Dark Energy Phenomenology
 a new parametrization 
Je-An Gu 顧哲安
臺灣大學梁次震宇宙學與粒子天文物理學研究中心
Leung Center for Cosmology and Particle Astrophysics (LeCosPA), NTU
Collaborators : Yu-Hsun Lo 羅鈺勳 @ NTHU
Qi-Shu Yan 晏啟樹 @ NTHU
2009/03/19 @ 清華大學
CONTENTS
 Introduction (basics, SN, SNAP)
 Supernova Data Analysis — Parametrization
 A New (model-indep) Parametrization
by Gu, Lo and Yan
 Test and Results
 Summary
Introduction
(Basic Knowledge, SN, SNAP)
Concordance:  = 0.73 , M = 0.27
Accelerating Expansion
Based on FLRW Cosmology
(homogeneous & isotropic)
absolute
magnitude M
luminosity
distance dL
F L
4 d L2
F: flux (energy/areatime)
L: luminosity (energy/time)
SN Ia Data: dL(z) [ i.e, dL,i(zi) ]
distance
modulus
  5 log10 (dL / 10pc )
redshift z
from Supernova Cosmology Project: http://www-supernova.lbl.gov/
1998 Distance Modulus
  mM
m( z )  m( z )  m ( z )

fiducial model
  m  M  5 log10 (dL / Mpc )  25
   0.7,  m  0.3
SCP
(Perlmutter et. al.)
(can hardly distinguish different models)
2006
Riess et al., ApJ 659, 98 (2007) [astro-ph/0611572]
  5 log10 (dL / 10pc )
Supernova / Acceleration Probe (SNAP)
observe ~2000 SNe in 2 years
(2366 in our analysis)
statistical uncertainty mag = 0.15 mag  7% uncertainty in dL
sys = 0.02 mag at z =1.5
z
= 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
Models: Dark Geometry
vs. Dark Energy
Einstein Equations
Gμν
＝
Geometry
Matter/Energy
↑
Dark Geometry
8πGNTμν
↑
Dark Matter / Energy
• Modification of Gravity
•  (from vacuum energy)
• Extra Dimensions
• Quintessence/Phantom
• Averaging Einstein Equations
for an inhomogeneous universe
(Non-FLRW)
(based on FLRW)
Supernova Data Analysis
— Parametrization / Fitting Formula —
Data vs. Models
Observations
Data
mapping out
the evolution history
(e.g. SNe Ia , BAO)
Models
Theories
Data Analysis
(e.g. 2 fitting)
M1
M2
M3
:
Quintessence
:
a dynamical scalar
field
:
M1 (tracker quint., exponential)


V    VA exp    A1 

M 

constraint s on parameters : VA , A1, M,0 ,0
M2 (tracker quint., power-law)
n
V    m 4n   A2 

constraint s on parameters : n, A2 , m,0 ,0


Reality : Many models survive
Observations
Data
mapping out
the evolution history
(e.g. SNe Ia , BAO)
Models
Theories
Data Analysis
(e.g. 2 fitting)
Data
Shortages:
– Tedious.
– Distinguish between models ??
– Reality: many models survive.
M1 (O)
M2 (O)
M3 (X)
M4 (X)
M5 (O)
M6 (O)
:
:
:
:
:
:
Dark Energy info NOT
clear.
Reality : Many models survive
 Instead of comparing models and data (thereby ruling out models),
Extract information about dark energy directly from data
by invoking model independent parametrizations.
Model-independent Analysis
Data
Dark Energy
Information
Constraints on
Parameters
analyzed
by
invoking
Parametrization
Fitting Formula
in
wa
1
0
Riess et al.
ApJ, 2007
Example
[e.g. wDE(z), DE(z)]
wDE(z)
-0.5
-1
-1.5
-1
 Linder, PRL, 2003
z
w0
w DE ( z )  w 0  w a (1  a )
 w 0  w a z 1  z 
 wDE ( z )  w 0  w1z
w1
DE (z) DE (0)
Wang & Mukherjee
ApJ 650, 2006
w0
w : equation of state, an important quantity characterizing the nature of an energy content.
It corresponds to how fast the energy density changes along with the expansion.
Answering questions about Dark Energy
 Instead of comparing models and data (thereby ruling out models),
Extract information about dark energy directly from data
by invoking model independent parametrizations,
thereby answering essential questions about dark energy.
Issue
?
How to choose
the parametrization ?
 Linder, PRL, 2003 : wDE (z)  w 0  w a (1 a)  w 0  w a z 1 z 
 Taylor expansion : wDE (z)  w 0  w1z
“standard
parametrization” ?
A New Parametrization
(proposed by Gu, Lo, and Yan)
New Parametrization
by Gu, Lo & Yan
SN Ia Data: dL(z) [ i.e, dL,i(zi) (and errors) ]
For a supernova located at r (t emit ) when it emitted photons
(at time t emit ) that we observe today,
a0
1 z 
,
a0  a(t 0 ) , t 0 : present time
a(t emit )
z dz 
a 1 da
( H : Hubble expansion rate )
a0 r ( z )  c  
,
H 
0 H ( z )
a a dt
z dz 
d L ( z )  a0  r ( z )  1  z   c 1  z   
0 H ( z )
Assume flat universe.
3H02
8G
2
2  total ( z )
H (z) 
 total ( z )  H0 
, c 
3
c
8G
 total ( z )  m ( z )  r ( z )   x ( z )
“m”: NR matter
– DM & baryons
“r”: radiation
“x”: dark energy
dz 
 z

