2015/03/31 by Laura Castelló Gomar, Mercedes Martín-Benito, Guillermo A. Mena Marugán · 114 citations
Mathematics · Physics and Astronomy · #Ansatz #Black Holes and Theoretical Physics #Classical mechanics #Cosmology and Gravitation Theories #Covariance #Gauge theory #Hamiltonian (control theory) #Hamiltonian constraint #Loop quantum gravity #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Quantum #Quantum field theory #Quantum gravity #Quantum mechanics #Scalar field #gr-qc
paper · pdf · doi:10.1088/1475-7516/2015/06/045
published in Journal of Cosmology and Astroparticle Physics 2015(06), 045 (Institute of Physics) · 35 pages. Published in JCAP. Some misprints corrected
openalex publication_date 2015/06/26 · arxiv created 2015/11/27 · arxiv updated 2015/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We consider cosmological perturbations around homogeneous and isotropic spacetimes minimally coupled to a scalar field and present a formulation which is designed to preserve covariance. We truncate the action at quadratic perturbative order and particularize our analysis to flat compact spatial sections and a field potential given by a mass term, although the formalism can be extended to other topologies and potentials. The perturbations are described in terms of Mukhanov-Sasaki gauge invariants, linear perturbative constraints, and variables canonically conjugate to them. This set is completed into a canonical one for the entire system, including the homogeneous degrees of freedom. We find the global Hamiltonian constraint of the model, in which the contribution of the homogeneous sector is corrected with a term quadratic in the perturbations, that can be identified as the Mukhanov-Sasaki Hamiltonian in our formulation. We then adopt a hybrid approach to quantize the model, combining a quantum representation of the homogeneous sector with a more standard field quantization of the perturbations. Covariance is guaranteed in this approach inasmuch as no gauge fixing is adopted. Next, we adopt a Born-Oppenheimer ansatz for physical states and show how to obtain a Schrödinger-like equation for the quantum evolution of the perturbations. This evolution is governed by the Mukhanov-Sasaki Hamiltonian, with the dependence on the homogeneous geometry evaluated at quantum expectation values, and with a time parameter defined also in terms of suitable expectation values on that geometry. Finally, we derive effective equations for the dynamics of the Mukhanov-Sasaki gauge invariants, that include quantum contributions, but have the same ultraviolet limit as the classical equations. They provide the master equation to extract predictions about the power spectrum of primordial scalar perturbations.