Effective dynamics of the hybrid quantization of the Gowdy $T^3$ universe

The quantum dynamics of the linearly polarized Gowdy $T^3$ model (compact inhomogeneous universes admitting linearly polarized gravitational waves) is analyzed within loop quantum cosmology by means of an effective dynamics. The analysis, performed via analytical and numerical methods, proves that the behavior found in the evolution of vacuum (homogeneous) Bianchi I universes is preserved qualitatively also in the presence of inhomogeneities. More precisely, the initial singularity is replaced by a big bounce which joins deterministically two large classical universes. In addition, we show that the size of the universe at the bounce is at least of the same order of magnitude (roughly speaking) as the size of the corresponding homogeneous universe obtained in the absence of gravitational waves. In particular, a precise lower bound for the ratio of these two sizes is found. Finally, the comparison of the amplitudes of the gravitational wave modes in the distant future and past shows that, statistically (i.e., for large samples of universes), the difference in amplitude is enhanced for nearly homogeneous universes, whereas this difference vanishes in inhomogeneity-dominated cases. The presented analysis constitutes the first systematic effective study of an inhomogeneous system within loop quantum cosmology, and it proves the robustness of the results obtained for homogeneous cosmologies in this context.

Figure 1

In this plot, we show all positive roots of Eq. (5.1) in a logarithmic scale for the particular values ${\Theta }_{\delta }=800l_p^2$ and ${\Theta }_{\sigma }=100l_p^2$. In (a) $\alpha =0.01$, whereas in (b) $\alpha =5\times {10}^{-4}$. The dashed line corresponds to the section $F={\Theta }_{\delta }{\Theta }_{\sigma }$.

Figure 2

The ratio between the minimal root and the value of the fixed homogeneous bounce $v_h={10}^5$ in terms of $\alpha$ and $\beta$.

Figure 3

The ratio $v_{\min }/v_h$ for a fixed value of $\beta =0.5$. In the first plot, $v_h$ appears in a logarithmic scale, but in a linear scale in the others. The second and third plots are in fact the same, only the perspective is modified.

Figure 4

The minimum bounce point $v_{\min }$ divided by its homogeneous counterpart $v_h$ for a fixed value of $\alpha =0.1$.

Figure 5

The ratio $v_{\min }/v_h$ for fixed values of $\alpha =0.1$ and $\beta =0.5$. This is a section of the previous plots which shows the infimum of $v_{\min }\approx 0.05v_h$.

Figure 6

The black (continuous) curves are dynamical effective trajectories which, after bouncing in all the three directions, converge to the classical trajectories displayed in (dashed) red. The initial data are $v_{\theta }={10}^6{l}_p^2$, $v_{\delta }={10}^4{l}_p^2$, and $v_{\sigma }={10}^4{l}_p^2$, with constants of motion ${\Theta }_{\delta }=300l_p^2$ and ${\Theta }_{\sigma }=100l_p^2$. Only five inhomogeneity modes have been excited. In order to obtain initial conditions for those inhomogeneities, the width of the Gaussian random number generator has been chosen to be ${10}^{-1}$. This has led to the initial values $H_0^{\xi }\approx 0.3846\hslash$ and $H_{\mathrm{int}}\approx 0.02719\hslash$ ($\alpha \approx 0.0707$). The two classical trajectories are related by a flip of sign in the constants ${\Theta }_{\delta }$ and ${\Theta }_{\sigma }$.

Figure 7

An example of the near-critical behavior of the evolution near the throat formed by different branches of roots. The system bounces and recollapses several times before going back to the region with large values of $v_{\theta }$.

Figure 8

Typical evolution of the amplitude $|a_m|$ of two modes of the inhomogeneities. During the evolution, away from the bounce, its mean value (obtained by subtracting the oscillations) is conserved. However, we see two distinct types of behavior for the change of this mean value trough the bounce in $v_{\theta }$. In case (a), the mean value changes by an amount which is almost one amplitude of the oscillations, whereas it remains practically unchanged in case (b).

Figure 9

The dependence of the change of the amplitude through the bounce on the initial phase shift of the mode.

Figure 10

In these plots we show, in a linear and a logarithmic scale respectively, the absolute and the relative change in the amplitude of the inhomogeneities $|a_m|$ produced when the bounce is crossed, represented against the ratio between the actual bounce point $v_b$ and its homogeneous counterpart $v_h$. We note that, for small $v_b/v_h$ (almost homogeneous universe), $\Delta |a_m|$ can exceed the unity, regime which corresponds to the situation when the inhomogeneities are amplified. On the other hand, for large $v_b/v_h$ (very inhomogeneous universe), $|\Delta |a_m||<{10}^{-4}$, and thus the amplitude of the inhomogeneity mode is approximately conserved. The horizontal line in the right plot corresponds to the data points where $|\Delta |a_m||<{10}^{-16}$, and could not be measured accurately owing to floating point precision limits.

Figure 11

The behavior of the amplification of the inhomogeneities through the bounce shown for the population of dynamical trajectories determined by the initial data that are generated with each of the algorithms specified in Sec. 7b, listed there as cases (i)–(vi). Remarkably, for all these cases, the data stay (approximately) concentrated in a specific region of the represented plane. We can distinguish the near-homogeneous sector (with data concentrated about the origin) and the inhomogeneity-dominated one (in a patch separated from the former by a sharp drop). The cases with $\sigma \propto v^{2/3}$ select a particular sector of that patch. The sharp horizontal line about $|\Delta |a_m||={10}^{-16}$ has the same origin as in Fig. 10.