1 Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture October 15 2010

1

Dark energy paramters

Andreas Albrecht (UC Davis)

U Chicago Physics 411 guest lecture

October 15 2010

2

How can one accelerate the universe?

3


1) A cosmological constant

43

3 3

a Gp

a

4



43

3 3

a Gp

a

AKA

43

3

a Gp

a

p

5



43

3 3

a Gp

a

AKA

43

3

a Gp

a

p 1

pw

6



AKA

43

3

a Gp

a

p

1p

w


2) Cosmic inflation

43

3 I I

a Gp

a

8


2) Cosmic inflation

43

3 I I

a Gp

a

V

9


2) Cosmic inflation

43

3 I I

a Gp

a

2

1I

I

pw

V

Dynamical

0

V



1p

w

2) Cosmic inflation

1I

I

pw


3) Modify Einstein Gravity

12

Focus on w a

13

Focus on w a

Cosmic scale factor = “time”

14

Focus on w a


DETF: 0 1aw a w w a

15

Focus on w a


DETF: 0 1aw a w w a

(Free parameters = ∞)

16

Focus on w a


DETF: 0 1aw a w w a

(Free parameters = ∞)

(Free parameters = 2)

17

The Dark Energy Task Force (DETF)

Created specific simulated data sets (Stage 2, Stage 3, Stage 4)

Assessed their impact on our knowledge of dark energy as modeled with the w0-wa parameters

0 1aw a w w a

18

The DETF stages (data models constructed for each one)

Stage 2: Underway

Stage 3: Medium size/term projects

Stage 4: Large longer term projects (ie JDEM, LST)

DETF modeled

• SN

•Weak Lensing

•Baryon Oscillation

•Cluster data

19

DETF Projections

Stage 3

Fig

ure

of m

erit

Impr

ovem

ent

over

S

tage

2

20

DETF Projections

Ground

Fig

ure

of m

erit

Impr

ovem

ent

over

S

tage

2

21

DETF Projections

Space

Fig

ure

of m

erit

Impr

ovem

ent

over

S

tage

2

22

DETF Projections

Ground + Space

Fig

ure

of m

erit

Impr

ovem

ent

over

S

tage

2

23

Followup questions:

In what ways might the choice of DE parameters have skewed the DETF results?

What impact can these data sets have on specific DE models (vs abstract parameters)?

To what extent can these data sets deliver discriminating power between specific DE models?

How is the DoE/ESA/NASA Science Working Group looking at these questions?

24

Followup questions:





25

0 0.5 1 1.5 2 2.5-4

-2

0

wSample w(z) curves in w

0-w

a space

0 0.5 1 1.5 2

-1

0

1

w

Sample w(z) curves for the PNGB models

0 0.5 1 1.5 2

-1

0

1

z

w

Sample w(z) curves for the EwP models

w0-wa can only do these

DE models can do this (and much more)

w

z

0( ) 1aw a w w a

How good is the w(a) ansatz?

26

0 0.5 1 1.5 2 2.5-4

-2

0

wSample w(z) curves in w

0-w

a space

0 0.5 1 1.5 2

-1

0

1

w

Sample w(z) curves for the PNGB models

0 0.5 1 1.5 2

-1

0

1

z

w

Sample w(z) curves for the EwP models

w0-wa can only do these

DE models can do this (and much more)

w

z

How good is the w(a) ansatz?

NB: Better than

0( ) 1aw a w w a

0( )w a w& flat

27

0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

a

w

Illustration of stepwiseparameterization of w(a)

Z=4 Z=0

28

0.2 0.4 0.6 0.8 10

0.05

0.1

a

w

Measure

( 1)w w Z=4 Z=0


29

0.2 0.4 0.6 0.8 10

0.05

0.1

a

w

discrete paramterizationmodel

Each bin height is a free parameter


30

0.2 0.4 0.6 0.8 10

0.05

0.1

a

w



Refine bins as much as needed


31

0.2 0.4 0.6 0.8 10

0.05

0.1

a

w




Z=4 Z=0


0.2 0.4 0.6 0.8 10

0.05

0.1

a

w


32




33

10-2

10-1

100

101

-1

0

1

z

Try N-D stepwise constant w(a)

w a

AA & G Bernstein 2006 (astro-ph/0608269 ). More detailed info can be found at http://www.physics.ucdavis.edu/Cosmology/albrecht/MoreInfo0608269/

