Cosmic recall and the scattering picture of loop quantum cosmology

The global dynamics of a homogeneous Universe in loop quantum cosmology is viewed as a scattering process of its geometrodynamical equivalent. This picture is applied to build a flexible (easy to generalize) and not restricted just to exactly solvable models method of verifying the preservation of the semiclassicality through the bounce. The devised method is next applied to two simple examples: (i) the isotropic Friedmann-Robertson-Walker universe, and (ii) the isotropic sector of the Bianchi I model. For both of them we show that the dispersions in the logarithm of the volume $\ln (v)$ and scalar field momentum $\ln (p_{\varphi })$ in the distant future and past are related via strong triangle inequalities. This implies, in particular, a strict preservation of the semiclassicality (in considered degrees of freedom) in both the cases (i) and (ii). Derived inequalities are general: valid for all the physical states within the considered models.

Figure 1

The behavior of ${\partial }_k\alpha$ for different quantization prescriptions compared against its large $k$ limit ${\alpha }^{\prime }(k)=-\ln (k)$. Figure (a) shows the shape of the derivative in the entire domain investigated numerically, whereas (b) presents the zoom for small values of $k$.

Figure 2

The values of the rescaled remnant function $\beta (k)≔({\alpha }^{\prime }(k)+\ln (k))k^2$ calculated numerically within the domain of reliability of the applied method. Figure (a) shows the result for different prescriptions, whereas (b) presents the dependence of calculated $\beta$ on the choice of $v_M$ on the example of the APS prescription. One can see that as $v_M$ increases $\beta (k)$ converges.

Figure 3

(a) The rescaled derivative $k{\partial }_k^2\alpha$ for different quantization prescriptions shown in the domain, where it could be determined numerically in the reliable manner. (b) The values of $k{\partial }_k^2\alpha$ for APS prescription evaluated on different sets of probing points with the upper bound ranging from ${10}^7$ to $2\times {10}^8$. The numerical solutions converge as the bound increases.