Coordinate time dependence in quantum gravity

The intuitive classical space-time picture breaks down in quantum gravity, which makes a comparison and the development of semiclassical techniques quite complicated. Using ingredients of the group averaging method to solve constraints one can nevertheless introduce a classical coordinate time into the quantum theory, and use it to investigate the way a semiclassical continuous description emerges from discrete quantum evolution. Applying this technique to test effective classical equations of loop cosmology and their implications for inflation and bounces, we show that the effective semiclassical theory is in good agreement with the quantum description even at short scales.

Figure 1

Expectation value of the volume (+) compared to the effective solution $V(t)$ (dashed curve) and the volume $⟨\widehat{a}{⟩}^3(t)$ computed from the expectation value of the scale factor (×) for the case of cosmological constant. The initial peak of the coherent state is chosen around $m_c=200$ with $\sigma =20$ and $N=500$. We take $\Lambda ={10}^{-3}$.

Figure 2

A spreading wave packet for $\Lambda ={10}^{-3}$, as in Fig. 1. At the back of the right-hand side one can see small oscillations building up when the wave reaches the boundary. The vertical axis shows the magnitude of the normalized wave function, which starts as a Gaussian at the front and evolves to the back.

Figure 3

Growth of standard deviation ($\tilde{\sigma }/{\tilde{\sigma }}_{\mathrm{i}\mathrm{n}}$) and skewness ($s$) for the cosmological constant case studied in Fig. 1. The coherent state remains almost peaked over the classical value due to small growth in skewness.

Figure 4

Expectation value of the volume (+) compared to the classical solution $V(t)$ (dashed curve) and the volume computed from the expectation value of the triad (×) for dust with $M=10$ and an initial peak around $m_c=100$ ($N=1000$, $\sigma =20$). The ambiguity parameter $j$ for the effective density is $j=400$, such that the density peaks at $m_{*}\approx j/2=200$ corresponding to a volume $V_{*}=(2\gamma {\ell }_{\mathrm{P}}^2{m}_{*}/3{)}^{3/2}\approx 180{\ell }_{\mathrm{P}}^3$, which is reached around $t=1.1$.

Figure 5

The evolution of standard deviation and skewness for the case in Fig. 4. The spread increases most strongly in the inflationary regime ($t<1.1$), while the wave packet is skewed stronger during unaccelerated expansion.

Figure 6

A spreading wave packet $|{\psi }_{t,i}|$ for the evolution shown in Fig. 4.

Figure 7

Time derivative of scale factors (+ corresponds to those computed from volume expectation value and × refers to those of scale factor expectation values) plotted in Fig. 4. The dashed curve shows the classical curve. The classical description does not match the evolution from quantum theory.

Figure 8

Volume expectation values (+ for $⟨\widehat{V}⟩$ and × for $⟨\widehat{a}{⟩}^3$) compared to the effective classical solution (dashed curve). Compared to the points in Fig. 4 the quantum behavior is different since the initial wave packet is now peaked on the effective rather than unmodified classical constraint surface (which decreases the initial $c_0$).

Figure 9

Time derivative of scale factors plotted in Fig. 8. The inset shows variation of Hubble rate with time ($H=\dot{a}/a$).

Figure 10

The time variation of standard deviation and skewness for dust with effective density modification.

Figure 11

Wave packet evolving toward the classical singularity and bouncing off, only penetrating negligibly to negative $m$ (to the right). The parameters are $m_c=100$, $N=500$, $\sigma =10$, $M=10$, $j=200$.

Figure 12

Volume expectation values for the wave packet in Fig. 11. The expectation values bounce at nonzero values, while the classical curve hits zero volume.

Figure 13

Volume expectation values for the wave packet initially peaked at the effective constraint surface compared to a numerical solution (solid curve) of the effective Friedmann equation (26). Since the corresponding wave packet, plotted in Fig. 15, moves closer to the classical singularity than in Fig. 11, the spread of wave function increases faster than in Fig. 12.

Figure 14

Increase in spread and skewness for backward evolution through a bounce as in Fig. 13.

Figure 15

Wave packet evolving toward the classical singularity and bouncing off as in Fig. 11, but initially peaked on the effective constraint surface at $m_c=80$. Since the bounce radius now is smaller, the leakage to negative $m$ is larger (see rightmost contour line).

Figure 16

Extrinsic curvature fluctuation $\Delta \dot{a}=\sqrt{⟨(\widehat{\dot{a}}{)}^2⟩-⟨\widehat{\dot{a}}{⟩}^2}$ corresponding to Fig. 13 where the operator for $\dot{a}$ is derived from the $c$ operator mapping a state ${\psi }_m$ to $\frac{1}{4}i({\psi }_{m+1}-{\psi }_{m-1})$.

Figure 17

Time derivative of the scale factor expectation values in Fig. 13 compared to the effective solution (solid curve). The dashed curve connects expectation values $⟨\widehat{\dot{a}}⟩$.