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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. - PowerPoint PPT Presentation
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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 @ NTHUQi-Shu Yan 晏啟樹 @ NTHU
2009/03/19 @ 清華大學
CONTENTS
Introduction (basics, SN, SNAP)
Supernova Data Analysis — Parametrization
Summary
A New (model-indep) Parametrization by Gu, Lo and Yan
Test and Results
Introduction
(Basic Knowledge, SN, SNAP)
Accelerating Expansion
(homogeneous & isotropic)
Based on FLRW Cosmology
Concordance: = 0.73 , M = 0.27
from Supernova Cosmology Project: http://www-supernova.lbl.gov/
redshift z
absolutemagnitude M
luminosity distance dL
2 4 LdLF
F: flux (energy/areatime)L: luminosity (energy/time)
SN Ia Data: dL(z) [ i.e, dL,i(zi) ]
distancemodulus )pc10/( 5 10 Ldlog
3.0 ,7.0
model fiducial
)()()(
m
zmzmzm
(can hardly distinguish different models)
SCP(Perlmutter et. al.)
Distance Modulus Mm 25)pc/( 5 10 MdlogMm L
1998
2006)pc10/( 5 10 Ldlog
Riess et al., ApJ 659, 98 (2007) [astro-ph/0611572]
Supernova / Acceleration Probe (SNAP)
z 0~0.2 0.2~1.2 1.2~1.4 1.4~1.7
# of SN 50 1800 50 15
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)
(2366 in our analysis)
(Non-FLRW)
Models: Dark Geometry vs. Dark Energy
Einstein Equations
Geometry Matter/Energy
Dark Geometry
↑Dark Matter / Energy
↑
Gμν = 8πGNTμν
• Modification of Gravity
• Averaging Einstein Equations
• Extra Dimensions
for an inhomogeneous universe
(from vacuum energy)
• Quintessence/Phantom
(based on FLRW)
Supernova Data Analysis
— Parametrization / Fitting Formula —
ObservationsData
Data Analysis
ModelsTheories
mapping out the evolution history
(e.g. SNe Ia , BAO)
M1
M2
M3
:::
(e.g. 2 fitting)
Data vs. Models
Quintessencea dynamical scalar field
M
AVV A1exp
M1 (tracker quint., exponential)
001 ,,,, :parameters on sconstraint MAVA
nn AmV 24
M2 (tracker quint., power-law)
002 ,,,, :parameters on sconstraint mAn
M1 (O)
M2 (O)
M3 (X)
M4 (X)
M5 (O)
M6 (O)
:
:
:
ObservationsData
Data Analysis
ModelsTheories
mapping out the evolution history
(e.g. SNe Ia , BAO) (e.g. 2 fitting)
Data
:
:
:Dark Energy info NOT clear.
Reality : Many models survive
Shortages:– Tedious.– Distinguish between models ??– Reality: many models survive.
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.
)0()( DEDE z
Data
ParametrizationFitting Formula
Dark Energy Information
Model-independent Analysis
Constraints on Parameters
analyzedby
invoking
in
Example
zzww
awwzw
a
a
1
)1()(
0
0DE
Linder, PRL, 2003
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.
0
-1
1
w0
wa
Riess et al.ApJ, 2007
-1
-1.5
-0.5
wDE(z)
z
[e.g. wDE(z), DE(z)]
zwwzw 10DE )( w1
w0
Wang & MukherjeeApJ 650, 2006
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 ?
zwwzw
zzwwawwzw aa
10DE
00DE
)( :expansion Taylor 1)1()( :2003 PRL, Linder,
“standard
parametrization” ?
