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What Figure of Merit Should We Use to Evaluate Dark Energy Projects?

What Figure of Merit Should We Use to Evaluate Dark Energy Projects?. Yun Wang STScI Dark Energy Symposium May 6, 2008. How We Probe Dark Energy. Cosmic expansion history H ( z ) or DE density  X ( z ): tells us whether DE is a cosmological constant

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What Figure of Merit Should We Use to Evaluate Dark Energy Projects?

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

  2. How We Probe Dark Energy • Cosmic expansion history H(z) or DE density X(z): tells us whether DE is a cosmological constant H2(z) = 8 G[m(z) + r(z) +X(z)]/3  k(1+z)2 • Cosmic large scale structure growth rate function fg(z), or growth history G(z): tells us whether general relativity is modified fg(z)=dln/dlna, G(z)=(z)/(0) =[m-m]/m Yun Wang, 5/6/2008

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

  4. DETF FoM • DETF figure of merit = 1/[area of 95% C.L. w0-wa error ellipse], for wX(a) = w0+(1-a)wa • Pivot Value of a: At a=ap, wp= w0 + (1-ap)wa. Making wpwa=0 gives1-ap= – w0wa/ wa2: DETF FoM = 1/[6.17(wa)(wp)] • FoMr= 1/[(wa)(wp)] • ap is different for each survey, thus wp refers to a different property of DE in each survey. Yun Wang, 5/6/2008

  5. 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) Yun Wang, 5/6/2008

  6. 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? Yun Wang, 5/6/2008

  7. Generalized FoM • For parameters {fi}: FoMr= 1/[det Cov(f1, f2, f3, …)]1/2 • Can be easily applied to both real and simulated data • DETF FoMr = 1/[(wa)(wp)] = 1/[det Cov(w0wa)]1/2 Wang (2008) Yun Wang, 5/6/2008

  8. What Parameters to Use: • Two considerations: • Simple, clear, intuitive physical meaning • Minimally correlated • 2 Parameter Test: {w0, w0.5} wX(a) = 3w0.5-2w0+3(w0-w0.5)a w0 = wX(z=0), w0.5 = wX(z=0.5) • 3 Parameter Test: {X0.5, X1.0, X1.5} value of X(z) = X(z)/X(z=0) at z = 0.5, 1.0, 1.5 simplest smooth interpolation: polynomial Wang (2008) Yun Wang, 5/6/2008

  9. WMAP5 (Komatsu et al. 2008) SNe (Riess et al. 2007 compilation of data) BAO (Eisenstein et al. 2005) Data w0w0.5 r(w0w0.5) FoMr 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 w0wa r(w0wa) FoMr 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) Yun Wang, 5/6/2008

  10. WMAP5+SNe+BAO (w0, w0.5) (w0, wa) Wang (2008) Yun Wang, 5/6/2008

  11. (w0,w0.5) versus (w0,wa): • Both are linear functions of cosmic scale factor a • Simple transformation: w0.5 = w0 + wa /3 • (w0, w0.5) are significantly less correlated than (w0, wa) • For current data, pdf of w0.5 is more Gaussian than the pdf of wa • z = 0.5is around the epoch when cosmic acceleration began Yun Wang, 5/6/2008

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

  13. 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) Yun Wang, 5/6/2008

  14. WMAP5+SNe+BAO X(z>1.5) X0.5X1.0 X1.5 FoMr X1.5 1.059+/-0.213 2.556+/-1.215 7.503+/-8.037 2.077 X1.5e(z-1.5) 1.091+/-0.195 2.436+/-1.121 6.533+/-7.351 2.402 X(z>1.5) r(X0.5X1.0) r(X0.5X1.5) r(X1.0X1.5) X1.5 -0.389 -0.666 0.906 X1.5e(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: {X0.5, X1.0, X1.5} value of X(z) = X(z)/X(z=0) at z = 0.5, 1.0, 1.5 Wang (2008) Yun Wang, 5/6/2008

  15. Example of Future Data galaxy redshift survey : 20,000 sq deg, 0.3 < z <2.1 to H = 22 dw0 dwar(w0wa) dwp 1/[pa] BAO/P(k) 0.101 0.319 -0.88 0.049 63.9 BAO/P(k)+fg(z) 0.047 0.192 -0.76 0.031 167.3 [BAO/P(k)+fg(z)]+Planck 0.046 0.118 -0.99 0.008 1089.9 dw0 dw0.5 r(w0w0.5) BAO/P(k) 0.101 0.052 0.16 BAO/P(k)+fg(z) 0.047 0.042 -0.023 [BAO/P(k)+fg(z)]+Planck 0.046 0.0097 0.73 Yun Wang, 5/6/2008

  16. Yun Wang, 5/6/2008

  17. Differentiating dark energy and modified gravity fg=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) Yun Wang, 5/6/2008

  18. Summary: • For parameters {fi}: FoMr= 1/[det Cov(f1, f2, f3, …)]1/2 • 2 Parameter Test: {w0, w0.5}, wX(a) = 3w0.5-2w0+3(w0-w0.5)a w0 = wX(z=0), w0.5 = wX(z=0.5) * w0.5 = w0 + wa /3 • 3 Parameter Test: {X0.5, X1.0, X1.5}, X(z > 1.5)= X1.5 X0.5, X1.0, X1.5: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. Yun Wang, 5/6/2008

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

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

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

  22. w(z) = w0+wa(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) Yun Wang, 5/6/2008

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