Entangled de Sitter from Stringy Axionic Bell pair I: An analysis using
  Bunch Davies vacuum
  In this work, we study the quantum entanglement and compute entanglement
entropy in de Sitter space for a bipartite quantum field theory driven by axion
originating from ${\bf Type~ IIB}$ string compactification on a Calabi Yau
three fold (${\bf CY^3}$) and in presence of ${\bf NS5}$ brane. For this
compuation, we consider a spherical surface ${\bf S}^2$, which divide the
spatial slice of de Sitter (${\bf dS_4}$) into exterior and interior sub
regions. We also consider the initial choice of vaccum to be Bunch Davies
state. First we derive the solution of the wave function of axion in a
hyperbolic open chart by constructing a suitable basis for Bunch Davies vacuum
state using Bogoliubov transformation. We then, derive the expression for
density matrix by tracing over the exterior region. This allows us to compute
entanglement entropy and R$\acute{e}$nyi entropy in $3+1$ dimension. Further we
quantify the UV finite contribution of entanglement entropy which contain the
physics of long range quantum correlations of our expanding universe. Finally,
our analysis compliments the necessary condition for the violation of Bell's
inequality in primordial cosmology due to the non vanishing entanglement
entropy for axionic Bell pair.
