Corrections to the Friedmann Equations from LQG for a Universe with a Free Scalar Field

TL;DR

This paper derives effective quantum-corrected Friedmann equations in loop quantum cosmology, showing that semiclassical states follow classical trajectories until quantum effects become significant near Planck densities.

Contribution

It introduces a method to project full quantum dynamics onto a finite-dimensional submanifold, deriving effective equations that incorporate leading quantum corrections in loop quantum cosmology.

Findings

Semiclassical states follow classical trajectories until high densities.

Effective equations accurately reproduce quantum deviation behavior.

Quantum corrections become significant near Planck density.

Abstract

In loop quantum cosmology the quantum dynamics is well understood. We approximate the full quantum dynamics in the infinite dimensional Hilbert space by projecting it on a finite dimensional submanifold thereof, spanned by suitably chosen semiclassical states. This submanifold is isomorphic with the classical phase space and the projected dynamical flow provides effective equations incorporating the leading quantum corrections to the classical equations of motion. Numerical work has been done using quantum states which are semiclassical at late times. These states follow the classical trajectory until the density is on the order of 1% of the Planck density then deviate strongly from the classical trajectory. The effective equations we obtain reproduce this behavior to surprising accuracy.