Gauge-invariant perturbations in hybrid quantum cosmology

Abstract

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.

Primordial tensor modes of the early Universe

“…We will mainly follow Refs. [21,22,27]. We consider a flat FRW spacetime with compact T 3 topology minimally coupled to a scalar field Φ subject to a quadratic potential (due to its special interest in observations) of mass m. The full action of the model is the time integral of the Lagrangian (see Ref.…”

“…Our analysis (as well as the one in Refs. [12,13,27]) will be based in the choice of ϕ as physical time in the unperturbed quantum theory 2 . This evolution picture in cosmology is commonly used since the constraint equation for the physical states Ψ 0 of the exact homogeneous model resembles a Klein-Gordon equation with ϕ playing the role of time.…”

“…If we now compute the inner product on the left-hand side of (27) with the bra associated to Γ ϕ , we obtain…”

“…[27] for scalar perturbations. Our analysis shows for the first time the explicit relation of the hybrid quantization with the dressed metric approach of Refs.…”

“…Besides, this hybrid quantization approach for cosmological perturbations admits a fully covariant description [27] for scalar inhomogeneities, but with strong differences with respect to the deformed algebra approach of Refs. [19,20].…”