  Much of the dynamical structure of the Kuiper belt can be explained if
Neptune migrated over several AU, and/or if Neptune was scattered to an
eccentric orbit during planetary instability. An outstanding problem with the
existing formation models is that the distribution of orbital inclinations they
predicted is narrower than the one inferred from observations. Here we perform
numerical simulations of Kuiper belt formation starting from an initial state
with Neptune at 20 < a_{N,0} < 30 AU and a dynamically cold outer disk
extending from beyond a_{N,0} to 30 AU. Neptune's orbit is migrated into the
disk on an e-folding timescale 1 <= tau <= 100 Myr. A small fraction (~10^{-3})
of the disk planetesimals become implanted into the Kuiper belt in the
simulations. By analyzing the orbital distribution of the implanted bodies in
different cases we find that the inclination constraint implies that tau >= 10
Myr and a_{N,0} <= 25 AU. The models with tau < 10 Myr do not satisfy the
inclination constraint, because there is not enough time for various dynamical
processes to raise inclinations. The slow migration of Neptune is consistent
with other Kuiper belt constraints, and with recently developed models of
planetary instability/migration. Neptune's eccentricity and inclination are
never large in these models (e_N<0.1, i_N<2 deg), as required to avoid
excessive orbital excitation in the >40 AU region, where the Cold Classicals
presumably formed.
