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In figure 1, we show the temperature dependence of u$_{1}$ and u$_{2}$ and compare these results with the leading order expressions c$_{1}$ and c$_{2}$ given by (25) (shown by the dashed lines). The thermodynamic functions needed to obtain these sound speeds are calculated using the microscopic theory of Nozi` eres and Schmitt-Rink (NSR) [25, 26]. We see that c$_{1}$ and c$_{2}$ are extremely good approximations at all temperatures. At temperatures T ≳ 0 . 4 T$_{c}$ , x = c 2 $_{2}$/c 2 1 is very small, even though ϵ$_{LP}$ ∼ O (1). At lower temperatures, while x is no longer small ( x = 1 / 3 at T = 0), ϵ$_{LP}$ becomes extremely small. Thus in both limits, we find that the correction term xϵ$_{LP}$ in (24) is negligible. 

The fact that u$_{1}$ ≃ c$_{1}$ and u$_{2}$ ≃ c$_{2}$ in a fluid with small thermal expansion such as the uni- tary Fermi gas means that first and second sound propagate at essentially the same velocities as pure density and temperature oscillations, as would arise when ϵ$_{LP}$ = 0. Surprisingly, this does not mean that first and second sound are uncoupled density and temperature oscillations. We discuss this in detail in B. 