A New Parametrization
(proposed by Gu, Lo, and Yan)
New Parametrizationby Gu, Lo & Yan
SN Ia Data: dL(z) [ i.e, dL,i(zi) (and errors) ]
z
L
z
emit
emit
emit
zH
zdzczzrazd
dt
da
aa
aH
zH
zdczra
ttaata
az
ttr
00
00
0000
)(11)()(
1 ,
)()(
time present : , )( , )(
1
today, observe wethat ) time (at photons emitted it when)( at located supernova a For
Assume flat universe.
z
z
zdzwzz
zzzzG
HzHz
GzH
0 xx4
r3
m
xrmtotal
20
cc
total20total
2
1)(13exp)0(1)0(1)0(
)()()()(8
3 ,
)()(
3
8)(
( H : Hubble expansion rate )
“m”: NR matter – DM & baryons “r”: radiation “x”: dark energy
New Parametrizationby Gu, Lo & Yan
)(1)()()()(0rmtotal zzzzz w
)()()()()( xrmtotal 0zzzzz w
)()()(
)()(
0rm
x
zzz
zz
w
The density fraction (z) is the very quantity to parametrize.
)0()0( and )0( state of equation constant a with sourceenergy an ofdensity energy the i.e.,
1)0()( (i)
xx0
13x
0
0
0
w
ww
ww
zz
)()()( :parts 2 into )( Separate xxx 0zzzz w
0)0()0()0( e/differenccorrection the :)( (ii) xxx 0 wz
[ instead of x(z) or wx(z) ]
0
0
0
0
13xrm
13xx
rm
x
1)0()()(
1)0()(
)()()(
)()(w
w
w
w
zzz
zz
zzz
zz
New Parametrizationby Gu, Lo & Yan
GeneralFeatures
zz
z
z
as 0)(
times earlier in 1)(
cases most in 1)(0)0(0)0(x
0)(
0)0( :)( for conditionsBoundary
z
20 0
variable) of (change 4tan
z
z
0)2(
0)0( :)( for conditionsBoundary
11
0 sincos2
1)( :series Fourier Invoke
nn
nn nBnAA
New Parametrizationby Gu, Lo & Yan
11
0 sincos2
1)( :series Fourier Invoke
nn
nn nBnAA
0 )0(
02
1 ConditionBoundary
1x0
10
nn
nn
nBww
AA
The lowest-order nontrivial parametrization:
0(z) ngguaranteei for required is 1 x A
New Parametrizationby Gu, Lo & Yan
)(1)()()( )()()()(
)()()()(
0
0
rm
xrm
xrmtotal
zzzzzzzz
zzzz
w
w
w0 : present value of DE EoSA “amplitude” of the correction x(z)
222
13x
4rr
3mm
14)(
1)0()(
1)0()(
1)0()(
0
0
zAzz
zz
zz
zz
ww
013x
4r
3m22
2
c
total 1)0(1)0(1)0(1
41
)( wzzzz
Azz
,,,, :parameters 0xrm Aw
c)0( ii i.e.,
In our analysis, we set 73.01 ,0 ,27.0 mxrm
2 parameters: w0 , A
Data] [cf. )()()(total zdzHz L
New Parametrizationby Gu, Lo & Yan
22
2
rm
x
1
4
)()()(
)()()(
0
0
z
Az
zzz
zzz
w
w
0
0
13x
4rr
3mm
1)0()(
1)0()(
1)0()(
ww zz
zz
zz
For the present epoch, ignore r(z).
