Dark Energy Phenomenology:
Quintessence Potential Reconstruction
Je-An Gu 顧哲安
National Center for Theoretical Sciences
Collaborators : Chien-Wen Chen 陳建文 @ NTU
Pisin Chen 陳丕燊 @ SLAC
New blood :
(in alphabetical order)
羅鈺勳 @ NTHU
Qi-Shu Yan 晏啟樹 @ NTHU
2007/10/16 @ NTHU
Content
 Introduction (basic knowledge, motivation; SN; SNAP)
 Supernova Data Analysis (parametrization / fitting formula)
 Quintessence -- potential reconstruction: general formulae
 Quintessence Potential Reconstruction
(from data via two parametrizations)
 Summary
Introduction
(Basic Knowledge, Motivation, SN, SNAP)
Concordance:  = 0.73 , M = 0.27
Accelerating Expansion
Based on FLRW Cosmology
(homogeneous & isotropic)
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
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)
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. astro-ph/0611572
  5 log10 (dL / Mpc )  25
2006
Riess et al. astro-ph/0611572
Supernova / Acceleration Probe (SNAP)
observe ~2000 SNe in 2 years
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
Supernova Data Analysis
( Parametrization / Fitting Formula )
Observations / Data
mapping out the evolution history
(e.g. SNe Ia , Baryon Acoustic Oscillation)
• N models and 1 data set
 N analyses
Phenomenology
Not so meaningful….
•  models and M data set
Data Analysis
• N models and M data set
 NM analyses
  analyses !!
Reality: many models survive.
Models / Theories
(of Dark Energy)
( Reality : Many models survive )
 Instead of comparing models and data (thereby ruling out models),
Extract physical information about dark energy from data
in model independent manner.
Two Basic Questions
about Dark Energy
which should be answered first
?
Is Dark Energy played by  ?
i.e. wDE = 1 ?
?
Is Dark Energy metamorphic ?
i.e. wDE = const. ?
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.
Observations / Data
mapping out the evolution history
(e.g. SNe Ia , Baryon Acoustic Oscillation)
analyzed
by invoking
Parametrization
Fitting Formula
(model independent ?)
Dark Energy Info
 wDE = 1 ?
 wDE = const. ?
Parametrization / Fitting Formula : one example
N
dL (z )   ci z i
(polynomial fit of dL)
{ dL(0) = 0  c0 = 0 }
i 1
 Best Fit: Minimizing the 2 function (function of ci’s)  constraints on ci’s
exp' t
fit


d
(
z
)

d
2
L
k
L ( zk )

 ({ci })   
d L ( z k )
k 0 

Nz
1  w DE
1  z 3 mH02 (1  z )2  2r  / r 3

3
mH02 (1  z )3  1/ r 2
2
 r (z)  dL (z) /(1 z) 
 i
  i c
 Error Evaluation: Gaussian error propagation: [from dL(z) to w(z)]
w w 
dL(z)  w(z)
w 2   
 ij
i j c i c j
 ij : covariant matrix of the simulated sample of ci
d L ( zk )   mag d L( zk )(ln 10) / 5

Two Parametrizations / Fits
3H02
Assume k  0 (flat) , i.e., 0  c 
, set a0  1
8G
1 z 
a0 1

a a
Fit 1 w  ( z )  w 0  w 1z
z
 w 0  w a (1  a )
Fit 2 w  ( z )  w 0  w a
1 z
(Linder)
( “ ” means dark energy )
Two Parametrizations / Fits
(1) w  w 0  w1z ;
1 z z
d (z) 
Ho 0
fit
L
z
 w 0  w a (1  a)
(2) w ( z )  w 0  w a
1 z
dz
 i
dz 

m (1  z)3   exp 3 1  w ( z)

0


1

z


z


  i c

 1  m 
 Best Fit: minimizing the 2 function (function of wi’s)  constraints on wi’s
Nz
d
 2 ({w i })   
k 0
exp't
L

