Redshifted 21-cm emission signal from the halos in Dark Ages
  The emission in the hyperfine structure 21 cm line of atomic hydrogen arising
in the halos with masses $\sim10^6-10^{10}$ M$_\odot$ from the Dark Ages in the
models with Warm Dark Matter (WDM) is analysed. The halos are assumed to be
formed from Gaussian density peaks of cosmological density perturbations at
$10\lesssim z\lesssim50$. Semi-analytical modelling of the formation of
individual spherical halos in multi-component models shows that gas in them has
the kinetic temperature in the range of $60-800$ K under adiabatic compression
of the collapsing halo, and the temperature of each halo depends on the time of
virialization. It is shown that inelastic collisions between neutral hydrogen
atoms are the dominant excitation mechanism for hyperfine structure levels,
which pulls the spin temperature closer to the kinetic temperature. The
brightness temperature of individual halos is in the range of 1-10 K, depending
on the mass of the halo and its virialization redshift, and increasing as these
two increase. The apparent angular radii of such halos are in the range
0.06-1.2 arcseconds, their surface number density decreasing exponentially from
a few per arcmin$^2$ for the lowest mass and redshift to nearly zero for higher
values. Assuming a 1 MHz observation bandwidth the surface number density of
the halo at various redshifts is evaluated as well as beam-averaged
differential antenna temperatures and fluxes of hydrogen emission from halos of
different masses. The beam-averaged signal strongly depends on the cut-off
scale in the mass function of dark ages halos that may be caused by
free-streaming of WDM particles. The finding is compared with the upper limits
on the amplitude of the power spectrum of the hydrogen 21-cm line fluctuations
derived from the recent observation data obtained with MWA and LOFAR.
