Hosoya proposed that the cosmic time during the inflationary

Hosoya [9] proposed that the cosmic time during the inflationary phase in slow-roll of the inflaton scalar field ϕ with the potential could be the inflaton field itself (or, to be precise, its logarithm, t ∼ ln (ϕ/ϕ0), where ϕ0 is the initial value of the field). In this regime, the master wave equation of quantum cosmology, or the Wheeler–DeWitt (WdW) equation [7,8], takes (in a homogeneous model of the Universe) the form of a Schrödinger equation with the specified cosmic time parameter. In this approximation the radius of the Universe a is a quantum dynamical variable described by the wave function ψ(t, a). At this point the initial state of the Universe at must be determined. Ref. [9] regards the whole real axis (with negative values included) as the domain of definition of the scale variable a. The ordering of non-commuting operators in the WdW equation was chosen accordingly, and a Gaussian wave packet was taken as an initial state of the Universe. Before we can move on from discussing the quantum epoch in the birth of the Universe to the inflationary phase of its exponential expansion, we feel it is necessary to refine our definition of the initial state of the Universe. First of all, let us limit the domain of definition of the scale factor: a ∈ [0, ∞), assuming it to be a radial variable in the geometry sector of the configuration space of the Universe. The operator ordering in the WdW equation and the corresponding measure of integration in the configuration space must be selected accordingly. In this selection, we shall keep in mind the hyperbolic structure of the WdW equation for a more general anisotropic model of the Universe (“the mixmaster model”) [8]. Let us define the initial state of the Universe in the present work as a state of minimal excitation of all physical degrees of freedom, including the anisotropy parameters. For this purpose let us use the principle of minimum XMU-MP-1 for these excitations. A generalization of the notion of energy in GR made by Witten as a positive definite functional for the case of an island mass distribution with the asymptotically flat space-time geometry [10,11] was discussed for the case of the closed Universe in [12].
Canonical quantization of a homogeneous anisotropic model of the Universe Let us begin with a homogeneous anisotropic model of the Universe (“the mixmaster model” [8]), in which the 3D metrics of a spatial slice is parameterized as follows: where are the dual 1-forms of the rotation group SO (3);
Therefore, the spatial 3D geometry of the Universe in the specified system of spherical coordinates with the angular variables (ψ, θ, φ) is described by two independent dynamical variables , which is precisely as much independent degrees of freedom per spatial point as there is in the general case [8]. The only remaining Hamiltonian constraint H (super-Hamiltonian) in this model defining its dynamics has the form:
The second contribution to the Hamiltonian (6) given by (8) corresponds to the physical degrees of freedom of the model (the anisotropy parameters). We added to it the potential energy of the inflaton field as well. The potential energy of the anisotropy has a complicated form [8], but for determining the initial state of the Universe, a harmonic approximation is sufficient:
Canonical quantization of the model is reduced to replacing the canonical momenta in (7) and (8) by the corresponding differential operators:
The problem of operator ordering is that in the expression (7) it may be done in different ways. To settle on a specific way of ordering, we should implement the idea that the scale factor a is a radial variable in the configuration space of the model which is equipped by a metrical structure determined by the quadratic form of the canonical momenta in the super-Hamiltonian (6)
Therefore, the covariant components of the DeWitt metrics in the configuration space form the diagonal matrix