1. What Figure of Merit Should We Use to Evaluate Dark Energy Projects? Yun Wang Yun Wang STScI Dark Energy Symposium STScI Dark Energy Symposium May 6, 2008

2. Yun Wang, 5/6/2008 How We Probe Dark Energy Cosmic expansion history H(z) or DE density  X (z):Cosmic expansion history H(z) or DE density  X (z): tells us whether DE is a cosmological constant H 2 (z) = 8  G[  m (z) +  r (z) +  X (z)]/3  k(1+z) 2 Cosmic large scale structure growth rate function f g (z), or growth history G(z):Cosmic large scale structure growth rate function f g (z), or growth history G(z): tells us whether general relativity is modified f g (z)=dln  /dlna, G(z)=  (z)/  (0)  =[  m -  m  ]/  m 

3. Yun Wang, 5/6/2008 Observational Methods for Dark Energy Search SNe Ia (Standard Candles):SNe Ia (Standard Candles): method through which DE has been discovered; independent of clustering of matter, probes H(z) Baryon Acoustic Oscillations (Standard Ruler):Baryon Acoustic Oscillations (Standard Ruler): calibrated by CMB, probes H(z). [The same data probe growth rate f g (z) as well, if bias b(z) and redshift distortion parameter  can be measured independently.] Weak Lensing Tomography and Cross- Correlation Cosmography:Weak Lensing Tomography and Cross- Correlation Cosmography: probes growth factor G(z), and H(z) Galaxy Cluster Statistics :Galaxy Cluster Statistics : probes H(z)

4. Yun Wang, 5/6/2008 DETF FoM DETF figure of merit = 1/[area of 95% C.L. w 0 -w a error ellipse], for w X (a) = w 0 +(1-a)w a Pivot Value of a: At a=a p, w p = w 0 + (1-a p )w a. Making  w p  w a  =0 gives 1-a p = –  w 0  w a  /  w a 2  : DETF FoM = 1/[6.17  (w a )  (w p )] FoM r = 1/[  (w a )  (w p )] a p is different for each survey, thus w p refers to a different property of DE in each survey.

5. Yun Wang, 5/6/2008 Albrecht & Bernstein (2007) defined FoM = 1/[  1  2 …  9 ] where  i is the width of the error ellipsoid along the axis defined by the i-th eigenvector of the Fisher matrix, and the 9 parameters are the parameters in a piecewise constant model of w(a)

6. Yun Wang, 5/6/2008 Given a set of DE parameters, what is the simplest, intuitive, and meaningful way to define a FoM?Given a set of DE parameters, what is the simplest, intuitive, and meaningful way to define a FoM? What are the sets of minimal DE parameters that we should use in comparing different DE projects?What are the sets of minimal DE parameters that we should use in comparing different DE projects?

7. Yun Wang, 5/6/2008 Generalized FoM For parameters {f i }: FoM r = 1/[det Cov(f 1, f 2, f 3, …)] 1/2 Can be easily applied to both real and simulated data DETF FoM r = 1/[  (w a )  (w p )] = 1/[det Cov(w 0 w a )] 1/2 Wang (2008)

8. Yun Wang, 5/6/2008 What Parameters to Use: Two considerations: –Simple, clear, intuitive physical meaning –Minimally correlated 2 Parameter Test: {w 0, w 0.5 } w X (a) = 3w 0.5 -2w 0 +3(w 0 -w 0.5 )a w 0 = w X (z=0), w 0.5 = w X (z=0.5) 3 Parameter Test: {X 0.5, X 1.0, X 1.5 } value of X(z) =  X (z)/  X (z=0) at z = 0.5, 1.0, 1.5 simplest smooth interpolation: polynomial Wang (2008)

9. Yun Wang, 5/6/2008 WMAP5 (Komatsu et al. 2008) SNe (Riess et al. 2007 compilation of data) BAO (Eisenstein et al. 2005) Data w 0 w 0.5 r(w 0 w 0.5 ) FoM r WMAP5 +SNe -1.08+/-0.60 -1.94+/-1.57 -0.40 1.2 WMAP5 +SNe+BAO -0.94+/-0.23 -0.95+/-0.21 -0.51 25.0 (factor of improvement in FoM: 21.6) Data w 0 w a r(w 0 w a ) FoM r WMAP5 +SNe -1.07+/-0.65 -2.96+/-6.76 -0.67 0.3 WMAP5 +SNe+BAO -0.94+/-0.23 -0.05+/-1.13 -0.88 8.3 (factor of improvement in FoM: 27.0) Wang (2008)

