Probing Dark Energy
via Supernova Observations :
How Parametrization alters Conclusion
Je-An Gu (顧哲安)
National Center for Theoretical Sciences
2007/03/08 @ NTHU
Content
™ Introduction
(Basic Knowledge; Motivation)
™ Fits (fitting formulae) to dL– z Relation [& wDE(z)]
(parametrization)
™ Reconstruction of wDE(z) and Fit Testing
™ Summary and Discussion
Introduction
(Motivation; Basic Knowledge, SNAP)
Accelerating Expansion
Based on FLRW Cosmology
(homogeneous & isotropic)
Supernova (SN)
Type Ia Supernova (SN Ia) :
– thermonulear explosion of carbon-oxide white dwarfs –
• Correlation between the peak luminosity and the decline rate
⇒ absolute magnitude M
⇒ luminosity distance dL
(distance precision: σmag = 0.15 mag → δdL/dL ~ 7%)
• Spectral information → redshift z
SN Ia Data: dL(z) [ i.e, dL,i(zi) ]
[ ~ x(t) ~ position (time) ]
Basic Knowledge
™ Luminosity distance dL
F: flux (energy/area×time)
L: luminosity (energy/time)
F= L
4π d L2
2
⎛
dr
2
2
2
2
2⎞
+ r dΩ ⎟⎟
FLRW Cosmology ds = −dt + a (t ) ⎜⎜
2
⎝ 1− k r
⎠
d L ( z ) = a0 r ( z )(1 + z ) = r ( z )(1 + z ) for a0 ≡ 1 (flat RW) ,
r(z) : coordinate distance
(flat RW)
r (z) = ∫
z
0
1
dz'
H ( z' )
(1+z = a0/a)
Basic Knowledge
™ Magnitude m
F2 / F1 = 100 ( m1 − m2 ) / 5
( i.e. F ∝ 100 − m / 5 )
™ Absolute Magnitude M = m (10pc)
( m −M ) / 5
2
100
= F10pc / F = (d L / 10pc )
™ Distance Modulus μ ≡ m − M
μ ≡ m − M = 5 log10 (d L / Mpc ) + 25
SN Ia Data: dL(z) → μ(z)
Δm( z ) = m( z ) − mΛ ( z )
1998
↑
fiducial model
Ω Λ = 0.7, Ω m = 0.3
SCP
(Perlmutter et. al.)
(can hardly distinguish different models)
2004
Fig.4 in astro-ph/0402512 [Riess et al., ApJ 607 (2004) 665]
Gold Sample (data set) [MLCS2k2 SN Ia Hubble diagram]
- Diamonds: ground based discoveries
- Filled symbols: HST-discovered SNe Ia
- Dashed line: best fit for a flat cosmology: ΩM=0.29 ΩΛ=0.71
2006
Riess et al. astro-ph/0611572
2006
Riess et al. astro-ph/0611572
Supernova / Acceleration Probe (SNAP)
observe ~2000 SNe in 2 years
statistical uncertainty σmag = 0.15 mag → 7% uncertainty in dL
σsys = 0.02 mag at z =1.5
σz
= 0.002 mag (negligible)
z
0~0.2
0.2~1.2
1.2~1.4
1.4~1.7
# of SN
50
1800
50
15
Observations
mapping out the evolution history
(e.g. SNe Ia , Baryon Acoustic Oscillation)
Phenomenology
Data Analysis
(invoking “fitting formula” / “parametrization”)
Model / Theory
Motivation
™ Potential impact of improved future SNe data on
our understanding of dark energy.
[ To obtain DE info, e.g. wDE(z) ]
™ Answering the questions: wDE = –1 ?
wDE′ = 0 ?
™ Showing the importance of a “good” fit to the dL– z
relation in order to draw conclusion on above issues.
Observations / Data
mapping out the evolution history
(e.g. SNe Ia , Baryon Acoustic Oscillation)
Fitting Formula / Parametrization
(model-independent ?)