( zk )  d ( zk ) 

d L ( zk )

fit
L
 Error Bar: Gaussian error propagation:
w  
2
i j
w w
w i w j
 ij   (1  z) i j  ij
i j
d L ( zk )   mag d L( zk )( ln10) / 5
2
In the parametrization part …..
Two Basic Questions
about Dark Energy which should be answered first
?
?
Is Dark Energy played by  ? Is Dark Energy metamorphic ?
i.e. wDE = 1 ?
i.e. wDE = const. ?
?
Which parametrization is capable of ruling out :
wDE = 1
wDE = const.
trivial incapable example:
CDM model
trivial incapable example:
const wDE model
(We can never know which model is correct.)
(What we can do is ruling out models.)
?
Quintessence Model
( Potential Reconstruction: general formulae)
Friedmann-Lemaitre-Robertson-Walker (FLRW) Cosmology
Homogeneous & Isotropic Universe :
2

dr
2
2
2
2
2
Robertson - Walker (RW) metric : ds  dt  a (t ) 
 r d 
2
 1 k r

2

k 8G
a

   2 
3
a a
a
4G

(   3p )
a
3
energy - momentum conservati on :
 
 
d a 3   pd a 3
for pi  w i  i with costant w i ,
 i  a 3(1w )
i
1
  0 , p     accelerati on : a  0
3
(Dark Energy)
Candidates: 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
• Averaging Einstein Equations
for an inhomogeneous universe
(Non-FLRW)
(based on FLRW)
FLRW + Quintessence
Quintessence: dynamical scalar field 


Action : S   dx 4 g  g        V  
2

1
 2
 1 2

3
H
 2    V    0
Field equation:
2
t
t a
1   
1
     2  2  V  
2  t  2a
2
energy density
and pressure :
1   
1
2
p     2    V  
2  t  6a
2
Slow evolution and weak spatial dependence
 V() dominates  w ~ 1  Acceleration
?
How to achieve it (naturally) ?
FLRW + Quintessence
Quintessence: dynamical scalar field 


Action : S   dx 4 g  g        V  
2

1
 2
 1 2

3
H
 2    V    0
Field equation:
2
t
t a
1   
1
     2  2  V  
2  t  2a
2
energy density
and pressure :
1   
1
2
p     2    V  
2  t  6a
2
Assume    t     K  V , p  K  V , where K  2
 w 
p

 K  V 
1
2
K  V 
Tracker Quintessence
M 4n
 Power-law :
V   
 Exponential :
V    V0 exp   
 M
n
V  

Quintessence Potential Reconstruction
(general formulae)
3H02
Assume k  0 (flat) , i.e., 0  c 
8G
1 z 
 i
w  z  , 
a0 1

a a
  i c
d
dz  
 z
da
 3(1  w  )

 z    0  exp 3  1  w  ( z )


a
1  z 
 0
8G
8G 
dz   
 z
2
3
M    
H z  
 0   m (1  z )   exp 3  1  w  ( z)
 
0

3
3
1 z 


1  w z z dz
 z   0   
0
H 2 z
1  z
z
V z  
1
1  w 
2
 z   0
, V z   V  z   0 
1 2
1
  K  1  w  
2
2
K  V  
K  V  p  w 

