  We present a detailed derivation of the recently suggested new type of
hill-top inflation [arXiv:1509.07270] originating from the microcanonical
density matrix initial conditions in cosmology driven by conformal field theory
(CFT). The cosmological instantons of topology $S^1\times S^3$, which set up
these initial conditions, have the shape of a garland with multiple periodic
oscillations of the scale factor of the spatial $S^3$-section. They describe
underbarrier oscillations of the inflaton and scale factor in the vicinity of
the inflaton potential maximum, which gives a sufficient amount of inflation
required by the known CMB data. We build the approximation of two coupled
harmonic oscillators for these garland instantons and show that they can
generate inflation consistent with the parameters of the CMB primordial power
spectrum in the non-minimal Higgs inflation model and in $R^2$ gravity. In
particular, the instanton solutions provide smallness of inflationary slow-roll
parameters $\epsilon$ and $\eta<0$ and their relation $\epsilon\sim\eta^2$
characteristic of these two models. We present the mechanism of formation of
hill-like inflaton potentials, which is based on logarithmic loop corrections
to the asymptotically shift-invariant tree level potentials of these models in
the Einstein frame. We also discuss the role of $R^2$-gravity as an
indispensable finite renormalization tool in the CFT driven cosmology, which
guarantees the non-dynamical (ghost free) nature of its scale factor and
special properties of its cosmological garland type instantons. Finally, as a
solution to the problem of hierarchy between the Planckian scale and the
inflation scale we discuss the concept of a hidden sector of conformal higher
spin fields.
