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 FoM= 1 / \sqrt { { det } { Cov } ( f _ { 1 } ,f _ { 2 } ,f _ { 3 } , ... ) } , where { Cov } ( f _ { 1 } ,f _ { 2 } ,f _ { 3 } , ... ) 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 _ { * } ) , \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 \pm 8 ( km/s ) Mpc ^ { -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 } -2 w _ { 0 } +3 ( w _ { 0 } - w _ { 0.5 } ) a ] is significantly smaller than that of ( w _ { 0 } ,w _ { a } ) [ with w _ { X } ( a ) = w _ { 0 } + ( 1 - a ) w _ { a } ] .
In order to obtain model-independent constraints on dark energy , we parametrize the dark energy 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 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 .