  We propose cosmological observations of Lyman-alpha and 21-cm signal to set
stringent constraints on freeze-in dark matter (FIDM). Explicitly we consider
Higgs (neutrino)-portal FIDM in the single-field context, which injects energy
into the intergalactic medium via its annihilation (decay). With respect to
Lyman-alpha the baseline ionization history is inferred from low redshift data
about astrophysical reionization, whereas for 21-cm signal the baseline values
of 21-cm power spectrum are obtained through a standard modeling of star
formation developed. We use numerical tools to derive the FIDM induced
deviations from these baseline values in high redshift region. Our results show
that (i) current Lyman-alpha data has already constrained the neutrino portal
FIDM mass to be less than 1.06 MeV, (ii) future Lyman-alpha data about the
intergalactic medium temperature with a 10(100)% precision at z~9-15 is
sufficient to exclude the Higgs (neutrino)-portal FIDM, and (iii) future SKA
sensitivity (1000 hrs) on the 21-cm power spectrum for reference wavenumber
k*=0.2h /Mpc at z~15-16 is also able to exclude the surviving neutrino-portal
FIDM mass window.
