Model-free interpretation of X-ray Thomson scattering measurements

X-ray Thomson scattering (XRTS) has emerged as a widely used diagnostics for extreme states of matter in a great variety of situations, and over a broad range of parameters. The standard approach for the interpretation of XRTS measurements is given by the forward modeling approach, where the electronic dynamic structure factor $S_{ee}(\mathbf{q},\omega)$ is computed from a suitable theoretical model, convolved with the combined source-and-instrument function, and then matched with the experimental observation, treating a-priori unknown parameters such as the mass density, temperature and ionization state as free fit parameters. Very recently, it has been suggested that this inherent model dependence can be avoided by analyzing XRTS spectra in terms of the imaginary-time correlation function (ITCF) $F_{ee}(\mathbf{q},\tau)$ [Dornheim \textit{et al.}, \textit{Nature Commun.}~\textbf{13}, 7911 (2022)], giving one model-free access to the temperature, normalization, Rayleigh weight, as well as a number of other properties. Here, we present a comprehensive review article on these developments, including accessible discussions of the method's theoretical background in terms of Feynman's imaginary-time path integral picture of statistical mechanics as well as its remaining limitations, in particular with respect to the source-and-instrument function of the experimental set-up. In addition, we discuss new chances for the further development of this framework by utilizing emerging capabilities for high-repetition XRTS experiments with meV resolution over spectral ranges of tens of eV at state-of-the-art x-ray free electron laser (XFEL) facilities such as the European XFEL in Germany.