  A proto-neutron star (PNS) is a newly formed compact object in a core
collapse supernova. In this Letter, the neutrino emission from the cooling
process of a PNS is investigated using two types of nuclear equation of state
(EOS). It is found that the neutrino signal is mainly determined by the
high-density EOS. The neutrino luminosity and mean energy are higher and the
cooling time scale is longer for the softer EOS. Meanwhile, the neutrino mean
energy and the cooling time scale are also affected by the low-density EOS
because of the difference in the population of heavy nuclei. Heavy nuclei have
a large scattering cross section with neutrinos owing to the coherent effects
and act as thermal insulation near the surface of a PNS. The neutrino mean
energy is higher and the cooling time scale is longer for an EOS with a large
symmetry energy at low densities, namely a small density derivative coefficient
of the symmetry energy, $L$.
