Matter-Geometry entanglement in quantum cosmology

We present a study of the evolution of entanglement entropy of matter and geometry in quantum cosmology. For a variety of Gaussian initial states and their linear combinations, and with evolution defined with respect to a relational time, we show numerically that (i) entanglement entropy increases rapidly at very early times, and subsequently saturates to a constant non-zero value, and (ii) that the saturation value of this entropy is a linear function of the energy associated to the quantum state: $S_{\text{ent}}^\psi = \gamma \langle \hat{H} \rangle_\psi$. These results suggest a remnant of quantum entanglement in the macroscopic Universe from the era of the Big Bang, independent of the initial state parameters, and a "First Law" associated with matter-gravity entanglement entropy in quantum gravity.