N
N
⎛
⎞
i
⎜⎜ e.g. d L ( z ) = ∑ c i z ; w φ = ∑ w i (1 + z )i ⎟⎟
i =o
i =0
⎠
⎝
Information about Physical Quantities
Characterizing our universe or DE models
to be reconstructed
(e.g. wφ , ρφ , statefinders{r,s} , ……)
Fits (Fitting Formulae)
to dL– z Relation
Two Fits
polynomial fit of dL
N
Fit 1
dL (z ) = ∑ ci z i
{ dL(0) = 0 ⇒ c0 = 0 }
i =0
Huterer & Turner, 1999, PRD
N
Fit 2 w φ ( z ) = ∑ w i (1 + z )
i =0
i
Maor, Brustein, & Steinhardt, 2001, PRL
Astier, astro-ph/0008306
Weller & Albrecht, 2001, PRL
Fit 1
N
dL (z ) = ∑ ci z i
i =0
(flat RW)
{ dL(0) = 0 ⇒ c0 = 0 }
(polynomial fit of dL)
ρ&φ
= −3H (1 + wφ )
ρφ
r (z) = ∫
a& 8πG
H= =
( ρ m + ρφ )
a
3
1+wφ ← H & ρφ
ρφ ← H (or r ′) & ρm(z)
[r (z ) = dL (z ) /(1 + z )]
(a0 ≡ 1)
a −1 = 1 + z
z
0
1
dz'
H ( z' )
(H = 1/r ′)
(well known)
Å [ data Î dL or ci ]
( Ωi ≡ ρi / ρc )
3
′
′
′
1 + z 3Ω H (1 + z ) + 2r / r
⇒ 1 + wφ =
3
2
3
Ω H (1 + z ) − 1/ r ′
2
m 0
2
m 0
2
Fit 2
N
w φ = ∑ w i (1 + z )i
i =0
1+ z
fit
dL (z) =
Ho
∫
(1 + z′)−3 / 2 d z′
z
0
Ω m + Ωφ (1 + z′)
(Ω
φ
3w 0
[
]
⎧ N
⎫
i
exp ⎨3 ∑ (1 + z′) − 1 w i / i ⎬
⎩ i =1
⎭
= 1− Ωm )
Other Fits / Parametrizations
• w(a)= w0+waz/(1+z)
Linder 2003, PRL 90, 091301
•
H ( x)
h( x ) =
= Ω m x 3 + A0 + A1 x + A2 x
H0
x = 1 + z , A0 + A1 + A2 = 1 − Ω m
Alam, Sahni, Saini & Starobinsky, astro-ph/0311364
Reconstruction of wDE (z)
Testing Parametrizations
( Weller & Albrecht 2001, PRD, astro-ph/0106079 )
Procedures of Testing Parametrizations
Background cosmology
[pick a DE Model with wDEth(z)]
↓
generate Data w.r.t. SNAP
(Monte Carlo simulation)
employ a Fit to
the dL(z) or w(z)
reconstruct wDEexp’t (z)
↓
compare wDEexp’t and wDEth
Fit 1
N
d ( z ) = ∑ ci z i
fit
L
background cosmology: ΛCDM { Ωm = 0.3, ΩΛ = 0.7 }
i =0
⎛ Ω ≡ ρi ⎞
⎜ i
ρ c ⎟⎠
⎝
1 + z 3Ω m H 02 (1 + z )2 + 2r ′′ / r ′3
1+ wφ =
3
Ω m H 02 (1 + z )3 − 1/ r ′2
¾Best Fit:
Minimizing the χ2 function:
⎛d
χ ({ c i }) = ∑ ⎜⎜
k =0 ⎝
N
2
z
exp' t
L
(function of ci’s)
(zk ) − d (zk ) ⎞
⎟⎟
δd L ( zk )
⎠
fit
L
2
¾Error Evaluation: Gaussian error propagation: [from dL(z) to w(z)]
dL(z) → w(z)
δw φ = ∑
2
i j
∂w φ ∂w φ
∂c i ∂c j
σ ij
σ ij : covariant matrix of the simulated sample of c i
δd L ( zk ) = σ mag d L ( zk )(ln10) / 5
N
dL (z ) = ∑ ci z i
N=3
N=3
i =0
N=5
wDE
N=4
z
N=4
mean value:
error bar:
N=3 Æ
N=4 Æ
N=5
wrong; N=4.5 Æ ok
best reconstruction for wφ
but only – 1.3 < wφ < – 0.7 at the 1σ level
Fit 2
N
w φ = ∑ w i (1 + z )i
i =0
1+ z z
fit
dL (z) =
H o ∫0
¾Best Fit:
(1 + z′)−3 / 2 dz′
Ω m + Ωφ (1 + z′)
3w 0
[
]
⎧ N
⎫
exp⎨3∑ (1 + z′)i − 1w i / i ⎬
⎩ i =1
⎭
(Ω
φ
minimizing the χ2 function:
⎛d
χ ({ w i }) = ∑ ⎜⎜
k =0 ⎝
Nz
2
exp' t
L
(zk ) − d (zk ) ⎞
⎟⎟
δd L (zk )
⎠
fit
L
¾Error Bar: Gaussian error propagation:
δw φ = ∑
2
i j
∂w φ ∂w φ
