Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture
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Transcript of Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture
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Dark energy paramters
Andreas Albrecht (UC Davis)
U Chicago Physics 411 guest lecture
October 15 2010
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How can one accelerate the universe?
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How can one accelerate the universe?
1) A cosmological constant
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3 3
a Gp
a
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How can one accelerate the universe?
1) A cosmological constant
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3 3
a Gp
a
AKA
43
3
a Gp
a
p
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How can one accelerate the universe?
1) A cosmological constant
43
3 3
a Gp
a
AKA
43
3
a Gp
a
p 1
pw
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How can one accelerate the universe?
1) A cosmological constant
AKA
43
3
a Gp
a
p
1p
w
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How can one accelerate the universe?
2) Cosmic inflation
43
3 I I
a Gp
a
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How can one accelerate the universe?
2) Cosmic inflation
43
3 I I
a Gp
a
V
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How can one accelerate the universe?
2) Cosmic inflation
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3 I I
a Gp
a
2
1I
I
pw
V
Dynamical
0
V
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How can one accelerate the universe?
1) A cosmological constant
1p
w
2) Cosmic inflation
1I
I
pw
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How can one accelerate the universe?
3) Modify Einstein Gravity
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Focus on w a
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Focus on w a
Cosmic scale factor = “time”
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Focus on w a
Cosmic scale factor = “time”
DETF: 0 1aw a w w a
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Focus on w a
Cosmic scale factor = “time”
DETF: 0 1aw a w w a
(Free parameters = ∞)
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Focus on w a
Cosmic scale factor = “time”
DETF: 0 1aw a w w a
(Free parameters = ∞)
(Free parameters = 2)
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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
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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
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DETF Projections
Stage 3
Fig
ure
of m
erit
Impr
ovem
ent
over
S
tage
2
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DETF Projections
Ground
Fig
ure
of m
erit
Impr
ovem
ent
over
S
tage
2
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DETF Projections
Space
Fig
ure
of m
erit
Impr
ovem
ent
over
S
tage
2
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DETF Projections
Ground + Space
Fig
ure
of m
erit
Impr
ovem
ent
over
S
tage
2
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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?
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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?
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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?
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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
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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
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0.2 0.4 0.6 0.8 10
0.05
0.1
a
w
Measure
( 1)w w Z=4 Z=0
Illustration of stepwiseparameterization of w(a)
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0.2 0.4 0.6 0.8 10
0.05
0.1
a
w
discrete paramterizationmodel
Each bin height is a free parameter
Illustration of stepwiseparameterization of w(a)
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0.2 0.4 0.6 0.8 10
0.05
0.1
a
w
discrete paramterizationmodel
Each bin height is a free parameter
Refine bins as much as needed
Illustration of stepwiseparameterization of w(a)
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0.2 0.4 0.6 0.8 10
0.05
0.1
a
w
discrete paramterizationmodel
Each bin height is a free parameter
Refine bins as much as needed
Z=4 Z=0
Illustration of stepwiseparameterization of w(a)
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0.2 0.4 0.6 0.8 10
0.05
0.1
a
w
discrete paramterizationmodel
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Each bin height is a free parameter
Refine bins as much as needed
Illustration of stepwiseparameterization of w(a)
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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
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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
Used by
Huterer & Turner; Huterer & Starkman; Knox et al; Crittenden & Pogosian Linder; Reiss et al; Krauss et al de Putter & Linder; Sullivan et al
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10-2
10-1
100
101
-1
0
1
z
Try N-D stepwise constant w(a)
w a
AA & G Bernstein 2006
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
Allows greater variety of w(a) behavior
Allows each experiment to “put its best foot forward”
Any signal rejects Λ
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10-2
10-1
100
101
-1
0
1
z
Try N-D stepwise constant w(a)
w a
AA & G Bernstein 2006
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
Allows greater variety of w(a) behavior
Allows each experiment to “put its best foot forward”
Any signal rejects Λ“Convergence”
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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
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Q: How do you describe error ellipsis in ND space?
A: In terms of N principle axes and corresponding N errors :
2D illustration:
if
i
1
2
Axis 1
Axis 2
1f
2f
Principle component analysis
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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
Q: How do you describe error ellipsis in ND space?
A: In terms of N principle axes and corresponding N errors :
if
i
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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
Q: How do you describe error ellipsis in ND space?
A: In terms of N principle axes and corresponding N errors :
if
i
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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
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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
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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 ND ellipses by principle axes and corresponding errorsWL Stage 4 Opt
“Convergence”z-=4 z =1.5 z =0.25 z =0
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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
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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)
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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).
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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).
![Page 48: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/48.jpg)
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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).
