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.