∂w i ∂w j
σ ij = ∑ (1 + z)i+ j σ ij
i j
δd L ( zk ) = σ mag d L ( zk )(ln10) / 5
2
= 1− Ωm )
Background Cosmology:
Periodic Potential
V (φ ) = V0 [1 + δ sin( βφ )] e − λφ
PNGB
Inverse Tracker
SUGRA
Brane
Periodic Potential
Pure
Exponential
Two Exponentials
Trapped Minimum Model
Exponential Tracker
wDE
Background Cosmology:
N=0
Periodic Potential
V (φ ) = V0 [1 + δ sin( βφ )] e
N=2
TH
− λφ
N=1
z
N=0
N=0 fit:
N=1 fit:
N=2 fit:
Error bars:
N=1
N=2
wφ = const cannot reproduce the evolving model
poor for z > 0.6
better
{N=1} ~ {N=2} for z < 0.7 ; {N=2} increases rapidly for z > 0.7
Distinguishing Λ and other models via N=1 fit
(Ωm=0.3)
wDE
z
Periodic Potential
Inverse Tracker
SUGRA
V (φ ) = V0 [1 + δ sin( βφ )]e − λφ
V (φ ) = M 4+α / φ α
V (φ ) = M 4 +α φ −α exp (φ / M pl )2 / 2
[
]
Background Cosmology:
Periodic Potential
Ωm=0.3
V (φ ) = V0 [1 + δ sin( βφ )] e − λφ
N=0
N=1
N=0, χ2 = 31 (poor)
N=1, χ2 = 0.47 (good enough)
N=2
N=2, χ2 = 7.3×10-3
Background Cosmology:
Periodic Potential
V (φ ) = V0 [1 + δ sin( βφ )] e − λφ
w, N=1
(Ωm=0.3)
w, N=2
dL, N=3
dL, N=2
best fit
Can we reconstruct an evolving wφ with SNe ?
N
N
i =0
i =0
~ zi
w φ ( z ) = ∑ w i (1 + z )i = ∑ w
i
⎛k ⎞
~
w i = ∑ ⎜⎜ ⎟⎟w k
k =0 ⎝ i ⎠
N
⎛ k ⎞⎛ l ⎞
Gaussian error propagatio n : δw = ∑ ⎜⎜ ⎟⎟⎜⎜ ⎟⎟σ kl
kl ⎝ i ⎠⎝ i ⎠
~ = w +w , w
~ =w
Use N = 1 fit : w
0
0
1
1
1
2
i
The evolution coefficients with error bars for the linear fit
~ +w
~z
wφ = w
0
1
“+” : evolution ; “–” : no evolution; “0” : marginal evolution
~
w
0
δw~0
~
w
1
δw~1
PNGB
Inverse Tracker
SUGRA
Brane
Periodic Potential
Pure
Exponential
Two Exponentials
Trapped Minimum Model
Exponential Tracker
~ +w
~z
wφ = w
0
1
Distinguish wφ = const and evolving wφ
68.3% confidence
99% confidence
SUGRA
SUGRA
wφ=const.
(-1< wφ <0)
Periodic
Periodic
Conclusion
¾ Fit 2
N
⎡
i⎤
⎢w φ = ∑ w i (1 + z ) ⎥
i =0
⎣
⎦
better
¾ SNAP data + {w, N=1} fit
ƒ distinguishing Λ and other models?
ƒ telling us whether wφ evolves?
Good
Not so good
Another Example
• Alam, Sahni, and Starobinsky, 2004, JCAP, astro-ph/0403687
“The case for Dynamical dark energy revisited ”
( Hey! Surprise! Constant wDE is disfavored!! )
• Jönsson, Goobar, Amanullah, and Bergström, 2004, JCAP, astro-ph/0404468
“No evidence for dark energy metamorphosis? ”
( You must be kidding me …… )
Alam, Sahni, and Starobinsky, 2004
ρ DE ( x ) / ρ 0c = A0 + A1x + A2 x
2
x ≡ 1+ z
( ~ a truncated Taylor series of the dark energy density )
A1x + 2 A2 x 2
• w DE ( x ) = −1 +
3( A0 + A1x + A2 x 2 )
• Flatness : Ωm + ΩDE = 1 Î A0 + A1 + A2 = 1−Ωm
Prior : Ωm = 0.3
( Ω i ≡ ρi / ρ c )
Alam, Sahni, and Starobinsky, 2004
• Parametrization :
wDE
ρ DE ( x )
= A0 + A1x + A2 x 2
ρ0c
• Prior :
flatness ; Ωm = 0.3
z
⇓
Constant wDE is disfavored.