 m (0)1  z   r (0)1  z    x (0) exp 3  1  w x ( z )

0

1 z 

3
4
New Parametrization
by Gu, Lo & Yan
Separate  x ( z ) into 2 parts :
 x ( z )  w ( z )  x ( z )
0
0
(i) w 0 ( z )   x (0)1  z 
i.e., the energy density of an energy source
with a constant equation of state w 0  w x (0) and w 0 (0)   x (0)
3 1w

(ii) x ( z ) : the correction /differenc e

w0
(0)   x (0)  x (0)  0

 total ( z )  m ( z )  r ( z )  w ( z )  x ( z )
0
x ( z )
 (z) 
 m ( z )   r ( z )  w ( z )
0
3 1w 
 x ( z )  w ( z )
 x ( z )   x (0)1  z 


m ( z )  r ( z )  w ( z ) m ( z )  r ( z )   x (0)1  z 31w
0
0
0
 total ( z )  m ( z )  r ( z )  w ( z ) 1   ( z )
0
The density fraction  (z) is the very quantity to parametrize.
[ instead of x(z) or wx(z) ]
0

New Parametrization
by Gu, Lo & Yan
General
Features
x (0)  0   (0)  0
 ( z )  1 in most cases
 ( z )  1 in earlier times
 ( z )  0 as z  
  (0 )  0
 Boundary conditions for  ( z ) : 
 ()  0
z  tan 4 
(change of variable)
0  z    0    2
  (0 )  0
 Boundary conditions for  ( ) : 
 (2 )  0


1
Invoke Fourier series :  ( )  A0   An cos n   Bn sin n
2
n 1
n 1
New Parametrization
by Gu, Lo & Yan


1
Invoke Fourier series :  ( )  A0   An cos n   Bn sin n
2
n 1
n 1

1
Boundary Condition 
A0   An  0
2
n 1
w 0  w x (0 ) 

 nB
n 1
n
0
The lowest-order nontrivial parametrization:
A  1 is required for guaranteei ng x (z)  0
New Parametrization
by Gu, Lo & Yan
total (z)  H(z)  dL (z) [cf. Data]
m ( z )  m (0)1  z 
4
r ( z )  r (0)1  z 
3 1w
w ( z )   x (0)1  z 
3
 total ( z )  m ( z )  r ( z )   x ( z )
 m ( z )  r ( z )  w ( z )  x ( z )
 m ( z )  r ( z )  w ( z ) 1   ( z )
0
0
i.e.,
0
 ( z )  4 Az
2
0

1  z 
2 2
i  i (0) c

 total ( z ) 
4 Az 2 
3
4
3 1w






 1 


(
0
)
1

z


(
0
)
1

z


(
0
)
1

z

m
r
x
2 2
c
 1  z  
parameters :
m, r , x ,w 0 , A
In our analysis, we set
0


w0 : present value of DE EoS
A  “amplitude” of the correction x(z)
m  0.27, r  0, x  1 m  0.73
 2 parameters: w0 , A
New Parametrization
by Gu, Lo & Yan
m ( z )  m (0)1  z 
4
r ( z )  r (0)1  z 
3 1w
w ( z )   x (0)1  z 
 x ( z )  w ( z )
4 Az 2
 (z) 

m ( z )  r ( z )  w ( z ) 1  z 2 2
3
0
0
For the present epoch, ignore r(z).
0


4 Az 2  m (0)

 3w 0  


 x ( z )  w 0 ( z )  x ( z )   x (0)1  z 
1

1

1

z


2 2 

(
0
)

