Virial halo mass function in the ${\it Planck}$ cosmology
  We study halo mass functions with high-resolution $N$-body simulations under
a $\Lambda$CDM cosmology. Our simulations adopt the cosmological model that is
consistent with recent measurements of the cosmic microwave backgrounds with
the ${\it Planck}$ satellite. We calibrate the halo mass functions for
$10^{8.5} \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} M_\mathrm{vir} /
(h^{-1}M_\odot) \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10^{15.0 - 0.45
\, z}$, where $M_\mathrm{vir}$ is the virial spherical overdensity mass and
redshift $z$ ranges from $0$ to $7$. The halo mass function in our simulations
can be fitted by a four-parameter model over a wide range of halo masses and
redshifts, while we require some redshift evolution of the fitting parameters.
Our new fitting formula of the mass function has a 5\%-level precision except
for the highest masses at $z\le 7$. Our model predicts that the analytic
prediction in Sheth $\&$ Tormen would overestimate the halo abundance at $z=6$
with $M_\mathrm{vir} = 10^{8.5-10}\, h^{-1}M_\odot$ by $20-30\%$. Our
calibrated halo mass function provides a baseline model to constrain warm dark
matter (WDM) by high-$z$ galaxy number counts. We compare a cumulative
luminosity function of galaxies at $z=6$ with the total halo abundance based on
our model and a recently proposed WDM correction. We find that WDM with its
mass lighter than $2.71\, \mathrm{keV}$ is incompatible with the observed
galaxy number density at a $2\sigma$ confidence level.
