  The neutrino field is commonly assumed to be isotropic and homogeneous in the
early universe. However, due to the large neutrino density, a small
perturbation of the isotropy of the neutrino field could potentially be
amplified by the non-linear flavor mixing caused by neutrino self-interactions.
We carry out the first numerical simulations of the neutrino flavor evolution
in a multi-angle anisotropic setting. Due to the computational challenges
involved, we adopt a simplified framework consisting of a homogeneous universe
with two angle bins -- left and right moving modes -- for neutrinos and
antineutrinos, together with an approximate form for the collision term which
goes beyond the commonly adopted damping approximation. By assuming a small
initial left-right asymmetry of $\mathcal{O}(10^{-15})$, we convincingly
demonstrate that flavor evolution can be affected in both mass orderings, with
implications on the effective number of thermally excited neutrino species
($N_{\mathrm{eff}}$). Notably, the correction to $N_{\rm eff}$ is comparable to
higher order corrections from finite temperature QED effects in normal
ordering. In addition, by assuming an initial lepton asymmetry in the neutrino
sector of the same order as the baryon one [$\mathcal{O}(10^{-9})$], we find
that the neutrino-antineutrino asymmetry grows by several orders of magnitude
for isotropic as well as anisotropic initial conditions. This work clearly
shows that it is imperative to critically revisit standard assumptions
concerning neutrino flavor mixing in the early universe, especially in the
light of possible implications on the cosmological observables.
