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The particle nature of dark matter remains a mystery. In this paper, weconsider two dark matter models---Late Forming Dark Matter (LFDM) andUltra-Light Axion (ULA) models---where the matter power spectra show noveleffects on small scales. The high redshift universe offers a powerful probe oftheir parameters. In particular, we study two cosmological observables: theneutral 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-bodysimulations with modified initial conditions, and generate reionization fieldsusing an excursion-set model. The N-body approximation is valid on the lengthand halo mass scales studied.  We show that LFDM and ULA models predict an increase in the HI power spectrumfrom 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 hydrogenfraction $\bar{x}_{HI}=0.5$ must be achieved by $z=8$, the reionization processallows 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 inferredfrom damped Lyman-$\alpha$ observations to the theoretical predictions of ourmodels 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 literatureusing different observables; we briefly discuss how these bounds compare withpossible constraints from the observation of luminosity function of galaxies athigh redshifts. In the case of ULAs, these constraints are also consistent witha solution to the cusp-core problem of CDM.

The effects of the small-scale DM power on the cosmological neutral hydrogen (\HI) distribution at high redshifts