Spikes in the mixmaster regime of $G_2$ cosmologies

We produce numerical evidence that spikes in the mixmaster regime of $G_2$ cosmologies are transient and recurring, supporting the conjecture that the generalized mixmaster behavior is asymptotically nonlocal where spikes occur. Higher-order spike transitions are observed to split into separate first-order spike transitions.

Figure 1

The spacetime diagram showing the $x=\mathrm{const}$ worldlines (vertical), the spatial hypersurfaces $\tau =T=\mathrm{const}$ (horizontal) with $\tau =T\to \infty$ at the singularity, the particle horizon (45° lines) of the observer in the center, and the $X=\mathrm{const}$ lines which are timelike inside the particle horizon but spacelike outside (dashed lines).

Figure 2

The Weyl scalar invariants (normalized by the Hubble scalar) for the numerical solution, the matching exact spike solution (with $w=5.8644$, ${\tau }_0=2.2205$), and their difference, during the first spike recurrence over the time interval [0.5, 3.5].

Figure 3

The Weyl scalar invariants (normalized by the Hubble scalar) for the numerical solution, the matching exact spike solution (with $w=1.8329$, ${\tau }_0=6.6029$), and their difference, during the second spike recurrence over the time interval [3.5, 9.5].

Figure 4

The orbit along the spike point $x=0$ and a nearby nonspike point $x=1$ projected onto the $({\Sigma }_{+}^H,{\Sigma }_{-}^H)$ plane for the perturbed spike simulation. The initial (at $T=0$) and final (at $T=16$) points are marked by the letters $i$ and $f$, respectively.

Figure 5

The orbit along the spike point $x=0$ and a nearby nonspike point $x=1$ projected onto the $({\Sigma }_{+}^H,{\Sigma }_{-}^H)$ plane for the generic initial data simulation. The initial (at $T=0$) and final (at $T=40$) points are marked by the letters $i$ and $f$, respectively.

Figure 6

The Weyl scalar invariants (normalized by the Hubble scalar) for the numerical solution, the matching exact spike solution (with $w=-3.7114$, ${\tau }_0=2.6461$), and their difference, during the first spike recurrence over the time interval [0, 5].

Figure 7

The Weyl scalar invariants (normalized by the Hubble scalar) for the numerical solution, the matching exact spike solution (with $w=-4.6228$, ${\tau }_0=38.1923$), and their difference, during the second spike recurrence over the time interval [35, 40].

Figure 8

The orbit along the spike point $x=0$ projected onto the $({\Sigma }_{+}^H,{\Sigma }_{-}^H)$ plane for the perturbed second-order (left figure) and third-order (right figure) spike simulation. The orbits follow the false and the true spike orbits, respectively. The initial (at $T=0$) and final (at $T=12$) points are marked by the letters $i$ and $f$, respectively.

Figure 9

The Weyl scalar invariants (normalized by the Hubble scalar) for the numerical solution, the matching exact false spike solution (with $w=5.5005$, ${\tau }_0=2.3329$), and their difference, during the first false spike recurrence over the time interval [1, 3].

Figure 10

The Weyl scalar invariants (normalized by the Hubble scalar) for the numerical solution, the matching exact false spike solution (with $w=1.5006$, ${\tau }_0=7.6904$), and their difference, during the second false spike recurrence over the time interval [6, 10].

Figure 11

The Weyl scalar invariants (normalized by the Hubble scalar) for the numerical solution, the matching exact spike solution (with $w=-3.5002$, ${\tau }_0=4.4985$), and their difference, during the first spike recurrence over the time interval [3, 6].