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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.

Future dark energy experiments that are significantly more ambitious than current ones are required to illuminate the nature of dark energy. In order to compare proposed future dark energy experiments in a useful manner, we need to choose the appropriate figure of merit (FoM) for dark energy constraints [50].
In this paper, we explore the optimization of FoM using current observational data from supernovae, galaxy clustering, and cosmic microwave background anisotropy (CMB) data. We describe our method in Sec.II, present our results in Sec.III, and conclude in Sec.IV.

II. METHOD
A. General definition for figure of merit When we estimate a set of parameters, {f i } (i=1, 2, ..., N ), from data, the most intuitive figure of merit is the N -dimensional volume enclosed by the 68% or 95% confidence level (C.L.) contours of the parameters. If the likelihood surfaces for all the parameters are Gaussian, this N -dimensional volume is proportional to the squre root of the covariance matrix of {f i }, det Cov(f 1 , f 2 , f 3 , ...). For N = 2, the 68% or 95% C.L. contours of f 1 and f 2 are ellipses, with enclosed area given by π det Cov(f 1 , f 2 ) multiplied by 2.30 or 6.17. Parametrizing dark energy equation of state as w X (a) = w 0 + (1 − a)w a [51], the Dark Energy Task Force (DETF) defined FoM to be the inverse of the area enclosed by the 95% C.L. contour of (w 0 , w a ) [52], i.e., where w p = w 0 − w a δw 0 δw a / δw 2 a , and σ(w i ) = δw 2 i . Note that σ(w a )σ(w p ) = det Cov(w 0 , w a ), thus the conversion to w p is not needed to calculate the FoM. For real data, the likelihood surfaces for the parameters {f i } are almost always non-Gaussian at the 95% C.L., thus defining the FoM as enclosed area or volume by the 95% C.L. contours of {f i } becomes problematic. We propose the definition for a relative generalized FoM given by where {f i } are the chosen set of dark energy parameters. This definition has the advantage of being easy to calculate for either real or simulated data. We have streamlined the definition to omit numerical factors since what matters is the relative FoM between different experiments.
Note that while this FoM definition has an intuitive physical interpretation, it rewards experiments that yield very correlated estimates of the dark energy parameters. This is especially true in applying the DETF FoM, since (w 0 , w a ) are always highly correlated. Hence the dark energy FoM [as defined in Eq. (2)] is most meaningful when the dark energy parameters {f i } are chosen such that they are minimally correlated with each other.

