Spectral geometry as a probe of quantum spacetime

Abstract

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 behaviour 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.

Asymptotic Safety, Fractals, and Cosmology

“…There the spectral dimension is even smaller than near the fixed point, namely d s = 4/3 in the case of 4 dimensions classically. Moreover, we shall argue that the (3-dimensional) results reported in [63] are in perfect accord with QEG, but that the shortest possible length scale achieved in the simulations is not yet close to the Planck length. Rather the Monte Carlo data probes the transition between the classical and the newly discovered "semi-classical" regime [68].…”

“…In particular, it is easily accessible in Monte Carlo simulations of the Causal Dynamical Triangulations (CDT) approach in d = 4 [61] and d = 3 [63] as well as in Euclidean Dynamical Triangulations (EDT) [67]. This feature allows a direct comparison between D CDT s (T ) and D EDT s (T ) obtained within the discrete approaches and D QEG s (T ) capturing the fractal properties of the QEG effective space-times.…”

“…This feature allows a direct comparison between D CDT s (T ) and D EDT s (T ) obtained within the discrete approaches and D QEG s (T ) capturing the fractal properties of the QEG effective space-times. In [68] we carried out this analysis for d = 3, using the Monte Carlo data obtained in [63] according to the following scheme: (i) First, we numerically construct a RG trajectory g k (g 0 , λ 0 ), λ k (g 0 , λ 0 ) depending on the initial conditions g 0 , λ 0 , by solving the flow equations (18).…”

“…[63] carried out CDT simulations for d = 3 macroscopic dimensions, which favor a value near d s = 2 on the shortest length-scale probed since, in this case, the QEG prediction for the fixed point region is the value d s = 3/2 [66]. Furthermore, the authors of ref.…”

“…Along a different line of investigations, the Causal Dynamical Triangulation (CDT) approach has been developed and first Monte-Carlo simulations were performed [59][60][61][62][63][64], see [65] for a recent review. In this framework one attempts to compute quantum gravity partition functions by numerically constructing the continuum limit of an appropriate statistical mechanics system.…”