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Untangling the connection between redshift space coordinates, a velocitymeasurement, and three dimensional real space coordinates, is a cosmologicalproblem that is often modeled through a linear understanding of thevelocity-position coupling. This linear information is better preserved in theLagrangian space picture of the matter density field. Through Lagrangian spacemeasurements, we can extract more information and make more accurate estimatesof the linear growth rate of the universe. In this paper, we address the linearmodelling of matter particle velocities through transfer functions, and indoing so examine to what degree the decrease in correlation with initialconditions may be contaminated by velocity-based nonlinearities. With athorough analysis of the monopole-quadrupole ratio, we find the best-fit valuefor the Eulerian velocity dispersion, $\sigma_p = 378.3$ km/s. The covarianceof the cosmological linear growth rate $f$, is estimated in the Eulerian andLagrangian cases. Comparing Lagrangian and Eulerian, we find that the error in$f$ improves by a factor of 3, without the need for nonlinear velocitydispersion modelling.

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