Effective constraints of loop quantum gravity

Within a perturbative cosmological regime of loop quantum gravity corrections to effective constraints are computed. This takes into account all inhomogeneous degrees of freedom relevant for scalar metric modes around flat space and results in explicit expressions for modified coefficients and of higher order terms. It also illustrates the role of different scales determining the relative magnitude of corrections. Our results demonstrate that loop quantum gravity has the correct classical limit, at least in its sector of cosmological perturbations around flat space, in the sense of perturbative effective theory.

Figure 1

Edges adjacent to a vertex $\mathbf{v}$ in a given direction $I$. For the edge oriented oppositely to the chosen one for direction $I$, the labels “$\mathbf{v}-\mathbf{I},I$” and “$\mathbf{v},-I$” can be chosen interchangeably, defining in this way negative values for the label $I$. We use boldface vector notation whenever points have to be shifted by unit vectors $\mathbf{I}$ along the lattice directions.

Figure 2

Elementary lattice plaquette with holonomies around a closed loop.

Figure 3

Four plaquettes adjacent to vertex $\mathbf{v}$ in the $(I,J)$-plane. The arrows indicate the directions in which the relevant holonomies are traversed.

Figure 4

Behavior of the correction function $\alpha$. It approaches one from above for large arguments. For small arguments, the function is increasing from zero and reaches a peak value larger than 1. Also shown is the limiting case $r=2$ which does not show a peak but a constant correction function for $\mu >1$.

Figure 5

Behavior of the correction function $\alpha$ for larger $j$. The general trend is similar to the case for $j=1/2$, but there are $[j]+1$ spikes at $\mu =1,\dots ,j$ for integer $j$ and $\mu =1/2,\dots ,j$ otherwise. To the right from the peak, the function is smooth.

Figure 6

Comparison between the correction function (55) for $j=10$ (punctured line) and $j=20$ (dotted line) and its approximation (56). The spikes are smeared out by the approximation.