N parameters are coefficients of the “top hat functions”

11

( ) 1 1 ,N

i i ii

w a w a wT a a

1,i iT a a

http://arxiv.org/abs/astro-ph/0608269

http://www.physics.ucdavis.edu/Cosmology/albrecht/MoreInfo0608269/

34

10-2

10-1

100

101

-1

0

1

z


w a

AA & G Bernstein 2006 (astro-ph/0608269 ). More detailed info can be found at http://www.physics.ucdavis.edu/Cosmology/albrecht/MoreInfo0608269/


11

( ) 1 1 ,N

i i ii

w a w a wT a a

1,i iT a a

Used by

Huterer & Turner; Huterer & Starkman; Knox et al; Crittenden & Pogosian Linder; Reiss et al; Krauss et al de Putter & Linder; Sullivan et al

http://arxiv.org/abs/astro-ph/0608269

http://www.physics.ucdavis.edu/Cosmology/albrecht/MoreInfo0608269/

35

10-2

10-1

100

101

-1

0

1

z


w a

AA & G Bernstein 2006


11

( ) 1 1 ,N

i i ii

w a w a wT a a

1,i iT a a

Allows greater variety of w(a) behavior

Allows each experiment to “put its best foot forward”

Any signal rejects Λ

36

10-2

10-1

100

101

-1

0

1

z


w a

AA & G Bernstein 2006


11

( ) 1 1 ,N

i i ii

w a w a wT a a

1,i iT a a

Allows greater variety of w(a) behavior

Allows each experiment to “put its best foot forward”

Any signal rejects Λ“Convergence”

37

2D illustration:

1

2

Axis 1

Axis 2

1f

2f

Q: How do you describe error ellipsis in ND space?

A: In terms of N principle axes and corresponding N errors :

if

i

38



2D illustration:

if

i

1

2

Axis 1

Axis 2

1f

2f

Principle component analysis

39

2D illustration:

1

2

Axis 1

Axis 2

1f

2f

NB: in general the s form a complete basis:

i ii

w c f

if

The are independently measured qualities with errors

ic

i



if

i

40

2D illustration:

1

2

Axis 1

Axis 2

1f

2f


i ii

w c f

if


ic

i



if

i

41

1 2 3 4 5 6 7 8 90

1

2

Stage 2 ; lin-a NGrid

= 9, zmax

= 4, Tag = 044301

i

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

1

2

3

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

4

5

6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

7

8

9

i

Prin

cipl

e A

xes

if i

a

Characterizing ND ellipses by principle axes and corresponding errorsDETF stage 2

z-=4 z =1.5 z =0.25 z =0

42

1 2 3 4 5 6 7 8 90

1

2

Stage 4 Space WL Opt; lin-a NGrid

= 9, zmax

= 4, Tag = 044301

i

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

1

2

3

4

5

6

7

8

9

i

Prin

cipl

e A

xes

if i

a

Characterizing ND ellipses by principle axes and corresponding errorsWL Stage 4 Opt

z-=4 z =1.5 z =0.25 z =0

43

0 2 4 6 8 10 12 14 16 180

1

2


= 16, zmax

= 4, Tag = 054301

i

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

1

2

3

4

5

6

7

8

9

i

Prin

cipl

e A

xes

if i

a

Characterizing ND ellipses by principle axes and corresponding errorsWL Stage 4 Opt

“Convergence”z-=4 z =1.5 z =0.25 z =0

44

BAOp BAOs SNp SNs WLp ALLp1

10

100

1e3

1e4

Stage 3

Bska Blst Slst Wska Wlst Aska Alst1

10

100

1e3

1e4

Stage 4 Ground

BAO SN WL S+W S+W+B1

10

100

1e3

1e4

Stage 4 Space

Grid Linear in a zmax = 4 scale: 0

1

10

100

1e3

1e4

Stage 4 Ground+Space

[SSBlstW lst] [BSSlstW lst] Alllst [SSWSBIIIs] SsW lst

DETF(-CL)

9D (-CL)

DETF/9DF

45


10

100

1e3

1e4

Stage 3


10

100

1e3

1e4

Stage 4 Ground


10

100

1e3

1e4

Stage 4 Space


1

10

100

1e3

1e4



DETF(-CL)

9D (-CL)

DETF/9DF

Stage 2 Stage 4 = 3 orders of magnitude (vs 1 for DETF)

Stage 2 Stage 3 = 1 order of magnitude (vs 0.5 for DETF)

46

Upshot of ND FoM:

1) DETF underestimates impact of expts

2) DETF underestimates relative value of Stage 4 vs Stage 3

3) The above can be understood approximately in terms of a simple rescaling (related to higher dimensional parameter space).

