Probing the small-scale primordial power spectrum via relic neutrinos and acoustic reheating

We show that the dissipation of small-scale perturbations through diffusion damping after neutrino decoupling lowers the present-day neutrino temperature compared to the expected value of $1.96\text{K}$. This reduces the relic neutrino abundance by an amount controlled by the integral of the primordial curvature power spectrum ${\mathrm{\Delta }}_{\mathcal{R}}^2(k)$. We find that a relic neutrino detection by PTOLEMY can set limits ${\mathrm{\Delta }}_{\mathcal{R}}^2(k)\lesssim \mathcal{O}(0.1)$ on scales $k\lesssim 3\times {10}^5{\text{Mpc}}^{-1}$, complementary to limits from Big Bang Nucleosynthesis, spectral distortions, pulsar timing arrays, and future dark ages 21-cm observations.