00
0
3
x
m22
213
xxx 1)0(
)0(1
1
411)0()()()( ww
w zz
Azzzzz
Linder zzwww
aa
aa ezzz
zwwawwzw
1313
xx
00x
01)0()(1
1)(
1z ,
)0(
)0(4
1 , 3212
3
)0(
)0(14131
)0(
)(
x
m13
200
x
m0
x
x
0
zAz
zzwwAzwz
w
1z ,
1 , 3522
3131
)0(
)(
313
22000
x
x
0
aa www
a
ez
zzwwwzwz
Test and Results
Background cosmology [pick a DE Model with wDE
th(z)]
generate Data
w.r.t. present or future observations (Monte Carlo simulation)
Procedures of Testing Parametrizations
employ a Fit to the dL(z) or w(z)
reconstruct wDEexp’t (z)
compare wDEexp’t and wDE
th
192 SNe Ia : real data and simulated data
A
w0
Gu, Lo & Yan
4.1x w 1 6.0
real
16.0,8.0,0.1,2.1,4.1x w
2.0x w
Info1 : Black vs. Blue(s)Info2 : Blue vs. Blue
overlap consistency
no overlap distinguishable
from 192 SNe Ia
192 SNe Ia : real data and simulated data
A
w0
Gu, Lo & Yan
4.1x w 1 6.0
real
6.0,8.0,0.1,2.1,4.1x w 22.0x w
Info1 : Black vs. Blue(s)Info2 : Blue vs. Blue
from 192 SNe Ia
192 SNe Ia : real data and simulated data
A
w0
Gu, Lo & Yan
4.1x w 1 6.0
real
6.0,8.0,0.1,2.1,4.1x w 32.0x w
Info1 : Black vs. Blue(s)Info2 : Blue vs. Blue
from 192 SNe Ia
192 SNe Ia : real data and simulated data
wa
w0
1
Linder
real
4.1x w1
6.0
Info1 : Black vs. Blue(s)Info2 : Blue vs. Blue
6.0,8.0,0.1,2.1,4.1x w2.0x w
from 192 SNe Ia
192 SNe Ia : real data and simulated data
wa
w0
Linder
real
4.1x w
1
6.0
2
Info1 : Black vs. Blue(s)Info2 : Blue vs. Blue
6.0,8.0,0.1,2.1,4.1x w2.0x w
from 192 SNe Ia
192 SNe Ia : real data and simulated data
wa
w0
Linder
real
4.1x w
1
6.0
3
Info1 : Black vs. Blue(s)Info2 : Blue vs. Blue
6.0,8.0,0.1,2.1,4.1x w2.0x w
from 192 SNe Ia
A
w0
wa
w0
1
Gu, Lo & Yan Linder
real
real
6.0,8.0,0.1,2.1,4.1x w
4.1x w 1 6.0
12.0x w
From 192 SNe Ia
Info1 : Black vs. Blue(s)Info2 : Blue vs. Blue
A
w0
wa
w0
Gu, Lo & Yan Linder
real
real
4.1x w 1 6.0
2 2
Info1 : Black vs. Blue(s)Info2 : Blue vs. Blue
6.0,8.0,0.1,2.1,4.1x w
2.0x w
From 192 SNe Ia
A
w0
wa
w0
Gu, Lo & Yan Linder
real
real
4.1x w 1 6.0
3 3
Info1 : Black vs. Blue(s)Info2 : Blue vs. Blue
6.0,8.0,0.1,2.1,4.1x w
2.0x w
From 192 SNe Ia
realdata
1
Info1 : Blue vs. White(s)
overlap consistency
80.0,85.0,90.0,95.0,00.1,05.1,10.1,15.1,20.1x w05.0x w
simulateddata
simulated
real
Gu, Lo & Yanfrom 192 SNe Ia
80.0,85.0,90.0,95.0,00.1,05.1,10.1,15.1,20.1x w05.0x w
Linder
1
from 192 SNe Ia
realdata
Info1 : Blue vs. White(s)
overlap consistency
simulateddata
simulated
real
A
w0
16.0,7.0,8.0,9.0,0.1,1.1,2.1,3.1,4.1x w
4.1x w 1 6.0
from2366 simulated SNe Ia
from192 real SNe Ia
1.0x w
2366 simulated SNe vs. 192 real SNeGu, Lo & Yan
2366 simulated SNe vs. 192 real SNe
w0
wa
Linder
from192 real SNe Ia
from2366 simulated SNe Ia
4.1x w 1 6.0
16.0,7.0,8.0,9.0,0.1,1.1,2.1,3.1,4.1x w
1.0x w
A
w0
4.1x w 1 6.0
2366 Simulated SN Ia Data
1
Gu, Lo & Yan
60.0,65.0,,00.1,,30.1,35.1,40.1x w
05.0x w
Info from comparing nearby contours
