Appearance of a classical mixmaster universe from the no-boundary quantum state

We investigate the appearance of the classical anisotropic universe from the no-boundary quantum state according to the prescription proposed by Hartle, Hawking, and Hertog. Our model is homogeneous, anisotropic, closed universes with a minimally coupled scalar field and cosmological constant. We found that there are an ensemble of classical Lorentzian histories with anisotropies and experience inflationary expansion at late time, and the probability of histories with anisotropies are lower than isotropic histories. Thus, the no-boundary condition may be able to explain the emergence of our Universe. If the classical late time histories are extended back, some become singular by the existence of initial anisotropies with large accelerations. However, we do not find any chaotic behavior of anisotropies near the initial singularity.

Figure 1

The real and imaginary part of the complex-Euclidean solution on the X axis. From top to bottom, scale factor $a$, scalar field $\varphi$, anisotropy ${\beta }_{+}$, and ${\beta }_{-}$. Left row is real part, right row is imaginary part of the variables. $\mu =3/4$, ${\varphi }_0=2$, $P_2=0.02$, $M_2=0.15$ and $X=0.871$, $\theta =0.0942$, ${\Theta }_p=3.01$, ${\Theta }_m=-0.133$.

Figure 2

The real and imaginary part of the complex-Euclidean solution on the Y axis. All imaginary parts become zero at large scale.

Figure 3

The real part of ${\beta }_{+}$ vs ${\beta }_{-}$. The universe point comes from high anisotropy to low anisotropy with some bounce. Note that this is a Euclidean solution, not a classical Lorentzian.

Figure 4

The real and imaginary part of the Euclidean action $I$.

Figure 5

Lorentzian history of two models. From top to bottom, $a$, $\varphi$, ${\beta }_{+}$, ${\beta }_{-}$. Left row is model A (high anisotropy), right row is model B (little anisotropy). We use the complex-Euclidean solution at $Y=4$ as an initial value at $t=4$. Model A: $\mu =3/4$, ${\varphi }_0=2$, $P_2=0.02$, $M_2=0.15$. Model B: $\mu =3/4$, ${\varphi }_0=2$, $P_2=0.002$, $M_2=0.015$

Figure 6

${\beta }_{+}$ vs ${\beta }_{-}$ in classical history. Left is model A, right is model B.

Figure 7

The real part of the Euclidean action. Left is model A, right is model B. The probability of model A is smaller than model B.

Figure 8

The area where the classical histories exist in the $P_2-M_2$ plane. White points represent the histories that have initial bounce, black ones have initial singularity. There is a critical line where the initial state changed. And there seems to be an upper limit in each $P_2$, $M_2$.

Figure 9

The relative probability distribution in $P_2-M_2$ plane. Left represents the same area with Fig. 8, right represents the area nearby origin (isotropic).

Figure 10

${\beta }_{+}$ vs ${\beta }_{-}$ at the bounce.

Figure 11

The relative probability distribution in the ${\beta }_{+}-{\beta }_{-}$ plane at the bounce. There is a high probability surface but no clear structure at low probability.

Figure 12

${\beta }_{+}$ vs ${\beta }_{-}$ at large scale in classical history. White region represents bouncing history, black region has initial singularity. Since the anisotropy is constant at large scale (see Fig. 5), this figure represents the observable one.

Figure 13

The relative probability distribution in ${\beta }_{+}-{\beta }_{-}$ at large scale in classical history. The probability distribution is peaked around the isotropic universe.