1 what can we learn about dark energy? andreas albrecht uc davis december 17 2008 ntu-davis meeting...

1

What can we learn about dark energy?

Andreas Albrecht

UC Davis

December 17 2008

NTU-Davis meetingNational Taiwan University

2

Background

3

Supernova

Preferred by data c. 2003

Amount of “ordinary” gravitating matter A

mount

of

w=

-1 m

att

er

(“D

ark

energ

y”)

“Ordinary” non accelerating matter

Cosmic acceleration

Accelerating matter is required to fit current data

4

Dark energy appears to be the dominant component of the physical

Universe, yet there is no persuasive theoretical explanation. The

acceleration of the Universe is, along with dark matter, the observed

phenomenon which most directly demonstrates that our fundamental

theories of particles and gravity are either incorrect or incomplete.

Most experts believe that nothing short of a revolution in our

understanding of fundamental physics* will be required to achieve a

full understanding of the cosmic acceleration. For these reasons, the

nature of dark energy ranks among the very most compelling of all

outstanding problems in physical science. These circumstances

demand an ambitious observational program to determine the dark

energy properties as well as possible.

From the Dark Energy Task Force report (2006)www.nsf.gov/mps/ast/detf.jsp,

astro-ph/0690591*My emphasis

http://www.nsf.gov/mps/ast/detf.jsp

5

How we think about the cosmic acceleration:

Solve GR for the scale factor a of the Universe (a=1 today):

Positive acceleration clearly requires

• (unlike any known constituent of the Universe) or

• a non-zero cosmological constant or

• an alteration to General Relativity.

/ 1/ 3w p

43

3 3

a Gp

a

6

Some general issues:

Properties:






/ 1/ 3w p

43

3 3

a Gp

a

/ 1/ 3w p

Two “familiar” ways to achieve acceleration:

1) Einstein’s cosmological constant and relatives

2) Whatever drove inflation: Dynamical, Scalar field?

1w

7







/ 1/ 3w p

43

3 3

a Gp

a

8







/ 1/ 3w p

43

3 3

a Gp

a

Theory allows a multitude of possiblefunction w(a). How should we model

measurements of w?

9

Dark energy appears to be the dominant component of the physical

Universe, yet there is no persuasive theoretical explanation. The

acceleration of the Universe is, along with dark matter, the observed

phenomenon which most directly demonstrates that our fundamental

theories of particles and gravity are either incorrect or incomplete.

Most experts believe that nothing short of a revolution in our

understanding of fundamental physics* will be required to achieve a

full understanding of the cosmic acceleration. For these reasons, the

nature of dark energy ranks among the very most compelling of all

outstanding problems in physical science. These circumstances

demand an ambitious observational program to determine the dark

energy properties as well as possible.

From the Dark Energy Task Force report (2006)www.nsf.gov/mps/ast/detf.jsp,

astro-ph/0690591*My emphasis

DETF = a HEPAP/AAAC subpanel to guide planning of future dark energy experiments

More info here

http://www.nsf.gov/mps/ast/detf.jsp

10

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

11

Followup questions:

In what ways might the choice of DE parameters biased 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?




12

Followup questions:





New work, relevant to setting a concrete threshold

for Stage 4




13




Followup questions:





NB: To make concretecomparisons this work ignores

various possible improvements to the DETF data models.

(see for example J Newman, H Zhan et al

& Schneider et al)

DETF

14




Followup questions:





15

DETF Review

16w

wa

w(a) = w0 + wa(1-a)w(a) = w0 + wa(1-a)

DETF figure of merit:Area

95% CL contour

(DETF parameterization… Linder)

17

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

18

DETF Projections

Stage 3

Fig

ure

of m

erit

Impr

ovem

ent

over

S

tage

2

19

DETF Projections

Ground

Fig

ure

of m

erit

Impr

ovem

ent

over

S

tage

2

20

DETF Projections

Space

Fig

ure

of m

erit

Impr

ovem

ent

over

S

tage

2

21

DETF Projections

Ground + Space

Fig

ure

of m

erit

Impr

ovem

ent

over

S

tage

2

22

Combination

Technique #2

Technique #1

A technical point: The role of correlations

23

From the DETF Executive Summary

One of our main findings is that no single technique can answer the outstanding questions about dark energy: combinations of at least two of these techniques must be used to fully realize the promise of future observations.

Already there are proposals for major, long-term (Stage IV) projects incorporating these techniques that have the promise of increasing our figure of merit by a factor of ten beyond the level it will reach with the conclusion of current experiments. What is urgently needed is a commitment to fund a program comprised of a selection of these projects. The selection should be made on the basis of critical evaluations of their costs, benefits, and risks.

