  We examine a framework with light new physics, which couples to the Standard
Model only via neutrino mixing. Taking the hints from the short-baseline
anomalies seriously and combining them with modern cosmological data and recent
IceCube measurements, we obtain surprisingly effective constraints on the
hidden force: keV $\lesssim M \lesssim0.3$ GeV for the mediator mass and
$g_{h}>10^{-6}-10^{-3}$ for the coupling constant. Flavor equilibration between
the hidden and active neutrinos can be delayed until temperatures of $\sim 1$
MeV, but not below $\sim 100$ keV. This scenario can be tested with
next-generation Cosmic Microwave Background, IceCube, and oscillation
experiments.
