Effects of a quantum or classical scalar field on the entanglement entropy of a pair of universes

Using the formalism of the third quantization in canonical quantum gravity, the entropy of entanglement of a pair of universes created in the multiverse from the vacuum has recently been calculated. Here, we examine the differences between considering a scalar field as quantum or classical on the entanglement entropy of three different pairs: de Sitter universes, flat stiff matter dominated universes, and closed universes with a scalar field. We find that the entanglement entropy is unchanged, zero, or dependent on the treatment of the scalar field, respectively.