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 $m_bsim 10^{-22}$eV. We recover results reported previously (Irv{s}iv{c} et al. 2017b; Armengaud et al. 2017) that the CDM model agrees better with the BOSS data than the FP$psi$DM model by a large margin, and we find the difference of total $chi^2$s is $120$ for $420$ data bins. These previous results demand a larger boson mass by a factor $>10$ to be consistent with the date and are in tension with the favoured value determined from local satellite galaxies. We however find that such tension is removed as some EA$psi$DM models predict Lyman-$alpha$ flux spectra agreeing better with the BOSS data than the CDM model, and the difference of total $chi^2$s can be as large as $24$ for the same bin number. This finding arises with no surprise since EA$psi$DM models have unique spectral shapes with spectral bumps in excess of the CDM power near the small-scale cutoff typical of $psi$DM linear matter power spectra as well as more extended cutoffs than FP$psi$DM (Zhang & Chiueh 2017a,b).
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