4) DETF FoM is fine for most purposes (ranking, value of combinations etc).

47

Upshot of ND FoM:





48

Upshot of ND FoM:





49

Upshot of ND FoM:





50

An example of the power of the principle component analysis:

Q: I’ve heard the claim that the DETF FoM is unfair to BAO, because w0-wa does not describe the high-z behavior to which BAO is particularly sensitive. Why does this not show up in the 9D analysis?

51


10

100

1e3

1e4

Stage 3


10

100

1e3

1e4

Stage 4 Ground


10

100

1e3

1e4

Stage 4 Space


1

10

100

1e3

1e4



DETF(-CL)

9D (-CL)

DETF/9DF

Specific Case:

52

1 2 3 4 5 6 7 8 90

1

2

Stage 4 Space BAO Opt; lin-a NGrid

= 9, zmax

= 4, Tag = 044301

i

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

1

2

3

4

5

6

7

8

9

BAO

z-=4 z =1.5 z =0.25 z =0

53

1 2 3 4 5 6 7 8 90

1

2

Stage 4 Space SN Opt; lin-a NGrid

= 9, zmax

= 4, Tag = 044301

i

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

1

2

3

4

5

6

7

8

9

SN

z-=4 z =1.5 z =0.25 z =0

54

1 2 3 4 5 6 7 8 90

1

2

Stage 4 Space BAO Opt; lin-a NGrid

= 9, zmax

= 4, Tag = 044301

i

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

1

2

3

4

5

6

7

8

9

BAODETF 1 2,

z-=4 z =1.5 z =0.25 z =0

55

SN

1 2 3 4 5 6 7 8 90

1

2


= 9, zmax

= 4, Tag = 044301

i

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

1

2

3

4

5

6

7

8

9

DETF 1 2,

z-=4 z =1.5 z =0.25 z =0

56

1 2 3 4 5 6 7 8 90

1

2


= 9, zmax

= 4, Tag = 044301

i

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

1

2

3

4

5

6

7

8

9

z-=4 z =1.5 z =0.25 z =0

SNw0-wa analysis shows two parameters measured on average as well as 3.5 of these

2/9

1 21

3.5eD

i

DETF

9D

57

Upshot of ND FoM:



3) The above can be understood approximately in terms of a simple rescaling


Inverts cost/FoMEstimatesS3 vs S4

58

Upshot of ND FoM:



3) The above can be understood approximately in terms of a simple rescaling


A nice way to gain insights into data (real or imagined)

59

Followup questions:





60

A: Only by an overall (possibly important) rescaling

Followup questions:





61

Followup questions:





62

How well do Dark Energy Task Force simulated data sets constrain specific scalar field quintessence models?

Augusta Abrahamse

Brandon Bozek

Michael Barnard

Mark Yashar

+AA

+ DETFSimulated data

+ Quintessence potentials + MCMC

See also Dutta & Sorbo 2006, Huterer and Turner 1999 & especially Huterer and Peiris 2006

63

The potentials

Exponential (Wetterich, Peebles & Ratra)

PNGB aka Axion (Frieman et al)

Exponential with prefactor (AA & Skordis)

Inverse Power Law (Ratra & Peebles, Steinhardt et al)

64

The potentials




0( )V V e

0( ) (cos( / ) 1)V V

2

0( )V V e

0( )m

V V


65

The potentials




0( )V V e

0( ) (cos( / ) 1)V V

2

0( )V V e

0( )m

V V


Stronger than average

motivations & interest

66

The potentials




0( )V V e

0( ) (cos( / ) 1)V V

2

0( )V V e

PRD 2008

Inverse Tracker (Ratra & Peebles, Steinhardt et al)

0( )m

V V

67

0.2 0.4 0.6 0.8 1-1

-0.9

-0.8

-0.7

-0.6

-0.5

a

w(a

)

PNGBEXPITAS

…they cover a variety of behavior.

