Redshift space distortions in Lagrangian space and the linear large scale velocity field of dark matter

Untangling the connection between redshift space coordinates, a velocity measurement, and three dimensional real space coordinates, is a cosmological problem that is often modeled through a linear understanding of the velocity-position coupling. This linear information is better preserved in the Lagrangian space picture of the matter density field. Through Lagrangian space measurements, we can extract more information and make more accurate estimates of the linear growth rate of the universe. In this paper, we address the linear modelling of matter particle velocities through transfer functions, and in doing so examine to what degree the decrease in correlation with initial conditions may be contaminated by velocity-based nonlinearities. With a thorough analysis of the monopole-quadrupole ratio, we find the best-fit values for the Eulerian velocity dispersion, $\sigma_p = 378.3$ km/s for a Lorentzian finger-of-God damping factor and $\sigma_p = 254.6$ km/s for a Gaussian one. The covariance of the cosmological linear growth rate $f$, is estimated in the Eulerian and Lagrangian cases. Comparing Lagrangian and Eulerian, we find that the error in $f$ improves by a factor of 3, without the need for nonlinear velocity dispersion modelling.