Axion excursions of the landscape during inflation
  Because of their quantum fluctuations, axion fields had a chance to
experience field excursions traversing many minima of their potentials during
inflation. We study this situation by analyzing the dynamics of an
axion-spectator field $\psi$, present during inflation, with a periodic
potential given by $v(\psi) = \Lambda^4 [1 - \cos (\psi / f)]$. By assuming
that the vacuum expectation value of the field is stabilized at one of its
minima, say $\psi = 0$, we compute every $n$-point correlation function of
$\psi$ up to first order in $\Lambda^4$ using the in-in formalism. This
computation allows us to identify the distribution function describing the
probability of measuring $\psi$ at a particular amplitude during inflation.
Because $\psi$ is able to tunnel between the barriers of the potential, we find
that the probability distribution function consists of a non-Gaussian
multimodal distribution such that the probability of measuring $\psi$ at a
minimum of $v(\psi)$ different from $\psi=0$ increases with time. As a result,
at the end of inflation, different patches of the Universe are characterized by
different values of the axion field amplitude, leading to important
cosmological phenomenology: (a) Isocurvature fluctuations induced by the axion
at the end of inflation could be highly non-Gaussian. (b) If the axion defines
the strength of standard model couplings, then one is led to a concrete
realization of the multiverse. (c) If the axion corresponds to dark matter, one
is led to the possibility that, within our observable Universe, dark matter
started with a nontrivial initial condition, implying novel signatures for
future surveys.
