  Understanding the emergence of classical behavior from a quantum theory is
vital to establishing the quantum origin for the temperature fluctuations
observed in the Cosmic Microwave Background (CMB). We show that a real-space
approach can comprehensively address the quantum-to-classical transition
problem in the leading order of curvature perturbations. To this end, we test
spatial bipartitions of quadratic systems for the interplay between three
different signatures of classical behavior: i) decoherence, ii) peaking of the
Wigner function about classical trajectories, and iii) relative suppression of
non-commutativity in observables. We extract these signatures from the
covariance matrix of a multi-mode Gaussian state and address them primarily in
terms of entanglement entropy and log-classicality. Through a phase-space
stability analysis of spatial sub-regions via their reduced Wigner function, we
ascertain that the underlying cause for the dominance of classicality
signatures is the occurrence of gapped inverted mode instabilities. While the
choice of conjugate variables enhances some of these signatures, decoherence
studied via entanglement entropy is the stronger and more reliable condition
for classicality to emerge. We demonstrate the absence of decoherence, which
preempts a quantum-to-classical transition of scalar fluctuations in an
expanding background in $(1+1)$-dimensions using two examples: i) a Tanh-like
expansion and ii) a de-Sitter expansion. We provide connection between log
classicality and particle number by studying the evolution of each normal mode
at late times. We then extend the analysis to leading order fluctuations in
$(3+1)-$dimensions to show that a quantum-to-classical transition occurs in the
de-Sitter expansion and discuss the relevance of our analysis in distinguishing
cosmological models.
