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Using cosmological particle hydrodynamical simulations and uniform ultraviolet backgrounds, we compare Lyman-$\alpha$ forest flux spectra predicted by the conventional cold dark matter (CDM) model, the free-particle wave dark matter (FP$\psi$DM) model and extreme-axion wave dark matter (EA$\psi$DM) models of different initial axion field angles against the BOSS Lyman-$\alpha$ forest absorption spectra with a fixed boson mass $m_b\sim 10^{-22}$eV.
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