De Sitter universe from causal dynamical triangulations without preferred foliation

We present a detailed analysis of a recently introduced version of causal dynamical triangulations that does not rely on a distinguished time slicing. Focusing on the case of $2+1$ spacetime dimensions, we analyze its geometric and causal properties, present details of the numerical setup, and explain how to extract “volume profiles.” Extensive Monte Carlo measurements of the system show the emergence of a de Sitter universe on large scales from the underlying quantum ensemble, similar to what was observed previously in standard causal dynamical triangulation quantum gravity. This provides evidence that the distinguished time slicing of the latter is not an essential part of its kinematical setup.

Figure 1

A disk in the flat Euclidean plane is triangulated in terms of six equilateral triangles, one of which is subsequently removed (left). Regluing the remaining triangles, one obtains a disk with positive curvature, characterized by a deficit angle $\theta =\pi /3$ and independent of the embedding space (right).

Figure 2

Two-dimensional triangulations: from Euclidean DT, consisting of equilateral triangles with Euclidean signature (left), and from Lorentzian CDT, constructed from Minkowskian triangles with one spacelike and two timelike links (right). In both cases, curvature is present at all interior vertices whose coordination number (number of triangles meeting at the vertex) is not six. Timelike links are shown in red and spacelike ones in blue.

Figure 3

Two adjacent spatial slices in a piece of foliated CDT triangulation in $2+1$ dimensions. Tetrahedra of both types are shown.

Figure 4

The four triangle types which can be constructed from using just two link lengths, spacelike (blue) and timelike (red). Only the two Minkowskian triangles (of types $sst$ and $tts$) at the center have the correct signature for triangulating two-dimensional Lorentzian spacetimes. The green lines inside the Lorentzian triangles indicate light rays through the corner vertices.

Figure 5

Three examples of vertex neighborhoods in $1+1$ dimensions. Causality is violated at the central vertex whenever the number of light cones is not two (left and center). At a causally well-behaved vertex, one crosses exactly two light cones (equivalently, four lightlike directions) when going around the vertex (right).

Figure 6

Two Lorentzian triangles with consistent assignments of future-pointing arrows to its edges, as explained in the text (left). The time orientation of a given triangle determines the time orientation of its direct neighbors (right).

Figure 7

A source and a sink of time in $1+1$ dimensions. In both cases, vertex causality is violated at the central vertex.

Figure 8

A bubble (left) and a strip configuration (right) in $1+1$ dimensions.

Figure 9

All tetrahedra in $2+1$ dimensions that can be constructed from time- and spacelike links of fixed squared length, allowing for any signature. The tetrahedra highlighted in yellow have the correct Lorentzian signature; for the $T_7$ tetrahedron, this only holds for $\alpha <1$. The link labeling shown for the first tetrahedron will be used for the other tetrahedra, too.

Figure 10

Dihedral angle at the link with label “2” of a $T_2$ tetrahedron (left). Star of a link, all of whose tetrahedra contribute dihedral angles at the link (center). Two-dimensional cut perpendicular to the link. In the case depicted, the space is Lorentzian, with green lines representing light rays originating at the center (right).

Figure 11

In a triangulation obeying vertex causality, intersecting the boundary of a spherical vertex neighborhood with the light cones at the vertex $V$ results in two disconnected circles (left). Part of the surface triangulation of a unit ball around $V$, showing timelike radial links as red, spacelike ones as blue, and light cone crossings as green dots (right).

Figure 12

The four fundamental tetrahedral building blocks, each equipped with one (out of two possible) time orientations.

Figure 13

(a) A ring of $T_2$ tetrahedra, forming a simple bubble. (b) By inserting tetrahedra of type $T_3$, we can form more complicated bubbles. (c) More general bubbles also contain other tetrahedra types. (d) A ring of $T_3$ tetrahedra gives rise to a pinching, where two spatial disks meet in a single vertex. (e) By adding two $T_2$ tetrahedra, the pinching is extended to a link. (f) Bubbles and pinchings can occur in combination (timelike links are omitted here).

