  Massive neutrinos suppress the growth of cosmic structure on small,
non-linear, scales. It is thus often proposed that using statistics beyond the
power spectrum can tighten constraints on the neutrino mass by extracting
additional information from these non-linear scales. We study the information
content regarding neutrino mass at the field level, quantifying how much of
this information arises from the difference in non-linear evolution between a
cosmology with 1 fluid (CDM) and 2 fluids (CDM + neutrinos). We do so by
running two $N$-body simulations, one with and one without massive neutrinos;
both with the same phases, and matching their linear power spectrum at a given,
low, redshift. This effectively isolates the information encoded in the linear
initial conditions from the non-linear cosmic evolution. We demonstrate that
for $k \lesssim 1\,h/{\rm Mpc}$, and for a single redshift, there is negligible
difference in the real-space CDM field between the two simulations. This
suggests that all the information regarding neutrino mass is in the linear
power spectrum set by the initial conditions. Thus any probe based on the CDM
field alone will have negligible constraining power beyond that which exists at
the linear level over the same range of scales. Consequently, any probe based
on the halo field will contain little information beyond the linear power. We
find similar results for the matter field responsible for weak lensing. We also
demonstrate that there may be much information beyond the power spectrum in the
3d matter field, however, this is not observable in modern surveys via dark
matter halos or weak lensing. Finally, we show that there is additional
information to be found in redshift space.
