}$ curvature and frame transitions. The paths are not unique; e.g., at the point $X$ two continuations are possible, a frame transition ${\mathcal{T}}_{R_1}$ or a curvature transition ${\mathcal{T}}_{N_1}$ (dashed line). However, the different paths rejoin at the point $Y$ after an additional curvature-frame transition. At the point $Y$ the chain continues with a (not shown) ${\mathcal{T}}_{R_3}$ transition.

Figure 12

A joint low or high velocity spike transition ${\mathcal{T}}_{\mathsf{J}\mathsf{o}}$ consists of a low velocity spike solution ${\mathcal{S}}_{\mathsf{L}\mathsf{o}}$ followed by a family of ${\mathcal{T}}_{R_1}$ frame transitions and the “second part” of a high velocity spike transition ${\mathcal{T}}_{\mathsf{H}\mathsf{i}}$. In (a) we depict the numerical evolution of four time lines associated with different values of $z$ from $z=0$ (spike surface trajectory) to $|z|\gg 1$ (BKL chain). Subfigure (b) shows the curves of constant time (dashed lines) for a ${\mathcal{T}}_{\mathsf{J}\mathsf{o}}$, which correspond to segments of the straight lines representing ${\mathcal{T}}_{N_1}$ transitions.

Figure 13

Examples of (parts of) spike chains. In (a) the continuous lines represent the spike time line ($z=0$ trajectory) of ${\mathcal{T}}_{\mathsf{H}\mathsf{i}}\mathrm{\text{−}}{\mathcal{T}}_{R_3}\mathrm{\text{−}}{\mathcal{T}}_{R_1}\mathrm{\text{−}}{\mathcal{T}}_{\mathsf{H}\mathsf{i}}\mathrm{\text{−}}{\mathcal{T}}_{R_3}\mathrm{\text{−}}{\mathcal{T}}_{R_1}\mathrm{\text{−}}{\mathcal{T}}_{R_3}$; the dashed lines are ${\mathcal{T}}_{R_1}\mathrm{\text{−}}{\mathcal{T}}_{R_3}\mathrm{\text{−}}{\mathcal{T}}_{R_1}$. In (b) we have ${\mathcal{T}}_{\mathsf{H}\mathsf{i}}\mathrm{\text{−}}{\mathcal{T}}_{R_3}\mathrm{\text{−}}{\mathcal{T}}_{\mathsf{J}\mathsf{o}}$ (where ${\mathcal{T}}_{\mathsf{J}\mathsf{o}}$ consists of three pieces: ${\mathcal{S}}_{\mathsf{L}\mathsf{o}}\mathrm{\text{−}}{\mathcal{T}}_{R_1}\mathrm{\text{−}}{\mathcal{T}}_{\mathsf{H}\mathsf{i}}$).

Figure 14

A Kasner point $O$ in sector (132) with $1/2<\check{u}<1$ is the initial point of a joint low or high velocity spike transition ${\mathcal{T}}_{\mathsf{J}\mathsf{o}}$ (where the continuous line represents the time line of the spike surface); however, there is also a ${\mathcal{T}}_{R_1}$ frame transition emerging from $O$, which can be continued with a ${\mathcal{T}}_{N_1}$ curvature transition and another ${\mathcal{T}}_{R_1}$ frame transition. The two trajectories meet at the same final Kasner point.