  In concordance cosmology, dark matter density perturbations generated by
inflation lead to nonlinear, virialized minihalos, into which baryons collapse
at redshift $z \sim 20$. We survey here novel baryon evolution produced by a
modification of the power spectrum from white noise density perturbations at
scales below $k \sim 10 h {Mpc}^{-1}$ (the smallest scales currently measured
with the Lyman-$\alpha$ forest). Exotic dark matter dynamics, such as would
arise from scalar dark matter with a late phase transition (similar to an
axion, but with lower mass), create such an amplification of small scale power.
The dark matter produced in such a phase transition collapses into minihalos,
with a size given by the dark matter mass within the horizon at the phase
transition. If the mass of the initial minihalos is larger than $\sim 10^{-3}
M_\odot$, the modified power spectrum is found to cause widespread baryon
collapse earlier than standard $\Lambda$CDM, leading to earlier gas heating. It
also results in higher spin temperature of the baryons in the 21 cm line
relative to $\Lambda$CDM at redshifts $z > 20$ if the mass of the minihalo is
larger than $1 M_\odot$. It is estimated that experiments probing 21 cm
radiation at high redshift will contribute a significant constraint on dark
matter models of this type for initial minihalos larger than $\sim 10 M_\odot$.
Early experiments reaching to $z\approx 15$ will constrain minihalos down to
$\sim 10^3 M_\odot$.
