Updated Axion CDM energy density
  We update cosmological bound on axion model. The contribution from the anharmonic effect and the newly introduced initial overshoot correction are considered. We present an explicit formula for the axion relic density in terms of the QCD scale Lambda_{QCD}, the current quark masses m_q's and the Peccei-Quinn scale F_a, including firstly introduced 1.85 factor which is from the initial overshoot. 
Ji-Haeng Huh