Chaos of Yang-Mills field in class A Bianchi spacetimes

Studying the Yang-Mills field and the gravitational field in class A Bianchi spacetimes, we find that chaotic behavior appears in the late phase (the asymptotic future). In this phase, the Yang-Mills field behaves as that in Minkowski spacetime, in which we can understand it by a potential picture, except for types VIII and IX. At the same time, in the initial phase (near the initial singularity), we numerically find that the behavior seems to approach the Kasner solution. However, we show that the Kasner circle is unstable and the Kasner solution is not an attractor. From an analysis of stability and numerical simulation, we find a Mixmaster-like behavior in Bianchi I spacetime. Although this result may provide a counterexample to the Belinskii, Khalatnikov, and Lifshitz (BKL) conjecture, we show that the BKL conjecture is still valid in Bianchi IX spacetime. We also analyze a multiplicative effect of two types of chaos, that is, chaos with the Yang-Mills field and that in vacuum Bianchi IX spacetime. Two types of chaos seem to coexist in the initial phase. However, the effect due to the Yang-Mills field is much smaller than that of the curvature term.

Figure 1

Two typical phases in the Bianchi I spacetime: (a) one is the initial phase ($\Sigma$ dominance) and (b) the other is the late phase (${\Omega }_{\mathrm{Y}\mathrm{M}}$ dominance), where $\Sigma$, ${\Omega }_{\mathrm{Y}\mathrm{M}}$, and $K$ denote the Hubble-normalized shear strength, the Hubble-normalized energy density of the Yang-Mills field, and the Hubble-normalized curvature, respectively. These are constrained by the generalized Friedmann equation: ${\Omega }_{\mathrm{Y}\mathrm{M}}+{\Sigma }^2+K=1$.

Figure 2

The contour of the Yang-Mills field potential $V(A,B,C)$.

Figure 3

Fractal basin boundaries of initial data in Bianchi I spacetime. We set three windows at $|A|=2.8$ (bright gray), $|B|=2.8$ (dark gray), and $|C|=2.8$ (black) in three channels along $A$, $B$, and $C$ axes, where the fates of final state of orbits are determined. Initial conditions are as follows: $\tau =0$, $H=1$, ${\Sigma }_{+}=0.02$, ${\Sigma }_{-}=0.001$, $N_1=N_2=N_3=0$, $a^{\prime }=0.3$, $b^{\prime }=0.5$, $c^{\prime }=0.7$. Further, (a) $A(=a)$ and $C(=c)$ vary from $0.6$ to $1.0$. (b) An enlargement map of a part of (a) [$0.73<A(=a)<0.75$ and $0.75<C(=c)<0.77$]. $B(=b)$ is obtained from Eq. (2.13). In each case, a grid is $200\times 200$.

Figure 4

The fractal dimension of the black region in Fig. 3a. The horizontal axis shows the logarithmic size of boxes (${log}_{10}L$), where $L$ is normalized by a grid size, while the vertical axis gives the logarithmic number of boxes (${log}_{10}{n}_L$) which contain black points. This figure shows the fractal dimension is $D=1.687$.

Figure 5

Time evolution of $E$: (a) Bianchi ${\mathrm{V}\mathrm{I}}_0$ spacetime. $E$ approaches constant for this chaotic system, in which a potential picture is valid. (b) Bianchi ${\mathrm{V}\mathrm{I}\mathrm{I}}_0$ spacetime. $E$ seems to converge within 20% fluctuations. (c) Bianchi VIII spacetime. $E$ seems not to converge.

Figure 6

Fractal basin boundaries of initial data in Bianchi ${\mathrm{V}\mathrm{I}\mathrm{I}}_0$ spacetime. We set three windows at $|A|=2.8$ (bright gray), $|B|=2.8$ (dark gray), and $|C|=2.8$ (black) in three channels along $A$, $B$, and $C$ axes, where the fates of final state of orbits are determined. Initial conditions are as follows: $\tau =0$, $H=1$, ${\Sigma }_{+}=0.02$, ${\Sigma }_{-}=0.001$, $N_1=0.2$, $N_2=0.3$, $N_3=0$, $a^{\prime }=0.3$, $b^{\prime }=0.5$, $c^{\prime }=0.7$. Further, (a) $A(=a)$ and $C(=c)$ vary from $0.6$ to $1.0$. (b) An enlargement map of a part of (a) [$0.88<A(=a)<0.9$ and $0.93<C(=c)<0.95$]. $B(=b)$ is given from Eq. (2.13). In each case, a grid is $200\times 200$.

Figure 7

Fractal basin boundaries of initial data in Bianchi VIII spacetime. We set three windows at $|A|=2.8$ (bright gray), $|B|=2.8$ (dark gray), and $|C|=2.8$ (black) in three channels along $A$, $B$, and $C$ axes, where the fates of final state of orbits are determined. Initial conditions are the following: $\tau =0$, $H=1$, ${\Sigma }_{+}=0.02$, ${\Sigma }_{-}=0.001$, $N_1=0.2$, $N_2=0.3$, $N_3=-0.2$, $a^{\prime }=0.3$, $b^{\prime }=0.5$, $c^{\prime }=0.7$. Further, (a) $A(=a)$ and $C(=c)$ vary from $0.6$ to $1.0$. (b) An enlargement map of a part of (a) [$0.88<A(=a)<0.9$ and $0.93<C(=c)<0.95$]. $B(=b)$ is given from Eq. (2.13). In each case, a grid is $200\times 200$.

Figure 8

An orbit which starts near some point A on the Kasner circle evolves into another point B on the circle in the past direction: (a) The orbit in the ${\Sigma }_{+}\mathrm{\text{−}}{\Sigma }_{-}$ plane. The initial values are $\Sigma =0.81$ and $\Psi =0$ (point A), which is close to the Kasner circle (solid circle). $\Psi$ eventually goes close to $\pi$ (point B, which is slightly less than $\pi$). (b) The time evolution $\Psi$ for the same orbit. The orbit endures for a long time near the Kasner circle, and eventually it moves to another point on the Kasner circle. (c) The same orbit in the ${\Sigma }_{+}\mathrm{\text{−}}(ab{)}^2/H^2$, ${\Sigma }_{+}\mathrm{\text{−}}(bc{)}^2/H^2$, ${\Sigma }_{+}\mathrm{\text{−}}(ca{)}^2/H^2$ planes. The initial values are $\Sigma =0.81$ and $(ab{)}^2/H^2=3.60\times {10}^{-7}$, $(bc{)}^2/H^2=5.76\times {10}^{-2}$, $(ca{)}^2/H^2=1.60\times {10}^{-7}$. $\Psi \sim 0$ results in the initial growth of $(bc{)}^2/H^2$, but $\Psi \sim \pi$ (slightly smaller than $\pi$) results in the decay of $(bc{)}^2/H^2$ and the growth of $(ab{)}^2/H^2$ in the final stage.