  Many works have concentrated on the observable signatures of the dark matter
being an ultralight axion-like particle (ALP). We concentrate on a particularly
dramatic signature in the late-time cosmological matter power spectrum that
occurs if the symmetry breaking that establishes the ALP happens after
inflation -- white-noise density fluctuations that dominate at small scales
over the adiabatic fluctuations from inflation. These fluctuations alter the
early history of nonlinear structure formation. We find that for symmetry
breaking scales of $f_A \sim 10^{13}-10^{15}$GeV, which requires a high
effective maximum temperature after inflation, ALP dark matter with particle
mass of $m_A \sim 10^{-13}-10^{-20}$eV could significantly change the number of
high-redshift dwarf galaxies, the reionization history, and the Ly$\alpha$
forest. We consider all three observables. We find that the Ly$\alpha$ forest
is the most constraining of current observables, excluding $f_A \gtrsim
10^{15}$GeV ($m_A \lesssim 10^{-17}$eV) in the simplest model for the ALP and
considerably lower values in models coupled to a hidden strongly interacting
sector ($f_A \gtrsim 10^{13}$GeV and $m_A \lesssim 10^{-13}$eV). Observations
that constrain the extremely high-redshift tail of reionization may disfavor
similar levels of isocurvature fluctuations as the forest. Future $z\sim 20-30$
21cm observations have the potential to improve these constraints further using
that the supersonic motions of the isocurvature-enhanced abundance of
$\sim10^4M_\odot$ halos would shock heat the baryons, sourcing large BAO
features.
