## 4 Model Simulations
As an example, we demonstrate this for the experimental data of the polystyrene sample without aluminum. This sample was considered as one with the lowest noise level in our experiment. The time interval and increment of the simulated signal \(\Delta q(\tau)/q_{0}\) were chosen the same as in our pump-probe experiment. One of the typical realizations of the \(\Delta q(\tau)/q_{0}\) with the signal level \(S=0.7\times 10^{-3}\) and noise level \(N=1.4\times 10^{-3}\) is shown in Figure 9a. From this figure it is difficult to conclude on the presence or absence of the signal. \begin{table}
\begin{tabular}{c c c c} \hline  & **PS without Al** & **PS with Al** & **Silica** \\ \hline Sphere diameter, nm & 420 & 376 & 238 \\ Longitudinal sound velocity \(c_{L}\), m/s & 2350 & 2350 & 5950 \\ Transverse sound velocity \(c_{T}\), m/s & 1210 & 1210 & 3760 \\ Frequency of the breathing mode, GHz & 4.97 & 5.54 & 19.06 \\ Noise level \(N\times 10^{-3}\) & 1.4 & 1.8 & 1.8 \\ Relative deviation in interparticle distance \(\Delta d/d_{0}\times 10^{-3}\) & 0.7 & 0.9 & 0.9 \\ Deviation in interparticle distance \(\Delta d\), nm & 0.29 & 0.35 & 0.22 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of PS and silica nanoparticles used in our model. The corresponding frequencies of the first spheroidal mode of colloidal particles were obtained from reference [30]. The noise level was derived from the experimental data. Relative and absolute deviation in the interparticle distance (\(\Delta d/d_{0}\) and \(\Delta d\)), which could be detected in our experiment with the probability of 60%, were calculated from the experimental data and simulations. Similar to the analysis of the experimental data, the Fourier spectra were calculated for the simulated signal with the noise (see Figure 9b). Next, the behavior of the Fourier component amplitude (\(A_{\omega}\)) corresponding to the first spheroidal mode frequency \(\omega\) = 4.97 GHz was studied. The criterion of detecting the signal was considered as:  
\[A_{\omega}>\overline{A}+3\sigma \tag{6}\]  
where \(\overline{A}\) is the mean and \(\sigma\) is the standard deviation of all Fourier components in the spectrum. The Fourier spectrum of the simulated data (see Figure 9a), as well as its \(3\sigma\) level are shown in Figure 9b. It can be seen that the contribution of the vibrational mode in the Fourier spectrum is higher than the criterion stipulated by Equation (6). Therefore, the algorithm detected the presence of the 4.97 GHz frequency signal. We calculated the probability of successful signal detection as a function of signal-to-noise ratio \(S/N\) in the range of \((S/N)\) from 0 to 1.5 with 0.01 increment. During the simulations the amplitude of the signal \(S\) was changing, while the noise level \(N\) was kept constant. The probability of detecting signal \(P(S/N)\) was defined as:  
\[P(S/N)=\underset{n\rightarrow\infty}{lim}\bigg{(}\frac{n_{detected}(S/N)}{n} \bigg{)} \tag{7}\]  
where \(n_{detected}(S/N)\) is a number of simulations with successful detection of signal (according to the criterion stipulated by Equation (6)) and \(n\) = 10,000 is the total number of simulations for each \(S/N\) step. Results of these simulations for the experimental data of the polystyrene sample without aluminum are shown in Figure 10. As one can see from Figure 10 that, according to our criterion stipulated by Equation (6), the signal is 100% detectable if it is on the level of the noise (\(S/N\) = 1). Figure 9: (**a**) Simulation of the signal \(\Delta q(\tau)/q_{0}\) and (**b**) Fourier transform of this signal. The average value of all Fourier components is indicated by red line. The dashed red line corresponds to level of \(3\sigma\) above the average value, where \(\sigma\) is a standard deviation and is determined from the distribution of Fourier components.