Toolkit for Scalar Fields in Universes with finite-dimensional Hilbert Space

The holographic principle suggests that the Hilbert space of quantum gravity is locally finite-dimensional. Motivated by this point-of-view, and its application to the observable Universe, we introduce a set of numerical and conceptual tools to describe scalar fields with finite-dimensional Hilbert spaces, and to study their behaviour in expanding cosmological backgrounds. These tools include accurate approximations to compute the vacuum energy of a field mode $\mathbf{k}$ as a function of the dimension $d_{\mathbf{k}}$ of the mode Hilbert space, as well as a parametric model for how that dimension varies with $|\mathbf{k}|$. We show that the maximum entropy of our construction momentarily scales like the boundary area of the observable Universe for some values of the parameters of that model. And we find that the maximum entropy generally follows a sub-volume scaling as long as $d_{\mathbf{k}}$ decreases with $|\mathbf{k}|$. We also demonstrate that the vacuum energy density of the finite-dimensional field is dynamical and decays between two constant epochs in our fiducial construction. These results rely on a number of non-trivial modelling choices, but our general framework may serve as a starting point for future investigations of the impact of finite-dimensionality of Hilbert space on cosmological physics.