Comments on the Entanglement Spectrum of de Sitter Space

TL;DR

This paper challenges the assumption that de Sitter space's entanglement spectrum is flat, showing that certain spectral properties and entropy formulas can be explained by specific matrix models without requiring a flat spectrum.

Contribution

It demonstrates that the de Sitter entanglement spectrum need not be flat and provides matrix models that replicate the entropy and spectral properties.

Findings

Expectation value of a random projection operator is proportional to the ratio of dimensions, independent of spectrum.

Asymptotic estimates for trace of the density matrix and modular Hamiltonian are compatible for certain spectra.

Matrix models can replicate the entropy formula and spectral properties of de Sitter space.

Abstract

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.