The dominant systematic uncertainty in the age determination of galactic globular clusters is the depth of the convection envelope of the stars. This parameter is partially degenerate with metallicity which is in turn degenerate with age. However, if the metal content, distance and extinction are known, the position and morphology of the red giant branch in a color-magnitude diagram are mostly sensitive to the value of the depth of the convective envelope. Therefore, using external, precise metallicity determinations this degeneracy and thus the systematic error in age, can be reduced. Alternatively, the morphology of the red giant branch of globular clusters color magnitude diagram can also be used to achieve the same. We demonstrate that globular cluster red giant branches are well fitted by values of the depth of the convection envelope consistent with those obtained for the Sun and this finding is robust to the adopted treatment of the stellar physics. With these findings, the uncertainty in the depth of the convection envelope is no longer the dominant contribution to the systematic error in the age determination of the oldest globular clusters, reducing it from $0.5$ to $0.23$ or $0.33$ Gyr, depending on the methodology adopted: i.e., whether resorting to external data (spectroscopic metallicity determinations) or relying solely on the morphology of the clusterss color-magnitude diagrams. This results in an age of the Universe $t_{rm U}=13.5^{+0.16}_{-0.14} {rm (stat.)} pm 0.23(0.33) ({rm sys.})$ at 68% confidence level, accounting for the formation time of globular clusters and its uncertainty. An uncertainty of 0.27(0.36) Gyr if added in quadrature. This agrees well with $13.8 pm 0.02$ Gyr, the cosmological model-dependent value inferred by the Planck mission assuming the $Lambda$CDM model.
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