10. Yun Wang, 5/6/2008 WMAP5+SNe+BAO (w 0, w 0.5 ) (w 0, w a ) Wang (2008)

11. Yun Wang, 5/6/2008 ( w 0, w 0.5 ) versus ( w 0, w a ): Both are linear functions of cosmic scale factor a Simple transformation: w 0.5 = w 0 + w a /3 (w 0, w 0.5 ) are significantly less correlated than (w 0, w a ) For current data, pdf of w 0.5 is more Gaussian than the pdf of w a z = 0.5 is around the epoch when cosmic acceleration began

12. Yun Wang, 5/6/2008 Wang & Mukherjee (2007) [See Wang & Tegmark (2005) for the method to derive uncorrelated estimate of H(z) using SNe.] H(z) = [da/dt]/a

13. Yun Wang, 5/6/2008 Model- independent constraints on dark energy (as proposed by Wang & Garnavich 2001) The upward trend in X(z) at z ~ 1 [first found by Wang & Mukherjee (2004) and Daly & Djorgovski (2004)] has persisted. Wang & Mukherjee (2007)

14. Yun Wang, 5/6/2008 WMAP5+SNe+BAO X(z>1.5) X 0.5 X 1.0 X 1.5 FoM r X 1.5 1.059+/-0.213 2.556+/-1.215 7.503+/-8.037 2.077 X 1.5 e  (z-1.5) 1.091+/-0.195 2.436+/-1.121 6.533+/-7.351 2.402 X(z>1.5) r(X 0.5 X 1.0 ) r(X 0.5 X 1.5 ) r(X 1.0 X 1.5 ) X 1.5 -0.389 -0.666 0.906 X 1.5 e  (z-1.5) -0.303 -0.609 0.895 * about the same as WMAP3+SNe+BAO, with the same upward trend in X(z) at z ~ 1. 3 Parameter Test: {X 0.5, X 1.0, X 1.5 } value of X(z) =  X (z)/  X (z=0) at z = 0.5, 1.0, 1.5 Wang (2008)

15. Yun Wang, 5/6/2008 Example of Future Data galaxy redshift survey : 20,000 sq deg, 0.3 < z <2.1 to H = 22 dw 0 dw a r(w 0 w a ) dw p 1/[  p  a ] BAO/P(k) 0.101 0.319 -0.88 0.049 63.9 BAO/P(k)+f g (z) 0.047 0.192 -0.76 0.031 167.3 [BAO/P(k)+f g (z)]+Planck 0.046 0.118 -0.99 0.008 1089.9 dw 0 dw 0.5 r(w 0 w 0.5 ) BAO/P(k) 0.101 0.052 0.16 BAO/P(k)+f g (z) 0.047 0.042 -0.023 [BAO/P(k)+f g (z)]+Planck 0.046 0.0097 0.73

16. Yun Wang, 5/6/2008

17. Differentiating dark energy and modified gravity f g =dln  /dlna  =(  m -  m  )/  m  *  b(z)/b(z) = 0.01 assumed for a magnitude-limited redshift survey covering 28,600 (deg) 2. Wang, arXiv:0710.3885 JCAP in press (2008)

18. Yun Wang, 5/6/2008 Summary: For parameters {f i }: FoM r = 1/[det Cov(f 1, f 2, f 3, …)] 1/2 2 Parameter Test: {w 0, w 0.5 }, w X (a) = 3w 0.5 -2w 0 +3(w 0 -w 0.5 )a w 0 = w X (z=0), w 0.5 = w X (z=0.5) * w 0.5 = w 0 + w a /3 3 Parameter Test: {X 0.5, X 1.0, X 1.5 }, X(z > 1.5)= X 1.5 X 0.5, X 1.0, X 1.5 : X(z) =  X (z)/  X (z=0) at z = 0.5, 1.0, 1.5 *model-independent; democratic treatment of low z and high z measurements.

19. Yun Wang, 5/6/2008 The End

20. Yun Wang, 5/6/2008 Redshift space distortions Large scale compression due to linear motions gives the Kaiser factor  =f g /b, f g =dlnG/dlna~  (a) 0.55 G(z)=  (z)/  (0)  (a)=  m / .

21. Yun Wang, 5/6/2008 Getting the most distant SNe Ia: critical for measuring the evolution in dark energy density: Wang & Lovelave (2001)

22. Yun Wang, 5/6/2008 w(z) = w 0 +w a (1-a) 1+z = 1/a z: cosmological redshift a: cosmic scale factor WMAP3 +182 SNe Ia (Riess et al. 2007, inc SNLS and nearby SNe) +SDSS BAO (Wang & Mukherjee 2007)