Determining the Equation of State of the Expanding Universe -Inverse Problem in Cosmology-

Even if the luminosity distance as a function of redshift is obtained accurately using, for example, Type Ia supenovae, the equation of state of the Universe cannot be determined uniquely but it depends on one free parameter $\Omega_{k0} ={k}/(a_0^2H_0^2)$ where $a_0$ and $H_0$ are the present scale factor and the Hubble parameter, respectively. This degeneracy might be resolved if, for example, the time variations of the redshift of quasars are measured as proposed recently by Loeb. Therefore the equation of state of the Universe (or the metric of the universe model) might be determined without any theoretical assumption on the matter content of the Universe in future.