Figure of merit for dark energy constraints from current observational data

In order to make useful comparisons of different dark energy experiments, it is important to choose the appropriate figure of merit (FoM) for dark energy constraints. Here we show that for a set of dark energy parameters ${{f}_{i}}$, it is most intuitive to define $\mathrm{FoM}=1/\sqrt{\mathrm{det}\mathrm{Cov}({f}_{1},{f}_{2},{f}_{3},\dots{})}$, where $\mathrm{Cov}({f}_{1},{f}_{2},{f}_{3},\dots{})$ is the covariance matrix of ${{f}_{i}}$. In order for this FoM to represent the dark energy constraints in an optimal manner, the dark energy parameters ${{f}_{i}}$ should have clear physical meaning and be minimally correlated. We demonstrate two useful choices of ${{f}_{i}}$ using 182 SNe Ia (from the HST/GOODS program, the first year Supernova Legacy Survey, and nearby SN Ia surveys), $[R({z}_{*}),{l}_{a}({z}_{*}),{\ensuremath{\Omega}}_{b}{h}^{2}]$ from the five year Wilkinson Microwave Anisotropy Probe observations, and Sloan Digital Sky Survey measurement of the baryon acoustic oscillation scale, assuming the Hubble Space Telescope prior of ${H}_{0}=72\ifmmode\pm\else\textpm\fi{}8\text{ }\text{ }(\mathrm{km}/\mathrm{s})\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$, and without assuming spatial flatness. We find that for a dark energy equation of state linear in the cosmic scale factor $a$, the correlation of $({w}_{0},{w}_{0.5})$ [${w}_{0}={w}_{X}(z=0)$, ${w}_{0.5}={w}_{X}(z=0.5)$, with ${w}_{X}(a)=3{w}_{0.5}\ensuremath{-}2{w}_{0}+3({w}_{0}\ensuremath{-}{w}_{0.5})a$] is significantly smaller than that of $({w}_{0},{w}_{a})$ [with ${w}_{X}(a)={w}_{0}+(1\ensuremath{-}a){w}_{a}$]. In order to obtain model-independent constraints on dark energy, we parametrize the dark energy density function $X(z)={\ensuremath{\rho}}_{X}(z)/{\ensuremath{\rho}}_{X}(0)$ 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 dark energy equation of state, current observational 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 dark energy 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 dark energy parameter space to either enable the discovery of dark energy evolution, or the conclusive evidence for a cosmological constant.