from 2366 simulated SNe Ia
2366 Simulated SN Ia Data
A
w0
2
Gu, Lo & Yan
60.0,65.0,,00.1,,30.1,35.1,40.1x w
4.1x w1
6.0
05.0x w
Info from comparing nearby contours
from 2366 simulated SNe Ia
2366 Simulated SN Ia Data
A
w0
Gu, Lo & Yan
4.1x w1
6.0
60.0,65.0,,00.1,,30.1,35.1,40.1x w 305.0x w
Info from comparing nearby contours
from 2366 simulated SNe Ia
2366 Simulated SN Ia Data
A
w0
Gu, Lo & Yan
4.1x w1
6.0
60.0,65.0,,00.1,,30.1,35.1,40.1x w 405.0x w
Info from comparing nearby contours
from 2366 simulated SNe Ia
2366 Simulated SN Ia Data
A
w0
Gu, Lo & Yan
4.1x w
1
6.0
60.0,65.0,,00.1,,30.1,35.1,40.1x w 505.0x w
Info from comparing nearby contours
from 2366 simulated SNe Ia
2366 Simulated SN Ia DataLinder
wa
w0
60.0,65.0,,00.1,,30.1,35.1,40.1x w
4.1x w 1 6.0
105.0x w
Info from comparing nearby contours
from 2366 simulated SNe Ia
2366 Simulated SN Ia DataLinder
wa
w0
4.1x w1
6.0
60.0,65.0,,00.1,,30.1,35.1,40.1x w 205.0x w
Info from comparing nearby contours
from 2366 simulated SNe Ia
2366 Simulated SN Ia DataLinder
wa
w0
4.1x w
16.0
60.0,65.0,,00.1,,30.1,35.1,40.1x w 305.0x w
Info from comparing nearby contours
from 2366 simulated SNe Ia
2366 Simulated SN Ia DataLinder
wa
w0
4.1x w
16.0
60.0,65.0,,00.1,,30.1,35.1,40.1x w 405.0x w
Info from comparing nearby contours
from 2366 simulated SNe Ia
2366 Simulated SN Ia DataLinder
wa
w0
4.1x w
1
6.0
60.0,65.0,,00.1,,30.1,35.1,40.1x w 505.0x w
Info from comparing nearby contours
from 2366 simulated SNe Ia
Summary
A parametrization (by utilizing Fourier series) is proposed.
This parametrization is particularly reasonable and controllable.
The current SN Ia data, including 192 SNe, can distinguish between constant-wx models with wx = 0.2 at 1 confidence level.
Summary
Via this parametrization:
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
It is systematical to include more terms and more parameters.
Thank you.
Supernova (SN) : mapping out the evolution herstory
Type Ia Supernova (SN Ia) : (standard candle)
– thermonulear explosion of carbon-oxide white dwarfs –
Correlation between the peak luminosity and the decline rate
absolute magnitude M
luminosity distance dL
(distance precision: mag = 0.15 mag dL/dL ~ 7%)
Spectral information redshift zSN Ia Data: dL(z) [ i.e, dL,i(zi) ] [ ~ x(t) ~ position (time) ]
2 4 LdLF F: flux (energy/areatime)
L: luminosity (energy/time)
Distance Modulus Mm 25)pc/( 5 10 MdlogMm L
(z)
history
pc 10
100 5/
mM
F m
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
2004
2006 Riess et al., ApJ 659, 98 (2007) [astro-ph/0611572]
M1 (X)
M2 (X)
M3 (O)
M4 (X)
M5 (X)
M6 (X)
:
:
:
ObservationsData
Data Analysis
ModelsTheories
mapping out the evolution history
(e.g. SNe Ia , BAO) (e.g. 2 fitting)
OK, if few models survive.
Data
:
:
:
Tedious work, worthwhile if few models survive, and accordinglyclear Dark Energy info obtained.