  The mean matter density within the turnaround radius, which is the boundary
that separates a nonexpanding structure from the Hubble flow, was recently
proposed as a novel cosmological probe. According to the spherical collapse
model, the evolution with cosmic time of this turnaround density, $\rm
\rho_{ta}(z)$, can be used to determine both $\rm \Omega_m$ and
$\Omega_\Lambda$, independently of any other currently used probe. The
properties of $\rm \rho_{ta}$ predicted by the spherical collapse model were
also shown to persist in the presence of full three-dimensional effects in $\rm
\Lambda$CDM N-body cosmological simulations when considering galaxy clusters at
the present time, $z=0$. However, a small offset was discovered between the
spherical-collapse prediction of the value of $\rho_{ta}$ at $z=0$ and its
value measured in simulations. In this letter, we explore whether this offset
evolves with cosmic time; whether it differs in different cosmologies; whether
its origin can be confidently identified; and whether it can be corrected. We
found that the offset does evolve slightly with redshift, and that it
correlates strongly with the deviation from spherical symmetry of the dark
matter halo distribution inside and outside of the turnaround radius. We used
an appropriate metric to quantify deviations in the environment of a structure
from spherical symmetry. We found that using this metric, we can construct a
sphericity-selected sample of halos for which the offset of $\rho_{ta}$ from
the spherical collapse prediction is zero, independently of redshift and
cosmology. We found that a sphericity-selected halo sample allows us to recover
the simulated cosmology, and we conclude that the turnaround density evolution
indeed encodes the cosmology in N-body simulations.
