Directional dependence of the local estimation of $H_0$ and the non
  perturbative effects of primordial curvature perturbations
  Recent measurements of the cosmic microwave background (CMB) radiation have
shown an apparent tension with the present value of the Hubble parameter
inferred from local observations of supernovae, which look closer, i.e.
brighter, than what is expected in a homogeneous model with a value of $H_0$
equal to the one estimated from CMB observations. We examine the possibility
that such a discrepancy is the consequence of the presence of a local
inhomogeneity seeded by primordial curvature perturbations, finding that a
negative peak of the order of less than two standard deviations could allow to
fit low red-shift supernovae observations without the need of using a value of
the Hubble parameter different from $H_0^{CMB}$. The type of inhomogeneity we
consider does not modify the distance to the last scattering, making it
compatible with the constraints of the PLANCK mission data. Our calculation is
fully relativistic and non perturbative, and for this reason shows important
effects which were missed in the previous investigations using relativistic
perturbations or Newtonian approximations, because the structures seeded by
primordial curvature perturbations can be today highly non linear, and
relativist Doppler terms cannot be neglected.
  Comparison with studies of local structure based on galaxy surveys and
luminosity density analysis reveals that the density profile we find could in
fact be compatible with the one obtained for the same region of sky where is
located most of the Cepheids used to calibrate the luminosity distance of the
supernovae employed for the local $H_0$ estimation, suggesting a possible
directional dependence and calibration bias which could be partially attributed
to the presence of the Sloan Great Wall.
