A stochastic quasi-classical wavefunction of the Universe from the third quantization procedure

(abbreviated) We study quantized solutions of WdW equation describing a closed FRW universe with a $Λ$ term and a set of massless scalar fields. We show that when $Λ\ll 1$ in the natural units and the standard $in$-vacuum state is considered, either wavefunction of the universe, $Ψ$, or its derivative with respect to the scale factor, $a$, behave as random quasi-classical fields at sufficiently large values of $a$, when $1 \ll a \ll e^{2\over 3Λ}$ or $a \gg e^{2\over 3Λ}$, respectively. Statistical r.m.s value of the wavefunction is proportional to the Hartle-Hawking wavefunction for a closed universe with a $Λ$ term. Alternatively, the behaviour of our system at large values of $a$ can be described in terms of a density matrix corresponding to a mixed state, which is directly determined by statistical properties of $Ψ$. It gives a non-trivial probability distribution over field velocities. We suppose that a similar behaviour of $Ψ$ can be found in all models exhibiting copious production of excitations with respect to $out$-vacuum state associated with classical trajectories at large values of $a$. Thus, the third quantization procedure may provide a 'boundary condition' for classical solutions of WdW equation.

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