Combining weak-lensing tomography and spectroscopic redshift surveys

ABSTRACT

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 the 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 analyse 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 per cent accuracy per Δz= 0.1 bin at z < 1 when ∼10 galaxies arcmin⁻² are measured in the lensing survey and all haloes with M > M_min= 10¹³ h⁻¹  M_⊙ have spectra. For the gravitational growth parameter parameter γ (, combining the lensing information with RSD analysis of non-transverse modes yields accuracy σ(γ) ≈ 0.01. Adding lensing information to the RSD survey improves σ(γ) by an amount equivalent to a 3 × (10 ×) increase in the RSD survey area when the spectroscopic survey extends down to halo mass 10^13.5 (10¹⁴) h⁻¹ M_⊙. We also find that the σ(γ) of overlapping surveys is equivalent to that of surveys 1.5–2× larger if they are separated on the sky. This gain is greatest when the spectroscopic mass threshold is 10¹³–10¹⁴ h⁻¹ M_⊙, similar to Luminous Red Galaxy 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 per cent.