Phenomenological dynamics of loop quantum cosmology in Kantowski-Sachs spacetime

The fundamental theory and the semiclassical description of loop quantum cosmology (LQC) have been studied in the Friedmann-Robertson-Walker and Bianchi I models. As an extension to include both anisotropy and intrinsic curvature, this paper investigates the cosmological model of Kantowski-Sachs spacetime with a free massless scalar field at the level of phenomenological dynamics with the LQC discreteness corrections. The LQC corrections are implemented in two different improved quantization schemes. In both schemes, the big bang and big crunch singularities of the classical solution are resolved and replaced by the big bounces when the area or volume scale factor approaches the critical values in the Planck regime measured by the reference of the scalar field momentum. Symmetries of scaling are also noted and suggest that the fundamental spatial scale (area gap) may give rise to a temporal scale. The bouncing scenarios are in an analogous fashion of the Bianchi I model, naturally extending the observations obtained previously.

Figure 1

Classical solution. The initial condition is given at ${\varphi }_0={\varphi }_{\max }$ with $p_b({\varphi }_0)=p_{b,\max }=\sqrt{K_c^2+K_{\varphi }^2}=5.\times {10}^5{\ell }_{\mathrm{Pl}}^2$ (${\ell }_{\mathrm{Pl}}≔\sqrt{G\hslash }$ ), $p_c({\varphi }_0)=4.\times {10}^5{\ell }_{\mathrm{Pl}}^2$, $K_c=3.\times {10}^5{\ell }_{\mathrm{Pl}}^2$, and $K_{\varphi }=4.\times {10}^5{\ell }_{\mathrm{Pl}}^2$ (i.e. $p_{\varphi }=4.\times {10}^5\hslash \sqrt{4\pi G}$). (a) $p_b(\varphi )$ and $p_c(\varphi )=g_{\Omega \Omega }(\varphi )$. (b) $g_{xx}(\varphi )$. (c) $\mathbf{V}(\varphi )$. (d) $K_b(\varphi )$, which asymptotically approaches $K_{b\pm }$, as indicated by dashed lines.

Figure 2

Solution in the $\overline{\mu }$-scheme phenomenological dynamics. With the same initial condition as given in Fig. 1 (and the Barbero-Immirzi parameter set to $\gamma =1$). (a) $p_b(\varphi )$ and $p_c(\varphi )=g_{\Omega \Omega }(\varphi )$. $p_b$ is periodic while $p_c$ only bounces 3 times and grows up (with wiggled motion) toward infinity in the far future and past. (b) $g_{xx}(\varphi )$. (c) ${\varrho }_b(\varphi )$, which signals the occurrence of big bounces of $p_b$ when approaching ${\varrho }_{b\pm ,\mathrm{crit}}$ given by (3.29) and indicated by dashed lines. (d) ${\varrho }_c(\varphi )$, which signals the occurrence of big bounces of $p_c$ when approaching ${\varrho }_{c,\mathrm{crit}}$ given by (3.24) and indicated by the dashed line. (e) $\cos ({\overline{\mu }}_{b}b)$: the periodic curve; $\cos ({\overline{\mu }}_{c}c)$: the curve which flips signs only 3 times (for 3 bounces) and eventually remains $-1$ in the future and $+1$ in the past. (f) ${\overline{K}}_b(\varphi )$, which becomes flat whenever $p_b$ undergoes the bounce. (g) Effective $K_c$ with the constants $\pm K_c$ indicated by dashed lines. [See (3.32).] (h) Effective $K_b$ with the constants $K_{b\pm }(\pm K_c)$ indicated by dashed lines. [See (3.33).]

Figure 3

Solution in the ${\overline{\mu }}^{\prime }$-scheme phenomenological dynamics. Same initial condition as given in Fig. 1 (and $\gamma =1$). (a) $p_b(\varphi )$ and $p_c(\varphi )=g_{\Omega \Omega }(\varphi )$. The bounces of $p_b$ and $p_c$ occur around the same moments. $p_b$ and $p_c$ both oscillate (more and more rapidly) toward the future and past: $p_b$ grows up while $p_c$ descends into deep Planck regime, at which the semiclassical description can no longer be trusted. (b) $g_{xx}(\varphi )$. (c) ${\rho }_{\varphi }(\varphi )$, which signals the occurrence of big bounces. The critical values given by (3.76, 3.77) with different effective $K_c$ are indicated by dashed lines. The epochs of big bounces are very close to the moments when the curve ${\rho }_{\varphi }$ intersects one of the dashed lines. (d) $\cos ({\overline{\mu }}_b^{\prime }b)$ and $\cos ({\overline{\mu }}_c^{\prime }c)$, fairly close to each other. (e) $f(\varphi )$. $f\approx 0$ in the classical cycle around $\varphi ={\varphi }_0$ but $f$ yields the time-varying signature given by (3.54) in the adjacent classical cycles. (f) ${\overline{K}}_b^{\prime }(\varphi )$, which remains almost constant across big bounces. (g) Effective $K_c$ with the constants given by (3.65) indicated by dashed lines. (h) Effective $K_b$ with the asymptotic values given by (3.66) indicated by dashed lines.

Figure 4

(a) The surface in pink is ${\square }_{\theta \varphi }$, the physical area of which is to be shrunk to $\Delta$ in the ${\overline{\mu }}^{\prime }$-scheme. (b) The surfaces in blue are ${⊡}_{\theta }$ and ${⊡}_{\varphi }$, the physical areas of which are to be shrunk to $\Delta$ in the $\overline{\mu }$-scheme.