  Cosmic background neutrinos ($C_{\nu}B)$ helicity composition is different
for Dirac or Majorana neutrinos making detectors based on $C_{\nu}B$ capture
sensitive to the nature of neutrinos. We calculate, for the first time, the
helicity changes of neutrinos crossing dark matter fields, to quantitatively
calculate this effect on the capture rate. We show that a fraction of neutrinos
change their helicity, regardless of them being deflected by a void or a dark
matter halo. The average signal from the 100 most massive voids or halos in a
Gpc$^3$ gives a prediction that if neutrinos are Dirac, the density of the
$C_{\nu} B$ background measured on Earth should be 48 cm${^{-3}}$ for
left-helical neutrinos, a decrease of 15% (53.6 cm${^{-3}}$; 5%) for a halo
(void) with respect to the standard calculation without including gravitational
effects due to large scale structures. In terms of the total capture rate in a
100 g tritium detector, this translates in $4.9^{+1.1}_{-0.8}$ neutrinos per
year for the Dirac case, as a function of the unknown neutrino mass scale, or
8.1 per year if neutrinos are Majorana. Thus although smaller than the factor
two for the non-relativistic case, it is still large enough to be detected and
it highlights the power of future $C_{\nu} B$ detectors, as an alternative to
neutrinoless double beta decay experiments, to discover the neutrino nature.
