  Determining the expansion of the universe, i.e. the Hubble constant (H0) to
better than 2% is required in order to understand the nature of dark energy.
However, the two most accurate methods to do it, the cosmic microwave
background and the distance scale ladder are inconsistent today, which is
refereed as the "tension". One of the key to resolve this tension is related to
the calibration of the period-luminosity (PL) of Cepheids : what is its
zero-point ? Is it metallicity dependent ? Can we reduce the dispersion of the
PL relation ? The eclipsing binaries method was recently used to determine the
distance to LMC with a 2.2% accuracy, which is crucial to constrain the
zero-point of the PL relation and to study the impact of metallicity. However,
this method is based on the hypothesis that stars are perfect blackbody, which
is not always the case as shown by inconsistencies in the surface brightness -
color relations of early type stars for instance, that are actually useful for
the distance determination of eclipsing binaries in distant galaxies (i.e. M31,
M33). On the other hand, it is now possible to apply the Baade-Wesselink method
to Cepheids in the Magellanic Clouds, however the value of the projection
factor and its dependence with the period of Cepheids remains a key issue.
Understanding the dynamical structure of the atmosphere of Cepheids (p-factor,
k-factor) and their environment cannot be circumvented in order to determine
the distance in the universe. This is even more true that the environment of
Cepheids could increase the dispersion of the PL relation. The next generation
of interferometers (MATISSE, SPICA), the Gaia parallaxes, in the context of
JWST, PLATO, and ELT will help to resolve these fundamental issues.
