Dong-il Hwang, Bum-Hoon Lee, Ewan D. Stewart, Dong-han Yeom, Heeseung Zoe
In this paper, we compare dispersions of a scalar field in Euclidean quantum gravity and stochastic inflation. We use Einstein gravity with a minimally coupled scalar field and a quadratic potential. We restrict our attention to small mass and small field cases. In Euclidean approach, we introduce the ground state wave function which is approximated by instantons. In stochastic approach, we introduce probability distribution of Hubble patches that can be approximated by a locally homogeneous universes up to a smoothing scale. We are assuming that the ground state wave function should correspond the stationary state of the probability distribution of the stochastic universe. By comparing the dispersion of both approaches, we conclude mainly three results. (1) For a statistical distribution with a certain value, we can find a corresponding instanton in the Euclidean side and it should be a complex-valued instanton. (2) The finite scale factor of the Euclidean approach corresponds the smoothing scale of the stochastic side; the universe is homogeneous up to the scale factor. (3) In addition, as the mass increases a critical value, both approaches break at the same time. Hence, generation of inhomogeneity in stochastic approach and the instability of classicality in Euclidean approach are related.
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http://arxiv.org/abs/1208.6563
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