Determining the Equation of State of the Expanding Universe Using a New Independent Variable

To determine the equation of state of the universe, we propose to use a newindependent variable $R\equiv (H_0/c)(d_L(z)/(1+z))$, where $H_0$ and $d_L(z)$are the present Hubble parameter and the luminosity distance, respectively. Forthe flat universe suggested from the observation of the anisotropy of cosmicmicrowave background, the density and the pressure are expressed as$\rho/\rho_0=4(df/dR)^2/f^6$ and $p/\rho_0=-4/3(d^2f/dR^2)/f^5$ where $\rho_0$is the present density and $f(R)=1/\sqrt{1+z(R)}$. In $(R, f)$ plane the signas well as the strength of the pressure is in proportion to the curvature ofthe curve $f(R)$. We propose to adopt a Pade-like expression of$f(R)=1/\sqrt{u}$ with $u\equiv 1+\sum\limits_{n=1}^{N}u_nR^n$. For flat$\Lambda$ model the expansion up to N=7 has at most an error $< 0.2%$ for $z <1.7$ and any value of $\Lambda$. We also propose a general method to determinethe equation of state of the universe which has $N-1$ free parameters. If thenumber of parameters are smaller than $N-1$, there is a consistency check ofthe equation of state so that we may confirm or refute each model.Comment: 12 pages, to be published in the Astrophysical Journa