Theoretical strategies for
constraining dark energy
: challenges and pitfalls
Seokcheon (sky) Lee
skylee@phys.sinica.edu.tw
Institute of Physics, Academia Sinica
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 1/18
Outline
Evidences for the current accelerating expansion
Geometrical tests : H(z), SNe, CMB, BAO
Dynamical tests : Linear growth factor (EG), Nonlinear
growth (SCM), Cluster numbers
Optimal strategies
Parametrizations of ω
Pitfalls
Future work : SZe, WL
Summary
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 2/18
Make sense ?
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 3/18
Geometrical Probes
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 4/18
Geometrical probes : SNe Type Ia
Dataset
SNLS1
Redshift Range
0.015 ≤ z ≤ 1.01
# of SN
115
Filtered subsets
SNLS, LR
Released
2005
Gold06
0.024 ≤ z ≤ 1.76
182
SNLS1, HST, SCP, HZSST
2006
ESSENCE
0.016 ≤ z ≤ 1.76
192
SNLS1, HST, ESSENCE
2007
Union
0.015 ≤ z ≤ 1.55
307
Gold06, ESSENCE
2008
Constitution
0.015 ≤ z ≤ 1.55
397
Union, CfA3
2009
SDSS
0.022 ≤ z ≤ 1.55
288
Nearby, SDSS-II, ESSENCE,
SNLS, HST
luminosity distance :
Rz
dz ′
dL (z) = c(1 + z) 0
H(z ′ )
,
where

3 + (1 − Ω
Ωm0 (1 + z)
H(z) = H0 
m0 )
Rz
× exp[3 0 (1 + ω(x))d ln(1 + x)]
distance modulus :
µth = mth − M
= 5 log10 [dL (z)] + 42.38



1
2
2009
Geometrical probes : SNe Type Ia
Dataset
SNLS1
Redshift Range
0.015 ≤ z ≤ 1.01
# of SN
115
Filtered subsets
SNLS, LR
Released
2005
Gold06
0.024 ≤ z ≤ 1.76
182
SNLS1, HST, SCP, HZSST
2006
ESSENCE
0.016 ≤ z ≤ 1.76
192
SNLS1, HST, ESSENCE
2007
Union
0.015 ≤ z ≤ 1.55
307
Gold06, ESSENCE
2008
Constitution
0.015 ≤ z ≤ 1.55
397
Union, CfA3
2009
SDSS
0.022 ≤ z ≤ 1.55
288
Nearby, SDSS-II, ESSENCE,
2009
SNLS, HST

3 + (1 − Ω
Ωm0 (1 + z)
H(z) = H0 
m0 )
distance modulus :
µth = mth − M
= 5 log10 [dL (z)] + 42.38
1.2
1.8
0.6
z


wHzL
Rz
× exp[3 0
(1 + ω(x))d ln(1 + x)]
0.6
1
2
1.2
z
1.8
2
1.5
1 ODEP W0 m=0.20
0.5
0
-0.5
-1
-1.5
0.6
z
2
1.5
1 Gold W0 m=0.30
0.5
0
-0.5
-1
-1.5
0.6
1.2
wHzL
2
1.5
1 SNLS W0 m=0.20
0.5
0
-0.5
-1
-1.5
1.8
2
1.5
1 SNLS W0 m=0.30
0.5
0
-0.5
-1
-1.5
0.6
1.2
z
1.2
1.8
z
wHzL
where
wHzL
,
wHzL
Rz
dz ′
dL (z) = c(1 + z) 0
H(z ′ )
2
1.5
1 Gold W0 m=0.20
0.5
0
-0.5
-1
-1.5
wHzL
luminosity distance :
1.8
2
1.5
1 ODEP W0 m=0.30
0.5
0
-0.5
-1
-1.5
0.6
1.2
1.8
z
Consistency with standard rulers : SNLS1 > SDSS > Constitution >
Union > ESSENCE > Golden06
Geometrical probes : SNe Type Ia
Dataset
SNLS1
Redshift Range
0.015 ≤ z ≤ 1.01
# of SN
115
Filtered subsets
SNLS, LR
Released
2005
Gold06
0.024 ≤ z ≤ 1.76
182
SNLS1, HST, SCP, HZSST
2006
ESSENCE
0.016 ≤ z ≤ 1.76
192
SNLS1, HST, ESSENCE
2007
Union
0.015 ≤ z ≤ 1.55
307
Gold06, ESSENCE
2008
Constitution
0.015 ≤ z ≤ 1.55
397
Union, CfA3
2009
SDSS
0.022 ≤ z ≤ 1.55
288
Nearby, SDSS-II, ESSENCE,
2009
SNLS, HST
42
44.5
44.0
40
luminosity distance :
Rz
dz ′
dL (z) = c(1 + z) 0
H(z ′ )
H(z) =

