Constraints on Relaxion Windows

We examine the low energy phenomenology of the relaxion solution to the weak scale hierarchy problem. Assuming that the Hubble friction is responsible for a dissipation of the relaxion energy, we identify the cosmological relaxion window which corresponds to the parameter region compatible with a given value of the acceptable number of inflationary $e$-foldings. We then discuss a variety of observational constraints on the relaxion window, including those from astrophysical and cosmological considerations. We find that majority of the parameter space with a relaxion mass $m_\phi\gtrsim 100$ eV or a relaxion decay constant $f\lesssim 10^7$ GeV is excluded by existing constraints. There is an interesting parameter region with $m_\phi\sim \,0.2-10$ GeV and $f\sim\, {\rm few}-200$ TeV, which is allowed by existing constraints, but can be probed soon by future beam dump experiments such as the SHiP experiment, or by improved EDM experiments.