On the importance of lensing for galaxy clustering in photometric and spectroscopic surveys

We study the importance of gravitational lensing in the modelling of the number counts of galaxies. We confirm previous results for photometric surveys, showing that lensing cannot be neglected in a survey like LSST since it would infer a significant shift of cosmological parameters. For a spectroscopic survey like SKA2, we find that neglecting lensing in the monopole, quadrupole and hexadecapole of the correlation function also induces an important shift of parameters. For ${\Lambda}$CDM parameters, the shift is moderate, of the order of 0.6${\sigma}$ or less. However, for a model-independent analysis, that measures the growth rate of structure in each redshift bin, neglecting lensing introduces a shift of up to 2.3${\sigma}$ at high redshift. Since the growth rate is directly used to test the theory of gravity, such a strong shift would wrongly be interpreted as the breakdown of General Relativity. This shows the importance of including lensing in the analysis of future surveys. On the other hand, for a survey like DESI, we find that lensing is not important, mainly due to the value of the magnification bias parameter of DESI, $s(z)$, which strongly reduces the lensing contribution at high redshift. We also propose a way of improving the analysis of spectroscopic surveys, by including the cross-correlations between different redshift bins (which is neglected in spectroscopic surveys) from the spectroscopic survey or from a different photometric sample. We show that including the cross-correlations in the SKA2 analysis does not improve the constraints. On the other hand replacing the cross-correlations from SKA2 by cross-correlations measured with LSST improves the constraints by 10 to 20 %. Interestingly, for ${\Lambda}$CDM parameters, we find that LSST and SKA2 are highly complementary, since they are affected differently by degeneracies between parameters.