  We focus on the complex relationship between the shape of dark matter halos
and the cosmological models underlying their formation. We used three realistic
cosmological models from the Dark Energy Universe Simulation suite. They have
significantly distinct cosmological parameters ($\Omega_m$, $\sigma_8$, and
$w$) but quasi-indistinguishable cosmic matter fields beyond the scale of DM
halos. Firstly, we developed a robust method to measure the FoF halos shape,
while avoiding numerical artifacts on shape measurements. These artifacts are
generally induced by the presence of substructures (resolution-dependent) or by
any spherical a priori in contradiction with the halos triaxiality. We obtain a
marked dependence of the halos shape, both on their mass and on the
cosmological model. However, when re-expressing the halos shape parameters in
terms of the non-linear fluctuations of the cosmic matter field, the
cosmological dependence disappears. This new fundamental cosmological
invariance is a direct consequence of the matter field non-linear dynamics: as
the universe evolves, the non-linear fluctuations of the cosmic field increase,
driving the halos towards sphericity. The deviation from sphericity, measured
by the prolaticity, triaxiality, and ellipticity of the dark matter halos, is
therefore entirely encapsulated in the non-linear power spectrum. From this
fundamental invariant relation, we can reconstruct with remarkable accuracy the
non-linear variance of the cosmic matter field and, consequently, the
non-linear power spectrum. We also re-find the $\sigma_8$ amplitude of the
cosmological model. Our results highlight, not only the nuanced relationship
between dark matter halo formation and the underlying cosmology, but also the
potential of dark matter halo shape analysis as a powerful tool for probing the
non-linear dynamics of the cosmic matter field.
