Quantum Nature of the Big Bang: An Analytical and Numerical Investigation

TL;DR

The paper tackles the quantum nature of the big bang in loop quantum cosmology for a flat, isotropic model with a massless scalar field, addressing singularity resolution, emergent time, and deterministic evolution through the Planck regime. It develops a nonperturbative, background-independent quantization where the Hamiltonian constraint becomes a self-adjoint difference operator with a fixed area gap, and uses the scalar field as emergent time to deparameterize the dynamics. By constructing a physical Hilbert space, Dirac observables, and semi-classical states, and performing extensive numerical simulations, the authors demonstrate that the classical big bang is replaced by a robust quantum bounce with deterministic evolution across the deep Planck region, in contrast to the Wheeler–DeWitt theory where the singularity persists. The work also provides a detailed comparison with the WDW theory, clarifies the role of the area gap, and establishes a solid framework that can be extended to more general models (anisotropies and potentials), offering foundational tools for studying Planck-scale physics in loop quantum cosmology.

Abstract

Analytical and numerical methods are developed to analyze the quantum nature of the big bang in the setting of loop quantum cosmology. They enable one to explore the effects of quantum geometry both on the gravitational and matter sectors and significantly extend the known results on the resolution of the big bang singularity. Specifically, the following results are established for the homogeneous isotropic model with a massless scalar field: i) the scalar field is shown to serve as an internal clock, thereby providing a detailed realization of the `emergent time' idea; ii) the physical Hilbert space, Dirac observables and semi-classical states are constructed rigorously; iii) the Hamiltonian constraint is solved numerically to show that the big bang is replaced by a big bounce. Thanks to the non-perturbative, background independent methods, unlike in other approaches the quantum evolution is deterministic across the deep Planck regime. Our constructions also provide a conceptual framework and technical tools which can be used in more general models. In this sense, they provide foundations for analyzing physical issues associated with the Planck regime of loop quantum cosmology as a whole.

Figures (19)

• Figure 1: Classical phase space trajectories are plotted in the $\phi, p\sim\mu$ -plane. For $\mu \ge 0$, there is a branch which starts with a big-bang (at $\mu =0$) and expands out and a branch which contracts into a big crunch (at $\mu =0$). Their mirror images appear in the $\mu \le 0$ half plane.
• Figure 2: Crosses denote the values of an eigenfunction $e_{\omega}(\mu)$ of $\Theta$ for $\varepsilon=0$ and $\omega=20$. The solid curve is the eigenfunction $\underbar{e}_\omega(\mu)$ of the WDW $\ul\Theta$ to which $e_\omega(\mu)$ approaches at large positive $\mu$. As $\mu$ increases, the set of points on ${\cal L}_\varepsilon$ becomes denser and fill the solid curve. For visual clarity only some of these points are shown for $\mu >100$.
• Figure 3: The exponential growth of $|e^{\pm}_{-|k|}|(\mu)$ in the 'genuinely quantum region' is shown for three different values of $\omega$, where $\omega$ is given by $\Theta\, e^{\pm}_{-|k|} = \omega^2\, e^{\pm}_{-|k|}$.
• Figure 4: The amplification factor $\lambda^{\pm}$ in the 'genuinely quantum region is shown as a function of the parameter $\varepsilon$ labeling the lattice and $\omega$.
• Figure 5: The function $a(\varepsilon)$ of Eq \ref{['eq:lambda-fit']} is plotted by connecting numerically calculated data points.