Ωm0 (1
H0 
43.5
,
Μ
where
distance modulus :
µth = mth − M
Μ
43.0
42.5
36
+ z)3 + (1 − Ωm0 )
Rz
× exp[3 0
(1 + ω(x))d ln(1 + x)]
38
1
2


42.0
41.5
34
0.0
0.1
0.2
0.3
0.4
41.0
0.4
z
Models :
0.5
0.6
0.7
0.8
0.9
z
ω
=
−1.2(orange),
−1(green),
−0.8(yellow), and (ω0 , ωa ) = (−0.897, −0.885)
for CPL (blue)
= 5 log10 [dL (z)] + 42.38
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 5/18
1.0
Geometrical probes : H(z)
H(z) from passively evolving galaxies data :
D.Stern et.al. [2010]
z
H(z)
0.09
0.17
0.27
0.4
0.48
69 ± 12
83 ± 8
77 ± 14
95 ± 17
97 ± 62
0.88
0.9
1.3
1.43
1.53
1.75
90 ± 40
117 ± 23
168 ± 17
177 ± 18
140 ± 14
202 ± 40
Geometrical probes : H(z)
5
250
4
200
DH
HHzL
H
150
3
H%L
2
1
100
0
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
z
H(z) from different models. Orange(PCA),
Red(linear), Blue(logarithmic), Magenta(CPL),
Skyblue(tanh)
Relative errors of different model w.r.t ΛCDM :
SL[2011]
1.5
z
2.0
2.5
Geometrical probes : H(z)
5
250
4
200
DH
HHzL
H
150
3
H%L
2
1
100
0
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
2.0
2.5
z
z
0.0
H(z) from different models. Orange(PCA),
Red(linear), Blue(logarithmic), Magenta(CPL),
z
1+z
Ωa +Ωb
-0.2
0.0
Ωa +Ωb Tanh@lnH1+zLD
-0.2
-0.4
-0.4
Skyblue(tanh)
-0.6
Ω
Ω
-0.8
-0.6
-0.8
Relative errors of different model w.r.t ΛCDM :
-1.0
-1.0
SL[2011]
-1.2
-1.2
-1.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-1.4
0.0
0.2
z
z and
Models : ω = ωa + ωb 1+z
ωa + ωb tanh[ln(1 + z)]
1-σ error
0.4
0.6
0.8
z
1.0
1.2
1.4
Geometrical probes : H(z)
5
250
4
200
DH
HHzL
H
150
3
H%L
2
1
100
0
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
2.0
2.5
z
z
6
H(z) from different models. Orange(PCA),
4
Red(linear), Blue(logarithmic), Magenta(CPL),
4
Skyblue(tanh)
2
2
Ωb
Ωb
Relative errors of different model w.r.t ΛCDM :
0
0
SL[2011]
-2
-2
-2.5
-2.0
-1.5
-1.0
Ωa
-0.5
0.0
0.5
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Ωa
Contour plots of ωa and ωb for the corresponding models 1 and 2-σ
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 6/18
Geometrical probes : CMB & BAO
CMB :
Shift parameter R : the ratio of the location of the first acoustic peak of a reference flat SCDM model to one of a fiducial
′
l1
model : J.R.Bond et.al. [1997] , RWMAP =
= 1.123 ± 0.03
l1
′
q
q
q(Ωr ,arec )
′
′
′
2
R = q
, where q ≡
Ωr + 1 −
arec + Ωr
H0 r(z)
Ωm0
Both CMB and BAO also provide the dynamical probes : ISW effect SL [MPLA,2008]
R η0
Θl (k, η0 ) = (2l + 1) ηrec
dηe−τ 2Φ̇jl [k(η0 − η)] and changing amplitude of BAO SL et.al. [PRDR,2010]
Geometrical probes : CMB & BAO
CMB :
Shift parameter R : the ratio of the location of the first acoustic peak of a reference flat SCDM model to one of a fiducial
′
l1
model : J.R.Bond et.al. [1997] , RWMAP =
= 1.123 ± 0.03
l1
′
q
q
q(Ωr ,arec )
′
′
′
2
R = q
, where q ≡
Ωr + 1 −
arec + Ωr
H0 r(z)
Ωm0
Both CMB and BAO also provide the dynamical probes : ISW effect SL [MPLA,2008]
R η0
Θl (k, η0 ) = (2l + 1) ηrec
