  Peculiar velocities of galaxies and halos can be reconstructed from their
spatial distribution alone. This technique is analogous to the baryon acoustic
oscillations (BAO) reconstruction, using the continuity equation to connect
density and velocity fields. The resulting reconstructed velocities can be used
to measure imprints of galaxy velocities on the cosmic microwave background
(CMB) like the kinematic Sunyaev-Zel'dovich (kSZ) effect or the moving lens
effect. As the precision of these measurements increases, characterizing the
performance of the velocity reconstruction becomes crucial to allow unbiased
and statistically optimal inference. In this paper, we quantify the relevant
performance metrics: the variance of the reconstructed velocities and their
correlation coefficient with the true velocities. We show that the relevant
velocities to reconstruct for kSZ and moving lens are actually the halo --
rather than galaxy -- velocities. We quantify the impact of redshift-space
distortions, photometric redshift errors, satellite galaxy fraction, incorrect
cosmological parameter assumptions and smoothing scale on the reconstruction
performance. We also investigate hybrid reconstruction methods, where
velocities inferred from spectroscopic samples are evaluated at the positions
of denser photometric samples. We find that using exclusively the photometric
sample is better than performing a hybrid analysis. The 2 Gpc$/h$ length
simulations from AbacusSummit with realistic galaxy samples for DESI and Rubin
LSST allow us to perform this analysis in a controlled setting. In the
companion paper Hadzhiyska et al. 2024, we further include the effects of
evolution along the light cone and give realistic performance estimates for
DESI luminous red galaxies (LRGs), emission line galaxies (ELGs), and Rubin
LSST-like samples.
