  We propose a new method to recover the cosmological initial conditions of the
presently observed galaxy distribution, which can serve to run constrained
simulations of the Local Universe. Our method, the Reverse Zeldovich
Approximation (RZA), can be applied to radial galaxy peculiar velocity data and
extends the previously used Constrained Realizations (CR) method by adding a
Lagrangian reconstruction step. The RZA method consists of applying the
Zeldovich approximation in reverse to galaxy peculiar velocities to estimate
the cosmic displacement field and the initial linear matter distribution from
which the present-day Local Universe evolved.We test our method with a mock
survey taken from a cosmological simulation. We show that the halo peculiar
velocities at z = 0 are close to the linear prediction of the Zeldovich
approximation, if a grouping is applied to the data to remove virial motions.
We find that the addition of RZA to the CR method significantly improves the
reconstruction of the initial conditions. The RZA is able to recover the
correct initial positions of the velocity tracers with a median error of only
1.36 Mpc/h in our test simulation. For realistic sparse and noisy data, this
median increases to 5 Mpc/h. This is a significant improvement over the
previous approach of neglecting the displacement field, which introduces errors
on a scale of 10 Mpc/h or even higher. Applying the RZA method to the upcoming
high-quality observational peculiar velocity catalogues will generate much more
precise constrained simulations of the Local Universe.
