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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.