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 (LOS) 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% for $M_{\rm min}\simeq
10^{14} {\rm M_{\odot}}/h$, and is more than 50% for haloes with $M_{\rm
min}\simeq 2\times 10^{13} {\rm M_{\odot}}/h$ at $r>$4 Mpc/$h$. Second, the
best-fit gravitational redshift 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 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.
