Recovering 3D clustering information with angular correlations

We study how to recover the full 3D clustering information of P(\vec{k},z), including redshift space distortions (RSD), from 2D tomography using the angular auto and cross spectra of different redshift bins C_\ell(z,z'). We focus on quasilinear scales where the minimum scale \lambda_{min} or corresponding maximum wavenumber k_{max}= 2\pi/\lambda_{min} is targeted to be between k_{max}={0.05-0.2} h/Mpc. For spectroscopic surveys, we find that we can recover the full 3D clustering information when the redshift bin width \Delta z used in the 2D tomography is similar to the targeted minimum scale, i.e. \Delta z ~ {0.6-0.8} \lambda_{min} H(z)/c which corresponds to \Delta z ~ 0.01-0.05 for z<1. This value of \Delta z is optimal in the sense that larger values of \Delta z lose information, while smaller values violate our minimum scale requirement. For a narrow-band photometric survey, with photo-z error \sigma_z=0.004, we find almost identical results to the spectroscopic survey because the photo-z error is smaller than the optimal bin width \sigma_z<\Delta z. For a typical broad-band photometric survey with \sigma_z=0.1, we have that \sigma_z>\Delta z and most radial information is intrinsically lost. The remaining information can be recovered from the 2D tomography if we use \Delta z ~ 2\sigma_z. While 3D and 2D analysis are shown here to be equivalent, the advantage of using angular positions and redshifts is that we do not need a fiducial cosmology to convert to 3D coordinates. This avoids assumptions and marginalization over the fiducial model. In addition, it becomes straight forward to combine RSD, clustering and weak lensing in 2D space.