Figure 14

The three generalized Pachner moves for CDT in $2+1$ dimensions: 2-6 move (left), 2-3 move (center), and 4-4 move (right). The links shown in green are removed or created by the corresponding move.

Figure 15

New Monte Carlo moves in nonfoliated CDT. Three link collapse moves: bubble move (top left), pinching move (bottom left), and link collapse move for spacelike links (top right). Also shown is the polar move (bottom right), with arrows indicating a time orientation. Spacelike links are blue, timelike ones are red, and purple links (top right) can be either space- or timelike.

Figure 16

The phase diagram of nonfoliated CDT quantum gravity is the region inside the strip $-3<\alpha <-1/2$, visualized in green. Our investigation has probed the phase space along the two axes drawn in the figure: at various $k$ values for constant $\alpha =-1$ and at various $\alpha$ values for constant $k=0$.

Figure 17

Measurement of the average vertex density $N_0/N_3$ as a function of the inverse gravitational coupling $k$, for $\alpha =-1$ and system size $N_3=40.000$. The dots represent actual measurements and the lines linear interpolations. The dashed lines mark the kinematical bounds for standard DT (upper line) and CDT (lower line).

Figure 18

The numbers of tetrahedra of each of the four types, averaged over the sampled triangulations, as function of the coupling $k$, at $\alpha =-1$ and $N_3=40.000$.

Figure 19

The numbers of tetrahedra of each of the four types, averaged over the sampled triangulations, as a function of the coupling $\alpha$, at $k=0$ and $N_3=40.000$.

Figure 20

Typical tetrahedron distribution of a strictly foliated CDT configuration as function of “tetrahedron time,” extracted from a simulation with foliation constraint enabled (left). Zooming in on the central region of the distribution, one obtains the graph shown on the right.

Figure 21

Tetrahedron distribution of a typical path integral history from a simulation of the generalized CDT model at $k=0.0$, $\alpha =-0.7$.

Figure 22

Tetrahedron distributions of typical path integral configurations from simulations at coupling $k=0.0$ and system size $N_3=40.000$, for various choices of the asymmetry parameter $\alpha$. Note that all geometries display some degree of being weakly foliated, which becomes weaker with increasing $|\alpha |$.

Figure 23

Average volume profile $⟨N_2(t)⟩$ as function of time $t$, measured at $\alpha =-1$ for various values of $k$. The scale of the vertical axis is the same for all plots. Some of the profiles have a tail which is not visible here but is shown in Fig. 24 below.

Figure 24

The last three average volume profiles from Fig. 23, with a different scaling of the time axis and a small upward shift away from the axis to exhibit the tails of the distributions.

Figure 25

Volume distribution $N_2(t)$ of a typical triangulation from a simulation at $(k,\alpha )=(0.4,-1)$ and $N_3=40.000$ in the phase of high vertex density. The plot shows individual measurement points.

Figure 26

Average volume profiles at $(k,\alpha )=(0.0,-1.0)$ for four different system sizes $N_3=40$, 80, 120 and $160k$, before (left) and after (right) rescaling the axes as indicated, with $d=2.91$ to achieve a best overlap, which is seen to be of excellent quality.

Figure 27

Estimates for the macroscopic dimension $d$ from finite-size scaling, from measurements at fixed $\alpha =-1.0$ (left plot) and fixed $k=0.0$ (right plot). Large dots represent measurements where the volume profiles overlap with excellent quality; the smaller dots stand for overlaps of lesser quality.

Figure 28

Rescaled average volume profiles at $(k,\alpha )=(0.0,-1.0)$, corresponding to dimension $d=2.91$ and best fit to the ${cos}^2$ ansatz.

Figure 29

Rescaled average volume profiles at $(k,\alpha )=(-0.8,-1.0)$, corresponding to dimension $d=2.98$ and best fit to the ${cos}^2$ ansatz.