Cosmologia Quântica Computacional: uso do método de Crank-Nicolson em (2+1) na busca de soluções para a equação de Wheeler-DeWitt

Abstract

In the study of Cosmology, the Theory of General Relativity (RG) presents singularities in extremes situations, such as when we go back to the initial instants of the “Big-Bang”, or rather, in the Planck Era. The introduction of quantum formalism seeks to eliminate these singularities. This formalism assumes the existence of a wave function Ψ that carries all the properties of the system, known as the wave function of the Universe. The proposed models based on quantum formalism have their dynamics described by the so-called Wheeler-DeWitt Equation (WDW). Several works address solutions for this equation using different material contents, such as the cosmological constant, perfect fluid in the forms of: radiation, dust, rigid matter, Chaplygin’s gas, among others. Recently, several models with different matter contents have been quantized. In cases where the effective potential Vef assumes the form of a potential barrier, the possibility of the universe emerging to the right of the barrier through a tunneling mechanism has been analyzed. Recent solutions have adopted the Crank-Nicolson (CN) (1+1) to obtain numerical solutions for the WDW Equation and subsequently calculate the tunneling probability. According to the literature, after the Planck Era (PE), the Universe should emerge into a phase of great expansion known as the “Inflationary Phase”. At this stage, the presence of a scalar field (inflation field) is assumed, which dominates any other matter, leading the Universe to a very accelerated expansion, in a short period of time. This work considers a model in Einstein’s RG with Friedmann-Lemaître-Robertson-Walker (FRW) geometry to represent a homogeneous and isotropic model. The material content will be a perfect fluid in the form of radiation and a scalar field conformally coupled to gravity. In this case, the scalar field potential will have a cosmological constant term to represent vacuum energy and a mass term. The objective is to describe the primordial universe, that is, the phase that precedes the inflationary era. To do this, the model in question will be quantized, so that the dynamics will be governed by the so-called WDW Equation. The computational solution for this model (2+1) requires an efficient and robust computational package. Efficient because the volume of data is enormous, so the algorithms used need to be O(n) and especially parallelized. Robust because long-term computation must anticipate faults over time, such as processing interruptions. A computational package capable of performing simulations on High Performance Computing (HPC) Clusters was built; tests have shown that the program can simulate the evolution of the WDW Equation with minimal error introduction. From this, it was possible to carry out simulations in the model, the results are significant and point to a transition from the quantum model to the classical one.