  It has been recently claimed that there is no correlation between the
distribution of galaxies and their peculiar velocities within the Local Volume
(LV), namely a sphere of R=7/h Mpc around the Local Group (LG). It has been
then stated that this implies that either locally dark matter is not
distributed in the same way as luminous matter, or peculiar velocities are not
due to fluctuations in mass. To test that statement a set of constrained N-body
cosmological simulations, designed to reproduce the main observed large scale
structure, have been analyzed. The simulations were performed within the
flat-Lambda, open and flat matter only CDM cosmogonies. Two unconstrained
simulations of the flat-Lambda and open CDM models were performed for
comparison. LG-like objects have been selected so as to mimic the real LG
environment. The local gravitational field due to all halos found within each
LV is compared with the exact gravitational field induced by all matter in the
simulation. We conclude that there is no correlation between the exact and the
local gravitational field obtained by pairwise newtonian forces between halos.
Moreover, the local gravitational field is uncorrelated with the peculiar
velocities of halos. The exact gravitational field has a linear correlation
with peculiar velocities but the proportionality constant relating the velocity
with gravitational field falls below the prediction of the linear theory. Upon
considering all matter inside the LVs, the exact and local gravitational
accelerations show a much better correlation, but with a considerable scatter
independent on the cosmological models. The main conclusion is that the lack of
correlation between the local gravitation and the peculiar velocity fields
around LG-like objects is naturally expected in the CDM cosmologies.
