Behavior of Bianchi type I spacetime by path integral approach in LQC

We investigate the expansion behavior of Bianchi type I spacetime with a massless scalar field and a cosmological constant using the path integral approach of loop quantum cosmology developed by Ashtekar, Campiglia, and Henderson [Phys. Rev. D 82, 124043 (2010)]. We obtain the effective action for this model and calculate the semiclassical solutions. Although classical general relativity is a good approximation at large scales, it is shown that the loop quantum effect induces a repulsive force at small scales to avoid the big bang singularity as in the isotropic cases. We investigate the evolution of anisotropy in detail and find that the anisotropy cannot increase indefinitely like in the classical case.

Figure 1

The directional scale factors $a_i$ and quantum effect $U_Q$ in case A. Upper left panel: The directional scale factors with parameter $\tau$ The $\mathrm{\text{solid line}}=a_1$, $\mathrm{\text{black stars}}=a_2$, and $\mathrm{\text{white boxes}}=a_3$. Upper right panel: Quantum effect $U_Q=H^2-\Sigma -8\pi G/3\rho -\Lambda /3$. Lower left panel: The directional scale factors $a_i$ with proper time $t$. Lower right panel: Quantum effect $U_Q$ with proper time $t$. It is shown that the quantum effect causes a repulsive force to avoid the big bang singularity.

Figure 2

The LQC variable $b$ and directional Hubble parameter $H_1$ in case A. Left panel: $2lb/\pi$. Right panel: The directional Hubble parameter $H_1$. The solid line denotes the LQC solution. Black stars denote the large scale limit Eq. (65). White boxes denote Eq. (65) $+\pi /l$.

Figure 3

The ${\dot{\theta }}_i$ and their classical trajectories in case A. Upper left panel: ${\dot{\theta }}_1$. Upper right panel: ${\dot{\theta }}_2$. Lower left panel: ${\dot{\theta }}_3$. The solid line denotes the LQC solution, and the black stars denote the classical trajectory ($\propto a^{-3}$) in all figures. Note that because of the quantum effect, LQC solutions become large in excess of the classical trajectories at small scales.

Figure 4

Anisotropy $\Sigma$ and shear scalar $\Sigma ·a^6$ in case A. Left panel: Anisotropy $\Sigma$ defined by Eq. (63), which is proportional to $a^{-6}$ in classical theory. Right panel: $\Sigma ·a^6$, which is constant in classical theory. The solid line denotes the LQC solution, and the black stars denote the classical trajectory. The anisotropy in LQC exceeds the classical one by the quantum effect.

Figure 5

Anisotropy $\Sigma$ and mean scale factor $a$ in cases A, A2, and A3. Left panel: Anisotropy $\Sigma$ defined by Eq. (63). Right panel: The mean scale factor $a=(a_1{a}_2{a}_3{)}^{1/3}$. The $\mathrm{\text{solid line}}=\mathrm{\text{case}}\mathrm{A}$, $\mathrm{\text{black stars}}=\mathrm{\text{case}}\mathrm{A}2$, and $\mathrm{\text{white boxes}}=\mathrm{\text{case}}\mathrm{A}3$. Although case A2 has a larger anisotropy than case A at initial conditions, the anisotropy of case A2 is smaller than that of case A at the bounce.

Figure 6

The scale factors $a_i$, quantum effect $U_Q$, and mean Hubble parameter in case B ($\Lambda >0$). Upper left panel: The directional scale factors with parameter $\tau$. The $\mathrm{\text{solid line}}=a_1$, $\mathrm{\text{black stars}}=a_2$, and $\mathrm{\text{white boxes}}=a_3$. Upper right panel: Quantum effect $U_Q=H^2-\Sigma -8\pi G/3\rho -\Lambda /3$. Lower left panel: The mean scale factor $a=V^{1/3}=(a_1{a}_2{a}_3{)}^{1/3}$. Lower right panel: The mean Hubble parameter $H$. In this case the directional scale factor $a_1$ is monotonically decreasing, and $a_3$ is rapidly increasing like the Kasner solution. However, the mean scale factor $a$, i.e. the volume of the cell, is bouncing, and there is no big bang singularity.

Figure 7

The LQC variable $b$ and the directional Hubble parameter $H_1$ in case B. Left panel: $2lb/\pi$. Right panel: The directional Hubble parameter $H_1$. The solid line denotes the LQC solution. Black stars denote the large scale limit Eq. (65). White boxes denote Eq. (65) $+\pi /l$.

Figure 8

The ${\dot{\theta }}_i$ and their classical trajectories in case B. Upper left panel: ${\dot{\theta }}_1$. Upper right panel: ${\dot{\theta }}_2$. Lower left panel: ${\dot{\theta }}_3$. The solid line denotes the LQC solution, and black stars denote the classical trajectory ($\propto a^{-3}$) in all figures. Unlike case A, LQC solutions are smaller than the classical trajectories at small scales.

Figure 9

Anisotropy $\Sigma$ and shear scalar $\Sigma ·a^6$ in case B. Left panel: Anisotropy $\Sigma$ defined by Eq. (63), which is proportional to $a^{-6}$ in classical theory. Right panel: $\Sigma ·a^6$, which is constant in classical theory. The solid line denotes the LQC solution, and the black stars denote the classical trajectory. Contrary to case A the anisotropy in LQC is suppressed by the quantum effect.

Figure 10

Solution with a negative cosmological constant $\Lambda <0$ (case C). Upper left panel: The directional scale factors. The $\mathrm{\text{solid line}}=a_1$, $\mathrm{\text{black stars}}=a_2$, and $\mathrm{\text{white boxes}}=a_3$. Upper right panel: The quantum effect $U_Q$. Lower left panel: The mean scale factor $a$. Lower right panel: The mean Hubble parameter $H$. Bottom left panel: The directional Hubble scale $H_i$. Because of the negative cosmological constant the scale factors collapse at large scales, so the Universe becomes cyclic.

Figure 11

Anisotropy $\Sigma$, shear scalar $\Sigma ·a^6$, directional Hubble parameter $H_1$, and parameter $b$ in case C. Upper left panel: Anisotropy $\Sigma$. The solid line denotes the LQC solution, and the black stars denote the classical trajectory. Upper right panel: Shear scalar $\Sigma ·a^6$, which is constant in classical theory. Lower left panel: The directional Hubble parameter $H_1$ and the large scale limit. The $\mathrm{\text{solid line}}=\mathrm{\text{the}}\mathrm{LQC}\mathrm{\text{solution}}$, $\mathrm{\text{black stars}}=\mathrm{Eq}.(65)$, $\mathrm{\text{white boxes}}=\mathrm{Eq}.(65)+\pi /l$, and $\mathrm{\text{white triangles}}=\mathrm{Eq}.(65)+2\pi /l$. Lower right panel: The LQC parameter $b$.