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To see that the compressibility sum rule is saturated by these contributions from first and second sound, we note that Z$_{1}$/u 2 1 + Z$_{2}$/u 2 = 1 /v 2 $_{T}$, as can be seen from (14). 

Since we are working in the low frequency two-fluid hydrodynamic region, we can use the fact that ω ≪ T . In this limit, the detailed balance factor in (19) can be approximated by 

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The first quantum correction, given by the 1 / 2 term in (21) in this expansion, is needed to satisfy the sum rules previously discussed. This approximation ( ω ≪ T ) is often referred to as the “classical limit” in textbooks discussing the dynamic structure factor, although in fact it describes both superfluid ( T < T$_{c}$ ) and normal fluid ( T > T$_{c}$ ) collisional hydrodynamics. 

Keeping only the leading order first term in (21), (19) simplifies to 

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This expression is valid for Bose and Fermi superfluid gases as well as superfluid $^{4}$He, that is, all superfluids described by an order parameter with a phase and amplitude. Equation (22) shows that in the low-frequency two-fluid hydrodynamic region, W$_{i}$ ≡ Z$_{i}$/u 2 i gives the weight of the i th mode. In contrast, the weight of the i th mode in Bragg scattering [see (4) and (17)] is given by B$_{i}$ , where 

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Comparing (22) with (17), we see that the hydrodynamic limit ( ω ≪ T ) of S ( q, ω ) involves an extra factor of 1 /ω . 

We see that while Im χ$_{nn}$ ( q, ω ) and Im χ$_{nn}$ ( q, ω ) /ω share the same poles, the relative weights of first and second sound are quite different in these two functions, with B$_{2}$/B$_{1}$ = ( W$_{2}$/W$_{1}$ )( u$_{2}$/u$_{1}$ ) generally being much smaller than W$_{2}$/W$_{1}$ due to the smallness of u$_{2}$/u$_{1}$ (see Fig. 1). 

The two-fluid expression in (22) is known from earlier discussions of Brillouin scattering in superfluid $^{4}$He (see section IV). In contrast to high-frequency inelastic neutron scattering, Brillouin scattering is carried out with light in the visible part of the spectrum ( ω ≪ T ). As pointed out above [see (22)], this means that Brillouin scattering measures the imaginary 

