Comments on the Entanglement Spectrum of de Sitter Space

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.