Three-dimensional weak gravitational lensing of the 21-cm radiation
  background
  We study weak gravitational lensing by the cosmic large-scale structure of
the 21-cm radiation background in the 3d-weak lensing formalism. The interplay
between source distance measured at finite resolution, visibility and lensing
terms is analysed in detail and the resulting total covariance $C_{\ell}(k,k')$
is derived. The effect of lensing correlates different multipoles through
convolution, breaking the statistical homogeneity of the 21-cm radiation
background. This homogeneity breaking can be exploited to reconstruct the
lensing field $\hat{\phi}_{\ell m}(\kappa)$ and noise lensing reconstruction
$N_{\ell}^{\hat{\phi}}$ by means of quadratic estimators. The effects related
to the actual measurement process (redshift precision and visibility terms)
change drastically the values of the off-diagonal terms of the total covariance
$C_{\ell}(k,k')$. It is expected that the detection of lensing effects on a
21-cm radiation background will require sensitive studies and high-resolution
observations by future low-frequency radio arrays such as the SKA survey.
