Figure of Merit for Dark Energy Constraints from Current Observational   Data
  Choosing the appropriate figure of merit (FoM) for dark energy (DE) constraints is key in comparing different DE experiments. Here we show that for a set of DE parameters {f_i}, it is most intuitive to define FoM = 1/\sqrt{Cov(f1,f2,f3,...)}, where Cov(f1,f2,f3,...) is the covariance matrix of {f_i}. The {f_i} should be minimally correlated. We demonstrate two useful choices of {f_i} using 182 SNe Ia (compiled by Riess et al. 2007), [R(z_*), l_a(z_*), \Omega_b h^2] from the five year Wilkinson Microwave Anisotropy Probe (WMAP) observations, and SDSS measurement of the baryon acoustic oscillation (BAO) scale, assuming the HST prior of H_0=72+/-8 km/s Mpc^{-1} and without assuming spatial flatness. We find that the correlation of (w_0,w_{0.5}) [w_0=w_X(z=0), w_{0.5}=w_X(z=0.5), w_X(a) = 3w_{0.5}-2w_0+3(w_0-w_{0.5})a] is significantly smaller than that of (w_0,w_a) [w_X(a)=w_0+(1-a)w_a]. In order to obtain model-independent constraints on DE, we parametrize the DE density function X(z)=\rho_X(z)/\rho_X(0) as a free function with X_{0.5}, X_{1.0}, and X_{1.5} [values of X(z) at z=0.5, 1.0, and 1.5] as free parameters estimated from data. If one assumes a linear DE equation of state, current data are consistent with a cosmological constant at 68% C.L. If one assumes X(z) to be a free function parametrized by (X_{0.5}, X_{1.0}, X_{1.5}), current data deviate from a cosmological constant at z=1 at 68% C.L., but are consistent with a cosmological constant at 95% C.L.. Future DE experiments will allow us to dramatically increase the FoM of constraints on (w_0,w_{0.5}) and of (X_{0.5}, X_{1.0}, X_{1.5}). This will significantly shrink the DE parameter space to enable the discovery of DE evolution, or the conclusive evidence for a cosmological constant. 
Yun Wang