Observations / Data
analyzed by invoking
Parametrization
[for dL(z) , (z) , w (z) , …etc.]
Dark Energy Info
  (z)
  (z) and w (z)
Quint. Reconstruction
  z   0 ,V z  
 V  z   0 
(1) w z   w 0  w1z
(2) w ( z )  w 0  w a z /(1  z )
Quintessence Reconstruction
( from data via 2 parametrizations of w )
Data and Parametrizations
Yun Wang and Pia Mukherjee,
Astrophys.J. 650 (2006) 1
[astro-poh/0604051].
(1) w z   w 0  w1z w   w1 
(2) w ( z )  w 0  w a z /(1  z )
68%
95%
Astier05: 1yr SNLS: Astron.Astrophys.447 (2006) 31-48 [astro-ph/0510447]
WMAP3: Spergel et al., Astrophys.J.Suppl.170 (2007) 377 [astro-ph/0603449]
SDSS(BAO): Eisenstein et al., Astrophys.J. 633 (2005) 560 [astro-ph/0501171]
Data and Parametrizations
Astier05 + WMAP3 + SDSS (68% confidence level)
(1) w z   w 0  w1z
w 0  1.151 ~ 0.829 , w1  0.751 ~ 0.394
(2) w ( z )  w 0  w a z /(1  z )
w 0  1.217 ~ 0.818 , w a  1.091 ~ 1.006
SNAP expectation (68% confidence level) (centered on CDM)
(1) w z   w 0  w1z
w 0  1.05 ~ 0.95 , w1  0.12 ~ 0.12
(2) w ( z )  w 0  w a z /(1  z )
w 0  1.07 ~ 0.93 , w a  0.08 ~ 0.08
For Quintessen ce :  1  w   1  require :
(1) w z   w 0  w1z
 1  w 0  1,  1  w 0  2w1  1
(2) w ( z )  w 0  w a z /(1  z )
 1  w 0  1,  1  w 0  w a  1
Quintessence Potential Reconstruction
(general formulae)
3H02
Assume k  0 (flat) , i.e., 0  c 
8G
1 z 
 i
w  z  , 
a0 1

a a
  i c
d
dz  
 z
da
 3(1  w  )

 z    0  exp 3  1  w  ( z )


a
1  z 
 0
8G
8G 
dz   
 z
2
3
M    
H z  
 0   m (1  z )   exp 3  1  w  ( z)
 
0

3
3
1 z 


1  w z z dz
 z   0   
0
H 2 z
1  z
z
V z  
1
1  w 
2
 z   0
, V z   V  z   0 
1 2
1
  K  1  w  
2
2
K  V  
K  V  p  w 

Quintessence Potential Reconstruction
(1) w z   w 0  w1z
(Astier05) w 0
(SNAP) w 0
(Quint.)
 1.151 ~ 0.829 , w1  0.751 ~ 0.394
 1.05 ~ 0.95 , w1  0.12 ~ 0.12
 1  w 0  1,  1  w 0  2w1  1
V
(in unit of 0)
1.4
1.2
0.5
  0
0.1
[ in unit of (8G/3)1/2 ]
0.8
M  0.27 ,   0.73
Quintessence Potential Reconstruction
(2) w ( z )  w 0  w a z /(1  z )
(Astier05) w 0
 1.217 ~ 0.818 , w a  1.091 ~ 1.006
 1.07 ~ 0.93 , w a  0.08 ~ 0.08
(Quint.)  1  w 0  1,  1  w 0  w a  1
(SNAP) w 0
V
(in unit of 0)
1.6
1.2
0.5
  0
0.1
[ in unit of (8G/3)1/2 ]
0.8
M  0.27 ,   0.73
Quintessence Potential Reconstruction
(1) w z   w 0  w1z
(Astier05) w 0
(2) w ( z )  w 0  w a z /(1  z )
(Astier05) w 0
 1.151 ~ 0.829 , w1  0.751 ~ 0.394
(SNAP) w 0  1.05 ~ 0.95 , w1  0.12 ~ 0.12
 1.217 ~ 0.818 , w a  1.091 ~ 1.006
(SNAP) w 0  1.07 ~ 0.93 , w a  0.08 ~ 0.08
V
(in unit of 0)
1.6
1.2
0.5
  0
0.1
[ in unit of (8G/3)1/2 ]
0.8
M  0.27 ,   0.73
Tracker Quintessence
V    VA exp    A 
M 

