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At low temperature, a normal gas of unpaired spin-1/2 fermions is one of the cleanest realizations of a Fermi liquid. It is described by Landau's theory, where no phenomenological parameters are needed as the quasiparticle interaction function can be computed perturbatively in powers of the scattering length $a$, the sole parameter of the short-range interparticle interactions. Obtaining an accurate solution of the transport equation nevertheless requires a careful treatment of the collision kernel, as the uncontrolled error made by the relaxation time approximations increases when the temperature $T$ drops below the Fermi temperature. Here, we study sound waves in the hydrodynamic regime up to second order in the Chapman-Enskog's expansion. We find that the frequency $\omega_q$ of the sound wave is shifted above its linear depart as $\omega_q=c_1 q(1+\alpha q^2\tau^2)$ where $c_1$ and $q$ are the speed and wavenumber of the wave and the typical collision time $\tau$ scales as $1/a^2T^2$. Besides the shear viscosity, the coefficient $\alpha$ is described by a single second-order collision time which we compute exactly from an analytical solution of the transport equation, resulting in a positive dispersion $\alpha>0$. Our results suggest that ultracold atomic Fermi gases are an ideal experimental system for quantitative tests of second order hydrodynamics., Comment: 7 pages, 3 figures

Landau's Fermi-liquid (FL) theory has been successful at the phenomenological description of the normal phase of many different Fermi systems. Using a dilute atomic Fermi fluid with tunable interactions, we investigate the microscopic basis of Landau's theory with a system describable from first principles. We study transport properties of an interacting Fermi gas by measuring its density response to a periodic external perturbation. In an ideal Fermi gas, we measure for the first time the celebrated Lindhard function. As the system is brought from the collisionless to the hydrodynamic regime, we observe the emergence of sound, and find that the experimental observations are quantitatively understood with a first-principle transport equation for the FL. When the system is more strongly interacting, we find deviations from such predictions. Finally, we observe the shape of the quasiparticle excitations directly from momentum-space tomography and see how it evolves from the collisionless to the collisional regime. Our study establishes this system as a clean platform for studying Landau's theory of the FL and paves the way for extending the theory to more exotic conditions, such as nonlinear dynamics and FLs with strong correlations in versatile settings.