  The particle nature of dark matter remains a mystery. In this paper, we
consider two dark matter models---Late Forming Dark Matter (LFDM) and
Ultra-Light Axion (ULA) models---where the matter power spectra show novel
effects on small scales. The high redshift universe offers a powerful probe of
their parameters. In particular, we study two cosmological observables: the
neutral hydrogen (HI) redshifted 21-cm signal from the epoch of reionization,
and the evolution of the collapsed fraction of HI in the redshift range $2 < z
< 5$. We model the theoretical predictions of the models using CDM-like N-body
simulations with modified initial conditions, and generate reionization fields
using an excursion-set model. The N-body approximation is valid on the length
and halo mass scales studied.
  We show that LFDM and ULA models predict an increase in the HI power spectrum
from the epoch of reionization by a factor between 2--10 for a range of scales
$0.1<k<4 \, \rm Mpc^{-1}$. Assuming a fiducial model where a neutral hydrogen
fraction $\bar{x}_{HI}=0.5$ must be achieved by $z=8$, the reionization process
allows us to put approximate bounds on the redshift of dark matter formation
$z_f > 4 \times 10^5$ (for LFDM) and the axion mass $m_a > 2.6 \times 10^{-23}
\, \rm eV$ (for ULA). The comparison of the collapsed mass fraction inferred
from damped Lyman-$\alpha$ observations to the theoretical predictions of our
models lead to the weaker bounds: $z_f > 2 \times 10^5$ and $m_a > 10^{-23} \,
\rm eV$. These bounds are consistent with other constraints in the literature
using different observables; we briefly discuss how these bounds compare with
possible constraints from the observation of luminosity function of galaxies at
high redshifts. In the case of ULAs, these constraints are also consistent with
a solution to the cusp-core problem of CDM.
