Curvature Constraints from the Causal Entropic Principle

Current cosmological observations indicate a preference for a cosmological constant that is drastically smaller than what can be explained by conventional particle physics. The Causal Entropic Principle (Bousso, {\it et al}.) provides an alternative approach to anthropic attempts to predict our observed value of the cosmological constant by calculating the entropy created within a causal diamond. We have extended this work to use the Causal Entropic Principle to predict the preferred curvature within the "multiverse". We have found that values larger than $\rho_k = 40\rho_m$ are disfavored by more than 99.99% and a peak value at $\rho_{\Lambda} = 7.9 \times 10^{-123}$ and $\rho_k =4.3 \rho_m$ for open universes. For universes that allow only positive curvature or both positive and negative curvature, we find a correlation between curvature and dark energy that leads to an extended region of preferred values. Our universe is found to be disfavored to an extent depending the priors on curvature. We also provide a comparison to previous anthropic constraints on open universes and discuss future directions for this work.