68

Challenges: Potential parameters can have very complicated (degenerate) relationships to observables

Resolved with good parameter choices (functional form and value range)

69

DETF stage 2

DETF stage 3

DETF stage 4

70

DETF stage 2

DETF stage 3

DETF stage 4

(S2/3)

(S2/10)

Upshot:

Story in scalar field parameter space very similar to DETF story in w0-wa space.

71

Followup questions:





72

A: Very similar to DETF results in w0-wa space

Followup questions:





73

Followup questions:





74

Michael Barnard PRD 2008

Followup questions:





75

Problem:

Each scalar field model is defined in its own parameter space. How should one quantify discriminating power among models?

Our answer:

Form each set of scalar field model parameter values, map the solution into w(a) eigenmode space, the space of uncorrelated observables.

Make the comparison in the space of uncorrelated observables.

76

1 2 3 4 5 6 7 8 90

1

2


= 9, zmax

= 4, Tag = 044301

i

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

f's

a

1

2

3

4

5

6

7

8

9

i

Prin

cipl

e A

xes

if i

a

Characterizing 9D ellipses by principle axes and corresponding errorsWL Stage 4 Opt

z-=4 z =1.5 z =0.25 z =0

1

2

Axis 1

Axis 2

1f

2f

i ii

w c f

77

●● ●

●

●

■■

■■

■

■ ■ ■ ■ ■

● Data■ Theory 1■ Theory 2

Concept: Uncorrelated data points (expressed in w(a) space)

0

1

2

0 5 10 15

X

Y

78

Starting point: MCMC chains giving distributions for each model at Stage 2.

79

DETF Stage 3 photo [Opt]

80


1 1/c

2 2/c

i ii

w c f

81

DETF Stage 3 photo [Opt] Distinct model locations

mode amplitude/σi “physical”

Modes (and σi’s) reflect specific expts.

1 1/c

2 2/c

82


1 1/c

2 2/c

83


3 3/c

4 4/c

84

Comments on model discrimination

•Principle component w(a) “modes” offer a space in which straightforward tests of discriminating power can be made.

•The DETF Stage 4 data is approaching the threshold of resolving the structure that our scalar field models form in the mode space.

85

Comments on model discrimination

•Principle component w(a) “modes” offer a space in which straightforward tests of discriminating power can be made.

•The DETF Stage 4 data is approaching the threshold of resolving the structure that our scalar field models form in the mode space.

86

Followup questions:





A:

• DETF Stage 3: Poor

• DETF Stage 4: Marginal… Excellent within reach

87

Followup questions:





A:



Structure in mode space

88

Followup questions:





A:



89

Followup questions:





90

DoE/ESA/NASA JDEM Science Working Group

Update agencies on figures of merit issues

formed Summer 08

finished ~now (moving on to SCG)

Use w-eigenmodes to get more complete picture

also quantify deviations from Einstein gravity

For today: Something we learned about normalizing modes

91


D Di i

i

w c f

if


ic

i

Define

/Di if f a

which obey continuum normalization:

D Di k j k ij

k

f a f a a then

where

i ii

w c f

Di ic c a

92

D Di i

i

w c f

Define

/Di if f a

which obey continuum normalization:

D Di k j k ij

k

f a f a a then

whereDi ic c a

Q: Why?

A: For lower modes, has typical grid independent “height” O(1), so one can more directly relate values of to one’s thinking (priors) on

Djf

Di i a

w

D Di i i i

i i

w c f c f

93

2 4 6 8 10 12 14 16 18 200

2

4

DETF= Stage 4 Space Opt All fk=6

= 1, Pr = 0

4210.50.20-2

0

2

z

Mode 1Mode 2

00.20.40.60.81-2

0

2

a

Mode 3

4210.50.20-2

0

2

z

Mode 4

i

Prin

cipl

e A

xes

if

94

4210.50.20-2

0

2

z


= 1, Pr = 0

Mode 5

00.20.40.60.81-2

0

2

a

Mode 6

4210.50.20-2

0

2

z

Mode 7

4210.50.20-2

0

2

z

Mode 8

Prin

cipl

e A

xes

if

95

Upshot: More modes are interesting (“well measured” in a grid invariant sense) than previously thought.

