Combining weak lensing tomography and spectroscopic redshift surveys
  Redshift space distortion (RSD) is a powerful way of measuring the growth of
structure and testing General Relativity, but it is limited by cosmic variance
and the degeneracy between galaxy bias b and the growth rate factor f. The
cross-correlation of lensing shear with the galaxy density field can in
principle measure b in a manner free from cosmic variance limits, breaking the
f-b degeneracy and allowing inference of the matter power spectrum from the
galaxy survey. We analyze the growth constraints from a realistic tomographic
weak lensing photo-z survey combined with a spectroscopic galaxy redshift
survey over the same sky area. For sky coverage f_sky=0.5, analysis of the
transverse modes measures b to 2-3% accuracy per \Delta z=0.1 bin at z<1 when
\sim10 galaxies per square arcmin are measured in the lensing survey and all
halos with M>M_min=10^13 h^{-1}M_solar have spectra. For the gravitational
growth parameter parameter \gamma (f=\Omega_m^{\gamma}), combining the lensing
information with RSD analysis of non-transverse modes yields accuracy
\sigma(\gamma)\sim 0.01. Adding lensing information to the RSD survey improves
\sigma(\gamma) by an amount equivalent to a 3 times (10 times) increase in RSD
survey area when the spectroscopic survey extends down to halo mass 10^13.5
(10^14) h^{-1} M_solar. We also find that the \sigma(\gamma) of overlapping
surveys is equivalent to that of surveys 1.5-2 times larger if they are
separated on the sky. This gain is greatest when the spectroscopic mass
threshold is 10^13 - 10^14 h^{-1} M_solar, similar to LRG surveys. The gain of
overlapping surveys is reduced for very deep or very shallow spectroscopic
surveys, but any practical surveys are more powerful when overlapped than when
separated. The gain of overlapped surveys is larger in the case when the
primordial power spectrum normalization is uncertain by >0.5%.
