Neutrino bound states and bound systems

Yukawa interactions of neutrinos with a new light scalar boson $\phi$ can lead to formation of stable bound states and bound systems of many neutrinos ($\nu$-clusters). For allowed values of the coupling $y$ and the scalar mass $m_\phi$, the bound state of two neutrinos would have the size larger than $10^{12}$ cm. Bound states with sub-cm sizes are possible for keV scale sterile neutrinos with coupling $y > 10^{-4}$. For $\nu$-clusters we study in detail the properties of final stable configurations. If there is an efficient cooling mechanism, these configurations are in the state of degenerate Fermi gas. We formulate and solve equations of the density distributions in $\nu$-clusters. In the non-relativistic case, they are reduced to the Lane-Emden equation. We find that (i) stable configurations exist for any number of neutrinos, $N$; (ii) there is a maximal central density $\sim 10^9$ cm$^{-3}$ determined by the neutrino mass; (iii) for a given $m_\phi$ there is a minimal value of $Ny^3$ for which stable configurations can be formed; (iv) for a given strength of interaction, $S_\phi = (ym_\nu/m_\phi)^2$, the minimal radius of $\nu$-clusters exists. We discuss the formation of $\nu$-clusters from relic neutrino background in the process of expansion and cooling of the Universe. One possibility realized for $S_\phi > 700$ is the development of instabilities in the $\nu$-background at $T < m_\nu$ which leads to its fragmentation. For allowed values of $y$, cooling of $\nu$-clusters due to $\phi$-bremsstrahlung and neutrino annihilation is negligible. The sizes of $\nu$-clusters may range from $\sim$ km to $\sim 5$ Mpc. Formation of clusters affects perspectives of detection of relic neutrinos.