Scale-dependent homogeneity measures for causal dynamical triangulations

I propose two scale-dependent measures of the homogeneity of the quantum geometry determined by an ensemble of causal triangulations. The first measure is volumetric, probing the growth of volume with graph geodesic distance. The second measure is spectral, probing the return probability of a random walk with diffusion time. Both of these measures, particularly the first, are closely related to those used to assess the homogeneity of our own universe on the basis of galaxy redshift surveys. I employ these measures to quantify the quantum spacetime homogeneity as well as the temporal evolution of quantum spatial homogeneity of ensembles of causal triangulations in the well-known physical phase. According to these measures, the quantum spacetime geometry exhibits some degree of inhomogeneity on sufficiently small scales and a high degree of homogeneity on sufficiently large scales. This inhomogeneity appears unrelated to the phenomenon of dynamical dimensional reduction. I also uncover evidence for power-law scaling of both the typical scale on which inhomogeneity occurs and the magnitude of inhomogeneity on this scale with the ensemble average spatial volume of the quantum spatial geometries.

Figure 1

Three types of causal 3-simplices employed in $(2+1)$-dimensional causal dynamical triangulations: (a) (3, 1) 3-simplex, (b) (2, 2) 3-simplex, (c) (1, 3) 3-simplex. The first number in the ordered pair indicates the number of vertices on the initial spacelike 3-surface ($t=0$), and the second number in the ordered pair indicates the number of vertices on the final spacelike 3-surface ($t=1$). I denote by $N_3^{(3,1)}$ the number of (3, 1) 3-simplices, by $N_3^{(2,2)}$ the number of (2, 2) 3-simplices, and by $N_3^{(1,3)}$ the number of (1, 3) 3-simplices in a causal triangulation. I have taken this figure from [18].

Figure 2

(a) Number $N_2^{\mathrm{SL}}$ of spacelike 2-simplices as a function of the discrete time coordinate $\tau$ for three causal triangulations from an ensemble characterized by $T=64$, $N_3=30850$, and $k_0=1.0$. (b) Number $N_2^{\mathrm{SL}}$ of spacelike 2-simplices as a function of the shifted discrete time coordinate $\tau$ for three causal triangulations from an ensemble characterized by $T=64$, $N_3=30850$, and $k_0=1.0$.

Figure 3

(a) Ensemble average number $⟨N_2^{\mathrm{SL}}⟩$ of spacelike 2-simplices as a function of the discrete time coordinate $\tau$ for an ensemble characterized by $T=64$, $N_3=30850$, and $k_0=1.0$. (b) Fit of the discretization (2.11) to the ensemble average number $⟨N_2^{\mathrm{SL}}⟩$ of spacelike 2-simplices as a function of the discrete time coordinate $\tau$ for an ensemble characterized by $T=64$, $N_3=30850$, and $k_0=1.0$. (c) Diagonal $⟨n_2^{\mathrm{SL}}(\tau )n_2^{\mathrm{SL}}(\tau ){⟩}^{1/2}$ of the ensemble average covariance of deviations $n_2^{\mathrm{SL}}(\tau )$ from the ensemble average $⟨N_2^{\mathrm{SL}}(\tau )⟩$ for an ensemble characterized by $T=64$, $N_3=30850$, and $k_0=1.0$.

Figure 4

Volumetric homogeneity measure ${\mathcal{H}}_V$ (dark, left axis) and Hausdorff dimension $D_{\mathrm{H}}$ (light, right axis) of the quantum spacetime geometry as a function of the graph geodesic distance $r$ for ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$. (a) $N_3=10850$ (magenta). (b) $N_3=30850$ (orange). (c) $N_3=61440$ (purple).

Figure 5

(a) Volumetric homogeneity measure ${\mathcal{H}}_V$ of the quantum spacetime geometry as a function of the graph geodesic distance $r$ for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$. (b) Finite size scaling analysis of the volumetric homogeneity measure ${\mathcal{H}}_V$ of the quantum spacetime geometry as a function of the graph geodesic distance $r$ for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$.

Figure 6

(a) The graph geodesic distance $r_{\max }$ as a function of the number $N_3$ of 3-simplices for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$. (b) The finite size scaled graph geodesic distance $r_{\max }/N_3^{1/3}$ as a function of the number $N_3$ of 3-simplices for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$. (c) The volumetric homogeneity measure ${\mathcal{H}}_V(r_{\max })$ of the quantum spacetime geometry as a function of the number $N_3$ of 3-simplices for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$.

Figure 7

Spectral homogeneity measure ${\mathcal{H}}_S$ (black, left axis) and spectral dimension $D_{\mathrm{S}}$ (grey, right axis) of the quantum spacetime geometry as a function of the diffusion time $\sigma$ for an ensemble of causal triangulations characterized by $T=64$ and $k_0=1.0$. (a) $N_3=10850$ (b) $N_3=30850$ (c) $N_3=61440$.

Figure 8

(a) Spectral homogeneity measure ${\mathcal{H}}_S$ of the quantum spacetime geometry as a function of the diffusion time $\sigma$ for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$. (b) Finite size scaling analysis of the spectral homogeneity measure ${\mathcal{H}}_S$ of the quantum spacetime geometry as a function of the diffusion time $\sigma$ for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$.

Figure 9

(a) Spectral dimension $D_S$ of the quantum spacetime geometry as a function of the diffusion time $\sigma$ for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$. (b) Finite size scaling analysis of the spectral dimension $D_S$ of the quantum spacetime geometry as a function of the diffusion time $\sigma$ for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$.