24

Followup questions:

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




25

Followup questions:





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

0( ) 1aw a w w a

How good is the w(a) ansatz?

27

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

28

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/

9 parameters are coefficients of the “top hat functions” 1,i iT a a

11

( ) 1 1 ,N

i i ii

w a w a wT a a

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

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

29

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/

9 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

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/

30

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

31

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”

32

Q: How do you describe error ellipsis in 9D space?

A: In terms of 9 principle axes and corresponding 9 errors :

2D illustration:

1

2

Axis 1

Axis 2

if

i

1f

2f

33



2D illustration:

if

i

1

2

Axis 1

Axis 2

1f

2f

Principle component analysis

34



2D illustration:

1

2

Axis 1

Axis 2

if

i

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

35



2D illustration:

1

2

Axis 1

Axis 2

if

i

1f

2f


i ii

w c f

if


ic

i

36

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 9D ellipses by principle axes and corresponding errorsDETF stage 2

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

37

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

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

38

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


“Convergence”z-=4 z =1.5 z =0.25 z =0

39

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

40


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)

41

Upshot of 9D 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).

42

Upshot of 9D FoM:





43

Upshot of 9D FoM:





44

Upshot of 9D FoM:





45

Upshot of 9D FoM:



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


Inverts cost/FoMEstimatesS3 vs S4

46

Upshot of 9D 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)

47

Followup questions:





48

A: Only by an overall (possibly important) rescaling

Followup questions:





49

Followup questions:





50

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

51

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)

52

The potentials




0( )V V e

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

2

0( )V V e

0( )m

V V


53

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

54

The potentials




0( )V V e

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

2

0( )V V e

ArXiv Dec 08, PRD in press

In prep.

Inverse Tracker (Ratra & Peebles, Steinhardt et al)

0( )m

V V

55

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.

56

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

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

57

DETF stage 2

DETF stage 3

DETF stage 4

58

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.

59

Followup questions:





60

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

Followup questions:





61

Followup questions:





62

Michael Barnard et al arXiv:0804.0413

Followup questions:





63

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.

64

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


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

1

2

Axis 1

Axis 2

1f

2f

i ii

w c f

65

●● ●

●

●

■■

■■

■

■ ■ ■ ■ ■

● Data■ Theory 1■ Theory 2

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

0

1

2

0 5 10 15

X

Y

66

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

67

DETF Stage 3 photo [Opt]

68


1 1/c

2 2/c

i ii

w c f

69

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

70


1 1/c

2 2/c

71


3 3/c

4 4/c

72

Eigenmodes:

Stage 3 Stage 4 gStage 4 s

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

73

Eigenmodes:

Stage 3 Stage 4 gStage 4 s

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

N.B. σi change too

74

DETF Stage 4 ground [Opt]

75


76

DETF Stage 4 space [Opt]

77


78

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)

79

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

80

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

81


82


83


84


85


86


87

Quantify discriminating power:

88

Stage 4 space Test Points

Characterize each model distribution by four “test points”

89


Characterize each model distribution by four “test points”

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

90


91

•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.

92

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

2

2

93

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

94

DETF Stage 3 photoTe

st P

oint

Mod

elComparison Model

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

95

DETF Stage 3 photoTe

st P

oint

Mod

elComparison Model

96

DETF Stage 4 groundTe

st P

oint

Mod

elComparison Model

97

DETF Stage 4 spaceTe

st P

oint

Mod

elComparison Model

98

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

99

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


100

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


101

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

102

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.

103

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.

104

Followup questions:





A:

• DETF Stage 3: Poor

• DETF Stage 4: Marginal… Excellent within reach

105

Followup questions:





A:



Structure in mode space

106

Followup questions:





A:



107

Followup questions:





108

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

109


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

110

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

111

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

112

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

113

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

114

A:


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





Summary


115

A:







Summary


116

A:







Summary


117

A:







Summary


118

A:







Summary


119

A:







Summary


120

A:







Summary


Interesting contributionto discussion of Stage 4

(if you believe scalar field modes)

121


i) Using w(a) eigenmodes

ii) Revealing value of higher modes

122

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

123

END

124

Additional Slides

125

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

1260 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

127

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?

128


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:

129

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

130

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

131

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

132

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

133

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

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i

DETF

9D

134

Stage 2

135

Stage 2

136

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Stage 2

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140

Stage 2

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