The stochastic relaxion

We revisit the original proposal of cosmological relaxation of the electroweak scale by Graham, Kaplan and Rajendran in which the Higgs mass is scanned during inflation by an axion field, the relaxion. We investigate the regime where the relaxion is subject to large fluctuations during inflation. We include the stochastic dynamics of the relaxion described by means of the Fokker-Planck formalism. We derive a new stopping condition for the relaxion taking into account transitions between the neighboring local minima of its potential. Relaxion fluctuations have important consequences even in the ``classical-beats-quantum" regime. We determine that for a large Hubble parameter during inflation, the random walk prevents the relaxion from getting trapped at the first minimum. The relaxion stops much further away, where the potential is less shallow. Interestingly, this essentially jeopardises the ``runaway relaxion" threat from finite density effects, restoring most of the relaxion parameter space. We also explore the ``quantum-beats-classical" regime, opening large new regions of parameter space. We investigate the consequences for both the QCD and non-QCD relaxion. The most striking implication is thatthe misalignment of the relaxion from its local minimum due to fluctuations can naturally explain the observed dark matter density in the universe