Constraints of the variation of fundamental couplings and sensitivity of
  the equation of state of dense matter
  We discuss the coupled variations of the gravitational, strong and
electroweak coupling constants and the current knowledge of the nuclear
equation of state based on heavy ion collision experiments and neutron star
mass-radius relationship. In particular we focus in our description on
phenomenological parameters, $R$, relating variations in the quantum
chromodynamics scale $\Lambda_{QCD}$ and the fine structure constant $\alpha$,
and $S$, relating variation of $v$, the Higgs vacuum expectation value and the
Yukawa couplings, $h$, in the quark sector. This parametrization is valid for
any model where gauge coupling unification occurs at some (unspecified) high
energy scale. From a physically motivated set of equations of state for dense
matter we obtain the constrained parameter phase space $(R, S)$ in high density
nuclear environments. This procedure is complementary to (although currently
less powerful than) those used in low density conditions. For variations of
$\Delta \alpha/\alpha=0.005$ we find that the obtained constrained parameter
lies on a strip region in the $(R, S)$ plane that partially overlaps some of
the allowed values of parameters derived from primordial abundances. This may
be of interest in the context of unification scenarios where a dense phase of
the universe may have existed at early times.
