Sterile neutrinos: direct mixing effects versus induced mass matrix of active neutrinos

Mixing of active neutrinos with sterile ones generate ``induced'' contributions to the mass matrix of active neutrinos $\sim m_S \sin^2\theta_{aS}$, where $m_S$ is the Majorana mass of the sterile neutrino and $\theta_{aS}$ is the active-sterile mixing angle. We study possible effects of the induced matrix which can modify substantially the implications of neutrino oscillation results. We have identified the regions of $m_S$ and $\sin^2\theta_{aS}$ where the induced matrix (i) provides the dominant structures, (ii) gives the sub-dominant effects and (iii) where its effects can be neglected. The induced matrix can be responsible for peculiar properties of the lepton mixing and neutrino mass spectrum, in particular, it can generate the tri-bimaximal mixing. We update and discuss bounds on the induced masses from laboratory measurements, astrophysics and cosmology. We find that substantial impact of the induced matrix is possible if $m_S \sim 0.1-1$ eV and $\sin^2\theta_{aS} \sim 10^{-3} - 10^{-2}$ or $m_S \geq 200$ MeV and $\sin^2\theta_{aS} \leq 10^{-9}$. The bounds can be relaxed in cosmological scenarios with low reheating temperature, if sterile neutrinos decay sufficiently fast, or their masses change with time.