dηe−τ 2Φ̇jl [k(η0 − η)] and changing amplitude of BAO SL et.al. [PRDR,2010]
BAO :
Radial size : AB = ∆r = ∆t = ∆z
a
H(z)
R
Transverse size : CD = r∆θ = ∆θ 0z dz
H(z)
Dilation scale :
"
#1
R z
2 z
3
dz
BAO
BAO
DV (z) =
0
H(z)
H(zBAO )
SDSS : zBAO ≃ 0.35 : D.J. Einstein et.al. [2005]
DV (zBAO ) = 1370 ± 64 Mpc
Similar to Alcock-Pazcynski (AP) test ∆z = H(z)r(z)
∆θ
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 7/18
Dynamical Probes
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 8/18
Linear growth factor
at sub-horizon scale, matter density perturbation δm (~
k, a) = δ0 (k)Dg (a) grows uniformly as long
2
d ln H
3 dD
3 Ωm0
as DE does not cluster : d D
+ a
2 +
5 f (k, a)D = 0
da
da − 2
da
a
3ω−1 3ω−1 h
i
Ω
3ω +
6ω a
2
a
F 1 − 1 , 1 + 1 , 3 − 1 , − m0
2
2ω 2
3ω 2
6ω
Ω0
de
h
i
Ωm0 3ω
1
1
1
1
c2 F −
,
,
+
,− 0 a
: SL et.al. [PRD2010, PLB2010]
3ω 2ω 2
6ω
Ωde
D(a) = c1
Ω
m0
Ω0
de
Linear growth factor
at sub-horizon scale, matter density perturbation δm (~
k, a) = δ0 (k)Dg (a) grows uniformly as long
2
d ln H
3 dD
3 Ωm0
as DE does not cluster : d D
+ a
2 +
5 f (k, a)D = 0
da
da − 2
da
a
3ω−1 3ω−1 h
i
Ω
3ω +
6ω a
2
a
F 1 − 1 , 1 + 1 , 3 − 1 , − m0
2
2ω 2
3ω 2
6ω
Ω0
de
h
i
Ωm0 3ω
1
1
1
1
c2 F −
,
,
+
,− 0 a
: SL et.al. [PRD2010, PLB2010]
3ω 2ω 2
6ω
Ωde
D(a) = c1
Ω
m0
Ω0
de
0.5
0.75
0.80
0.75
0.65
0.70
-0.5
f
D
0.0
0.70
Ω
0.60
0.65
0.55
0.60
0.50
0.55
0.45
0.50
-1.0
0.5
0.6
0.7
0.8
0.9
1.0
-1.5
0.5
0.6
0.7
a
0.8
a
0.9
1.0
-2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0
Wm
f (a) = d ln D ≡ Ωm (a)γ SL et.al. [2009]
d ln a
i
h
3
Γ[ 11 ]/Γ[ 5 ]
0 )2
6
6
ln − 3 Ω0
+(Ω
m
2 m
0
F [ 3 , 5 , 11 ,−Ω0
0
de /Ωm ]
2 6 6
γL =
ln Ω0
m
EG =
Ωm0
SL et.al. [in preparation], SL [JCAP 2011]
f (a)
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 9/18
Spherical collapse model
R̈ = − 4πG ρ
Ṙ
cluster + (1 + 3ω)ρdec , ρ̇cluster + 3 R ρcluster = 0
R
3
ρ
ρ̇dec + 3(1 + ω) Ṙ ρdec = αΓ where Γ = 3(1 + ω) Ṙ − ȧ ρdec with 0 ≤ α ≤ 1 ζ ≡ cluster R
R
a
ρm
ta
spherical collapse model
3ω−1 3ω−1 h
i
Ω
3ω +
6ω a
2
F 1 − 1 , 1 + 1 , 3 − 1 , − m0
a
2
2ω 2
3ω 2
6ω
Ω0
de
h
i
Ω
c2 F − 1 , 1 , 1 + 1 , − m0
a3ω : SL et.al. [JCAP2010, PLB2010]
3ω 2ω 2
6ω
Ω0
de
D(a) = c1
Ω
m0
Ω0
de
Spherical collapse model
R̈ = − 4πG ρ
Ṙ
cluster + (1 + 3ω)ρdec , ρ̇cluster + 3 R ρcluster = 0
R
3
ρ
ρ̇dec + 3(1 + ω) Ṙ ρdec = αΓ where Γ = 3(1 + ω) Ṙ − ȧ ρdec with 0 ≤ α ≤ 1 ζ ≡ cluster R
R
a
ρm
ta
spherical collapse model
3ω−1 3ω−1 h
i
Ω
3ω +
6ω a
2
F 1 − 1 , 1 + 1 , 3 − 1 , − m0
a
2
2ω 2
3ω 2
6ω
Ω0
de
h
i
Ω
c2 F − 1 , 1 , 1 + 1 , − m0
a3ω : SL et.al. [JCAP2010, PLB2010]
3ω 2ω 2
6ω
Ω0
de
D(a) = c1
Ω
m0
Ω0
de
2.6
2.2
2.4
2.1
1.20
à
æ
ì
æ
Ζ
3Π 2
J 4 N
à
1.18
2.2
Ζ
2.0
3Π 2
J 4 N
æ
2.0
à
1.16
1.9
Τ
æ
à
æ
1.14
1.8
à
æ
1.8
1.12
1.6
à
æ
à
æ à
1.10
1.7
æ
1.4
0.15
0.20
0.25
0.30
0.35
1.08
0.40
0.2
0.4
W0m
0.6
0.8
1.0
0.0
0.1
0.2
zta
0.3
0.4
0.5
y
2 −0.724+0.157Ωmta +α(1+ωde )(1+3ωde )(0.064−0.368Ωmta )
ζsk = 3π Ωmta
SL et.al. [JCAP 2010]
4










