Nonperturbative quantum de Sitter universe

The dynamical generation of a four-dimensional classical universe from nothing but fundamental quantum excitations at the Planck scale is a long-standing challenge to theoretical physicists. A candidate theory of quantum gravity which achieves this goal without invoking exotic ingredients or excessive fine-tuning is based on the nonperturbative and background-independent technique of causal dynamical triangulations. We demonstrate in detail how in this approach a macroscopic de Sitter universe, accompanied by small quantum fluctuations, emerges from the full gravitational path integral, and how the effective action determining its dynamics can be reconstructed uniquely from Monte Carlo data. We also provide evidence that it may be possible to penetrate to the sub-Planckian regime, where the Planck length is large compared to the lattice spacing of the underlying regularization of geometry.

Figure 1

Background geometry $⟨N_3(i)⟩$: Monte Carlo measurements for fixed $N_4^{(4,1)}=160.000$ ($N_4=362.000$) and best fit (12) yield indistinguishable curves at given plot resolution. The bars indicate the average size of quantum fluctuations.

Figure 2

The measured average shape $⟨N_3(i)⟩$ of the quantum universe at $\Delta =0.6$, for ${\kappa }_0=2.2$ (broader distribution) and ${\kappa }_0=3.6$ (narrower distribution), taken at $N_4^{(4,1)}=160.000$.

Figure 3

The measured average shape $⟨N_3(i)⟩$ of the quantum universe at ${\kappa }_0=2.2$, for $\Delta =0.6$ (broad distribution) and $\Delta =0.2$ (narrow distribution), both taken at $N_4^{(4,1)}=160.000$.

Figure 4

The directly measured expectation values ${\overline{N}}_3(i)$ (thick gray curves), compared to the averages ${\overline{N}}_3(i)$ reconstructed from the measured covariance matrix $\widehat{C}$ (thin black curves), for ${\kappa }_0=2.2$ and $\Delta =0.6$, at various fixed volumes $N_4^{(4,1)}$. The twofold symmetry of the interpolated curves around the central symmetry axis results from an explicit symmetrization of the collected data.

Figure 5

Reconstruction of the second derivative $U^{\prime \prime }({\overline{N}}_3(i))$ from the coefficients $u_i$, for ${\kappa }_0=2.2$ and $\Delta =0.6$ and $N_4^{(4,1)}=160.000$.

Figure 6

The second derivative $-U^{\prime \prime }(N_3)$ as measured for $N_4^{(4,1)}=160.000$ and ${\kappa }_0=2.2$ and $\Delta =0.6$.

Figure 7

The dimensionless second derivative $u=N_4^{5/4}{U}^{\prime \prime }(N_3)$ plotted against ${\nu }^{-5/3}$, where $\nu =N_3/N_4^{3/4}$ is the dimensionless spatial volume, for $N_4^{(4,1)}=40.000$, 80.000, and 160.000, ${\kappa }_0=2.2$ and $\Delta =0.6$. One expects a universal straight line near the origin (i.e. for large volumes) if the power law $U(N_3)\propto N^{1/3}$ is correct.

Figure 8

Analysis of the quantum fluctuations of Fig. 1: diagonal entries $F(t,t{)}^{1/2}$ of the universal scaling function $F$ from (40), for $N_4^{(4,1)}=20.000$, 40.000, 80.000, and 160.000.

Figure 9

Comparison of the two highest even eigenvectors of the covariance matrix $C(t,t^{\prime })$ measured directly (gray curves) with the two lowest even eigenvectors of $M^{-1}(t,t^{\prime })$, calculated semiclassically (black curves).

Figure 10

Comparison of data for the extended part of the universe: measured $C(t,t^{\prime })$ (upper panel) versus $M^{-1}(t,t^{\prime })$ obtained from analytical calculation (lower panel). The agreement is good, and would have been even better had we included only the even modes.

Figure 11

Testing relation (50) for the bare coupling constants ${\kappa }_0=2.2$ and $\Delta =0.6$, at four-volume $N_4^{(4,1)}=160.000$. Data have been collected for spatial slices at various distances close to the center of volume.

Figure 12

The measured effective coupling constant $k_1$ as a function of the bare ${\kappa }_0$ (top panel, $\Delta =0.6$ fixed) and the asymmetry $\Delta$ (bottom panel, ${\kappa }_0=2.2$ fixed). The marked point near the middle of the data points sampled is the point $({\kappa }_0,\Delta )=(2.2,0.6)$ where most measurements in the remainder of the paper were taken. The other marked points are those closest to the two phase transitions, to the “branched-polymer phase” (top panel) and the “crumpled phase” (bottom panel).