What Figure of Merit What Figure of Merit Should We Use to Evaluate Should We Use to Evaluate

Dark Energy Projects?Dark Energy Projects?

Yun Wang Yun Wang

STScI Dark Energy SymposiumSTScI Dark Energy Symposium

May 6, 2008May 6, 2008

Yun Wang, 5/6/2008

How We Probe Dark EnergyHow We Probe Dark Energy

• Cosmic expansion history HCosmic expansion history H((zz) or DE density ) or DE density XX((zz):):

tells us whether DE is a cosmological constanttells 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 fCosmic large scale structure growth rate function fgg((zz), ), or or

growth history Ggrowth history G((zz):):

tells us whether general relativity is modifiedtells us whether general relativity is modified

fg(z)=dln/dlna, G(z)=(z)/(0)

=[m-m]/m

Yun Wang, 5/6/2008

Observational Methods for Observational Methods for Dark Energy SearchDark 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 fg(z) as well, if bias b(z) and redshift distortion parameter can be measured independently.]

• Weak Lensing Tomography and Cross-Weak Lensing Tomography and Cross-Correlation Cosmography: Correlation Cosmography: probes growth factor G(z), and H(z)

• Galaxy Cluster StatisticsGalaxy Cluster Statistics: : probes H(z)

Yun Wang, 5/6/2008

DETF FoMDETF 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 wpwa=0 gives 1-ap= – w0wa/ 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

• Albrecht & Bernstein (2007) defined

FoM = 1/[12…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

• Given a set of DE parameters, what is the Given a set of DE parameters, what is the simplest, intuitive, and meaningful way to simplest, intuitive, and meaningful way to define a FoM?define a FoM?

• What are the sets of minimal DE What are the sets of minimal DE parameters that we should use in parameters that we should use in comparing different DE projects?comparing different DE projects?

Yun Wang, 5/6/2008

Generalized FoMGeneralized 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

What Parameters to What Parameters to Use: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, 1.5simplest smooth interpolation: polynomial

Wang (2008)

Yun Wang, 5/6/2008

WMAP5 (Komatsu et al. 2008)SNe (Riess et al. 2007 compilation of data)BAO (Eisenstein et al. 2005)

Data w0 w0.5 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 w0 wa 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

WMAP5+SNe+BAOWMAP5+SNe+BAO

(w0, w0.5) (w0, wa) Wang (2008)

Yun Wang, 5/6/2008

((ww00,,ww0.50.5) versus () versus (ww00,,wwaa):):

• 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 = 0.5 is around the epoch when cosmic acceleration began

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

Yun Wang, 5/6/2008

Model-Model-independent independent constraints constraints

on dark on dark energyenergy(as proposed by (as proposed by

Wang & Garnavich 2001)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

WMAP5+SNe+BAOWMAP5+SNe+BAOX(z>1.5) X0.5 X1.0 X1.5 FoMr X1.5 1.059+/-0.213 2.556+/-1.215 7.503+/-8.037 2.077X1.5e(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

Example of Future DataExample of Future Datagalaxy redshift survey : 20,000 sq deg, 0.3 < z <2.1 to H = 22

dw0 dwa r(w0wa) dwp 1/[pa]

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

Yun Wang, 5/6/2008

Differentiating Differentiating dark energy dark energy

and and modified modified gravitygravity

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

Summary: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)aw0 = 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

The EndThe End

Yun Wang, 5/6/2008

Redshift space distortionsRedshift 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

Getting the most distant SNe Getting the most distant SNe Ia:Ia: critical for measuring the evolution in dark energy density:

Wang & Lovelave (2001)

Yun Wang, 5/6/2008

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)