96

A:


• DETF Stage 4: Marginal… Excellent within reach (AA)





Summary


97

A:







Summary


98

A:







Summary


99

A:







Summary


100

A:







Summary


101

A:







Summary


102

A:







Summary


103


i) Using w(a) eigenmodes

ii) Revealing value of higher modes

104

2 4 6 8 10 12 14 16 18 200

2

4


= 1, Pr = 0

4210.50.20-2

0

2

z

Mode 1Mode 2

00.20.40.60.81-2

0

2

a

Mode 3

4210.50.20-2

0

2

z

Mode 4

i

Prin

cipl

e A

xes

if

105

END

106

Additional Slides

1070 5 10

10-3

10-2

10-1

100

101

102

mode

aver

age

proj

ectio

n

PNGB meanExp. meanIT meanAS meanPNGB maxExp. maxIT maxAS max

108

0 2 4 6 8 10

10-3

10-2

10-1

100

101

102

mode

ave

rag

e p

roje

ctio

n

PNGB meanExp. meanIT meanAS meanPNGB maxExp. maxIT maxAS max

109

Stage 2

110

Stage 2

111

Stage 2

112

Stage 2

113

Stage 2

114

Stage 2

115

Stage 2

116


3 3/c

4 4/c

117

Eigenmodes:

Stage 3 Stage 4 gStage 4 s

z=4 z=2 z=1 z=0.5 z=0

118

Eigenmodes:

Stage 3 Stage 4 gStage 4 s

z=4 z=2 z=1 z=0.5 z=0

N.B. σi change too

119

DETF Stage 4 ground [Opt]

120


121

DETF Stage 4 space [Opt]

122


123

0.2 0.4 0.6 0.8 1-1

-0.9

-0.8

-0.7

-0.6

-0.5

a

w(a

)

PNGBEXPITAS

The different kinds of curves correspond to different “trajectories” in mode space (similar to FT’s)

124

DETF Stage 4 ground

Data that reveals a universe with dark energy given by “ “ will have finite minimum “distances” to other quintessence models

powerful discrimination is possible.

2

125

Consider discriminating power of each experiment (look at units on axes)

126


127


128


129


130


131


132

Quantify discriminating power:

133

Stage 4 space Test Points

Characterize each model distribution by four “test points”

134


Characterize each model distribution by four “test points”

(Priors: Type 1 optimized for conservative results re discriminating power.)

135


136

•Measured the χ2 from each one of the test points (from the “test model”) to all other chain points (in the “comparison model”).

•Only the first three modes were used in the calculation.

•Ordered said χ2‘s by value, which allows us to plot them as a function of what fraction of the points have a given value or lower.

•Looked for the smallest values for a given model to model comparison.

137

Model Separation in Mode Space

Fraction of compared model within given χ2 of test model’s test point

Test point 4

Test point 1

Where the curve meets the axis, the compared model is ruled out by that χ2 by an observation of the test point.This is the separation seen in the mode plots.

99% confidence at 11.36

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2

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Model Separation in Mode Space

Fraction of compared model within given χ2 of test model’s test point

Test point 4

Test point 1

Where the curve meets the axis, the compared model is ruled out by that χ2 by an observation of the test point.This is the separation seen in the mode plots.

99% confidence at 11.36

…is this gap

This gap…

2

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DETF Stage 3 photoTe

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[4 models] X [4 models] X [4 test points]

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DETF Stage 3 photoTe

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DETF Stage 4 groundTe

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DETF Stage 4 spaceTe

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PNGB PNGB Exp IT AS

Point 1 0.001 0.001 0.1 0.2

Point 2 0.002 0.01 0.5 1.8

Point 3 0.004 0.04 1.2 6.2

Point 4 0.01 0.04 1.6 10.0

Exp

Point 1 0.004 0.001 0.1 0.4

Point 2 0.01 0.001 0.4 1.8

Point 3 0.03 0.001 0.7 4.3

Point 4 0.1 0.01 1.1 9.1

IT

Point 1 0.2 0.1 0.001 0.2

Point 2 0.5 0.4 0.0004 0.7

Point 3 1.0 0.7 0.001 3.3

Point 4 2.7 1.8 0.01 16.4

AS

Point 1 0.1 0.1 0.1 0.0001

Point 2 0.2 0.1 0.1 0.0001

Point 3 0.2 0.2 0.1 0.0002

Point 4 0.6 0.5 0.2 0.001

DETF Stage 3 photo

A tabulation of χ2 for each graph where the curve crosses the x-axis (= gap)For the three parameters used here, 95% confidence χ2 = 7.82,99% χ2 = 11.36.Light orange > 95% rejectionDark orange > 99% rejection

