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  • being a square root of the

    2018-11-02

    being a square root of the volume V/2π2, will be considered as arbitrary real . Now, according to our choice of the cosmic time (6) which is canonically conjugate to the kinetic energy of the scalar field (5), the WDW equation in the slow-roll regime can be replaced by the time-dependent Schrödinger equation: where
    Here, following Ref. [9], we have neglected the spatial curvature term in Eq. (8). Proto-inflation quantum state of the universe of concern to us is the quantum state of the universe at the moment of the cosmic time, when the inflaton scalar field takes its maximal value φ0. In the classical theory, the initial proto-inflation scale factor of the universe should be taken as the end point of the Vilenkin tunnel path determined as the non-zero solution of the equation
    The spatial curvature term is important in this definition of the initial radius of the universe before inflation:
    According to Eq. (11), the initial pre-inflation radius of the universe is inversely proportional to the square root of a gravitational energy density gV(φ0) related to the initial scalar field value φ0. Therefore, the initial value φ0 of the inflaton scalar field is related both to the beginning of the cosmic time and to the energy found for the universe before the inflation. One can read Eq. (10) as a balance between the gravitational energy of the vacuum condensate of the initial scalar field and an \"elastic energy\" of the spatial curvature of the universe [18] in the initial state. This simple classical energy balance is a ryanodine version of the gravitational constraints in a closed universe. In the general case, it can be formulated as a variant of the positive energy theorem for a closed universe [19], which has the form of an equality of two strictly positive quantities. One of them, the square of an eigenvalue of a 3D Dirac operator in a spatial slice, which includes the \"elastic energy\" of the curvature, we relate to an energy of space. The second one is a positive definite energy of all physical degrees of freedom of a closed universe, including the transversal degrees of freedom of gravitational field. In Refs. [14, 16] this positive energy theorem is used to define the ground state of the universe in quantum cosmology, with a minimal energy of space and, respectively, a minimal energy of its matter content. It is this ground state that we will take in the present work as a proto-inflation initial state of the universe. In the simple minisuperspace model given by the Hamiltonian constraint (3) the first term represents the energy of space, and the second one the energy of the scalar field. In the classical theory, the constraint implies that both energies compensate each other. In the definition of a ground state of the universe, given in Refs. [14, 16], the equality of the quantum average values of these energies takes into account an additional condition in the minimal principle. Principle of the space energy minimum.The quantitymust be extremal with respect to the variations of a quantum state of the universe ψ(x) and the Lagrangian multiplier L. Here and is the operator of the energy of space. Being a positive definite, the operator of the space energy (14) has a minimal positive average value. The corresponding state of minimal energy can be approximated by a probe function, which would be taken as a Gauss wave packet (as well as that in Ref. [9]): where χ0 is a variation parameter. For this state,. This parameter is determined by the quantum constraint equation which gives an estimation of the pre-inflation volume of the universe: where is the Planck length. As usual, apart from the ground state, there exists a set of extremal solutions, which obey the equation where plus the quantum constraint Eq. (16) for the Lagrangian multiplier L. Solutions of Eq. (18) are eigenstates of a quantum harmonic oscillator, on the condition that :
    From Eq. (19), we obtain the energy spectrum of space