  Motivated by recent work suggesting observably large spacetime fluctuations
in the causal development of an empty region of flat space, we conjecture that
these metric fluctuations can be quantitatively described in terms of a
conformal field theory of near-horizon vacuum states. One consequence of this
conjecture is that fluctuations in the modular Hamiltonian $\Delta K$ of a
causal diamond are equal to the entanglement entropy: $\langle \Delta K^2
\rangle = \langle K \rangle = \frac{A(\Sigma_{d-2})}{4 G_d}$, where
$A(\Sigma_{d-2})$ is the area of the entangling surface in $d$ dimensions. Our
conjecture applies to flat space, the cosmological horizon of dS, and AdS
Ryu-Takayanagi diamonds, but not to large finite area diamonds in the bulk of
AdS. We focus on three pieces of quantitative evidence, from a Randall-Sundrum
II braneworld, from the conformal description of black hole horizons, and from
the fluid-gravity correspondence. Our hypothesis also suggests that a broader
range of formal results can be brought to bear on observables in flat and dS
spaces.
