Cosmic variance of $H_0$ in light of forthcoming high-redshift surveys
  Forthcoming surveys will extend the understanding of cosmological large scale
structures up to unprecedented redshift. According to this perspective, we
present a fully relativistic framework to evaluate the impact of stochastic
inhomogeneities on the determination of the Hubble constant. To this aim, we
work within linear perturbation theory and relate the fluctuations of the
luminosity distance-redshift relation, in the Cosmic Concordance model, to the
intrinsic uncertainty associated to the measurement of $H_0$ from high-redshift
surveys ($0.15\le z\le3.85$). We first present the detailed derivation of the
luminosity distance-redshift relation 2-point correlation function and then
provide analytical results for all the involved relativistic effects, such as
peculiar velocity, lensing, time delay and (integrated) Sachs-Wolfe, and their
angular spectra. Hence, we apply our analytical results to the study of
high-redshift Hubble diagram, according to what has been recently claimed in
literature. Following the specific of Euclid Deep Survey and LSST, we conclude
that the cosmic variance associated with the measurement of the Hubble constant
is at most of 0.1 %. Our work extends the analysis already done in literature
for closer sources, where only peculiar velocity has been taken into account.
We then conclude that deep surveys will provide an estimation of the $H_0$
which will be more precise than the one obtained from local sources, at least
in regard of the intrinsic uncertainty related to a stochastic distribution of
inhomogeneities.
