Constraint on Matter Power Spectrum on $10^6-10^9M_\odot$ Scales from
  ${\large\tau_e}$
  An analysis of the physics-rich endgame of reionization at $z=5.7$ is
performed, utilizing jointly the observations of the Ly$\alpha$ forest, the
mean free path of ionizing photons, the luminosity function of galaxies and new
physical insight. We find that an upper limit on ${\rm \tau_e}$ provides a
constraint on the minimum mean free path (of ionizing photons) that is
primarily due to dwarf galaxies, which in turn yields a new and yet the
strongest constraint on the matter power spectrum on $10^6-10^9M_\odot$ scales.
With the latest Planck measurements of ${\rm \tau_e = 0.055 \pm 0.009}$, we can
place an upper limit of $(8.9\times 10^6, 3.8\times 10^7, 4.2\times
10^8)M_\odot$ on the lower cutoff mass of the halo mass function, or equivalent
a lower limit on warm dark matter particle mass ${\rm m_x \ge (15.1, 9.8,
4.6)keV}$ or on sterile neutrino mass ${\rm m_s \ge (161, 90, 33)keV}$, at $(1,
1.4, 2.2)\sigma$ confidence level, respectively.
