Spectral geometry as a probe of quantum spacetime

Employing standard results from spectral geometry, we provide strong evidence that in the classical limit the ground state of three-dimensional causal dynamical triangulations is de Sitter spacetime. This result is obtained by measuring the expectation value of the spectral dimension on the ensemble of geometries defined by these models, and comparing its large-scale behavior to that of a sphere (Euclidean de Sitter). From the same measurement we are also able to confirm the phenomenon of dynamical dimensional reduction observed in this and other approaches to quantum gravity—the first time this has been done for three-dimensional causal dynamical triangulations. In this case, the value for the short-scale limit of the spectral dimension that we find is approximately 2. We comment on the relevance of these results for the comparison to asymptotic safety and Hořava-Lifshitz gravity, among other approaches to quantum gravity.

Figure 1

The spectral dimension data against scaled diffusion time $\tilde{\sigma }=\sigma N^{-2/3}$. Data for $N=50\mathrm{k}$, 100 k, and 200 k is shown (the larger the value of $N$ is, the larger the maximum of the function is). Error bars are suppressed on most data points for clarity (this is done throughout the paper). The curves should converge to a limiting curve as $N\to \infty$. At these values of $N$ it there is good evidence of convergence to a continuum limit: the curves for $N=100\mathrm{k}$ and $N=200\mathrm{k}$ agree within error for $\tilde{\sigma }>0.5$, as do the $N=50\mathrm{k}$ and $N=100\mathrm{k}$ curves.

Figure 2

The spectral dimension data from CDT simulations with $N=200\mathrm{k}$ against scaled diffusion time $\tilde{\sigma }=\sigma N^{-2/3}$, plotted in black, fitted to the spectral dimension plot for the scaled sphere with $r=1.20$, $s=1.96$, superimposed in a lighter color. The two-parameter fit agrees with the data for $\tilde{\sigma }\gtrsim 1.5$. At very low times the “quantum correction” to the posited classical geometry is negative, but it becomes positive in an intermediate range.

Figure 3

The overlap between the CDT data for $D_s(\tilde{\sigma },N)$ and that for the scaled sphere with $r=1.20$, $a=1.96$, as a function of $1/N$, with errors. Data for $N=70\mathrm{k}$, 100 k, 140 k, and 200 k is used here. The difference between the two functions is measured as the integrated absolute difference between them in the range $0<\tilde{\sigma }<7.4$. As $1/N\to 0$ the difference comes closer to 0. The results are consistent with the convergence of the CDT data with the scaled sphere spectral dimension function as $N\to \infty$. (At smaller values of $N$ than 70 k, there is competition between the positive and negative quantum corrections that obscures the convergence.)

Figure 4

The Spectral dimension data with number of simplices $N=70\mathrm{k}$, at small diffusion times, is plotted with error bars in black. The exponential fit is superimposed in a lighter color.

Figure 5

The Spectral dimension data with number of simplices $N=200\mathrm{k}$, at small diffusion times, is plotted with error bars in black. The exponential fit is superimposed in a lighter color.

Figure 6

An embedding in ${\mathbb{R}}^3$ of the $\theta =\pi /2$ section of the three-dimensional stretched sphere, with $r=1.2$ and $s=1.96$.