The apparent distribution of large-scale structures in the universe is sensitive to the velocity/potential of the sources as well as the potential along the line-of-sight through the mapping from real space to redshift space ( redshift-space distortions , RSD ) .
Since odd multipoles of the halo cross-correlation function vanish when considering standard Doppler RSD , the dipole is a sensitive probe of relativistic and wide-angle effects .
We build a catalogue of ten million haloes ( Milky-Way size to galaxy-cluster size ) from the full-sky light-cone of a new ‘ ‘ RayGalGroupSims '' N-body simulation which covers a volume of ( 2.625 ~ { } h ^ { -1 } Gpc ) ^ { 3 } with 4096 ^ { 3 } particles .
Using ray-tracing techniques , we find the null geodesics connecting all the sources to the observer .
We then self-consistently derive all the relativistic contributions ( in the weak-field approximation ) to RSD : Doppler , transverse Doppler , gravitational , lensing and integrated Sachs-Wolfe .
It allows us , for the first time , to disentangle all contributions to the dipole from linear to non-linear scales .
At large scale , we recover the linear predictions dominated by a contribution from the divergence of neighbouring line-of-sights .
While the linear theory remains a reasonable approximation of the velocity contribution to the dipole at non-linear scales it fails to reproduce the potential contribution below 30 - 60 ~ { } h ^ { -1 } Mpc ( depending on the halo mass ) .
At scales smaller than \sim 10 ~ { } h ^ { -1 } Mpc , the dipole is dominated by the asymmetry caused by the gravitational redshift .
The transition between the two regimes is mass dependent as well .
We also identify a new non-trivial contribution from the non-linear coupling between potential and velocity terms .