A Reasonable Ab Initio Cosmological Constant Without Holography

Abstract

We give a well-motivated explanation for the origin of dark energy, claiming that it arises from a small residual negative scalar-curvature present even in empty spacetime. The vacuum has this residual curvature because spacetime is fundamentally discrete and there are more ways for a discrete geometry to have negative curvature than positive. We explicitly compute this effect in the well-known {\em dynamical triangulations} (DT) model for quantum gravity and the predicted cosmological constant $\Lambda$ agrees with observation. We begin by almost completely characterizing the DT-model's vacuum energies in dimension three. Remarkably, the energy gap between states comes in increments of [\Delta\mathcal{A} =\frac{\ell}{8\mathcal{V}}] in natural units, where $\ell$ is the "Planck length" in the model and $\mathcal{V}$ is the volume of the universe. Then, using only vacua in the $N$ energy levels nearest zero, where $N$ is the universe's radius in units of $\ell$, we apply our model to the current co-moving spatial volume to get $|\Lambda| \approx 10^{-123}$. This result comes with a rigorous proof and does not depend on any holographic principle or carefully tuned parameters. Our only unknown is the relative entropy of the low-energy states, which sets the sign of $\Lambda$. Numerical evidence strongly suggests that spacetime entropy in the DT-model is a decreasing function of scalar-curvature, so the model also predicts the correct sign for $\Lambda$.