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In this paper we present detailed M\"ossbauer spectroscopy study ofstructural and magnetic properties of the undoped parent compoundCaFe$_2$As$_2$ single crystal. By fitting the temperature dependence of thehyperfine magnetic field we show that the magneto-structural phase transitionis clearly first-order in nature and we also deduced the compressibility of oursample to be $1.67\times10^{-2}\,GPa^{-1}$. Within the Landau's theory of phasetransition, we further argue that the observed phase transition may stem fromthe strong magneto-structural coupling effect. Temperature dependence of theLamb-M\"ossbauer factor show that the paramagnetic phase and theantiferromagnetic phase exhibit similar lattice dynamics in high frequencymodes with very close Debye temperatures, $\Theta_D \sim$270\,K.

Magneto-structural coupling and harmonic lattice dynamics in CaFe$_2$As$_2$ probed by M\"ossbauer spectroscopy