One primary assumption in this derivation is the equivalence of the frequency-dependent viscosity to a so-called memory function:
where η(s) is the Laplace frequency-dependent viscosity and ς(s) is the memory function. As a special case example, ς(s) is a delta-function at s=0 for purely viscous fluids since they do not store energy (i.e. they do not possess any elasticity). Substituting this into the well-known Stokes-Einstein equation leads to a relation between the colloidal particles' mean-squared-displacement (measured by dynamic light scattering) and the complex shear modulus of the fluid, G*(ω) (after conversion to the Fourier frequency domain):
The authors note in the end of the paper that it's unknown why light scattering techniques should produce the shear modulus of the fluid since they measure elements along the diagonal of the system's linear response tensor, whereas the shear moduli are contained in the off-diagonal elements.
They also note (with explanations I don't quite understand) that "...the light scattering may not provide a quantitatively exact measure of the elastic moduli; nevertheless, as our results show, the overall trends are correctly captured, and the agreement is very good." (emphasis mine)
