## 2 Measurements of the _THz_ Spectrum of Noble Gases up to the 1990s  
### The Spectral Line-Shape in the Continuous Limit  
As mentioned in the introductory section, the variable of interest for this review is the spectrum of density fluctuations, or dynamic structure factor:  
\[S(Q,\omega)=\int_{V}d\overrightarrow{r}\int_{-\infty}^{+\infty}\left\langle \delta\rho(\overrightarrow{r},t)\delta\rho^{*}(\overrightarrow{r},0)\right\rangle exp \left[i(\overrightarrow{Q}\cdot\overrightarrow{r}-\omega t)\right]dt \tag{1}\]  
with \(\delta\rho(\overrightarrow{r},t)\) being the space \((\overrightarrow{r})\) and time (_t_) dependent density fluctuation of the target sample having volume \(V\). In the quasi-macroscopic regime, \(S(Q,\omega)\) can be described using the hydrodynamic theory of continuous media [3]. This approach stems from the formal expression of the conservation laws of the (density of) mass, momentum and energy of the fluid plus two constitutive laws: the Navier-Stokes' and heat transport's equations. As extensively discussed in several monographies [3; 22], the derived spectral shape is dominated by few long-lived collective modes, customarily referred to as hydrodynamic modes. These modes form a triplet well approximated by the following expression:\[S\left(Q,\omega\right)=\frac{S(Q)}{\pi}\left\{\frac{A_{h}z_{h}}{ \omega^{2}+z_{h}^{2}}+A_{S}\left[\frac{z_{S}}{\left(\omega-\omega_{S}\right)^{2 }+z_{S}^{2}}+\frac{z_{S}}{\left(\omega+\omega_{S}\right)^{2}+z_{S}^{2}}\right] \right.+\] \[\left.+A_{S}b\left[\frac{\left(\omega-\omega_{S}\right)}{\left( \omega-\omega_{S}\right)^{2}+z_{S}^{2}}-\frac{\left(\omega+\omega_{S}\right)} {\left(\omega+\omega_{S}\right)^{2}+z_{S}^{2}}\right]\right\}\]  
with \(b=\left[A_{h}z_{h}/\left(1-A_{h}\right)+z_{s}\right]/\omega_{s}\) and \(I_{s}=\left(1-I_{h}\right)/2\). One readily recognizes the three main components of the spectral shape (from left):  
1. The heat diffusion (Rayleigh) central peak, which relates to entropy fluctuations diffusing at constant pressure (\(P\)). 2. The two Brillouin acoustic side peaks, connected to \(P\)-fluctuations propagating at constant entropy. 3. An additional contribution (term \(\propto b\)) asymmetric around the Brillouin peaks position having negative tails. This term distorts the Lorentzian terms 1) and 2) ultimately enabling the convergence of the spectral moments \(\int_{-\infty}^{+\infty}\omega^{n}S(Q,\omega)d\omega\), with \(n\leq 2\). The hydrodynamic profile in Equation (2) contains shape parameters whose \(Q\)-dependence is not explicitly considered here. However, at low \(Q\)s the latter can be expressed through a polynomial \(Q\)-expansion (see, e.g., Equations (42)-(45) of [23]), which at the lowest order reads:  
\[\omega_{s}=c_{s}Q \tag{3a}\] \[z_{s}=\left[\left(\gamma-1\right)D_{T}+v_{L}\right]Q^{2}/2\] (3b) \[z_{h}=D_{T}Q^{2} \tag{3c}\]  
Here \(c_{s}\), \(D_{T}\) and \(v_{L}\) are, respectively, the adiabatic sound velocity the thermal diffusivity and the longitudinal kinematic viscosity. It should be recognized that Equation (3a-c) are valid only if \(D_{T}Q^{2},\Gamma Q^{2}<<c_{s}Q\), that is if the lifetime of hydrodynamic modes is much longer than the acoustic period. Under these conditions, the hydrodynamic modes are long-lived excitations appearing in the spectrum as well-resolved and sharp peaks, customarily referred to as Rayleigh-Brillouin triplet. Experimental methods such as Brillouin light scattering (\(BLS\)) provide a measurement of Rayleigh-Brillouin triplet by typically spanning the \(10^{-3}\)-\(10^{-2}\) nm\({}^{-1}\)\(Q\)-range. An example of the simplified hydrodynamic spectrum is presented in Figure 1 along with the three separate components discussed above. Figure 1: Typical shape of the Rayleigh-Brillouin triplet measured by Brillouin light scattering. The separate contributions to the total shape are represented by line of different color, as indicated in the legend.