Towards a better knowledge of the nuclear equation of state from the isoscalar breathing mode

The measurements of the isoscalar giant monopole resonance (GMR), also called the breathing mode, are analyzed with respect to their constraints on the quantity $M_c$, e.g. the density dependence of the nuclear incompressibility around the so-called crossing density $\rho_c$=0.1 fm$^{-3}$. The correlation between the centroid of the GMR, $E_\mathrm{GMR}$, and $M_c$ is shown to be more accurate than the one between $E_\mathrm{GMR}$ and the incompressibility modulus at saturation density, $K_\infty$, giving rise to an improved determination on the nuclear equation of state. The relationship between $M_c$ and $K_\infty$ is given as a function of the skewness parameter $Q_\infty$ associated to the density dependence of the equation of state. The large variation of $Q_\infty$ among different energy density functionnals directly impacts the knowledge of $K_\infty$: a better knowledge of $Q_\infty$ is required to deduce more accurately $K_\infty$. Using the Local Density Approximation, a simple and accurate expression relating $E_\mathrm{GMR}$ and the quantity $M_c$ is derived and successfully compared to the fully microscopic predictions.