B. Dark energy parametrization
We study constraints on a 2-parameter dark energy equation of state w X (z) linear in a, as well the dark energy density function X(z) ≡ ρ X (z)/ρ X (0) as a free function at z ≤ 1.5.
The 2-parameter w X (z) is given by where w 0 = w X (z = 0), and w c = w X (z = z c ). Eq.(3) corresponds to a dark energy density function If we choose a c = 1 + σ 2 (w 0 )/σ 2 (w 0 w a ), then (w 0 , w c ) are uncorrelated. For current data, z c ∼ 0.3. Choosing a c to make (w 0 , w c ) uncorrelated has the disadvantage that a c is different for different data sets.
We recommend choosing z c = 0.5; it is sufficiently close to z c ∼ 0.3 that the correlation of w 0 and w 0.5 = w X (z = 0.5) is relatively small. It is straightforward to show that if |σ 2 (w 0 w a )/[σ(w 0 )σ(w a )]| < 1, (w 0 , w c ) are less correlated than (w 0 , w a ) if This is always satisfied for z c = 0.5. Choosing z c = 0.5, the correlation of (w 0 , w 0.5 ) is smaller than that of (w 0 , w a ) by about a factor of 2 for the combined SNe, BAO, and CMB data considered in this paper. Fixing z c has the significant advantage of allowing the comparison of the same dark energy property for different data sets. For our results for the 2-parameter dark energy model, we use Eq.(3) with a c = 1/(1 + 0.5) = 2/3 (z c = 0.5). Thus In order to obtain model-independent constraints on dark energy, we parametrize the dark energy density function X(z) = ρ X (z)/ρ X (0) as a free functions 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. At z > 1.5, we choose either X(z) = X 1.5 , or X(z) = X 1.5 e α(z−1.5) (with α as an additional parameter to be estimated from data). Our results are insensitive to the assumption about X(z) at z > 1.5 (other than that dark energy becomes insignificant at z > 1.5). As more data become available at z > 1.5, we can include X 2.0 = X(z = 2.0), X 2.5 = X(z = 2.5), and X 3.0 = X(z = 3.0) as estimated parameters, as well as inserting more estimated X(z) values at z < 1.5. Early dark energy (significant at high z) is not required by current data, and leads to contradiction with observed cosmic structure formation [69], unless a cutoff is imposed.
The constraints on X 0.5 , X 1.0 , and X 1.5 are insensitive to the interpolation used in deriving X(z) elsewhere. The simplest smooth interpolation is given by a polynomial: where z = z/1.5.
Wang & Mukherjee 2007 [56] showed that the CMB shift parameters together with Ω b h 2 , provide an efficient summary of CMB data as far as dark energy constraints go. We use the covariance matrix of [R(z * ), l a (z * ), Ω b h 2 ] from the five year WMAP data (Table 11 of [57]), with z * given by fitting formulae from Hu & Sugiyama (1996) [58]: where CMB data are included in our analysis by adding the following term to the χ 2 of a given model with p 1 = R(z * ), p 2 = l a (z * ), and p 3 = Ω b h 2 : where p data i are the maximum likelyhood values given in Table 10 of [57].
SN Ia data give the luminosity distance as a function of redshift, d L (z) = (1 + z) r(z). We use 182 SNe Ia from the HST/GOODS program [59] and the first year SNLS [60], together with nearby SN Ia data, as compiled by [59]. We do not include the ESSENCE data [61], as these are not yet derived using the same method as thosed used in [59]. Combining SN Ia data derived using different analysis techniques leads to systematic effects in the estimated SN distance moduli [61,62]. Appendix A of [56] describes in detail how we use SN Ia data (fluxaveraged to reduce lensing-like systematic effects [62,63,64] and marginalized over H 0 ) in this paper.
We also use the SDSS baryon acoustic oscillation (BAO) scale measurement by adding the following term to the χ 2 of a model: where A is defined as and A BAO = 0.469 (n S /0.98) −0.35 , σ A = 0.017, and z BAO = 0.35 (independent of a dark energy model) [65]. We take the scalar spectral index n S = 0.96 as measured by WMAP five year observations [57]. For Gaussian distributed measurements, the likelihood function L ∝ e −χ 2 /2 , with where χ 2 CMB is given in Eq.(16), χ 2 SN e is given in Appendix A of [56], and χ 2 BAO is given in Eq. (17). The current BBN constraints [66], S = 0.942 ± 0.030 (N ν = 2.30 +0. 35 −0.34 ) rule out the standard model of particle physics (S = 1, N ν = 3) at 1σ [66]. Given the uncertainties involved in deriving the BBN constraints, we relax the standard deviation of S by a factor of two, so that the standard model of particle physics is allowed at 1σ. We find that the resultant BBN constraints do not have measurable effect on our dark energy constraints.
For all the dark energy constraints from combining the different data sets presented in this paper, we marginalize the SN Ia data over H 0 in flux-averaging statistics [56], and impose a prior of H 0 = 72 ± 8 (km/s)Mpc −1 from the HST Cepheid variable star observations [67].
We run a Monte Carlo Markov Chain (MCMC) based on the MCMC engine of [68] to obtain O(10 6 ) samples for each set of results presented in this paper. The parameters used are (Ω k , Ω m , h, Ω b h 2 , p DE ). The dark energy parameter set is described in Sec.IIB. We assumed flat priors for all the parameters, and allowed ranges of the parameters wide enough such that further increasing the allowed ranges has no impact on the results. The chains typically have worst e-values (the variance(mean)/mean(variance) of 1/2 chains) much smaller than 0.005, indicating convergence. The chains are subsequently appropriately thinned to ensure independent samples. Fig.1 shows the 68% and 95% C.L. contours of (w 0 , w 0.5 ) (upper panel) and (w 0 , w a ) (lower panel) from WMAP 5 year measurement of [R(z * ), l a (z * ), Ω b h 2 ], and 182 SNe Ia (from the HST/GOODS program, the first year Supernova Legacy Survey, and nearby SN Ia surveys), with and without the SDSS measurement of the baryon acoustic oscillation (BAO) scale. We have assumed the HST prior of H 0 = 72 ± 8 (km/s)Mpc −1 , and allowed Ω k to vary. Table 1 shows the mean, rms variance, and correlation coefficients of (w 0 , w 0.5 ) and (w 0 , w a ), as well as the relative dark energy FoM r defined in Eq. (2). Note that Pearson's correlation coefficient ρ xy = σ 2 (xy)/[σ(x)σ(y)]. Adding the SDSS BAO scale measurement improves the FoM r by a factor of 21.5 for (w 0 , w 0.5 ), and by a factor of 27.0 for (w 0 , w a ). Since (w 0 , w 0.5 ) are significantly less correlated than (w 0 , w a ), the improvement factor in FoM r of (w 0 , w 0.5 ) is a more reliable indicator of the impact of adding the SDSS BAO scale measurement. Fig.2 shows the one dimensional marginalized probability distributions (pdf) of (Ω m , h,  [56].

