Comments on the Entanglement Spectrum of de Sitter Space

Understand

We argue that the Schwarzschild-de Sitter black hole entropy formula does not imply that the entanglement spectrum of the vacuum density matrix of de Sitter space is flat.

• Specifically, we show that the expectation value of a random projection operator of dimension $d\gg 1$, on a Hilbert space of dimension $D\gg d$ and in a density matrix $\rho = e^{-K}$ with strictly positive spectrum, is $\frac{d}{D}\left(1 + o(\frac{1}{\sqrt{d}})\right)$, independent of the spectrum of the density matrix.
• In addition, for a suitable class of spectra the asymptotic estimates ${\rm Tr} (\rho K) \sim {\rm ln}\ D - o(1)$ and $ {\rm Tr} [\rho (K - \langle K\rangle)^2] = a \langle K \rangle$ are compatible for any order one constant $a$.
• We discuss a simple family of matrix models and projections that can replicate such modular Hamiltonians and the SdS entropy formula.