Testing the parametrization : ρ DE ( x ) / ρ 0c = A0 + A1x + A2 x 2
Parametrization vs. Reality
⎧Reality : fR ( x ) ~ sin x
⎨
⎩Parametrization : fP ( x ) = a0 + a1x
In this case, it is not surprising to have something wrong.
⎧Reality : fR ( x ) = constant
⎨
⎩Parametrization : fP ( x ) = a0 + a1x
In this case, fp(x) can exactly realize fR(x).
Naively, this parametrization should be good to obtain info.
Jönsson, Goobar, Amanullah, & Bergström, 2004
Testing the parametrization : ρ DE ( x ) / ρ 0c = A0 + A1x + A2 x 2
(which can exactly realize ρΛ)
Fiducial model : flat ΛCDM
(exactly realized by the above)
Î Simulated SN data
Î Fitting data via ρ DE ( x ) / ρ 0c = A0 + A1x + A2 x 2
Î Obtaining information about parameters As
and physical quantities, e.g., wDE(z).
Î Comparing with the flat-ΛCDM fiducial model
Jönsson, Goobar, Amanullah, & Bergström, 2004
Jönsson, Goobar, Amanullah, & Bergström, 2004
Jönsson, Goobar, Amanullah, & Bergström, 2004
Data
• Parametrization :
analyzed
by invoking
ρ DE ( x )
= A0 + A1x + A2 x 2
ρ0c
Physical Information
Conclusion
very unstable
Very sensitive to
the fine details of
the data
Summary and Discussion
Observations / Data
mapping out the evolution history
(e.g. SNe Ia , Baryon Acoustic Oscillation)
Fitting Formula / Parametrization
analyzed
by invoking
(model-independent ?)
N
N
⎛
⎞
i
⎜⎜ e.g. d L ( z ) = ∑ c i z ; w φ = ∑ w i (1 + z )i ⎟⎟
i =o
i =0
⎠
⎝
Information about Physical Quantities
Characterizing our universe or DE models
to be reconstructed
(e.g. wφ , ρφ , statefinders{r,s} , ……)
Summary & Discussion
• Different parametrizations may give different conclusions.
• A suitable parametrization for answering the question
whether w = constant is yet to be worked out.
[ Linder 2003 (PRL 90, 091301): w(a)= w0+waz/(1+z) ]
• For answering different questions or for obtaining different
physical information, we may need different parametrizations.
Happy Women’s Day !!
International Women’s Day (March 8)
http://www.un.org/ecosocdev/geninfo/women/womday97.htm
Pure exponentia l : V (φ ) = V0e − λφ
4
−1
V0 = 10 −120 M pl , λ = M pl , φ (0) = 0.135M pl , φ&(0) = 0
Pseudo - Nambu - Goldstone - boson (PNGB) : V (φ ) = M 4 [cos(φ / f ) + 1]
4
M 4 = 1.001× 10 −120 M pl , f = 0.1 M pl , φ (0) = 1.184 × 10 −4 M pl , φ&(0) = 0
Cosmologic al tracker solutions : V (φ ) = M 4 +α / φ α
M = 9.09 × 10 −31M pl
V (φ ) = M 4e M / φ
Supergravi ty (SUGRA) potential :
M
4 +α
φα
⎡1 ⎛ φ
exp⎢ ⎜
⎢ 2 ⎜⎝ M pl
⎣
M = 2.11× 10 −12 M pl , α = 6
⎞
⎟
⎟
⎠
2
⎤
⎥
⎥
⎦
M = 1.611× 10 −8 M pl
α = 11
PNGB
Inverse Tracker
SUGRA
Brane
Periodic Potential
Pure
Exponential
Two Exponentials
Trapped Minimum Model
Exponential Tracker
Dark Energy Phenomenology :
How Parametrization alters Conclusion
2007/03/08 @ NTHU