Implications of the cosmological 21-cm absorption profile for
  high-redshift star formation and deep JWST surveys
  Apart from its anomalously large depth, the cosmological 21-cm absorption
signal measured by the EDGES collaboration also has a shape that is distinctly
different from theoretical predictions. This peculiar shape requires the
establishment of a cosmic Ly $\alpha$ background within $25\,$Myr at redshift
$z\sim 20$ and a cosmic X-ray background within $50\,$Myr at redshift $z\sim
15$. Models with non-traditional components such as super-adiabatic baryonic
cooling or an excess radio background explain the depth of the observed
profile, but still conspicuously fail to explain its shape. In this paper, we
quantify the requirements imposed by the EDGES measurement on sources of Ly
$\alpha$ and X-ray photons at cosmic dawn. In extreme cases, the Ly $\alpha$
and X-ray emissivities require to be enhanced by up to an order of magnitude
relative to traditional models. Furthermore, this enhancement needs to be
active only for a short duration. We find that under conventional assumptions
for the cosmic star formation density, standard stellar populations are
incapable of meeting these conditions. Only highly unusual models of massive
metal-free stars seem to provide a possible mechanism. Conversely, if the
sources of Ly $\alpha$ and X-ray photons are compelled to have standard
properties, the EDGES measurement puts strong demands on the cosmic star
formation rate density. This provides interesting falsifiable predictions for
high-redshift galaxy surveys enabled by JWST. We derive predictions for galaxy
UV luminosity functions and number densities, and show that a deep JWST survey
with a limiting UV magnitude of $m_\mathrm{UV,lim}=32$ would potentially be
able to rule out the predictions enforced by the EDGES measurement.
