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  • and the corresponding values of

    2018-11-02

    and the corresponding values of the universe\'s volume: with , which implies the constraint Eq. (16). Corresponding eigenfunctions are the following [20]:
    Inflation as the birth of the space quanta So then, we suppose that the state of minimal excitation of space (15) is the initial proto-inflation state of the universe. It is not stationary for the Hamiltonian (13) and, therefore, will evolve with time. Let us consider the ryanodine manufacturer at small times when the inflaton scalar field can be considered to be constant (). The kernel of the evolution operator in this case would be calculated in the same way as that for the ordinary harmonic oscillator [21]. It becomes as follows: where
    Here is the action of the geometrical variable x(t) calculated on a classical trajectory with the end points (x0, 0) and (x1, T); it equals
    Evolution of the initial ground state with time is described in the form
    Some simple, but rather great calculations in volume give the following:
    According to Eq. (27), the packet width grows at large times exponentially with high accuracy:
    Such behavior also results in the average volume of the universe: which confirms the inflation scenario, but it is of interest to describe this behavior in the context of the birth of space quanta introduced earlier. Here we shall restrict ourselves to calculating the probability of the universe to remain at the initial ground state with time T. The amplitude of the outcome equals where
    The last multiplier Q(T) in Eq. (30) remains limited with time, as well as the third one, D(T). It is the second multiplier F(T) that provides the exponential damping of the amplitude:
    Since the space evolution described by the Schrödinger Eq. (8) is unitary, this result indicates the fast growth of the space quanta number. According to Eq. (30), this number increases exponentially:
    Having started with an initial state of the universe of minimal excitation at a given value of the inflaton scalar field φ0, we find out its expansion as the birth of the space energy quanta. Now, our interest is in the birth of matter from the original ground state. The presence of matter, different from the inflaton scalar field, can be recorded in our model as an additional part of the Hamiltonian constraint (3); for example, for a scalar field ϕ. Starting with its ground state, we do not obtain any sensible birth of its ryanodine manufacturer quanta at the inflation stage, when the potential energy of the inflaton scalar field φ dominates. However, according to the following quantum constraint equation the energy of matter is bound to be sufficiently high at the end of the inflation stage, when φ ≈ 0, in order to compensate the energy of space .
    Summary
    Acknowledgment