  Light thermal relics of the hot big bang, often quantified by the parameter
$N_\mathrm{eff}$, are one of the primary targets of cosmological measurements.
At present, the energy density in such relics is constrained to be less than
ten percent of the total energy density in radiation. Upcoming cosmic microwave
background (CMB) experiments, however, have the potential to measure the
radiation density at the one-percent level, which is close to well-motivated
theoretical targets. In this paper, we explore to what degree the CMB
observations can be enhanced by future large-scale structure surveys. We
carefully isolate the information encoded in the shape of the galaxy power
spectrum and in the spectrum of baryon acoustic oscillations (BAO). We find
that measurements of the shape of the power spectrum can significantly improve
on current and near-term CMB experiments. We also show that the phase shift of
the BAO spectrum induced by relic neutrinos can be detected at high
significance in future experiments.