√ 2
δlin (zvir ) = 3 ( ζ) 3
5
ρ
∆(zvir ) = cluster
ρm












1
3 + 9π √
4
8
ζ





zvir
x
= ζ vir
yvir










3






















2
3
2
= 3 (6 + 9π) 3 ≃ 1.58
20
= 18π 2
x3
vir
→ 147 instead of 178
x
4(F 1 , 3 , 5 ,− vir )2
2 2 2
Qta










NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 10/18
Cluster number
linear perturbation of DE : δQ̈ + 3HδQ̇ + (k2 + V,QQ )δQ = − 1
ḣQ̇0
2


D(a) 
2 (k)


linear power spectrum of δm : P (k, a) = AQ kns TQ
D(a0 )
2
2 2
ns +3
Q = 2π δH (c/H0 )
, where A
,
−1
δH = 2.05 × 10−5 α0 (Ωm )c1 +c2 ln Ωm exp[c3 (ns − 1) + c4 (ns − 1)2 ] with
c1 = −0.789|ω|0.0754−0.211 ln |ω| , c2 = −0.118 − 0.0727ω , c3 = −1.037 , c4 = −0.138 ,
α = (−ω)s ,
!
s = (0.012 − 0.036ω − 0.017ω −1 ) 1 − Ωm (a) + (0.098 + 0.029ω − 0.085ω −1 ) ln Ωm (a) : Ma et.al.
[1999], SL et.al. [2010]
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 11/18
Cluster number
linear perturbation of DE : δQ̈ + 3HδQ̇ + (k2 + V,QQ )δQ = − 1
ḣQ̇0
2