Figure 10

(a) The diffusion time ${\sigma }_{\text{infl}}$ as a function of the number $N_3$ of 3-simplices for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$. (b) The finite size scaled diffusion time ${\sigma }_{\text{infl}}/N_3^{2/3}$ as a function of the number $N_3$ of 3-simplices for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$. (c) The spectral homogeneity measure ${\mathcal{H}}_S(r{\sigma }_{\text{infl}})$ of the quantum spacetime geometry as a function of the number $N_3$ of 3-simplices for three ensembles of causal triangulations characterized by $T=64$ and $k_0=1.0$.

Figure 11

Ensemble average number $⟨N_2^{\mathrm{SL}}⟩$ of spacelike 2-simplices within the central accumulation as a function of the discrete time coordinate $\tau$ for an ensemble of causal triangulations characterized by $T=64$, $N_3=10850$, $k_0=1.0$. The colors red, green, blue, dark blue, dark green, and dark red correspond respectively to the discrete time coordinate values $\tau =-15/2$, $\tau =-9/2$, $\tau =-3/2$, $\tau =+3/2$, $\tau =+9/2$, and $\tau =+15/2$.

Figure 12

Volumetric homogeneity measure $H_V$ (dark, left axis) and Hausdorff dimension $d_{\mathrm{H}}$ (light, right axis) of the quantum spatial geometry as a function of the graph geodesic distance $r$ for an ensemble of causal triangulations characterized by $T=64$, $N_3=10850$, $k_0=1.0$. (a) Leaf of the distinguished foliation at $\tau =-15/2$ (red). (b) Leaf of the distinguished foliation at $\tau =-9/2$ (green). (c) Leaf of the distinguished foliation at $\tau =-3/2$ (blue). (d) Leaf of the distinguished foliation at $\tau =+15/2$ (dark red). (e) Leaf of the distinguished foliation at $\tau =+9/2$ (dark green). (f) Leaf of the distinguished foliation at $\tau =+3/2$ (dark blue).

Figure 13

Volumetric homogeneity measure $H_V$ of the quantum spatial geometry as a function of the graph geodesic distance $r$ for an ensemble of causal triangulations characterized by $T=64$, $N_3=10850$, $k_0=1.0$.

Figure 14

(a) Logarithm of the graph geodesic distance $r_{\max }$ as a function of the logarithm of the ensemble average number $⟨N_2^{\mathrm{SL}}⟩$ of spacelike 2-simplices for an ensemble of causal triangulations characterized by $T=64$, $N_3=10850$, $k_0=1.0$. (b) Linear fit for the discrete time coordinate values $\tau =-15/2$, $\tau =-9/2$, and $\tau =-3/2$ (left) and linear fit for the discrete time coordinate values $\tau =+3/2$, $\tau =+9/2$, and $\tau =+15/2$ (right).

Figure 15

(a) Logarithm of the homogeneity measure $H_V(r_{\max })$ as a function of the logarithm of the ensemble average number $⟨N_2^{\mathrm{SL}}(\tau )⟩$ of spacelike 2-simplices for an ensemble of causal triangulations characterized by $T=64$, $N_3=10850$, $k_0=1.0$. (b) Linear fit for the discrete time coordinate values $\tau =-15/2$, $\tau =-9/2$, and $\tau =-3/2$ (left) and linear fit for the discrete time coordinate values $\tau =+3/2$, $\tau =+9/2$, and $\tau =+15/2$ (right).

Figure 16

Spectral homogeneity measure $H_S$ (dark, left axis) and spectral dimension $d_{\mathrm{S}}$ (light, right axis) of the quantum spatial geometry as a function of the diffusion time $\sigma$ for an ensemble of causal triangulations characterized by $T=64$, $N_3=10850$, $k_0=1.0$. (a) Leaf of the distinguished foliation at $\tau =-15/2$ (red). (b) Leaf of the distinguished foliation at $\tau =-9/2$ (green). (c) Leaf of the distinguished foliation at $\tau =-3/2$ (blue). (d) Leaf of the distinguished foliation at $\tau =+15/2$ (dark red). (e) Leaf of the distinguished foliation at $\tau =+9/2$ (dark green). (f) Leaf of the distinguished foliation at $\tau =+3/2$ (dark blue).

Figure 17

Spectral homogeneity measure $H_S$ of the quantum spatial geometry as a function of the diffusion time $\sigma$ for an ensemble of causal triangulations characterized by $T=64$, $N_3=10850$, $k_0=1.0$.

Figure 18

Logarithm of the diffusion time ${\sigma }_{\text{infl}}$ as a function of the logarithm of the ensemble average number $⟨N_2^{\mathrm{SL}}⟩$ of spacelike 2-simplices for an ensemble of causal triangulations characterized by $T=64$, $N_3=10850$, $k_0=1.0$.

Figure 19

(a) Logarithm of the homogeneity measure $H_V({\sigma }_{\text{infl}})$ as a function of the logarithm of the ensemble average number $⟨N_2^{\mathrm{SL}}⟩$ of spacelike 2-simplices for an ensemble of causal triangulations characterized by $T=64$, $N_3=10850$, $k_0=1.0$. (b) Linear fit for the discrete time coordinate values $\tau =-15/2$, $\tau =-9/2$, and $\tau =-3/2$ corresponding to increasing discrete spatial 2-volume (left) and for the discrete time coordinate values $\tau =+3/2$, $\tau =+9/2$, and $\tau =+15/2$ corresponding to decreasing discrete spatial 2-volume (right).