Constraints on Alternate Universes: Stars and habitable planets with different fundamental constants

This paper develops constraints on the values of the fundamental constants that allow universes to be habitable. We focus on the fine structure constant $\alpha$ and the gravitational structure constant $\alpha_G$, and find the region in the $\alpha$-$\alpha_G$ plane that supports working stars and habitable planets. This work is motivated, in part, by the possibility that different versions of the laws of physics could be realized within other universes. The following constraints are enforced: [A] long-lived stable nuclear burning stars exist, [B] planetary surface temperatures are hot enough to support chemical reactions, [C] stellar lifetimes are long enough to allow biological evolution, [D] planets are massive enough to maintain atmospheres, [E] planets are small enough in mass to remain non-degenerate, [F] planets are massive enough to support sufficiently complex biospheres, [G] planets are smaller in mass than their host stars, and [H] stars are smaller in mass than their host galaxies. This paper delineates the portion of the $\alpha$-$\alpha_G$ plane that satisfies all of these constraints. The results indicate that viable universes --- with working stars and habitable planets --- can exist within a parameter space where the structure constants $\alpha$ and $\alpha_G$ vary by several orders of magnitude. These constraints also provide upper bounds on the structure constants ($\alpha,\alpha_G$) and their ratio. We find the limit $\alpha_G/\alpha<10^{-34}$, which shows that habitable universes must have a large hierarchy between the strengths of the gravitational force and the electromagnetic force.