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Transcript of 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
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:
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
/ 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
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
8
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
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
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?
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
12
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?
New work, relevant to setting a concrete threshold
for Stage 4
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
13
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
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?
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
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
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?
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?
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?
25
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?
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
29
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”
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
30
10-2
10-1
100
101
-1
0
1
z
Try N-D stepwise constant w(a)
w a
AA & G Bernstein 2006
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
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
Try N-D stepwise constant w(a)
w a
AA & G Bernstein 2006
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
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
Q: How do you describe error ellipsis in 9D space?
A: In terms of 9 principle axes and corresponding 9 errors :
2D illustration:
if
i
1
2
Axis 1
Axis 2
1f
2f
Principle component analysis
34
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
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
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
NB: in general the s form a complete basis:
i ii
w c f
if
The are independently measured qualities with errors
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
Stage 4 Space WL Opt; lin-a NGrid
= 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 9D ellipses by principle axes and corresponding errorsWL Stage 4 Opt
“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
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
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:
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).
43
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).
44
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).
45
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
4) DETF FoM is fine for most purposes (ranking, value of combinations etc).
Inverts cost/FoMEstimatesS3 vs S4
46
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
4) DETF FoM is fine for most purposes (ranking, value of combinations etc).
A nice way to gain insights into data (real or imagined)
47
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?
48
A: Only by an overall (possibly important) rescaling
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?
49
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?
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
Exponential (Wetterich, Peebles & Ratra)
PNGB aka Axion (Frieman et al)
Exponential with prefactor (AA & Skordis)
0( )V V e
0( ) (cos( / ) 1)V V
2
0( )V V e
0( )m
V V
Inverse Power Law (Ratra & Peebles, Steinhardt et al)
53
The potentials
Exponential (Wetterich, Peebles & Ratra)
PNGB aka Axion (Frieman et al)
Exponential with prefactor (AA & Skordis)
0( )V V e
0( ) (cos( / ) 1)V V
2
0( )V V e
0( )m
V V
Inverse Power Law (Ratra & Peebles, Steinhardt et al)
Stronger than average
motivations & interest
54
The potentials
Exponential (Wetterich, Peebles & Ratra)
PNGB aka Axion (Frieman et al)
Exponential with prefactor (AA & Skordis)
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:
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?
60
A: Very similar to DETF results in w0-wa space
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?
61
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?
62
Michael Barnard et al arXiv:0804.0413
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?
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
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
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
DETF Stage 3 photo [Opt]
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
DETF Stage 3 photo [Opt]
1 1/c
2 2/c
71
DETF Stage 3 photo [Opt]
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
DETF Stage 4 ground [Opt]
76
DETF Stage 4 space [Opt]
77
DETF Stage 4 space [Opt]
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
DETF Stage 3 photo [Opt]
82
DETF Stage 3 photo [Opt]
83
DETF Stage 4 ground [Opt]
84
DETF Stage 4 ground [Opt]
85
DETF Stage 4 space [Opt]
86
DETF Stage 4 space [Opt]
87
Quantify discriminating power:
88
Stage 4 space Test Points
Characterize each model distribution by four “test points”
89
Stage 4 space Test Points
Characterize each model distribution by four “test points”
(Priors: Type 1 optimized for conservative results re discriminating power.)
90
Stage 4 space Test Points
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
Blue: Ignore these because PNGB & Exp hopelessly similar, plus self-comparisons
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
Blue: Ignore these because PNGB & Exp hopelessly similar, plus self-comparisons
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:
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?
A:
• DETF Stage 3: Poor
• DETF Stage 4: Marginal… Excellent within reach
105
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?
A:
• DETF Stage 3: Poor
• DETF Stage 4: Marginal… Excellent within reach
Structure in mode space
106
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?
A:
• DETF Stage 3: Poor
• DETF Stage 4: Marginal… Excellent within reach
107
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?
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
NB: in general the s form a complete basis:
D Di i
i
w c f
if
The are independently measured qualities with errors
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
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e A
xes
if
112
4210.50.20-2
0
2
z
DETF= Stage 4 Space Opt All fk=6
= 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 3: Poor
• DETF Stage 4: Marginal… Excellent within reach (AA)
In what ways might the choice of DE parameters have skewed the DETF results?
A: Only by an overall (possibly important) rescaling
What impact can these data sets have on specific DE models (vs abstract parameters)?
A: Very similar to DETF results in w0-wa space
Summary
To what extent can these data sets deliver discriminating power between specific DE models?
115
A:
• DETF Stage 3: Poor
• DETF Stage 4: Marginal… Excellent within reach (AA)
In what ways might the choice of DE parameters have skewed the DETF results?
A: Only by an overall (possibly important) rescaling
What impact can these data sets have on specific DE models (vs abstract parameters)?
A: Very similar to DETF results in w0-wa space
Summary
To what extent can these data sets deliver discriminating power between specific DE models?
116
A:
• DETF Stage 3: Poor
• DETF Stage 4: Marginal… Excellent within reach (AA)
In what ways might the choice of DE parameters have skewed the DETF results?
A: Only by an overall (possibly important) rescaling
What impact can these data sets have on specific DE models (vs abstract parameters)?
A: Very similar to DETF results in w0-wa space
Summary
To what extent can these data sets deliver discriminating power between specific DE models?
117
A:
• DETF Stage 3: Poor
• DETF Stage 4: Marginal… Excellent within reach (AA)
In what ways might the choice of DE parameters have skewed the DETF results?
A: Only by an overall (possibly important) rescaling
What impact can these data sets have on specific DE models (vs abstract parameters)?
A: Very similar to DETF results in w0-wa space
Summary
To what extent can these data sets deliver discriminating power between specific DE models?
118
A:
• DETF Stage 3: Poor
• DETF Stage 4: Marginal… Excellent within reach (AA)
In what ways might the choice of DE parameters have skewed the DETF results?
A: Only by an overall (possibly important) rescaling
What impact can these data sets have on specific DE models (vs abstract parameters)?
A: Very similar to DETF results in w0-wa space
Summary
To what extent can these data sets deliver discriminating power between specific DE models?
119
A:
• DETF Stage 3: Poor
• DETF Stage 4: Marginal… Excellent within reach (AA)
In what ways might the choice of DE parameters have skewed the DETF results?
A: Only by an overall (possibly important) rescaling
What impact can these data sets have on specific DE models (vs abstract parameters)?
A: Very similar to DETF results in w0-wa space
Summary
To what extent can these data sets deliver discriminating power between specific DE models?
120
A:
• DETF Stage 3: Poor
• DETF Stage 4: Marginal… Excellent within reach (AA)
In what ways might the choice of DE parameters have skewed the DETF results?
A: Only by an overall (possibly important) rescaling
What impact can these data sets have on specific DE models (vs abstract parameters)?
A: Very similar to DETF results in w0-wa space
Summary
To what extent can these data sets deliver discriminating power between specific DE models?
Interesting contributionto discussion of Stage 4
(if you believe scalar field modes)
121
How is the DoE/ESA/NASA Science Working Group looking at these questions?
i) Using w(a) eigenmodes
ii) Revealing value of higher modes
122
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
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
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
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
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
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
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
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
134
Stage 2
135
Stage 2
136
Stage 2
137
Stage 2
138
Stage 2
139
Stage 2
140
Stage 2