The phenomenology of Quantum Bounce in the Klein-Gordon, Wheeler-DeWitt and Dirac formalisms

TL;DR

The paper investigates how Quantum Bounce phenomena manifest across three formalisms—Klein-Gordon, Wheeler-DeWitt, and Dirac—for early-Universe cosmology, using relational time as a clock and a time-dependent potential to induce transitions between contracting and expanding branches. By modeling the isotropic universe as a Klein-Gordon scattering problem, it derives a quantum bounce probability and analyzes volume expectations in Wheeler-DeWitt, revealing imprints in the volume while maintaining unitarity. Using a Dirac treatment of the Bianchi II model, it demonstrates a quantum Kasner transition as a reflection off a potential wall, suggesting a path toward a quantum BKL map for chaotic cosmology. Overall, the framework provides a unified probabilistic description of primordial dynamics across formalisms, with scalar-like versus fermionic-like behaviors depending on the anisotropy description.

Abstract

We investigate the different meanings that the concept of Quantum Bounce acquires in various formalisms. The original idea refers to the phenomenology that appears in the Klein-Gordon framework when homogeneous cosmologies are considered. In that case, the Quantum Bounce describes the quantum scattering between a collapsing and an expanding Universe branch, and therefore provides a quantum description of the semiclassical Big Bounce mechanism. Here, we show that the proposal of the Quantum Big Bounce is well-grounded, thanks to the computation of the volume operator mean values and its standard deviation in the Wheeler-DeWitt framework for the isotropic case. Then, we analyze the Bianchi models in the Dirac approach, now showing that the Quantum Bounce concept can be implemented to describe the Kasner transitions of the Belinski-Khalatnikov-Lifshitz map at a quantum level. In summary, the quantum scattering framework borrowed from particle physics can serve as a good model for different cosmological scenarios, which can exhibit scalar-like or fermionic-like behaviours depending on how the anisotropies are described in the dynamics.

Figures (6)

• Figure 1: Plot of $|S(\bar{k'},\bar{k})|^2$ as function of $\bar{k}'$, where $\bar{k},\bar{k}'$ are respectively the wavenumbers of the initial and final state of the Universe. From the left, $\bar{k}=-4,0,4$. Credits: QB2.
• Figure 2: Plot of $\langle\hat{V}\rangle$ and $\Delta\hat{V}$ in function of $\phi$ computed on $\Psi(\alpha,\phi)$.
• Figure 3: Plot of $\langle\hat{V}\rangle$ with error bars $\Delta\hat{V}$ in function of $\phi$, computed on $\Psi(\alpha,\phi)$ (blue trajectory) and $\Psi^*(\alpha,\phi)$ (purple trajectory).
• Figure 4: Plot of the Klein-Gordon density $\mathcal{N}(\alpha)$ associated to the wavepacket $\Psi(\alpha,\phi)$ for different values of time.
• Figure 5: Quantum Big Bounce probability density $\mathcal{P}_{\bar{k}'}$ in the presence of the potential $U(\phi)=2/(\phi^2+\epsilon)$, with $\epsilon=10^{-100}$. From the left, we fix $\bar{k}=-6,-2,2,6$.