Pulsating stars and eclipsing binaries as distances indicators in the universe

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.