Nonlocal quantum effects in cosmology: Quantum memory, nonlocal FLRW equations, and singularity avoidance

We discuss cosmological effects of the quantum loops of massless particles, which lead to temporal nonlocalities in the equations of motion governing the scale factor $a(t)$. For the effects discussed here, loops cause the evolution of $a(t)$ to depend on the memory of the curvature in the past with a weight that scales initially as $1/(t-t^{\prime })$. As one of our primary examples, we discuss the situation with a large number of light particles, such that these effects occur in a region where gravity may still be treated classically. However, we also describe the effect of quantum graviton loops and the full set of Standard Model particles. We show that these effects decrease with time in an expanding phase, leading to classical behavior at late time. In a contracting phase, within our approximations the quantum results can lead to a bouncelike behavior at scales below the Planck mass, avoiding the singularities required classically by the Hawking-Penrose theorems. For conformally invariant fields, such as the Standard Model with a conformally coupled Higgs, this result is purely nonlocal and parameter independent.

Figure 1

Vacuum polarization graph.

Figure 2

The evolution of the scale factor and its time derivative in an expanding dust-filled universe for $\mathrm{N}=10$.

Figure 3

The evolution of the scale factor and its time derivative in an expanding dust-filled universe for $\mathrm{N}=100$.

Figure 4

The evolution of the scale factor and its time derivative in an expanding dust-filled universe with $N={10}^3$ scalar fields.

Figure 5

The evolution of the scale factor and its time derivative in an expanding dust-filled universe with quantum graviton loops.

Figure 6

The evolution of the scale factor and its time derivative in an expanding radiation-filled universe with $N={10}^3$ scalar fields.

Figure 7

Collapsing dust-filled universe with ${\mu }_R=1$ and a single scalar field. The time derivative of the scale factor quickly stops diverging when the quantum correction becomes active.

Figure 8

Varying both the scale $\mu$ and the number of scalar particles $N_S$ in a collapsing dust-filled universe. The plots from left to right involve ($N_S=1$, ${\mu }_R=1$), ($N_S={10}^2$, ${\mu }_R=0.1$) and ($N_S={10}^4$, ${\mu }_R=0.01$). Note the change of scale along the time axis in the figures. The results illustrate the similarity of the quantum corrections with an energy scale that scales as $E\sim M_P/\sqrt{N}$.

Figure 9

Varying the scale ${\mu }_R$ in a collapsing dust-filled universe, with ${\mu }_R=0.1$ on the left and ${\mu }_R=10$ on the right.

Figure 10

The effect of graviton loops on a dust-filled universe. These have ${\mu }_R=0.1$ on the left and ${\mu }_R=1$ on the right.

Figure 11

Collapsing dust-filled universe with the Standard Model particles and a conformally coupled Higgs. The result is purely nonlocal and hence independent of any scale ${\mu }_R$.

Figure 12

Collapsing radiation-filled universe with gravitons only considered.

Figure 13

Collapsing radiation-filled universe with all the Standard Model particles included, as well as graviton loops.