III. RESULTS
We find that in spite of the different pdfs for Ω b h 2 , using the [R(z * ), l a (z * ), Ω b h 2 ] and [R(z * ), l a (z * ), z * ] measurements give about the same constraints on (Ω m , h, Ω k , w 0 , w a ). Using the [R(z * ), l a (z * ), Ω b h 2 ] measurement with z * fixed at 1090.4 gives slightly tighter constraints on (w 0 , w a ). In combination with the supernova and BAO data, the WMAP 5 year data improve constraints on (w 0 , w a ) slightly compared to the WMAP 3 year data, while tightening the constraints on Ω k and h. Fig.3 shows the constraints on the dark energy density function X(z) = ρ X (z)/ρ X (0) parametrized by Eq.(8), with X(z) at z > 1.5 given by either X(z) = X 1.5 or X(z) = X 1.5 exp[α(z − 1.5)]. Note that the assumption about dark energy at z > 1.5 has only a weak effect on the dark energy constraints at z ≤ 1.5. Note that taking X(z > 1.5) = X 1.5 gives slightly less stringent constraints on dark energy at z ≤ 1.5. This is because parametrizing dark energy at z > 1.5 with an extra parameter requires choosing the early dark energy parametrization such that it is not degenerate with cos- mic curvature; this is why Ω k is not well constrained if we choose X(z > 1.5) = X 1.5 (1 + z) α , but Ω k is well constrained if we choose X(z > 1.5) = X 1.5 exp[α(z − 1.5)] [56]. The latter helps break the degeneracy of Ω k with X(z), thus leading to much tighter constraints on Ω k and slightly tighter constraints on X(z) at z ≤ 1.5 (see Fig.3). This suggests that the more conservative approach in constraining dark energy is to assume that X(z > 1.5) = X 1.5 . Table 2 shows the mean, rms variance, and correlation coefficients of (X 0.5 , X 1.0 , X 1.5 ), as well as the relative dark energy FoM r defined in Eq.(2).

IV. SUMMARY AND DISCUSSION
In order to compare current and future dark energy experiments on the same footing, we have introduced a simple and straightforward definition for the Figure- parameters, Eq.(2), that is easily applicable to both real and simulated data.
We recommend the adoption of two dark energy parametrizations in comparing different experiments: (1) A dark energy equation of state w X (z) linear in a, with its values at z = 0 and z = 0.5, (w 0 , w 0.5 ), as parameters estimated from data [see Eq. (7)]. We find that (w 0 , w 0.5 ) are significantly less correlated than (w 0 , w a ) [see Table 1 and Fig.1], hence the factor of improve-ment in the FoM r [as defined in Eq.(2)] for (w 0 , w 0.5 ) is a more reliable indicator of the improvement in dark energy constraints than the factor of improvement of FoM r for (w 0 , w a ). (2) The dark energy density function X(z) = ρ X (z)/ρ X (0) parametrized by its values at z = 0.5, 1.0, and 1.5, (X 0.5 , X 1.0 , X 1.5 ), for z ≤ 1.5 [see Eq.(8)], and X(z > 1.5) = X 1.5 . We find that this flat cutoff in X(z) gives more conservative constraints on X(z) than parametrizing early dark energy with an extra parameter such that cosmic curvature is constrained (see Fig.3).
We have demonstrated the use of the FoM r [see Eq.
(2)] for these two dark energy parametrizations [see Eq. (7) and Eq. (8) If one assumes dark energy density to be a free function parametrized by its values at z = 0.5, 1.0, and 1.5, then dark energy deviates from a cosmological constant at z = 1.0 at 68% C.L., but is consistent with a cosmological constant at 95% C.L. (see Fig.3). This illustrates the importance of using the model-independent parametrization in probing dark energy. Measuring X(z) as a free function from data allows us to detect epochs of variation in dark energy density. It also allows us to con-strain a broader class of dark energy models than represented by w X (z); for example, dark energy models in which X(z) becomes negative in the past or future, which are excluded by fiat if one only measures w X (z) since X(z) = exp{ z 0 dz ′ 3[1 + w X (z ′ )]/(1 + z ′ )} [23]. The two parameter dark energy equation of state model (linear in a) implies strong assumptions about dark energy, and is not sensitive to a transient variation in dark energy; thus it is most useful in comparing forecasts for future dark energy experiments under the simplest assumptions.
Future dark energy experiments from both ground and space [52,73,74,75,76,77], together with CMB data from Planck [78], will dramatically improve our ability to probe dark energy, and eventually shed light on the nature of dark energy. Using both a linear dark energy equation of state [parameterized by (w 0 , w 0.5 )] and dark energy density function X(z) as a free function [parametrized by (X 0.5 , X 1.0 , X 1.5 )] provides a simple and balanced approach to exploring dark energy. Proposed future dark energy experiments should be evaluated by comparing their FoM r for both (w 0 , w 0.5 ) and (X 0.5 , X 1.0 , X 1.5 ) to that of current data.