Fermions in hybrid loop quantum cosmology

Abstract

This work pioneers the quantization of primordial fermion perturbations in hybrid Loop Quantum Cosmology (LQC). We consider a Dirac field coupled to a spatially flat, homogeneous, and isotropic cosmology, sourced by a scalar inflaton, and treat the Dirac field as a perturbation. We describe the inhomogeneities of this field in terms of creation and annihilation variables, chosen to admit a unitary evolution if the Dirac fermion were treated as a test field. Considering instead the full system, we truncate its action at quadratic perturbative order and construct a canonical formulation. In particular this implies that, in the global Hamiltonian constraint of the model, the contribution of the homogeneous sector is corrected with a quadratic perturbative term. We then adopt the hybrid LQC approach to quantize the full model, combining the loop representation of the homogeneous geometry with the Fock quantization of the inhomogeneities. We assume a Born-Oppenheimer ansatz for physical states and show how to obtain a Schrödinger equation for the quantum evolution of the perturbations, where the role of time is played by the homogeneous inflaton. We prove that the resulting quantum evolution of the Dirac field is indeed unitary, despite the fact that the underlying homogeneous geometry has been quantized as well. Remarkably, in such evolution, the fermion field couples to an infinite sequence of quantum moments of the homogeneous geometry. Moreover, the evolved Fock vacuum of our fermion perturbations is shown to be an exact solution of the Schrödinger equation. Finally, we discuss in detail the quantum backreaction that the fermion field introduces in the global Hamiltonian constraint. For completeness, our quantum study includes since the beginning (gauge-invariant) scalar and tensor perturbations, that were studied in previous works.

Time-dependent mass of cosmological perturbations in the hybrid and dressed metric approaches to loop quantum cosmology

“…In particular, the homogeneous part of these states, Γða; ϕÞ, where a generically denotes dependence on the homogeneous geometry, may be chosen as an exact solution to the, homogeneous, quantum FLRW model, and in this paper we will take it this way. However, let us comment that this choice is in principle not needed, and in fact one may consider other possibilities for Γ that incorporate the presence of some quantum backreaction of the perturbations onto the homogeneous sector of the model [19,63]. With this ansatz at hand, and provided that Γ is sufficiently peaked on the homogeneous geometry for all values of ϕ, then the imposition of the zero-mode of the Hamiltonian constraint leads to the requirement that the outcome of certain operators, acting exclusively on the Mukhanov-Sasaki and the tensor parts of the wave function, must be zero.…”

The Vacuum State of Primordial Fluctuations in Hybrid Loop Quantum Cosmology

“…A physically appealing possibility is to use this freedom, and hence remove (at least part of) the ambiguity, looking for annihilation and creation-like variables such that their Hamiltonian admits a quantum representation with good properties. In fact, there already exist some preliminary investigations about this issue in the technically simpler case of fermionic perturbations around flat FLRW spacetimes, within the hybrid LQC approach [31,53]. For a Dirac field in this cosmological scenario, one can also carry out a characterization of the annihilation and creation-like variables that define invariant vacua under the symmetry transformations of the system and display a unitarily implementable dynamics when considered on a classical background, all this while retaining a standard convention about the respective identification of particles and antiparticles [45].…”

“…Two of these approaches do not modify the dispersion relations of the perturbations, providing in particular a proper ultraviolet behavior that is compatible with the observed power spectra. These are the hybrid [20,[23][24][25][26][27][28][29][30][31] and dressed metric approaches [21,[32][33][34][35][36][37]. Although there are plenty of similarities between these two approaches, the main distinctions follow from a slightly different strategy for the quantization of the perturbations [38].…”

Quantum Linear Scalar Fields with Time Dependent Potentials: Overview and Applications to Cosmology

“…within the Hybrid Quantization Approach [32] in order to deal with (both scalar and fermionic) perturbations in quantum cosmology (see, for instance, Refs. [100][101][102][103][104][105][106][107]).…”