Advances in High Energy Physics

Volume 2017 (2017), Article ID 4015145, 5 pages

https://doi.org/10.1155/2017/4015145

## Big Bang as a Critical Point

Institute of Physics, Jagiellonian University, Ul. Łojasiewicza 11, 30-348 Kraków, Poland

Correspondence should be addressed to Jakub Mielczarek; lp.ude.ju@kerazcleim.bukaj

Received 12 May 2017; Accepted 4 July 2017; Published 22 August 2017

Academic Editor: Jerzy Kowalski-Glikman

Copyright © 2017 Jakub Mielczarek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

This article addresses the issue of possible gravitational phase transitions in the early universe. We suggest that a second-order phase transition observed in the Causal Dynamical Triangulations approach to quantum gravity may have a cosmological relevance. The phase transition interpolates between a nongeometric crumpled phase of gravity and an extended phase with classical properties. Transition of this kind has been postulated earlier in the context of geometrogenesis in the Quantum Graphity approach to quantum gravity. We show that critical behavior may also be associated with a signature change in Loop Quantum Cosmology, which occurs as a result of quantum deformation of the hypersurface deformation algebra. In the considered cases, classical space-time originates at the critical point associated with a second-order phase transition. Relation between the gravitational phase transitions and the corresponding change of symmetry is underlined.

#### 1. Introduction

Accumulating results of theoretical investigations indicate that the gravitational field exists in different phases. First indications supporting such an idea came from considerations of three-dimensional Euclidean quantum gravity [1]. By means of Monte Carlo simulations it was possible to explore the configuration of the gravitational field under various conditions. For four-dimensional Euclidean gravity, gravity exhibits two phases: the* crumpled* phase and the* branched polymer* phase [2]. This result has since been generalized to the case of four-dimensional gravity with an imposed causality condition, formulation known as Causal Dynamical Triangulations (CDT). The causality condition turned out to be essential for the correct phase structure of gravity, leading to emergence of the four-dimensional space-time [3]. Generation of such extended phase in the Euclidean approach introducing a nontrivial path integral measure remains an interesting possibility [4]. Furthermore, similarly to a phase structure of the Lifshitz scalar [5], the critical surface of CDT has been divided into three regions separated by the first- and second-order transition lines [6]. Interestingly, a theory describing gravity at a triple point (Lifshitz point) of the phase diagram has been constructed and shown to be power-counting renormalizable [7]. Further evidence for the nontrivial phase structure of gravity comes from Quantum Graphity [8]. This approach utilizes the idea of* geometrogenesis:* a transition between geometric and nongeometric phases of gravity.

The basic question one can ask, assuming the existence of the different phases of gravity, is where can the other phases be found? A natural place to search for them is high curvature regions such as interiors of the black holes and the early universe. Because of a horizon, a possibility of relating phase change inside of black holes with astronomical observations is a difficult task. Nevertheless, gravitational phase transitions occurring under the black hole horizon, including the signature change transition, have been a subject of theoretical studies (see, e.g., [9, 10]). Perhaps empirically more promising is a search for signatures of the gravitational phase transitions which took place in the early universe. We will focus on this direction here.

So far, there has been very little attention devoted to this issue in the literature. Most studies of the phase transitions in the early universe were dedicated to the matter sector, rather than gravity [11, 12]. Among the few studies on the gravitational phase transitions in the early universe, the work of [13, 14] is especially noteworthy. In [13] a specific model of geometrogenesis, through a second-order phase transition, has been proposed. It was shown that, by assuming the holographic principle to be fulfilled in the high temperature phase, it is possible to generate a power spectrum of primordial perturbations that is in agreement with observations. In [14] the cosmological relevance of second-order phase transitions is discussed. Arguments supporting generation of “inflationary” power spectrum from critical behavior of the gravitational field have been presented.

In what follows we attract attention to the fact that a second-order gravitational phase transition has recently been observed within Causal Dynamical Triangulations [15]. The transition takes place exactly between the phases of the form discussed in [8, 13]. Therefore, CDT gives a concrete realization of the scenario of geometrogenesis. We also show that gravitational phase transition may be associated with the deformation of general covariance, recently observed in the context of Loop Quantum Cosmology (LQC). In both cases, the phase transition is of second order, suggesting a critical nature of the emergence of classical space-time in the early universe.

#### 2. Causal Dynamical Triangulations

Analysis performed within four-dimensional CDT with a positive cosmological constant indicates the presence of three different phases of the gravitational field, called A, B, and C [6]. The phases are separated by the first- (A-C) and second-order (B-C) transition lines presumably intersecting at the triple point. The order of the A-B phase transition has not been determined so far.

At large scales, phase C forms the four-dimensional de Sitter space [16]. The phase is, however, not fully classical since it exhibits dimensional reduction to two dimensions at short scales [17]. This can be shown by investigating properties of the spectral dimension, defined via a diffusion process. Nevertheless, the phase C can be associated with the “usual" phase of gravity. The two remaining phases are fundamentally different from this phase. Phase A is characterized by a vanishing interaction between adjacent time slices. Phase B, resembling the* crumpled* phase in Euclidean gravity, is characterized by a large (tending to infinity in the volume limit) Hausdorff and spectral dimension. Phase B shares features of the high temperature phase postulated in Quantum Graphity. Moreover, this phase is separated with the low energetic phase C by the second-order phase transition. This is in one-to-one correspondence to the Quantum Graphity case. Based on this observation, we hypothesize the following.

*Hypothesis 1. *In the early universe, there was a second- (or higher) order phase transition from the high temperature nongeometric phase to the low temperature geometric phase of gravity. The transition is associated with a change of the connectivity structure between the elementary chunks of space.

The change of connectivity can be inferred from the considerations of the spectral dimensions of the phases B and C. In order to see it explicitly let us consider a toy model of the universe composed of the chunks of space. They will be represented by the nodes of a graph. A structure of adjacency is represented by the links.

In phase C, which is a geometric phase, the degree of vertices is low. In our toy model it equals and the resulting space is represented by the Ring graph (see Figure 1(a)). The spectral dimension of this graph can be found by determining spectrum (eigenvalues ) of the Laplace operator , where is an adjacency matrix and is a degree matrix. By using the expression for the trace of the heat kernel one can find thatwhere is a diffusion time.