  Very different processes characterize the decoupling of neutrinos to form the
cosmic neutrino background (C$\nu$B) and the much later decoupling of photons
from thermal equilibrium to form the cosmic microwave background (CMB). The
C$\nu$B emerges from the fuzzy, energy-dependent neutrinosphere and encodes the
physics operating in the early universe in the temperature range $T\sim
10\,{\rm MeV}$ to $T\sim10\,{\rm keV}$. This is the epoch where beyond Standard
Model (BSM) physics may be influential in setting the light element abundances
and the necessarily distorted fossil neutrino energy spectra. Here we use
techniques honed in extensive CMB studies to analyze the C$\nu$B as calculated
in detailed neutrino energy transport and nuclear reaction simulations. Our
moment method, relative entropy, and differential visibility approach can
leverage future high precision CMB and primordial abundance measurements to
provide new insights into the C$\nu$B and any BSM physics it encodes. We
demonstrate that the evolution of the energy spectrum of the C$\nu$B throughout
the weak decoupling epoch is accurately captured in the Standard Model by only
three parameters per species, a non-trivial conclusion given the deviation from
thermal equilibrium. Furthermore, we can interpret each of the three parameters
as physical characteristics of a non-equilibrium system. The success of our
compact description within the Standard Model motivates its use also in BSM
scenarios. We demonstrate how observations of primordial light element
abundances can be used to place constraints on the C$\nu$B energy spectrum,
deriving response functions that can be applied for general C$\nu$B spectral
distortions. Combined with the description of those deviations that we develop
here, our methods provide a convenient and powerful framework to constrain the
impact of BSM physics on the C$\nu$B.
