Reconstructing the Thermal Sunyaev-Zel'dovich Effect in 3D
  The thermal Sunyaev-Zel'dovich (tSZ) effect measures the line-of-sight
projection of the thermal pressure of free electrons and lacks any redshift
information. By cross correlating the tSZ effect with an external cosmological
tracer we can recover a good fraction of this lost information. Weak lensing
(WL) is thought to provide an unbiased probe of the dark Universe, with many WL
surveys having sky coverage that overlaps with tSZ surveys. Generalising the
tomographic approach, we advocate the use of the spherical Fourier-Bessel (sFB)
expansion to perform an analysis of the cross-correlation between the projected
(2D) tSZ Compton $y$-parameter maps and 3D weak lensing convergence maps. We
use redshift dependent linear biasing and the halo model as a tool to
investigate the tSZ-WL cross-correlations in 3D. We use the Press-Schechter
(PS) and the Sheth-Tormen (ST) mass-functions in our calculations, finding that
the results are quite sensitive to detailed modelling. We provide detailed
analysis of surveys with photometric and spectroscopic redshifts. The
signal-to-noise (S/N) of the cross-spectra $\mathcal{C}_{\ell} (k)$ for
individual 3D modes, defined by the radial and tangential wave numbers
$(k;\ell)$, remains comparable to, but below, unity though optimal binning is
expected to improve this. The results presented can be generalised to analyse
other CMB secondaries, such as the kinetic Sunyaev-Zel'dovich (kSZ) effect.
