The vacuum state in hybrid loop quantum cosmology

Abstract

In generic curved spacetimes, the absence of a natural vacuum state introduces an ambiguity that can undermine the physical relevance of predictions from any field quantization. In the context of inhomogeneous fields propagating in homogeneous, isotropic, but otherwise general cosmological spacetimes, this problem obstructs the extraction of robust predictions. This obstruction is aggravated in applications to cosmology of candidates to a quantum theory of gravity, where even the cosmological background where the fields propagate is treated as a quantum entity, or at most as an effective spacetime. One example is provided by the hybrid approach to quantum cosmology, in which a quantum mechanical description of the cosmological degrees of freedom, usually within Loop Quantum Cosmology (LQC), is combined with a more conventional Fock quantization of the infinite number of degrees of freedom that account for the inhomogeneities. In this context, we investigate in this thesis physical criteria to successfully remove the ambiguity of choice of vacuum state for two different types of fields in hybrid LQC: fermionic fields treated as perturbations, and primordial scalar and tensor perturbations leading to non-oscillating (NO) power spectra. For fermions, we first restrict ourselves to a family of vacua which leads to a unitarily implementable quantum Heisenberg evolution. Then, we manage to further restrict this choice by considering the asymptotic limit of large Fourier wavenumbers in the mode decomposition of the Dirac field and demanding there a convergent quantum backreaction. Further restrictions in this limit also guarantee that the fermionic contribution to the Hamiltonian be a well defined quantum operator on the dense subset of the fermionic Fock space which is spanned by the n-particle/antiparticle states. Finally, we use the entire available asymptotic freedom in what respects the definition of a vacuum state to eliminate from the fermionic Hamiltonian any term which creates or annihilates pairs of particles, at any given order in the asymptotic limit. We compare the vacuum selected by these physical criteria with fermionic adiabatic states, which had previously been proposed as potential vacua in cosmology. Actually, we prove that all adiabatic states allow a unitarily implementable quantum evolution. Furthermore, all of them but the zeroth order adiabatic state give rise to a finite backreaction. To finish our study of the fermionic vacuum, we apply the suggested asymptotic diagonalization procedure in a de Sitter Universe, showing that it picks out a unique vacuum state, which in fact coincides with the wellknown Bunch-davies vacuum. In addition to the problem of fermions in cosmology, we also discuss the possible choice of a vacuum state for scalar and tensor cosmological perturbations in LQC by demanding an NO power spectrum. This type of NO vacuum was originally introduced by numeric means to avoid the rapid oscillations in the spectrum found in the literature for other states of the perturbations, oscillations that could result in an amplification of power when averaged over bins of Fourier wavenumbers. We provide some analytic insights into why these oscillations may in fact be an artifact of the choice of vacuum state and how they can blur the actual quantum geometry effects in observational predictions if they are not avoided. We also give some analytical conditions that are necessary on a vacuum state if it is of NO type, and prove that in the ultraviolet asymptotic limit this class of vacua satisfy the asymptotic diagonalization proposal. Finally, we compare these necessary NO conditions with a construction for the vacuum put forward recently by Ashtekar and Gupt. This construction should select the state which is maximally classic at the end of inflation from a ball of states that is picked out by the so-called Quantum Homogeneity and Isotropy Hypothesis (QHIH). However, we find a loose step in the proposed construction which allows that the Ashtekar-Gupt vacuum to exist outside of the QHIH ball. In fact, we prove numerically that, in a kinetically dominated short-lived inflationary scenario typically considered in LQC, the Ashtekar-Gupt vacuum lies outside of the QHIH ball. Nonetheless, we show that the NO necessary conditions and the QHIH are mutually non-exclusive in this scenario