  The flavor composition of TeV--PeV astrophysical neutrinos, i.e., the
proportion of neutrinos of different flavors in their flux, is a versatile
probe of high-energy astrophysics and fundamental physics. Because flavor
identification is challenging and the number of detected high-energy
astrophysical neutrinos is limited, so far measurements of the flavor
composition have represented an average over the range of observed neutrino
energies. Yet, this washes out the potential existence of changes in the flavor
composition with energy and weakens our sensitivity to the many models that
posit them. For the first time, we measure the energy dependence of the flavor
composition, looking for a transition from low to high energies. Our
present-day measurements, based on the 7.5-year public sample of IceCube
High-Energy Starting Events (HESE), find no evidence of a flavor transition.
The observation of HESE and through-going muons jointly by next-generation
neutrino telescopes Baikal-GVD, IceCube-Gen2, KM3NeT, P-ONE, TAMBO, and TRIDENT
may identify a flavor transition around 200TeV by 2030. By 2040, we could infer
the flavor composition with which neutrinos are produced with enough precision
to establish the transition from neutrino production via the full pion decay
chain at low energies to muon-damped pion decay at high energies.