![Page 49: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/49.jpg)
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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).
![Page 50: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/50.jpg)
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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?
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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:
![Page 52: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/52.jpg)
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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
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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
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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
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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
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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
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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
4) DETF FoM is fine for most purposes (ranking, value of combinations etc).
Inverts cost/FoMEstimatesS3 vs S4
![Page 58: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/58.jpg)
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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
4) DETF FoM is fine for most purposes (ranking, value of combinations etc).
A nice way to gain insights into data (real or imagined)
![Page 59: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/59.jpg)
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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?
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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?
![Page 61: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/61.jpg)
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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?
![Page 62: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/62.jpg)
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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
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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)
![Page 64: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/64.jpg)
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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)
![Page 65: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/65.jpg)
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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
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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
PRD 2008
Inverse Tracker (Ratra & Peebles, Steinhardt et al)
0( )m
V V
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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.
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Challenges: Potential parameters can have very complicated (degenerate) relationships to observables
Resolved with good parameter choices (functional form and value range)
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DETF stage 2
DETF stage 3
DETF stage 4
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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.
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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?
![Page 72: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/72.jpg)
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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?
![Page 73: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/73.jpg)
73
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?
![Page 74: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/74.jpg)
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Michael Barnard PRD 2008
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?
![Page 75: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/75.jpg)
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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.
![Page 76: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/76.jpg)
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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
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●● ●
●
●
■■
■■
■
■ ■ ■ ■ ■
● Data■ Theory 1■ Theory 2
Concept: Uncorrelated data points (expressed in w(a) space)
0
1
2
0 5 10 15
X
Y
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Starting point: MCMC chains giving distributions for each model at Stage 2.
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DETF Stage 3 photo [Opt]
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DETF Stage 3 photo [Opt]
1 1/c
2 2/c
i ii
w c f
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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
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DETF Stage 3 photo [Opt]
1 1/c
2 2/c
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DETF Stage 3 photo [Opt]
3 3/c
4 4/c
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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.
![Page 85: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/85.jpg)
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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.
![Page 86: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/86.jpg)
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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
![Page 87: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/87.jpg)
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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
![Page 88: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/88.jpg)
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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
![Page 89: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/89.jpg)
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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?
![Page 90: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/90.jpg)
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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
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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
![Page 92: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/92.jpg)
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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
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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
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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
![Page 95: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/95.jpg)
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Upshot: More modes are interesting (“well measured” in a grid invariant sense) than previously thought.
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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?
![Page 97: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/97.jpg)
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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?
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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?
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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?
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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?
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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?
![Page 102: Dark energy paramters Andreas Albrecht (UC Davis) U Chicago Physics 411 guest lecture](https://reader035.fdocuments.in/reader035/viewer/2022062314/56814650550346895db360e0/html5/thumbnails/102.jpg)
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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?
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How is the DoE/ESA/NASA Science Working Group looking at these questions?
i) Using w(a) eigenmodes
ii) Revealing value of higher modes
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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
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END
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Additional Slides
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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
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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
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Stage 2
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Stage 2
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Stage 2
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Stage 2
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Stage 2
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Stage 2
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Stage 2
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DETF Stage 3 photo [Opt]
3 3/c
4 4/c
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Eigenmodes:
Stage 3 Stage 4 gStage 4 s
z=4 z=2 z=1 z=0.5 z=0
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Eigenmodes:
Stage 3 Stage 4 gStage 4 s
z=4 z=2 z=1 z=0.5 z=0
N.B. σi change too
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DETF Stage 4 ground [Opt]
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DETF Stage 4 ground [Opt]
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DETF Stage 4 space [Opt]
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DETF Stage 4 space [Opt]
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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)
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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
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Consider discriminating power of each experiment (look at units on axes)
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DETF Stage 3 photo [Opt]
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DETF Stage 3 photo [Opt]
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DETF Stage 4 ground [Opt]
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DETF Stage 4 ground [Opt]
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DETF Stage 4 space [Opt]
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DETF Stage 4 space [Opt]
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Quantify discriminating power:
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Stage 4 space Test Points
Characterize each model distribution by four “test points”
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Stage 4 space Test Points
Characterize each model distribution by four “test points”
(Priors: Type 1 optimized for conservative results re discriminating power.)
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Stage 4 space Test Points
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•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.
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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
2
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
st P
oint
Mod
elComparison Model
[4 models] X [4 models] X [4 test points]
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DETF Stage 3 photoTe
st P
oint
Mod
elComparison Model
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DETF Stage 4 groundTe
st P
oint
Mod
elComparison Model
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DETF Stage 4 spaceTe
st P
oint
Mod
elComparison Model
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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
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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
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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
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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