D(a) 
2 (k)


linear power spectrum of δm : P (k, a) = AQ kns TQ
D(a0 )
2
2 2
ns +3
Q = 2π δH (c/H0 )
, where A
,
−1
δH = 2.05 × 10−5 α0 (Ωm )c1 +c2 ln Ωm exp[c3 (ns − 1) + c4 (ns − 1)2 ] with
c1 = −0.789|ω|0.0754−0.211 ln |ω| , c2 = −0.118 − 0.0727ω , c3 = −1.037 , c4 = −0.138 ,
α = (−ω)s ,
!
s = (0.012 − 0.036ω − 0.017ω −1 ) 1 − Ωm (a) + (0.098 + 0.029ω − 0.085ω −1 ) ln Ωm (a) : Ma et.al.
[1999], SL et.al. [2010]
4
2.0
1.0
4
1 ´ 10
5000
ΣHM, z=0L
0.9
2000
Σ8
PHkL @Hh-1 MpcL3 D
2 ´ 10
1000
0.8
500
0.7
200
100
1.5
1.0
0.6
0.5
0.001
0.01
0.1
1
-1.1
-1.0
-0.9
k @h Mpc-1 D
-0.8
-0.7
-0.6
-0.5
9
2 (a) ≡
rms linear mass fluctuation : σR
2
δM
=
M (R,a) ω = −1.1 (brown), −1.0 (red), −0.8 (blue)
σ(M, z) ≃ (−ω)0.72+0.36ω 3.90 − 0.215 log
h
10
11
12
13
14
15
16
Log@MHh-1 MŸ LD
ΩQ
1 R ∞ k2 P (k, a)W (kR)2 dk
2π 2 0
M
h−1 M⊙
i D (z) g
Dg (z0 )
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 11/18
Cluster number
linear perturbation of DE : δQ̈ + 3HδQ̇ + (k2 + V,QQ )δQ = − 1
ḣQ̇0
2


D(a) 
2 (k)


linear power spectrum of δm : P (k, a) = AQ kns TQ
D(a0 )
2
2 2
ns +3
Q = 2π δH (c/H0 )
, where A
,
−1
δH = 2.05 × 10−5 α0 (Ωm )c1 +c2 ln Ωm exp[c3 (ns − 1) + c4 (ns − 1)2 ] with
c1 = −0.789|ω|0.0754−0.211 ln |ω| , c2 = −0.118 − 0.0727ω , c3 = −1.037 , c4 = −0.138 ,
α = (−ω)s ,
!
s = (0.012 − 0.036ω − 0.017ω −1 ) 1 − Ωm (a) + (0.098 + 0.029ω − 0.085ω −1 ) ln Ωm (a) : Ma et.al.
[1999], SL et.al. [2010]
0.8
0.7
0.6
0.6
Dg
VHzL @HcH0 -1 L3 D
0.8
0.5
0.4
0.4
0.2
0.3
0.2
0.0
0.0
0.5
1.0
1.5
2.0
z
0
1
2
3
4
z
′ cdt ′
′
V (z) = 0 4πd2
A (z ) dz (z )dz
Rz
linear growth factor : Dg
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 11/18
Cluster number
linear perturbation of DE : δQ̈ + 3HδQ̇ + (k2 + V,QQ )δQ = − 1
ḣQ̇0
2


