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This paper develops constraints on the values of the fundamental constantsthat allow universes to be habitable. We focus on the fine structure constant$\alpha$ and the gravitational structure constant $\alpha_G$, and find theregion in the $\alpha$-$\alpha_G$ plane that supports working stars andhabitable planets. This work is motivated, in part, by the possibility thatdifferent versions of the laws of physics could be realized within otheruniverses. The following constraints are enforced: [A] long-lived stablenuclear burning stars exist, [B] planetary surface temperatures are hot enoughto support chemical reactions, [C] stellar lifetimes are long enough to allowbiological evolution, [D] planets are massive enough to maintain atmospheres,[E] planets are small enough in mass to remain non-degenerate, [F] planets aremassive enough to support sufficiently complex biospheres, [G] planets aresmaller in mass than their host stars, and [H] stars are smaller in mass thantheir host galaxies. This paper delineates the portion of the$\alpha$-$\alpha_G$ plane that satisfies all of these constraints. The resultsindicate 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 provideupper bounds on the structure constants ($\alpha,\alpha_G$) and their ratio. Wefind the limit $\alpha_G/\alpha<10^{-34}$, which shows that habitable universesmust have a large hierarchy between the strengths of the gravitational forceand the electromagnetic force.

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