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Many works have concentrated on the observable signatures of the dark matterbeing an ultralight axion-like particle (ALP). We concentrate on a particularlydramatic signature in the late-time cosmological matter power spectrum thatoccurs if the symmetry breaking that establishes the ALP happens afterinflation -- white-noise density fluctuations that dominate at small scalesover the adiabatic fluctuations from inflation. These fluctuations alter theearly history of nonlinear structure formation. We find that for symmetrybreaking scales of $f_A \sim 10^{13}-10^{15}$GeV, which requires a higheffective maximum temperature after inflation, ALP dark matter with particlemass of $m_A \sim 10^{-13}-10^{-20}$eV could significantly change the number ofhigh-redshift dwarf galaxies, the reionization history, and the Ly$\alpha$forest. We consider all three observables. We find that the Ly$\alpha$ forestis the most constraining of current observables, excluding $f_A \gtrsim10^{15}$GeV ($m_A \lesssim 10^{-17}$eV) in the simplest model for the ALP andconsiderably lower values in models coupled to a hidden strongly interactingsector ($f_A \gtrsim 10^{13}$GeV and $m_A \lesssim 10^{-13}$eV). Observationsthat constrain the extremely high-redshift tail of reionization may disfavorsimilar levels of isocurvature fluctuations as the forest. Future $z\sim 20-30$21cm observations have the potential to improve these constraints further usingthat the supersonic motions of the isocurvature-enhanced abundance of$\sim10^4M_\odot$ halos would shock heat the baryons, sourcing large BAOfeatures.

Early Structure Formation Constraints on the Ultra-Light Axion in the Post-Inflation Scenario