D(a) 
2 (k)


linear power spectrum of δm : P (k, a) = AQ kns TQ
D(a0 )
2
2 2
ns +3
Q = 2π δH (c/H0 )
, where A
,
−1
δH = 2.05 × 10−5 α0 (Ωm )c1 +c2 ln Ωm exp[c3 (ns − 1) + c4 (ns − 1)2 ] with
c1 = −0.789|ω|0.0754−0.211 ln |ω| , c2 = −0.118 − 0.0727ω , c3 = −1.037 , c4 = −0.138 ,
α = (−ω)s ,
!
s = (0.012 − 0.036ω − 0.017ω −1 ) 1 − Ωm (a) + (0.098 + 0.029ω − 0.085ω −1 ) ln Ωm (a) : Ma et.al.
-5.5
ææ
æ
-6.0
æ
-6.5
ò
-7.0
ò
ò
-7.5
-8.0
14.0
14.2
14.4
14.6
14.8
15.0
log10 M@MŸ h-1 D
15.2
15.4
log10 n H> M L @Hh Mpc-1 L3 D
-5.0
Hn∆c =1.58 -n∆c =1.69 Ln∆c =1.58 H%L
log10 n H> M L @Hh Mpc-1 L3 D
[1999], SL et.al. [2010]
100
80
60
40
20
0
9
10
11
12
13
14
15
0
-2
-4
-6
-8
16
9
log10 M@MŸ h-1 D
10
11
12
13
14
15
16
log10 M@MŸ h-1 D
comoving number density for different z = 0, 0.5, 1.0, 2.0 (from top to bottom) when ω = −1 and δc = 1.58 Data from R.G. Carlberg et.al.
[1997] errors of n when δc = 1.69 PS : dn(M, z) =
r
0 2 2 ρm d ln σ δc exp − δc dM
π M 2 d ln M σ
2σ 2
s
n(> M ) from PS with δc = 1.58 (blue) and ST with 1.69 (red) fST (σ) = A
2
2b exp − bδc
π
2σ 2
1+
σ 2 p δc
2
σ
bδc
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 11/18
Cluster number
linear perturbation of DE : δQ̈ + 3HδQ̇ + (k2 + V,QQ )δQ = − 1
ḣQ̇0
2


D(a) 
2 (k)


linear power spectrum of δm : P (k, a) = AQ kns TQ
D(a0 )
2
2 2
ns +3
Q = 2π δH (c/H0 )
, where A
,
−1
δH = 2.05 × 10−5 α0 (Ωm )c1 +c2 ln Ωm exp[c3 (ns − 1) + c4 (ns − 1)2 ] with
c1 = −0.789|ω|0.0754−0.211 ln |ω| , c2 = −0.118 − 0.0727ω , c3 = −1.037 , c4 = −0.138 ,
α = (−ω)s ,
!
s = (0.012 − 0.036ω − 0.017ω −1 ) 1 − Ωm (a) + (0.098 + 0.029ω − 0.085ω −1 ) ln Ωm (a) : Ma et.al.
0.30
0.25
fHΣL
0.20
0.15
0.10
0.05
0.00
80
0
Dn H> M L H%L
log10 n H> M L @Hh Mpc-1 L3 D
[1999], SL et.al. [2010]
-2
-4
-6
1.5
2.0
2.5
40
20
0
-8
9
1.0
60
10
11
3.0
12
13
14
15
9
16
10
1Σ
11
12
13
14
15
16
log10 M@MŸ h-1 D
log10 M@MŸ h D
-1
different mass functions : fST , fM anera , fLN , fmod (from top to bottom) , Using original :
fLN (σ, z) = 0.32
s

