  We study theoretically the electron spin noise in quantum dots under
non-equilibrium conditions caused by the pumping by a train of circularly
polarized optical pulses. In such a situation, the nuclear spins are known to
adjust in such a way, that the electron spin precession frequencies become
multiples of the pump pulse repetition frequency. This so called phase
synchronization effect was uncovered in [Science {\bf 317}, 1896 (2007)] and
termed nuclei-induced frequency focusing of electron spin coherence. Using the
classical approach to the central spin model we evaluate the nuclear spin
distribution function and the electron spin noise spectrum. We show that the
electron spin noise spectrum consists of sharp peaks corresponding to the phase
synchronization conditions and directly reveal the distribution of the nuclear
spins. We discuss the effects of nuclear spin relaxation after the pumping is
over and analyze the corresponding evolution of nuclear spin distributions and
electron spin noise spectra.
