Recovering 3D clustering information with angular correlations

We study how to recover the full 3D clustering information ofP(k, z), including redshift space distortions (RSD), from 2D tomography using the angular auto- and cross-spectra of different redshift bins C_ℓ(z, z′). We focus on quasi-linear scales where the minimum scale λ_min or corresponding maximum wavenumber is targeted to be in the range . For spectroscopic surveys, we find that we can recover the full 3D clustering information when the redshift bin width Δz used in the 2D tomography is similar to the targeted minimum scale, i.e. which corresponds to for z < 1. This value of Δz is optimal in the sense that larger values of Δz lose information, while smaller values violate our minimum-scale requirement. For a narrow-band photometric survey, with photo-z error σ_z = 0.004, we find almost identical results to the spectroscopic survey because the photo-z error is smaller than the optimal bin width σ_z < Δz. For a typical broad-band photometric survey with σ_z = 0.1, we have that σ_z > Δz and most radial information is intrinsically lost. The remaining information can be recovered from the 2D tomography if we use Δz ≃ 2σ_z. While 3D and 2D analyses 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 straightforward to combine RSD, clustering and weak lensing in 2D space.