1 z 
x




 
 2
 m (0 )  3







1

3
1

w
z

4
A
1


1

w
2

3
w
z  1

 
0
0
0 z   ,

 x (z) 
 x (0 )  2
 


 x (0)  31w 0 
 (0 )
z
 4A  m
 z   , z  1

 x (0 )
3 1w 0 
Linder
w x ( z )  w 0  w a 1  a   w 0  w a
 x ( z )   x (0)1  z 
3 1w 0 w a 


z
1 z
e 3w a z 1 z 
 x ( z )  1  31  w 0 z  2  5w 0  3w 02  w a z 2   , z  1

2
 x (0)  z 31w w e 3w   , z  1


3
0
a
a
0

Test and Results
Procedures of Testing Parametrizations
Background cosmology
[pick a DE Model with wDEth(z)]

generate Data
w.r.t. present or future observations
(Monte Carlo simulation)
employ a Fit to
the dL(z) or w(z)
reconstruct wDEexp’t (z)

compare wDEexp’t and wDEth
Gu, Lo & Yan
192 SNe Ia : real data and simulated data
w x  1.4,1.2,1.0,0.8,0.6
w x  0.2
1
Info1 : Black vs. Blue(s) overlap  consistency
Info2 : Blue vs. Blue
no overlap  distinguishable
A
real
w x  1.4
 1  0.6
from 192 SNe Ia
w0
Gu, Lo & Yan
192 SNe Ia : real data and simulated data
w x  1.4,1.2,1.0,0.8,0.6
w x  0.2
real
A
w x  1.4
1
from 192 SNe Ia
Info1 : Black vs. Blue(s)
Info2 : Blue vs. Blue
w0
 0.6
2
Gu, Lo & Yan
192 SNe Ia : real data and simulated data
w x  1.4,1.2,1.0,0.8,0.6
w x  0.2
real
A
w x  1.4  1
from 192 SNe Ia
Info1 : Black vs. Blue(s)
Info2 : Blue vs. Blue
w0
 0.6
3
Linder
192 SNe Ia : real data and simulated data
w x  1.4,1.2,1.0,0.8,0.6
w x  0.2
from 192 SNe Ia
real
wa
 0.6
w x  1.4
Info1 : Black vs. Blue(s)
Info2 : Blue vs. Blue
1
w0
1
Linder
192 SNe Ia : real data and simulated data
w x  1.4,1.2,1.0,0.8,0.6
w x  0.2
real
from 192 SNe Ia
wa
 0.6
1
w x  1.4
Info1 : Black vs. Blue(s)
Info2 : Blue vs. Blue
w0
2
Linder
192 SNe Ia : real data and simulated data
real
w x  1.4,1.2,1.0,0.8,0.6
w x  0.2
from 192 SNe Ia
wa
 0.6
1
w x  1.4
Info1 : Black vs. Blue(s)
Info2 : Blue vs. Blue
w0
3
From 192 SNe Ia
Gu, Lo & Yan
Linder
w x  1.4,1.2,1.0,0.8,0.6
w x  0.2
1
1
real
real
A
wa
w x  1.4
 1  0.6
w0
Info1 : Black vs. Blue(s)
Info2 : Blue vs. Blue
w0
From 192 SNe Ia
Gu, Lo & Yan
Linder
w x  1.4,1.2,1.0,0.8,0.6
w x  0.2
2
2
real
real
A
wa
w x  1.4
w0
1
 0.6
Info1 : Black vs. Blue(s)
Info2 : Blue vs. Blue
w0
From 192 SNe Ia
Gu, Lo & Yan
Linder
w x  1.4,1.2,1.0,0.8,0.6
w x  0.2
3
real
3
real
A
wa
w x  1.4
w0
1
 0.6
Info1 : Black vs. Blue(s)
Info2 : Blue vs. Blue
w0
w x  1.20,1.15,1.10,1.05,1.00,0.95,0.90,0.85,0.80
Gu, Lo & Yan
w x  0.05
from 192 SNe Ia
simulated
1
real simulated
data
data
Info1 : Blue vs. White(s)
overlap  consistency
real
w x  1.20,1.15,1.10,1.05,1.00,0.95,0.90,0.85,0.80
Linder
w x  0.05
from 192 SNe Ia
1
simulated
real simulated
data
data
Info1 : Blue vs. White(s)
overlap  consistency
real
Gu, Lo & Yan
2366 simulated SNe vs. 192 real SNe
w x  1.4,1.3,1.2,1.1,1.0,0.9,0.8,0.7,0.6
w x  0.1
from
192 real SNe Ia
A
from
2366 simulated SNe Ia
w x  1.4
w0
1
 0.6
1
Linder
2366 simulated SNe vs. 192 real SNe
w x  1.4,1.3,1.2,1.1,1.0,0.9,0.8,0.7,0.6
w x  0.1
from
192 real SNe Ia
wa
from
2366 simulated SNe Ia
w x  1.4
w0
1
 0.6
1
Gu, Lo & Yan
2366 Simulated SN Ia Data
w x  1.40,1.35,1.30,,1.00,,0.65,0.60
w x  0.05
A
w x  1.4
1
 0.6
from 2366 simulated SNe Ia
Info from comparing nearby contours
