Distances and ages of NGC 6397, NGC 6752 and 47 Tuc

New improved distances and absolute ages for the Galactic globular clusters NGC 6397, NGC 6752, and 47 Tuc are obtained using the Main Sequence Fitting Method. We derived accurate estimates of reddening and metal abundance for these three clusters using a strictly differential procedure, where the Johnson $B-V$ and Strömgren $b-y$ colours and UVES high resolution spectra of turn-off stars and early subgiants belonging to the clusters were compared to similar data for field subdwarfs with accurate parallaxes measured by Hipparcos. The use of a reddening free temperature indicator (the profile of Hα) allowed us to reduce the error bars in reddening determinations to about 0.005 mag, and in metal abundances to 0.04 dex, in the scales defined by the local subdwarfs. Error bars in distances are then reduced to about 0.07 mag for each cluster, yielding ages with typical random errors of about 1 Gyr. We find that NGC 6397 and NGC 6752 have ages of $13.9\pm 1.1$ and $13.8\pm 1.1$ Gyr respectively, when standard isochrones without microscopic diffusion are used, while 47 Tuc is probably about 2.6 Gyr younger, in agreement with results obtained by other techniques sensitive to relative ages. If we use models that include the effects of sedimentation due to microscopic diffusion in agreement with our observations of NGC 6397, and take into account various sources of possible systematic errors with a statistical approach, we conclude that the age of the oldest globular clusters in the Galaxy is $13.4\pm 0.8\pm 0.6$ Gyr, where the first error bar accounts for random effects, and the second one for systematic errors. This age estimate is fully compatible with the very recent results from WMAP, and indicates that the oldest Galactic globular clusters formed within the first 1.7 Gyr after the Big Bang, corresponding to a redshift of $z\geq 2.5$, in a standard ΛCDM model. The epoch of formation of the (inner halo) globular clusters lasted about 2.6 Gyr, ending at a time corresponding to a redshift of $z\geq 1.3$. On the other hand, our new age estimate once combined with values of H 0 given by WMAP and by the HST Key Project, provides a robust upper limit at 95% level of confidence of $\Omega_{\rm M}<0.57$, independently of type Ia SNe, and strongly supports the need for a dark energy. The new cluster distances lead to new estimates of the horizontal branch luminosity, that may be used to derive the zero point of the relation between the horizontal branch absolute magnitude and metallicity: we obtain $M_V(HB)=(0.22\pm 0.05)({\rm [Fe/H]}+1.5)+(0.56\pm 0.07)$. This zero point is 0.03 mag shorter than obtained by Carretta et al. ([CITE]) and within the error bar it agrees with, but it is more precise than most of the previous individual determinations of the RR Lyrae absolute magnitude. When combined with the apparent average luminosity of the RR Lyrae stars in the LMC by Clementini et al. ([CITE]), this zero point provides a new estimate of the distance modulus to the LMC: $(m-M)_0=18.50\pm 0.09$.