C${\cal C}$osmological G${\cal G}$eometric P${\cal P}$hase F${\cal F}$rom P${\cal P}$ure Q${\cal Q}$uantum S${\cal S}$tates: A Study without/ with having Bell's Inequality Violation

In this paper, using the concept of Lewis Riesenfeld invariant quantum operator method for finding continuous eigenvalues mechanical wave functions we derive analytical expressions cosmological geometric phase, which is commonly identified to be Pancharatnam Berry phase from primordial perturbation scenario. We compute two possible physical situations, (1) absence Bell's inequality violation and (2) presence having contributions in sub Hubble region ( − k τ ≫ 1 $-k\tau \gg 1$ ), super ≪ \ll ) at horizon crossing point = massless field m / H $m/{\cal H}\ll partially ∼ H}\sim massive/heavy H}\gg background theory spatially flat quasi De Sitter geometry. The prime motivation work investigate various unknown features universe. To give realistic interpretation derived theoretical results express everything initially terms slowly varying conformal time dependent parameters, then connect with observation further observables, are spectral index/tilt scalar mode power spectrum n ζ $n_{\zeta }$ tensor-to-scalar ratio (r). Finally, identification helps us provide stringent numerical constraints on confronts well recent observation.