Gravitational redshift and asymmetric redshift-space distortions for stacked clusters

We derive the expression for the observed redshift in the weak field limit in the observer's past light cone, including all relativistic terms up to second order in velocity. We then apply it to compute the cluster-galaxy cross-correlation functions (CGCF) using N-body simulations. The CGCF is asymmetric along the line of sight owing to the presence of the small second-order terms such as the gravitational redshift (GRedshift). We identify two systematics in the modelling of the GRedshift signal in stacked clusters. First, it is affected by the morphology of dark matter haloes and the large-scale cosmic-web. The non- spherical distribution of galaxies around the central halo and the presence of neighbouring clusters systematically reduce the GRedshift signal. This bias is approximately 20 per cent for M-min similar or equal to 10(14)M(circle dot) h(-1), and is more than 50 per cent for haloes with M-min similar or equal to 2 x 10(13)M(circle dot) h(-1) at r > 4 Mpc h(-1). Secondly, the best-fitting GRedshift profiles as well as the profiles of all other relativistic terms are found to be significantly different in velocity space compared to their real space versions. We find that the relativistic Doppler redshift effect, like other second-order effects, is subdominant to the GRedshift signal. We discuss some subtleties relating to these effects in velocity space. We also find that the S/N of the GRedshift signal increases with decreasing halo mass.