w0
1
Gu, Lo & Yan
2366 Simulated SN Ia Data
w x  1.40,1.35,1.30,,1.00,,0.65,0.60
w x  0.05
A
w x  1.4
1
 0.6
from 2366 simulated SNe Ia
Info from comparing nearby contours
w0
2
Gu, Lo & Yan
2366 Simulated SN Ia Data
w x  1.40,1.35,1.30,,1.00,,0.65,0.60
w x  0.05
A
w x  1.4
1
 0.6
from 2366 simulated SNe Ia
Info from comparing nearby contours
w0
3
Gu, Lo & Yan
2366 Simulated SN Ia Data
w x  1.40,1.35,1.30,,1.00,,0.65,0.60
w x  0.05
A
w x  1.4
1
 0.6
from 2366 simulated SNe Ia
Info from comparing nearby contours
w0
4
Gu, Lo & Yan
2366 Simulated SN Ia Data
w x  1.40,1.35,1.30,,1.00,,0.65,0.60
w x  0.05
A
w x  1.4
1
 0.6
from 2366 simulated SNe Ia
Info from comparing nearby contours
w0
5
Linder
2366 Simulated SN Ia Data
w x  1.40,1.35,1.30,,1.00,,0.65,0.60
w x  0.05
wa
w x  1.4
1
 0.6
from 2366 simulated SNe Ia
Info from comparing nearby contours
w0
1
Linder
2366 Simulated SN Ia Data
w x  1.40,1.35,1.30,,1.00,,0.65,0.60
w x  0.05
wa
w x  1.4
1
 0.6
from 2366 simulated SNe Ia
Info from comparing nearby contours
w0
2
Linder
2366 Simulated SN Ia Data
w x  1.40,1.35,1.30,,1.00,,0.65,0.60
w x  0.05
wa
1
 0.6
w x  1.4
from 2366 simulated SNe Ia
Info from comparing nearby contours
w0
3
Linder
2366 Simulated SN Ia Data
w x  1.40,1.35,1.30,,1.00,,0.65,0.60
w x  0.05
wa
 0.6
1
w x  1.4
from 2366 simulated SNe Ia
Info from comparing nearby contours
w0
4
Linder
2366 Simulated SN Ia Data
w x  1.40,1.35,1.30,,1.00,,0.65,0.60
w x  0.05
wa
 0.6
1
w x  1.4
from 2366 simulated SNe Ia
Info from comparing nearby contours
w0
5
Summary
Summary
 A parametrization (by utilizing Fourier series) is proposed.
 This parametrization is particularly reasonable and controllable.
 It is systematical to include more terms and more parameters.
Via this parametrization:
 The current SN Ia data, including 192 SNe, can distinguish
between constant-wx models with wx = 0.2 at 1 confidence level.
 The future SN Ia data, if including 2366 SNe, can distinguish
between constant-wx models with wx = 0.05 at 3 confidence level,
and the models with wx = 0.1 at 5 confidence level
Thank you.
Supernova (SN) :
mapping out the evolution herstory
Type Ia Supernova (SN Ia) :
history
(standard candle)
– thermonulear explosion of carbon-oxide white dwarfs –
 Correlation between the peak luminosity and the decline rate
 absolute magnitude M
 luminosity distance dL
F L
4 d
2
L
F: flux (energy/areatime)
L: luminosity (energy/time)
(distance precision: mag = 0.15 mag  dL/dL ~ 7%)
F  100  m / 5
 Spectral information  redshift z
SN Ia Data: dL(z) [ i.e, dL,i(zi) ]
 (z)
M  m10 pc 
[ ~ x(t) ~ position (time) ]
Distance Modulus
  mM
  m  M  5 log10 (dL / Mpc )  25
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: M=0.29 =0.71
2006
Riess et al., ApJ 659, 98 (2007) [astro-ph/0611572]
OK, if few models survive.
Observations
Data
mapping out
the evolution history
(e.g. SNe Ia , BAO)
Models
Theories
Data Analysis
(e.g. 2 fitting)
Data
Tedious work,
worthwhile if few models survive,
and accordingly
clear Dark Energy info obtained.
:
:
:
M1
M2
M3
M4
M5
M6
:
:
:
(X)
(X)
(O)
(X)
(X)
(X)