Charge density wave order and fluctuations above T$_{CDW}$ and below superconducting T$_c$ in the kagome metal CsV$_{3}$Sb$_{5}$

The phase diagram of the kagome metal family \ch{AV3Sb5} (A = K, Rb, Cs) features both superconductivity and charge density wave (CDW) instabilities, which have generated tremendous recent attention. Nonetheless, significant questions remain. In particular, the temperature evolution and demise of the CDW state has not been extensively studied, and little is known about the co-existence of the CDW with superconductivity at low temperatures. We report an x-ray scattering study of \ch{CsV3Sb5} over a broad range of temperatures from 300 K to $\sim$ 2 K, below the onset of its superconductivity at $\textit{T}_\text{c}\sim$ 2.9 K. Order parameter measurements of the $2\times2\times2$ CDW structure show an unusual and extended linear temperature dependence onsetting at $T^*$ $\sim$ 160 K, much higher than the susceptibility anomaly associated with CDW order at $\textit{T}_\text{CDW}=94$ K. This implies strong CDW fluctuations exist to $\sim1.7\times\textit{T}_\text{CDW}$. The CDW order parameter is observed to be constant from $T=16$ K to 2 K, implying that the CDW and superconducting order co-exist below $\textit{T}_\text{c}$, and, at ambient pressure, any possible competition between the two order parameters is manifested at temperatures well below $\textit{T}_\text{c}$, if at all. Anomalies in the temperature dependence in the lattice parameters coincide with $\textit{T}_\text{CDW}$ for $\textit{c}(T)$ and with $T^*$ for $a(T)$.