Reconstructing the Universe

We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale, and at the same time provides a nontrivial consistency check of the method of causal dynamical triangulations. A closer look at the quantum geometry reveals a number of highly nonclassical aspects, including a dynamical reduction of spacetime to two dimensions on short scales and a fractal structure of slices of constant time.

Figure 1

The two fundamental building blocks of causal dynamically triangulated gravity. The flat four-simplex of type (4,1) on the left has four of its vertices at time $\tau$ and one at time $\tau +1$, and analogously for the (3,2)-simplex on the right. The “gap” between two consecutive spatial slices of constant integer time is filled by copies of these simplicial building blocks and their time-reversed counterparts, the (1,4)- and the (2,3)-simplices.

Figure 2

Unnormalized distribution of the number $N_4^{(3,2)}$ of four-dimensional simplices of type (3,2), at fixed numbers of four-simplices of type (4,1), for $N_4^{(4,1)}=40,80$ and 160 k (left to right) at ${\kappa }_0=2.2$ and $\Delta =0.6$.

Figure 3

A qualitative sketch of the phase diagram of four-dimensional causal dynamical triangulations as function of the gravitational coupling ${\kappa }_0$ and the asymmetry parameter $\Delta$.

Figure 4

Monte Carlo snapshot of a typical universe in phase A (${\kappa }_0=5.0$, $\Delta =0$), of discrete volume $N_4=45.5$ k and total time extent (vertical direction) $t=20$. In this and the following two figures, the circumference at integer proper time $\tau$ is chosen proportional to the spatial three-volume $V_3(\tau )$. The surface represents an interpolation between adjacent spatial volumes, without capturing the actual 4d connectivity between neighboring spatial slices.

Figure 5

Monte Carlo snapshot of a typical universe in phase B (${\kappa }_0=1.6$), of discrete volume $N_4=22.25$ k and total time extent $t=20$. The entire universe has collapsed into a slice of minimal time extension.

Figure 6

Monte Carlo snapshot of a typical universe in phase C (${\kappa }_0=2.2$) of discrete volume $N_4=91.1$ k and total time extent $t=40$.

Figure 7

The scaling of the volume-volume distribution, as function of the rescaled time variable $x=\delta /({\tilde{N}}_4^{\mathrm{eff}}{)}^{1/4}$. Data points come from system sizes ${\tilde{N}}_4=10,20,40,80$ and 160 k at ${\kappa }_0=2.2$, $\Delta =0.6$ and $t=80$.

Figure 8

Estimating the best scaling dimension $D_H$ from optimal overlaps of rescaled volume-volume distributions. The plot variable is the average error of the overlap as function of $D_H$.

Figure 9

The spectral dimension $D_S(\sigma )$ of causal dynamical triangulations as a function of the diffusion time $\sigma$, which is a direct measure of the distance scale probed. The measurements were taken at volumes ${\tilde{N}}_4=20$ k (bottom curve), 40 k and 80 k (top curve), and for ${\kappa }_0=2.2$, $\Delta =0.6$ and $t=80$.

Figure 10

The spectral dimension $D_S$ of the universe as function of the diffusion time $\sigma$, measured for ${\kappa }_0=2.2$, $\Delta =0.6$ and $t=80$, and a spacetime volume $N_4=181$ k. The averaged measurements lie along the central curve, together with a superimposed best fit $D_S(\sigma )=4.02-119/(54+\sigma )$. The two outer curves represent error bars.

Figure 11

Extracting the scaling dimension $D_2$ by finite-size scaling, in order to determine the kinetic term in the effective action of the scale factor.

Figure 12

The distribution $P_{N_3}(z)$ of rescaled volume differences $z$ between adjacent spatial slices, for three-volumes $N_3=20,40,80$ and 160 k.

Figure 13

The scaling of the volume-volume distribution, as function of the rescaled time variable $x=\delta /({\tilde{N}}_4^{\mathrm{eff}}{)}^{1/4}$, taken from Fig. 7, but now superimposed with the correlator obtained when using the minisuperspace action (40) to generate a distribution of three-geometries (black curve).

Figure 14

Log-log plot of the average linear geodesic size $⟨r{⟩}_{N_3}$ versus the three-volume $N_3$, measured for $\Delta =0.4$. The straight line corresponds to a Hausdorff dimension $d_h=3$. Similar measurements for $\Delta =0.5$ and $0.6$ yield virtually indistiguishable results.

Figure 15

The logarithmic derivative of the return probability for ${\kappa }_0=2.2$ and $\Delta =0.4$, and spatial volumes $N_3$ between 500 and 2000. The measurement was performed for a system with $N_4=91.1k$ and $t=40$. The increasing and decreasing lines correspond to even and odd times, and illustrate the different behavior for short diffusion times $\sigma$.

Figure 16

The distribution of the number of baby universes with minimal neck as a function of their rescaled size $x$, for ${\kappa }_0=2.2$ and $\Delta =0.4$. Distributions are normalized by the total number of baby universes in constant-time slices with $N_3>400$. This number grows with the total spacetime volume $N_4$, which is 45.5 k (bottom), 91 k (middle) and 181 k (top curve).

Figure 17

Distribution $P(n)$ of the coordination numbers $n$ of vertices in a three-dimensional spatial slice, for ${\kappa }_0=2.2$ and $\Delta =0.4$. The straight line represents a numerical fit to formula (53), with $n_0=2$.

Figure 18

Distribution of the largest coordination number $n_{\max }$ as a function of the spatial volume $N_3$, for ${\kappa }_0=2.2$, $\Delta =0.4$, $t=40$ and $N_4=91.1k$.

Figure 19

Log-log plot of the average linear geodesic size $⟨r{⟩}_N$ versus the volume $N$ of a thick slice. The straight line corresponds to a Hausdorff dimension $d_H=4$.

Figure 20

The logarithmic derivative of the return probability for ${\kappa }_0=2.2$ and $\Delta =0.6$ and thick slices of volumes $N$ between 900 and 2000. The measurement was performed for a system with $N_4=161k$ and $t=80$. Curves extrapolating between even evolution times decrease rapidly at small $\sigma$, whereas curves of odd times stay above $d_S=2$ at small times.