The Luminosity Distance, the Equation of State, and the Geometry of the Universe

The second derivative of the luminosity distance with respect to the redshift is written in terms of the deceleration parameter $q_0$. We point out that the third derivative contains the information regarding the sound speed of cosmic matter as well as the curvature of the universe. We restrict physically possible parameter ranges of the coefficients. It is found that there is a relation between the coefficients in a flat universe model with matter such that $c_{s0}(1+w_{\rm x0})=0$ ($c_{s0}$ is the total sound speed of the matter component and $w_{\rm x0}$=$p_{\rm x0}/\rho_{\rm x0}$).