1 dV
 const .
characteristic :
V d
 Exponential :
 Power-law :
V    M
4 n
  A
n
( n < 0 for Tracker )
1 dV
 nM ( 4n ) / nV 1/ n
V d

 dV dV  d  dV 
characteristic : n  1  V 




 d dz  dz  d 
1
1
Quintessence Potential Reconstruction
(1) w z   w 0  w1z
(Astier05) w 0
(SNAP) w 0
d  1 dV 


dz  V d 
(Quint.)
 1.151 ~ 0.829 , w1  0.751 ~ 0.394
 1.05 ~ 0.95 , w1  0.12 ~ 0.12
 1  w 0  1,  1  w 0  2w1  1
1
2
z
1
2
V : in unit of 0
  0 :
M  0.27 ,   0.73
in unit of 8G / 3
1/ 2
Quintessence Potential Reconstruction
(2) w ( z )  w 0  w a z /(1  z )
(Astier05) w 0
 1.217 ~ 0.818 , w a  1.091 ~ 1.006
 1.07 ~ 0.93 , w a  0.08 ~ 0.08
(Quint.)  1  w 0  1,  1  w 0  w a  1
(SNAP) w 0
d  1 dV 


dz  V d 
1
2
z
2
4
6
V : in unit of 0
  0 :
M  0.27 ,   0.73
in unit of 8G / 3
1/ 2
Quintessence Potential Reconstruction
(1) w z   w 0  w1z
(Astier05) w 0
(2) w ( z )  w 0  w a z /(1  z )
(Astier05) w 0
d  1 dV 


dz  V d 
 1.151 ~ 0.829 , w1  0.751 ~ 0.394
(SNAP) w 0  1.05 ~ 0.95 , w1  0.12 ~ 0.12
 1.217 ~ 0.818 , w a  1.091 ~ 1.006
(SNAP) w 0  1.07 ~ 0.93 , w a  0.08 ~ 0.08
Exponential V() disfavored.
1
2
z
1
2
3
V : in unit of 0
4
M  0.27 ,   0.73
  0 :
in unit of 8G / 3
1/ 2
Quintessence Potential Reconstruction
(1) w z   w 0  w1z
(Astier05) w 0
(2) w ( z )  w 0  w a z /(1  z )
(Astier05) w 0
d  1 dV 


dz  V d 
0.4
 1.151 ~ 0.829 , w1  0.751 ~ 0.394
(SNAP) w 0  1.05 ~ 0.95 , w1  0.12 ~ 0.12
 1.217 ~ 0.818 , w a  1.091 ~ 1.006
(SNAP) w 0  1.07 ~ 0.93 , w a  0.08 ~ 0.08
Exponential V() disfavored.
0.2
1
2
z
0.2
0.4
V : in unit of 0
  0 :
M  0.27 ,   0.73
in unit of 8G / 3
1/ 2
Tracker Quintessence
V    VA exp    A 
M 

1 dV
 const .
characteristic :
V d
 Exponential :
 Power-law :
V    M
4 n
  A
n
( n < 0 for Tracker )
1 dV
 nM ( 4n ) / nV 1/ n
V d

 dV dV  d  dV 
characteristic : n  1  V 




 d dz  dz  d 
1
1
Quintessence Potential Reconstruction
(1) w z   w 0  w1z
dn z 
dz
(Astier05) w 0
(SNAP) w 0
(Quint.)
 1.151 ~ 0.829 , w1  0.751 ~ 0.394
 1.05 ~ 0.95 , w1  0.12 ~ 0.12
 1  w 0  1,  1  w 0  2w1  1
0.2
1
2
z
0.2
1
M  0.27 ,   0.73

d  dV  
V

 

dz
d



n z   1 
dV dV 


d dz 

1
Quintessence Potential Reconstruction
(2) w ( z )  w 0  w a z /(1  z )
dn z 
dz
(Astier05) w 0
 1.217 ~ 0.818 , w a  1.091 ~ 1.006
 1.07 ~ 0.93 , w a  0.08 ~ 0.08
(Quint.)  1  w 0  1,  1  w 0  w a  1
(SNAP) w 0
2
1
1
2
z
1
2
3
M  0.27 ,   0.73

d  dV  
V

 

