Characteristics of the new phase in CDT

Abstract

The approach of Causal Dynamical Triangulations (CDT), a candidate theory of nonperturbative quantum gravity in 4D, turns out to have a rich phase structure. We investigate the recently discovered bifurcation phase and relate some of its characteristics to the presence of singular vertices of very high order. The transition lines separating this phase from the “time-collapsed” B-phase and the de Sitter phase are of great interest when searching for physical scaling limits. The work presented here sheds light on the mechanisms behind these transitions. First, we study how the B– transition signal depends on the volume fixing implemented in the simulations, and find results compatible with the previously determined second-order character of the transition. The transition persists in a transfer matrix formulation, where the system’s time extension is taken to be minimal. Second, we relate the new – transition to the appearance of singular vertices, which leads to a direct physical interpretation in terms of a breaking of the homogeneity and isotropy observed in the de Sitter phase when crossing from to the bifurcation phase .

Spectrum of the Laplace-Beltrami operator and the phase structure of causal dynamical triangulations

“…Indeed, the average spatial volume distribution in the C dS phase is in good agreement with the prediction for a de Sitter Universe, having a S 4 geometry after analytical continuation to the Euclidean space [28]. The bifurcation phase, instead, is characterized by the presence of two different classes of slices which alternate each other in the slice time t [5,6].…”

“…In order to take advantage of finite-size scaling methods (i.e., extrapolation of results to the infinite volume limit), it is convenient to control the volume by performing a Legendre transformation from the parameter triple ðk 4 ; k 0 ; ΔÞ to the triple ðV; k 0 ; ΔÞ, where the parameter k 4 is traded for a target volume V. In practice, this is implemented by a fine tuning of the parameter k 4 to a value that makes the total spacetime or spatial volumes 5 2 , one can investigate the properties of configurations sampled at different values of the remaining free parameters k 0 and Δ.…”

“…The general phase structure of CDT which is found in the k 0 -Δ plane is thoroughly discussed in the literature [1,5,6,27]. Here we will only recall some useful facts.…”

“…(3): Regge discretization of the continuous action, computation of volumes and dihedral angles, Wick rotation and the use of topological relations between simplex types. 5 The total spatial volume of a configuration is the number of spatial tetrahedra, which equals…”

“…Finite-size scaling computations using these observables as order parameters suggest a first-order nature for the A-C dS transition, while the B-C b transition appears to be of second order [27]. The definition of observables employed as order parameters for the C b -C dS transition is more involved [5,6]: in the C b phase, one of the two classes of spatial slices is characterized by the presence of vertices with very high coordination number; also in this case there are hints for a second-order transition, even if results might depend on the topology chosen for the spatial slices [7].…”