Recent results from the SNO, Super-Kamiokande, and Borexino experiments do not show the expected upturn of the energy spectrum of events (the ratio $R\equiv N_{\mathrm{obs}}/N_{\mathrm{SSM}}$) at low energies. At the same time, cosmological observations testify for the possible existence of additional relativistic degrees of freedom in the early Universe: $\Delta N_{\mathrm{eff}}=1–2$. These facts strengthen the case of a very light sterile neutrino, ${\nu }_s$, with $\Delta m_{01}^2\sim (0.7–2)\times {10}^{-5}{\mathrm{eV}}^2$, which mixes weakly with the active neutrinos. The ${\nu }_s$ mixing in the mass eigenstate ${\nu }_1$ characterized by ${sin}^{2}2\alpha \sim {10}^{-3}$ can explain an absence of the upturn. The mixing of ${\nu }_s$ in the eigenstate ${\nu }_3$ with ${sin}^2\beta \sim 0.1$ leads to production of ${\nu }_s$ via oscillations in the Universe and to additional contribution $\Delta N_{\mathrm{eff}}\approx 0.7–1$ before the big bang nucleosynthesis and later. Such a mixing can be tested in forthcoming experiments with the atmospheric neutrinos, as well as in future accelerator long baseline experiments. It has substantial impact on conversion of the supernova neutrinos.

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