[Paper Review] Emergence of Loop Quantum Cosmology from Loop Quantum Gravity: Lowest Order in h

Chun-Yen Lin|arXiv (Cornell University)|Nov 7, 2011

Noncommutative and Quantum Gravity Theories13 references3 citations

TL;DR

This paper derives loop quantum cosmology (LQC) from loop quantum gravity (LQG) by constructing a physical Hilbert space using matter fields as clocks and spatial frames. It shows that, in the $O(\hbar^0)$ approximation, the emergent dynamics reproduce spatially flat, homogeneous, and isotropic FRW cosmology with a big bounce, validating LQC as an effective limit of fundamental LQG dynamics.

ABSTRACT

To derive loop quantum cosmology from loop quantum gravity, I apply the model given in \cite{lin1} to a system with coupled gravitational and matter fields. The matter sector consists of a scalar field $phi$ serving as a cosmological clock, and other fields ${psi}$ providing physical spatial coordinates and frames. The physical Hilbert space of the model is constructed from the kinematical Hilbert space of loop quantum gravity, and the local observables in the physical Hilbert space are constructed using the matter coordinates and frames. A specific coherent physical state is then chosen, whose expectation values of the local observables give rise to homogeneous, isotropic and spatially flat gravitational and $phi$ fields at a late clock time. The equations governing these fields may be derived using the symmetry of the physical Hilbert space. When the matter back reactions from ${psi}$ are negligible, the result gives a specific loop quantum cosmological model in the $O(hbar^0)$ approximation, with calculable higher order corrections.

Motivation & Objective

• To derive loop quantum cosmology (LQC) from the fundamental framework of loop quantum gravity (LQG), establishing its foundational legitimacy.
• To construct a physical Hilbert space in LQG using matter fields as dynamical spacetime coordinates and frames.
• To demonstrate that coherent states in this framework yield emergent classical, homogeneous, isotropic, and spatially flat cosmological dynamics at late times.
• To show that the $O(\hbar^0)$ effective dynamics match a specific LQC model, including the big bounce and slow-roll inflation.
• To lay the groundwork for computing higher-order quantum corrections beyond $O(\hbar^0)$.

Proposed method

• Constructs the physical Hilbert space $\mathbb{H}$ from the kinematical Hilbert space of LQG, using a modified Hamiltonian constraint operator.
• Employs a massless scalar field $\phi$ as a cosmological clock and other matter fields $\{\psi\}$ as spatial coordinates and frames to define local observables.
• Uses coherent states in $\mathbb{H}$ to compute expectation values of observables that reproduce classical gravitational and scalar field dynamics.
• Imposes symmetry conditions on the physical Hilbert space to derive effective equations governing the emergent cosmological evolution.
• Applies the $O(\hbar^0)$ approximation to extract the effective dynamics, recovering FRW cosmology with a quantum bounce.
• Evaluates the scale of the big bounce using known cosmological parameters and the Immirzi parameter $\gamma \approx \ln(2)/\pi$.

Experimental results

Research questions

• RQ1 Can loop quantum cosmology emerge from the fundamental principles of loop quantum gravity through a consistent construction of physical states and observables?
• RQ2 What is the role of matter fields as clocks and spatial frames in defining physical observables in a background-independent quantum gravity framework?
• RQ3 Does the $O(\hbar^0)$ effective dynamics of the constructed model reproduce the standard FRW cosmology with a big bounce?
• RQ4 At what energy scale or volume does the big bounce occur in this emergent cosmology, and is it consistent with Planck-scale physics?
• RQ5 What are the higher-order quantum corrections beyond $O(\hbar^0)$, and how do they affect the emergent cosmological dynamics?

Key findings

• The $O(\hbar^0)$ effective dynamics of the model reproduce spatially flat, homogeneous, and isotropic FRW cosmology in the large-scale limit.
• The model exhibits a quantum bounce replacing the initial singularity, with the bounce occurring at a volume $d^3\boldsymbol{E}^{3/2}(T_c) \approx 10^{-156}\,\text{m}^3$, well below the Planck volume $V_p \approx 10^{-105}\,\text{m}^3$.
• The critical energy density for the bounce is $\boldsymbol{\rho}_c = \left(\frac{2}{27}\kappa^4\gamma^6\right)^{-1/2}(d^3P_1)^{-1}$, consistent with LQC expectations.
• The scalar field energy density at the current epoch is $\boldsymbol{\rho}_\phi(T_1) \approx 10^{-9}\,\text{J/m}^3$, matching observed dark energy density.
• Higher-order $\hbar$ corrections are significant near the bounce, indicating the need for explicit coherent state construction to compute them.
• The model provides a derivation of LQC from LQG, attributing its predictions to fundamental quantum gravity principles.