The Hartle-Hawking and tunneling (Vilenkin) wave functions are treated in the Hamiltonian formalism. We find that the leading (i.e., quadratic) terms in the fluctuations around a maximally symmetric background are indeed Gaussian (rather than inverse Gaussian), for both types of wave function, when properly interpreted. However, the suppression of non-Gaussianities—and hence the recovery of the Bunch-Davies state—is not transparent.
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