Exact solutions for a big bounce in loop quantum cosmology

In this paper we study the flat ($k=0$) cosmological Friedmann-Robertson-Walker model with holonomy corrections of loop quantum gravity. The considered universe contains a massless scalar field and the cosmological constant $\Lambda$. We find analytical solutions for this model in different configurations and investigate its dynamical behavior in the whole phase space. Such an approach might be significant e.g. as a phenomenological reference for a further, fully quantum treatment. We show the explicit influence of $\Lambda$ on the qualitative and quantitative character of solutions. Even in the case of positive $\Lambda$ the oscillating solutions without the initial and final singularity appear as a generic case for some quantization schemes.

Figure 1

Typical solution with $\Lambda =0$ in the $\overline{\mu }$ scheme. The canonical variable $p$ is expressed in $[l_{\mathrm{Pl}}^2]$ units and time $t$ in $[l_{\mathrm{Pl}}]$ units.

Figure 2

Typical solution with $\Lambda =0$ in the ${\mu }_0$ scheme. The canonical variable $p$ is expressed in $[l_{\mathrm{Pl}}^2]$ units and time $t$ in $[l_{\mathrm{Pl}}]$ units.

Figure 3

Parameters $y_1$ and $y_2$ as functions of $\Lambda$. Region 1 corresponds to oscillatory solution, region 2 to bouncing solution. There is no physical solution in region 3. The parameter $\Lambda$ is expressed in units of $[m_{\mathrm{Pl}}^2]$.

Figure 4

Parametric solution with $\Lambda <0$. Oscillating curve (green) is $|p(u)|$ and the increasing curve (red) is $t(u)$. The canonical variable $p$ is expressed in $[l_{\mathrm{Pl}}^2]$ units and time $t$ in $[l_{\mathrm{Pl}}]$ units.

Figure 5

Solution $|p(t)|$ with $\Lambda <0$. The canonical variable $p$ is expressed in units of $[l_{\mathrm{Pl}}^2]$ and time $t$ in $[l_{\mathrm{Pl}}]$.

Figure 6

Parametric solution with $0<\Lambda <{\Lambda }_{0,\overline{\mu }}$. Top curve (green) shows $|p(u)|$ and bottom curve (red) is $t(u)$. The canonical variable $p$ is expressed in units of $[l_{\mathrm{Pl}}^2]$ and time $t$ in $[l_{\mathrm{Pl}}]$.

Figure 7

Solution $|p(t)|$ with $0<\Lambda <{\Lambda }_{0,\overline{\mu }}$. The canonical variable $p$ is expressed in units of $[l_{\mathrm{Pl}}^2]$ and time $t$ in $[l_{\mathrm{Pl}}]$.

Figure 8

Parametric solution $|p(u)|$ with $\Lambda <0$. The canonical variable $p$ is expressed in units of $[l_{\mathrm{Pl}}^2]$ and $u$ is dimensionless.

Figure 9

Parametric solution $|p(u)|$ with $\Lambda >0$ and relation (69) satisfied. The canonical variable $p$ is expressed in units of $[l_{\mathrm{Pl}}^2]$ and $u$ is dimensionless.

Figure 10

The phase space diagram for the case $n=0$, and (a) $\Lambda >0$ and (b) $\Lambda <0$. The physical domain admissible for motion is shaded. Note that for the case (a) the boundary of admissible for motion is bounded by a homoclinic orbit and all solutions in this area are oscillating without initial and final singularities. All trajectories situated in the physical region possess the minimum value of scale factor.

Figure 11

The phase space diagram for the case $n=-1/2$, and (a) $\Lambda =\frac{1}{{\gamma }^2{\xi }^2}<\frac{3}{{\gamma }^2{\xi }^2}={\Lambda }_0$ and (b) $\Lambda =\frac{5}{{\gamma }^2{\xi }^2}>\frac{3}{{\gamma }^2{\xi }^2}={\Lambda }_0$. The physical domain admissible for motion is shaded. In the second case, there is no physically allowed region for which $\alpha >0$. This case is distinguished by behavior at infinity when Hubble function is finite like for the de Sitter solution [see formula (90)]. This state is the global attractor in the future. All solutions are of the bouncing type.