Modeling relativistic contributions to the halo power spectrum dipole

We study the power spectrum dipole of an N-body simulation which includes relativistic effects through ray-tracing and covers the low redshift Universe up to $z_{\rm max} = 0.465$ (RayGalGroup simulation). We model relativistic corrections as well as wide-angle, evolution, window and lightcone effects. Our model includes all relativistic corrections up to third-order including third-order bias expansion. We consider all terms which depend linearly on $\mathcal{H}/k$ (weak field approximation). We also study the impact of 1-loop corrections to the matter power spectrum for the gravitational redshift and transverse Doppler effect. We found wide-angle and window function effects to significantly contribute to the dipole signal. When accounting for all contributions, our dipole model can accurately capture the gravitational redshift and Doppler terms up to the smallest scales included in our comparison ($k=0.48\,h{\rm Mpc}^{-1}$), while our model for the transverse Doppler term is less accurate. We find the Doppler term to be the dominant signal for this low redshift sample. We use Fisher matrix forecasts to study the potential for the future Dark Energy Spectroscopic Instrument (DESI) to detect relativistic contributions to the power spectrum dipole. A conservative estimate suggests that the DESI-BGS sample should be able to have a detection of at least $4.4\sigma$, while more optimistic estimates find detections of up to $10\sigma$. Detecting these effects in the galaxy distribution allows new tests of gravity on the largest scales, providing an interesting additional science case for galaxy survey experiments.