  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 clusters's 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.
