Vacua and correlators in hyperbolic de Sitter space

Understand

We study the power - and bi -spectrum of vacuum fluctuations in a hyperbolic section of de Sitter space, comparing two states of physical interest: the Bunch-Davies and hyperbolic vacuum.

• We introduce a one -parameter family of de Sitter hyperbolic sections and their natural vacua, and identify a limit in which it reduces to the planar section and the corresponding Bunch -Davies vacuum state.
• Selecting the Bunch -Davies vacuum for a massless scalar field implies a mixed reduced density matrix in a hyperbolic section of de Sitter space.
• We stress that in the Bunch -Davies state the hyperbolic de Sitter $n$-point correlation functions have to match the planar de Sitter $n$-point correlation functions.