2
0.67δc
2(0.67)
exp
−
π
2σ 2


1
+



σ2
2
0.67δc

0.32 




δc
σ
n(> M ) with fLN for ω = −1 at z = 0, 0.5, 1, 2 (from top to bottom) relative errors of n(> M ) between ω = −1.1 and −1.0
at z = 0(solid) and 1(dashed) and between ω = −0.8 and −1.0 at z = 0(dot-dashed) and 1(long dashed)
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 11/18
Optimal Strategies
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 12/18
Parametrization of ω
Absence of compelling model requires the parametrization of ω :
Principal component analysis (PCA) : ω =
P
i ωi Θ(z − zi ) Tegmark et.al. [ApJ 1997]
so-called CPL parametrization : ω = ωa + ωb z
1+z
ω = ωa + ωb tanh[ln[1 + z]] SL [2011]
Fisher matrix :
P
Flm = −
P
Oi −O(zi ,~
p) ∂ 2 O(zi ,~
p)
+
2
∂p
∂p
m
σi
l
p) ∂O(zi ,~
p)
1 ∂O(zi ,~
f
≡ F
lm
∂pl
∂pm
σi2
P
p) ∂O(zi ,~
p)
1 ∂O(zi ,~
≃
2
∂p
∂p
m
σi
l
when ρde is a direct variable of O :
2 R z
P 1 ∂O(zi ,~
p)
i ∂ω d ln(1 + x) R zi ∂ω d ln(1 + x) ≡
e
F
3
lm =
0 ∂pm
0 ∂pl
∂ ln[ρde (zi ,~
p)]
σi2
R zi ∂ω
P 2 R zi ∂ω
Gi
0 ∂pl d ln(1 + x)
0 ∂pm d ln(1 + x)
2 G2 ln[1 + z ] R zj f (x)d ln[1 + x] − ln[1 + z ] R zi f (x)d ln[1 + x] 2 =
G
i 0
j 0
j=i+1 i j
i=1
 2