Blue: Ignore these because PNGB & Exp hopelessly similar, plus self-comparisons

144

PNGB PNGB Exp IT AS

Point 1 0.001 0.005 0.3 0.9

Point 2 0.002 0.04 2.4 7.6

Point 3 0.004 0.2 6.0 18.8

Point 4 0.01 0.2 8.0 26.5

Exp

Point 1 0.01 0.001 0.4 1.6

Point 2 0.04 0.002 2.1 7.8

Point 3 0.01 0.003 3.8 14.5

Point 4 0.03 0.01 6.0 24.4

IT

Point 1 1.1 0.9 0.002 1.2

Point 2 3.2 2.6 0.001 3.6

Point 3 6.7 5.2 0.002 8.3

Point 4 18.7 13.6 0.04 30.1

AS

Point 1 2.4 1.4 0.5 0.001

Point 2 2.3 2.1 0.8 0.001

Point 3 3.3 3.1 1.2 0.001

Point 4 7.4 7.0 2.6 0.001

DETF Stage 4 ground

A tabulation of χ2 for each graph where the curve crosses the x-axis (= gap).For the three parameters used here, 95% confidence χ2 = 7.82,99% χ2 = 11.36.Light orange > 95% rejectionDark orange > 99% rejection


145

PNGB PNGB Exp IT AS

Point 1 0.01 0.01 0.4 1.6

Point 2 0.01 0.05 3.2 13.0

Point 3 0.02 0.2 8.2 30.0

Point 4 0.04 0.2 10.9 37.4

Exp

Point 1 0.02 0.002 0.6 2.8

Point 2 0.05 0.003 2.9 13.6

Point 3 0.1 0.01 5.2 24.5

Point 4 0.3 0.02 8.4 33.2

IT

Point 1 1.5 1.3 0.005 2.2

Point 2 4.6 3.8 0.002 8.2

Point 3 9.7 7.7 0.003 9.4

Point 4 27.8 20.8 0.1 57.3

AS

Point 1 3.2 3.0 1.1 0.002

Point 2 4.9 4.6 1.8 0.003

Point 3 10.9 10.4 4.3 0.01

Point 4 26.5 25.1 10.6 0.01

DETF Stage 4 space

A tabulation of χ2 for each graph where the curve crosses the x-axis (= gap)For the three parameters used here, 95% confidence χ2 = 7.82,99% χ2 = 11.36.Light orange > 95% rejectionDark orange > 99% rejection


146

PNGB PNGB Exp IT AS

Point 1 0.01 0.01 .09 3.6

Point 2 0.01 0.1 7.3 29.1

Point 3 0.04 0.4 18.4 67.5

Point 4 0.09 0.4 24.1 84.1

Exp

Point 1 0.04 0.01 1.4 6.4

Point 2 0.1 0.01 6.6 30.7

Point 3 0.3 0.01 11.8 55.1

Point 4 0.7 0.05 18.8 74.6

IT

Point 1 3.5 2.8 0.01 4.9

Point 2 10.4 8.5 0.01 18.4

Point 3 21.9 17.4 0.01 21.1

Point 4 62.4 46.9 0.2 129.0

AS

Point 1 7.2 6.8 2.5 0.004

Point 2 10.9 10.3 4.0 0.01

Point 3 24.6 23.3 9.8 0.01

Point 4 59.7 56.6 23.9 0.01

DETF Stage 4 space2/3 Error/mode

Many believe it is realistic for Stage 4 ground and/or space to do this well or even considerably better. (see slide 5)

A tabulation of χ2 for each graph where the curve crosses the x-axis (= gap).For the three parameters used here, 95% confidence χ2 = 7.82,99% χ2 = 11.36.Light orange > 95% rejectionDark orange > 99% rejection

Documents

1 Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture October 15 2010