The RSD Sorting Hat: Unmixing Radial Scales in Projection
  Future data sets will enable cross-correlations between redshift space
distortions (RSD) and weak lensing (WL). While photometric lensing and
clustering cross-correlations have provided some of the tightest cosmological
constraints to date, it is not well understood how to optimally perform similar
RSD/WL joint analyses in a lossless way. RSD is typically measured in $3D$
redshift space, but WL is inherently a projected signal, making angular
statistics a natural choice for the combined analysis. Thus, we determine the
amount of RSD information that can be extracted using projected statistics.
Specifically we perform a Fisher analysis to forecast constraints and model
bias comparing two different Fingers-of-God (FoG) models using both, the $3D$
power spectrum, $P(k, \mu)$, and tomographic $C(\ell)$. We find that because
na\"ive tomographic projection mixes large scales with poorly modelled
nonlinear radial modes, it does not provide competitive constraints to the $3D$
RSD power spectrum without the model bias becoming unacceptably large. This is
true even in the limit of narrow tomographic bins. In light of this we propose
a new radial weighting scheme which unmixes radial RSD scales in projection
yielding competitive constraints to the $3D$ RSD power spectrum, while keeping
the model bias small. This work lays the groundwork for optimal joint analyses
of RSD and cosmic shear.
