  In the recent article Phys. Rev. D 100, no. 4, 043533 (2019) a compact phase
space generalization of the flat de Sitter cosmology has been proposed. The
main advantages of the compactification is that physical quantities are
bounded, and the quantum theory is characterized by finite dimensional Hilbert
space. Furthermore, by considering the $\mathbb{S}^2$ phase space, quantum
description is constructed with the use $SU(2)$ representation theory. The
purpose of this article is to apply effective methods to extract semi-classical
regime of the quantum dynamics. The analysis is performed both without prior
solving of the quantum constraint and by extracting physical Hamiltonian of the
model. At the effective level, the results of the two procedures are shown to
be equivalent. We find a nontrivial behavior of the fluctuations around the
recollapse of the universe, which is distinct from what is found after
quantization with the standard flat phase space. The behavior is reflected at
the level of the modified Friedmann equation with quantum back-reaction
effects, which is derived. Finally, an unexpected relation between the quantum
fluctuations of the cosmological sector and the holographic Bousso bound is
shown.
