On Vacuum Fluctuations in Quantum Gravity and Interferometer Arm Fluctuations

We propose a simple model of spacetime vacuum fluctuations motivated by AdS/CFT, where the vacuum is described by a thermal density matrix, $\rho = \frac{e^{-K}}{\mbox{Tr}(e^{-K})}$ with $K$ the modular Hamiltonian. In AdS/CFT, both the expectation value of $K$ and its fluctuations $\langle \Delta K^2 \rangle$ have been calculated; both obey an area law identical to the Bekenstein-Hawking area law of black hole mechanics: $\langle K \rangle = \langle \Delta K^2 \rangle = \frac{A}{4 G_N}$, where $A$ is the area of an (extremal) entangling surface. It has also been shown that $\Delta K$ gravitates in AdS, and hence generates metric fluctuations. These theoretical results are intriguing, but it is not known how to precisely extend such ideas about holographic quantum gravity to ordinary flat space. We take the approach of considering whether experimental signatures in metric fluctuations could determine properties of the vacuum of quantum gravity in flat space. In particular, we propose a theoretical model motived by the AdS/CFT calculations that reproduces the most important features of modular Hamiltonian fluctuations; the model consists of a high occupation number bosonic degree of freedom. We show that if this theory couples through ordinary gravitational couplings to the mirrors in an interferometer with strain sensitivity similar to what will be available for gravitational waves, vacuum fluctuations could be observable.