Semiclassical states, effective dynamics, and classical emergence in loop quantum cosmology

We construct physical semiclassical states annihilated by the Hamiltonian constraint operator in the framework of loop quantum cosmology as a method of systematically determining the regime and validity of the semiclassical limit of the quantum theory. Our results indicate that the evolution can be effectively described using continuous classical equations of motion with nonperturbative corrections down to near the Planck scale below which the Universe can only be described by the discrete quantum constraint. These results, for the first time, provide concrete evidence of the emergence of classicality in loop quantum cosmology and also clearly demarcate the domain of validity of different effective theories. We prove the validity of modified Friedmann dynamics incorporating discrete quantum geometry effects which can lead to various new phenomenological applications. Furthermore the understanding of semiclassical states allows for a framework for interpreting the quantum wave functions and understanding questions of a semiclassical nature within the quantum theory of loop quantum cosmology.

Figure 1

Evolution of the coherent state for $J=1/2$ and $P_{\varphi }=10l_p$. The coherent state remains sharply peaked and follows the classical trajectory given by Eq. (7). Amplification at scales close to $a=0$ are due to discrete quantum effects.

Figure 2

The trajectory of the peak of the coherent state ($+$ line) is compared with classical theory (solid line) for $J=1/2$ and $P_{\varphi }=10l_p$. The classical trajectory agrees extremely well with the quantum curve until the smallest discrete step in the scale factor.

Figure 3

Comparison of evolution via the quantum constraint ($+$ line), ET-I and ET-II (dashed line), and classical theory (solid line) for $J=500$, $P_{\varphi }=100l_p$. Classical theory breaks down for $a<a_{*}$, whereas ET-I and ET-II are valid until $a_0$.

Figure 4

Evolution for $J=500$, $P_{\varphi }=1600l_p$. The classical (solid line) and ET-I trajectory (dashed line) deviate from quantum evolution ($+$ line) for scale factors at which perturbative corrections become significant. ET-II agrees with evolution from the quantum constraint until $a_0$. Inset: ET-II corrections become significant when the matter density (solid line) approaches the critical density (dashed line).