ln[1 + zi ]zj − zi ln[1 + zj ]
if f (z) = z




1+z 

2
PN −1 PN
j
1 ln[1 + z ] ln[1 + z ] ln
2 2
if f (z) = ln[1 + z]
i
j
j=i+1 Gi Gj 
2
1+z
i=1

i


zj
2

zi

z

ln[1 + zj ] −
ln[1 + zi ]
if f (z) =

1+zi
1+zj
1+z
f)
det(F
H =
PN −1 PN
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 13/18
Parametrization of ω
0.2
0.0
0.0
Ωa +Ωb z
0.0
Ωa +Ωb ln@1+zD
-0.2
-0.2
z
1+z
Ωa +Ωb
-0.2
0.0
Ωa +Ωb Tanh@lnH1+zLD
-0.2
-0.4
-0.4
-0.4
-0.4
Ω -0.6
Ω
-0.6
-0.6
Ω
-0.8
-0.8
Ω
-0.8
-0.6
-0.8
-1.0
-1.0
-1.0
-1.0
-1.2
-1.2
-1.2
-1.2
-1.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-1.4
1.4
0.0
0.2
0.4
0.6
z
0.8
1.0
1.2
-1.4
1.4
0.0
0.2
0.4
0.6
z
0.8
1.0
1.2
-1.4
1.4
0.0
0.2
0.4
z
4
0.6
0.8
1.0
1.2
1.4
z
6
2
4
3
4
2
1
2
2
1
Ωb
Ωb
Ωb
Ωb
0
0
0
0
-1
-1
-2
-2
-2
-2
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
-2.5
-2.0
-1.5
Ωa
σω =
-1.0
-0.5
0.0
-2.5
-2.0
-1.5
Ωa
Pn
l=1
∂ω
∂ωl
2
Cll + 2
Pn
Caa + Cbb f (z)2 + 2Cab f (z)
-1.0
-0.5
0.0
0.5
-2.5
Ωa
l=1
Pn
m=l+1
∂ω
∂ωl
-2.0
-1.5
-1.0
-0.5
0.0
Ωa
∂ω
∂ωm
Clm =
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 14/18
Parametrization of ω
0.2
0.0
0.0
Ωa +Ωb z
0.0
Ωa +Ωb ln@1+zD
-0.2
-0.2
z
1+z
Ωa +Ωb
-0.2
0.0
Ωa +Ωb Tanh@lnH1+zLD
-0.2
-0.4
-0.4
-0.4
-0.4
Ω -0.6
Ω
-0.6
-0.6
Ω
-0.8
-0.8
Ω
-0.8
-0.6
-0.8
-1.0
-1.0
-1.0
-1.0
-1.2
-1.2
-1.2
-1.2
-1.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-1.4
1.4
0.0
0.2
0.4
z
0.6
0.8
1.0
1.2
-1.4
1.4
0.0
0.2
0.4
0.6
z
0.8
1.0
1.2
-1.4
1.4
0.0
0.2
0.4
z
4
0.6
0.8
1.0
1.2
1.4
z
6
2
4
3
4
2
1
2
2
1
Ωb
Ωb
Ωb
Ωb
0
0
0
0
-1
-1
-2
-2
-2
-2
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
-2.5
-2.0
-1.5
Ωa
f (z)
-1.0
0.0
-0.5
-2.5
-2.0
-1.5
Ωa
P Oi −O(zi ,~p) ∂ 2 O(zi ,~p)
σi2
∂ωa ∂ωb
P
p)
p) ∂O(zi ,~
1 ∂O(zi ,~
∂ωb
σi2 ∂ωa
-1.0
-0.5
Ωa
0.0
0.5
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Ωa
ωa∗
ωb∗
χ2min
det(F )
Ref
(a, a)
(a, b)
(b, b)
(a, a)
(a, b)
(b, b)
z
0.19
-0.66
-1.04
92.51
52.19
31.29
-0.834
0.211
9.486
149.3
[?]
ln[1 + z]
-0.12
-0.46
-0.45
93.14
38.52
16.72
-0.875
0.388
9.446
65.07
[?]
z
1+z
-0.34
-0.33
-0.21
93.66
29.65
9.75
-0.916
0.638
9.412
31.19
[?, ?]
tanh[ln[1 + z]]
-0.34
-0.41
-0.31
93.63
34.66
13.38
-0.906
0.520
9.413
46.35
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 14/18
Pitfalls
e:
determinant of F
f) =
det(F
PN −n+1 PN −n+2
i=1
j=i+1
···
2
2 G2 · · · G2 ε
G
W
(z
)W
(z
)
·
·
·
W
(z
)
a
n
i
ij···l
b j
l
l=n i j
l
PN
N number of components : N = N (N − 1)(N − 2) · · · (N − n + 1)/n!
n
n
PCA number of components decreases
-0.75
-1.10
-0.80
-1.15
-0.85
-0.90
Ωi
Ω i -1.20
-0.95
-1.00
-1.25
-1.05
-1.10
-1.30
0.0
0.5
1.0
1.5
2.0
0.0
z
a) The fiducial model is ω = −1.1 + 0.5
0.5
1.0
1.5
2.0
z
z (dashed) and the obtained values of ω s from PCA with H data (dotted),
i
1+z
DL data (dot-dashed), and H + DL data (solid). Error bars are obtained from the analysis of H + DL data.
z (dashed) and the obtained values of ω s from PCA with H data (dotted),
b) The fiducial model is ω = −1.1 − 0.3 1+z
i
DL data (dot-dashed), and H + DL data (solid)
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 15/18
Future works
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 16/18
′
Sunyaev-Zel dovich effect
angular power spectrum of the SZe using halo formalism:
2 R zmax dz dV R Mmax dM dn(M,z) |ỹ (M, z)|2
Cl = gν
l
0
dz Mmin
dM
gν : spectral function of SZe (-2 in Rayleigh-Jeans limi)
dn(M, z)/dM : comoving DM halo mass function
ỹl (M, z) : 2D Fourier transform of the projected Compton y-parameter
redshift distribution of Cl for a given l :
d ln Cl
=
d ln z
R
R
z dV
dM dn |ỹl |2
dz
dM
R
dz dV
dM dn |ỹl |2
dz
dM
haloes at z ∼ 1 determined Cl at l ∼ 3000 E. Komatsu et.al. [2002]
haloes at z ∼ 2 determined Cl at l ∼ 10000
mass distribution of Cl for a given l :
d ln Cl
=
d ln M
R
R
dz dV dn |ỹl |2
dz dM
R
dz dV
dM dn |ỹl |2
dz
dM
M
haloes 1014 h−1 M⊙ < M < 1014 h−1 M⊙ dominate Cl at l ∼ 3000 with peak at 3 × 1014 h−1 M⊙
l = 1000 peak at 5 × 1014 h−1 M⊙
l = 10000 peak at 1014 h−1 M⊙
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 17/18
Who am I ?
Who am I ?
After too much adoption
from other animals, she
′
even can t catch a fly.
How about in Cosmology
? (Growth factor,
Nonlinear model, mass
function, · · · )
Need to be consistent
NCTS HEP Journal Club : Theoretical strategies for constraining DE : sky Lee May 31, 2011 – p. 18/18