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.