  Current cosmological data drop an interesting hint about the existence of
sub-eV sterile neutrinos, which should be a part of the cosmic neutrino
background (C$\nu$B). We point out that such light sterile neutrinos may leave
a distinct imprint on the electron energy spectrum in the capture of relic
electron neutrinos by means of radioactive beta-decaying nuclei. We examine
possible signals of sterile neutrinos relative to active neutrinos,
characterized by their masses and sensitive to their number densities, in the
reaction $\nu^{}_e + ~^3{\rm H} \to ~^3{\rm He} + e^-$ against the
corresponding tritium beta decay. We stress that this kind of direct laboratory
detection of the C$\nu$B and its sterile component might not be hopeless in the
long term.
