Effective density matrix for vacua in asymptotically flat gravity

We explicitly construct the density matrix associated to the vacuum state of a large spherically symmetric causal diamond of area $A$ in four-dimensional asymptotically flat gravity. We achieve this using the soft effective action, which characterizes the low-energy gravitational degrees of freedom that arise in the long-distance limit of the Einstein-Hilbert action and consists of both the soft graviton mode and the Goldstone mode arising from the spontaneous breaking of supertranslation symmetry. Integrating out the soft graviton mode, we obtain an effective action for purely the Goldstone mode, from which we extract the density matrix and therefore the modular Hamiltonian $K_s$ associated to the vacuum state. As a corollary, we explicitly compute the mean and variance of $K_s$, finding $\langle \Delta {K}_s^2 \rangle = A/\epsilon_{\text{UV}}^2$, with $\epsilon_{\text{UV}}$ being a length-scale UV cutoff on the celestial sphere.