Solution space of Bianchi type I spacetime in loop quantum cosmology

We investigate the detailed structure of the solution space of the Bianchi type I spacetime with a massless scalar field in loop quantum cosmology (LQC). To analyze the dynamics we use the effective Hamiltonian written by new gauge-invariant variables which have a more direct physical interpretation. Since the effective equations of motion contain a nontrivial quantum correction, we may regard the universes as classical when the dynamics is well-described by the classical Friedmann equation. In addition to solutions which evolve into the classical universes, we find cyclic and stationary solutions which are dominated by the quantum effect for all time. We also show the asymmetric behavior and completely classify the Kasner transition across the bounce moment. We calculate the “probability” of having a classical isotropic universe by using the measure defined by the volume form in the solution space. Although we find that isotropic classical universes are disfavored in our model with a massless scalar field, we argue that LQC can explain the present isotropy if we consider a more realistic model.

Figure 1

Bounce region: $\mathrm{\text{Horizontal axis}}=C_{1}l/\pi$, $\mathrm{\text{longitudinal axis}}=C_{2}l/\pi$. Only this region has values of $b$ that satisfy the bounce condition.

Figure 2

Recollapse region $C_{1}l/\pi -C_{2}l/\pi$. Left: The recollapse region. Only this region has values of $b$ that satisfy the recollapse condition. Right: The combination of the bounce and the recollapse regions. The hexagon pattern is constructed.

Figure 3

The trajectories in the $(C_{1}l/\pi -C_{2}l/\pi )$ plane. Left: Some examples that start at the bounce. Right: All trajectories which bounce in $[0:4]\times [0:4]$.

Figure 4

The trajectories in phase space: $\mathrm{\text{Horizontal axis}}=(C_1=C_2)$ line, $\mathrm{\text{longitudinal axis}}=2bl/\pi$. The upper half of the curves denotes after-bounce trajectories, and the lower half denotes before-bounce trajectories. The three curves on the left have only one bounce, while the other curves denote cyclic universes.

Figure 5

The “only one bounce” region and contour of $M<1.5$ at $Q=0.01$. The $\mathrm{\text{horizontal axis}}=C_{1}l/\pi$, and the $\mathrm{\text{longitudinal axis}}=C_{2}l/\pi$. The contours denote $M=0.1$, 0.5, 1.0, 1.5 sequentially from the inside.

Figure 6

The region with directional Hubble parameters $H_i>0$. Left: After the bounce and $Q=0.01$. The upper region corresponds to $H_1>0$ and $H_3>0$, the middle region (dark region in gray scale) to all $H_i>0$, and the lower region to $H_2>0$ and $H_3>0$. Right: Before the bounce and $Q=0.01$. The upper region corresponds to $H_3>0$. $H_1$ and $H_2$ are negative in all regions. Lower: The solid line denotes $|\delta |=1/2$ and the dots denote $|\delta |=1/\sqrt{3}$.

Figure 7

The volume form ${\widehat{\omega }}_V$ in the $C_1\mathrm{\text{−}}{C}_2$ plane in the case $\Lambda =0$. We have ignored the coefficient ${\nu }^2/16B_1{B}_2$.

Figure 8

The probability of $M<\alpha$. The horizontal axis is $\alpha$ and the longitudinal axis is the “probability.” The red circles denote the probability using the measure of the solution space ${\widehat{\omega }}_V$, and the green triangles denote wrong estimates that only consider the area in the bounce region.

Figure 9

The classical solution space of Bianchi type I spacetime with a massless scalar field. The horizontal axis is ${\nu }_1$ and the vertical axis is ${\nu }_2$. The dotted lines denote ${\nu }_i=1/3$. The area inside the right triangle corresponds to “Point-type,” the shaded region outside the triangle corresponds to “Cigar-type,” and the outline of the triangle corresponds to “Barrel-type.” The ellipse inside the triangle denotes $|\delta |=1/2$.

Figure 10

Solution with the bounce at $(C_{1}l/\pi ,C_{2}l/\pi )=(0.2,1.9)$. Note that we use the harmonic time $\tau$ to plot the figure. Upper left: Volume of the cell $\nu$. We set $\nu =10$ at the bounce. Upper right: Directional Hubble parameters $H_i$. From top to bottom: $H_2$, $H_3$, $H_1$. Middle left: Energy density of the anisotropy $\Sigma$ (upper line) and matter ${\rho }_{\mathrm{mat}}$ (lower line). Middle right: Quantum effect $U_Q$. Lower left: The quantumness parameter $Q$.

Figure 11

Physical quantities at $Q=0.01$ after the bounce. Upper left: The matter energy density $\rho$. Upper right: The anisotropy $\Sigma$. Middle left: The quantum effect $U_q$. Middle right: The cell volume $\nu$. We set $\nu =10$ at the bounce. Lower right: The variable $(C_{1}l/\pi ,C_{2}l/\pi )$ at $Q<0.01$ starting from $[0:1]\times [0:1]$. Note that all $(C_{1}l/\pi ,C_{2}l/\pi )$ except for the last one are at the bounce.

Figure 12

Physical quantities at $Q=0.01$ before the bounce. Upper left: The matter energy density $\rho$. Upper right: The anisotropy $\Sigma$. Middle left: The quantum effect $U_q$. Middle right: The cell volume $\nu$. We set $\nu =10$ at the bounce. Lower left: The variable $(C_{1}l/\pi ,C_{2}l/\pi )$ at $Q<0.01$ starting from $[0:1]\times [0:1]$. Note that all $(C_{1}l/\pi ,C_{2}l/\pi )$ except for the last one are at the bounce.

Figure 13

The bounce and recollapse regions with $\Lambda =8.68\times {10}^{-1}$. Upper left: The bounce region. The red region denotes the $\Lambda =0$ case, and the blue region (dark region in gray scale) denotes the $\Lambda >0$ case. Upper right: The recollapse region. The red region denotes the $\Lambda =0$ case, and the blue region denotes the $\Lambda >0$ case. Lower left: The bounce and recollapse regions with $\Lambda >0$.

Figure 14

The bounce and recollapse regions with $\Lambda =-8.68\times {10}^{-1}$. Upper left: The bounce region. The blue region (dark region in gray scale) denotes the $\Lambda =0$ case, and the red region denotes the $\Lambda <0$ case. Upper right: The recollapse region. The blue region denotes the $\Lambda =0$ case, and the red region denotes the $\Lambda <0$ case. Lower left: The bounce and recollapse regions with $\Lambda <0$. Note that there is an intersection between the bounce and recollapse regions.