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 new independent variable R = (H-0/c)[d(L)(z)/(1 + z)], where H-0 and d(L)(z) a re the present Hubble parameter and the luminosity distance, respectively. For the flat universe suggested by the observation of the anisotropy of cos mic microwave backgrounds, the density and the pressure are expressed as rh o/rho (0) = 4(df/dR)(2)/f(6) and p/rho (0) = -4/3(d(2)f/dR(2))/f(5), where rho (0) is the present density and f(R) = 1/[1 + z(R)](1/2). In the (R,f) p lane the sign as well as the strength of the pressure is in proportion to t he curvature of the curve f(R). We propose to adopt a Pade-like expression of f(R) = 1/u(1/2) with u = 1 + Sigma (N)(n=1) u(n)R(n). For the flat Lambd a models, the expansion up to N = 7 has at most an error less than 0.2% for z < 1.7 and any value of <Lambda>. We also propose a general method to det ermine the equation of state of the universe that has N - 1 free parameters . If the number of parameters are smaller than N - 1, there is a consistenc y check of the equation of state so that we may confirm or refute each mode l.