Cosmological beam dump: constraints on dark scalars mixed with the Higgs
  boson
  Precision cosmology provides a sensitive probe of extremely weakly coupled
states due to thermal freeze-in production, with subsequent decays impacting
physics during well-tested cosmological epochs. We explore the cosmological
implications of the freeze-in production of a new scalar $S$ via the
super-renormalizable Higgs portal. If the mass of $S$ is at or below the
electroweak scale, peak freeze-in production occurs during the electroweak
epoch. We improve the calculation of the freeze-in abundance by including all
relevant QCD and electroweak production channels. The resulting abundance and
subsequent decay of $S$ is constrained by a combination of X-ray data, cosmic
microwave background anisotropies and spectral distortions, $N_{\rm eff}$, and
the consistency of BBN with observations. These probes constrain technically
natural couplings for such scalars from $m_S \sim$ keV all the way to $m_S \sim
100$ GeV. The ensuing constraints are similar in spirit to typical beam bump
limits, but extend to much smaller couplings, down to mixing angles as small as
$\theta_{Sh} \sim 10^{-16}$, and to masses all the way to the electroweak
scale.
