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Viscosity spectral functions of the dilute Fermi gas in kinetic theory

We compute the viscosity spectral function of the dilute Fermi gas for different values of the s-wave scattering length $a$, including the unitarity limit $a\to\infty$. We perform the calculation in kinetic theory by studying the response to a non-trivial background metric. We find the expected structure consisting of a diffusive peak in the transverse shear channel and a sound peak in the longitudinal channel. At zero momentum the width of the diffusive peak is $ω_0\simeq (2ε)/(3η)$ where $ε$ is the energy density and $η$ is the shear viscosity. At finite momentum the spectral function approaches the collisionless limit and the width is of order $ω_0\sim k(T/m)^{1/2}$.