Tunelamento quântico em modelos cosmológicos na teoria de Hořava-Lifshitz

Abstract

In this thesis, two cosmological models are examined from the perspective of Quantum Cosmology using the gravitational theory of Hořava-Lifshitz. Both models feature a perfect fluid as their material content. The first model is comprised of radiation (ω = 1/3), while the second model is filled with dust (ω = 0). These models are assembled using the Friedmann-Lemaître-Robertson-Walker metric with a positive curvature constant (k = +1). The dynamics of the models are described using the Hamiltonian formalism, with the ADM and Schutz formalisms utilized to derive the total Hamiltonian of the models. Classical solutions for the scale factor are obtained by constructing a phase space (a, pa), where a(t) represents the scale factor and pa(t) is its canonical conjugate momentum. A differential equation for the scale factor is then obtained, revealing the dynamics of the scale factor and demonstrating the absence of a singularity. As the objective of this work is to investigate the birth of the universe through the quantum tunnelling process, the models are quantized. This is achieved by introducing the wavefunction of the universe and applying Dirac’s formalism for constrained systems, leading to the Wheeler-DeWitt equation for the model under examination. The equation is further solved using a finite difference method and also by the WKB method. Then, the tunneling probabilities were obtained using the numerical solution, which is called T Pint, and the WKB solution, which is called T PWKB. Finally, an investigation is conducted to determine how the energy E and the HL theory parameters gc, gΛ, gr, and gs influence the tunnelling probabilities T Pint and T PWKB.