dz
d



n z   1 
dV dV 


d dz 

1
Quintessence Potential Reconstruction
(1) w z   w 0  w1z
(Astier05) w 0
(2) w ( z )  w 0  w a z /(1  z )
(Astier05) w 0
dn z 
dz
 1.151 ~ 0.829 , w1  0.751 ~ 0.394
(SNAP) w 0  1.05 ~ 0.95 , w1  0.12 ~ 0.12
 1.217 ~ 0.818 , w a  1.091 ~ 1.006
(SNAP) w 0  1.07 ~ 0.93 , w a  0.08 ~ 0.08
1
1
2
z
1
2
Power-law V() consistent with data.
M  0.27 ,   0.73

d  dV  
V

 

dz
d



n z   1 
dV dV 


d dz 

1
Quintessence Potential Reconstruction
(1) w z   w 0  w1z
(Astier05) w 0
(2) w ( z )  w 0  w a z /(1  z )
(Astier05) w 0
dn z 
dz
 1.151 ~ 0.829 , w1  0.751 ~ 0.394
(SNAP) w 0  1.05 ~ 0.95 , w1  0.12 ~ 0.12
 1.217 ~ 0.818 , w a  1.091 ~ 1.006
(SNAP) w 0  1.07 ~ 0.93 , w a  0.08 ~ 0.08
0.5
1
2
z
0.5
1
Power-law V() consistent with data.
M  0.27 ,   0.73

d  dV  
V

 

dz
d



n z   1 
dV dV 


d dz 

1
Quintessence Potential Reconstruction
(1) w z   w 0  w1z
(Astier05) w 0
(SNAP) w 0
nz 
(Quint.)
 1.151 ~ 0.829 , w1  0.751 ~ 0.394
 1.05 ~ 0.95 , w1  0.12 ~ 0.12
 1  w 0  1,  1  w 0  2w1  1
For power-law V(), 0.75 < n < 0.
1
2
z
0.2
1
M  0.27 ,   0.73

d  dV  
V

 

dz
d



n z   1 
dV dV 


d dz 

1
Quintessence Potential Reconstruction
(2) w ( z )  w 0  w a z /(1  z )
(Astier05) w 0
 1.217 ~ 0.818 , w a  1.091 ~ 1.006
 1.07 ~ 0.93 , w a  0.08 ~ 0.08
(Quint.)  1  w 0  1,  1  w 0  w a  1
(SNAP) w 0
nz 
1
For power-law V(), 1 < n < 0.
1
2
z
1
M  0.27 ,   0.73

d  dV  
V

 

dz
d



n z   1 
dV dV 


d dz 

1
Quintessence Potential Reconstruction
(1) w z   w 0  w1z
(Astier05) w 0
(2) w ( z )  w 0  w a z /(1  z )
(Astier05) w 0
nz 
1
 1.151 ~ 0.829 , w1  0.751 ~ 0.394
(SNAP) w 0  1.05 ~ 0.95 , w1  0.12 ~ 0.12
 1.217 ~ 0.818 , w a  1.091 ~ 1.006
(SNAP) w 0  1.07 ~ 0.93 , w a  0.08 ~ 0.08
For power-law V(), 0.75 < n < 0.
For power-law V(), 1 < n < 0.
1
1
M  0.27 ,   0.73
2
z

d  dV  
V

 

dz
d



n z   1 
dV dV 


d dz 

1
Summary
Summary
 Formulae for quintessence V() reconstruction presented.
 Quintessence V() reconstructed by recent data.
(SNLS SN, WMAP CMB, SDSS LSS-BAO)
 A model-indep approach to comparing (ruling out)
Quintessence models is proposed,
which involves characteristics of potentials.
For example, V/V for exponential and n(z) for power-law V().
Their derivatives w.r.t. z should vanish, as a consistency criterion.
 Exponential V() disfavored.
 For power-law V() , (1) –0.75 < n < 0 ; (2) –1 < n < 0 .