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    Mathematical proof that the universe originated from nothing

    That the universe began as the result of a quantum fluctuation in a vacuum is a definite mathematical possibility, especially because of the postulates of the Heisenberg uncertainty principle:

    A cosmological model is proposed in which the universe is created by quantum tunneling from literally nothing into a de Sitter space. After the tunneling, the model evolves along the lines of the inflationary scenario. This model does not have a big-bang singularity and does not require any initial or boundary conditions.

    In this paper I would like to suggest a new cosmological scenario in which the universe is spontaneously created from literally nothing, and which is free from the difficulties I mentioned in the preceding paragraph. This scenario does not require any changes in the fundamental equations of physics; it only gives a new interpretation to a well-known cosmological solution.

    We shall consider a model of interacting gravitational and matter fields. The matter content of the model can be taken to be that of some grand unified theory (GUT). The absolute minimum of the effective potential is reached when the Higgs field Φ responsible for the GUT symmetry breaking acquires a vacuum expectation value, (Φ) = σ << mp. The symmetric vacuum state, (Φ) = 0, has a nonzero energy density, ρv. For a Coleman-Weinberg potential,
    ρv ~ g4σ4 , (l)
    where g is the gauge coupling.

    Suppose that the universe starts in the symmetric vacuum state and is described by a closed Robertson-Walker metric
    ds2 = dt2 – a2(t)[dr2/(1 – r2) + r2 dr2dΩ2] . (2)

    The scale factor a(t) can be found from the evolution equation
    (da/dt)2 + 1 = 8/3πGpva2 , (3)
    The solution of this equation is the de Sitter space,
    a(t) = H-1 cosh(Ht), (4)
    where H = (8πGpv/3)1/2. It describes a universe which is contracting at t < 0, reaches its minimum size (amin = H-1) at t = 0, and is expanding at t > 0. This behaviour is analogous to that of a particle bouncing off a potential barrier at a = H-1. (Here a plays the role of the particle coordinate.) We know that in quantum mechanics particles can tunnel through potential barriers. This suggests that the birth of the universe might be a quantum tunneling effect. Then the universe has emerged having a finite size (a = H-1) and zero "velocity" (da/dt = 0); its following evolution is described by eq. (4) with t > 0.

    Sidney Coleman [9] has taught us that a semiclassical description of quantum tunneling is given by the bounce solution of euclidean field equations (that is, of the field equations with t changed to -it). Normally, bounce solutions are used to describe the decay of a quasistable state. If the decaying state is at the bottom of a potential well at x = xl, then the bounce solution starts with x = x1 at t -> -oo bounces off the classical turning point at the end of the barrier, and returns to x = x1 at t -> + oo.

    The euclidean version of eq. (3) is -(da/dt)2 + 1 = H2a2, and the solution is
    a(t) = H-1 cos(Ht). (5)

    Eqs. (2) and (5) describe a four-sphere, S4. This is the well-known de Sitter instanton [10]. The solution (5) does bounce at the classical turning point (a = H-1); however, it does not approach any initial state at t -> +/-oo. In fact, S4 is a compact space, and the solution (5) is defined only for |t| < π/2 H. The instanton (5) can be interpreted as describing the tunneling to de Sitter space (4) from nothing. Then the birth of the universe is symbolically represented in fig. 1(a).


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