Universal lower bound on the axion decay constant from free streaming effects

We show that enhancement of the axion relic abundance compared to the standard misalignment contribution generically leads to the production of nonzero momentum axion modes, resulting in warm dark matter behavior and enhanced isocurvature perturbations. It leads to universal constraints on the axion parameter space that are independent of detailed model assumptions and cosmological history. For models enhancing relic abundance with gradient axion modes, observations of the Lyman-$\alpha$ forest impose a lower bound on the axion decay constant, $f_a \gtrsim 10^{15} {\rm GeV}\,(10^{-18}{\rm eV}/m_a)$, from the free-streaming effect. For models relying on the delay of coherent axion oscillations, we obtain a slightly weaker bound, $f_a \gtrsim 10^{14} {\rm GeV}\,(10^{-18}{\rm eV}/m_a)$. We make relatively conservative choices to establish these universal bounds but also provide scaling parameters that can be calibrated for stronger constraints in concrete models and updated as observations improve.