Dark Energy Phenomenology a new parametrization

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


(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


)(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


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


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


)(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


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


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


from 192 SNe Ia


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


from 192 SNe Ia


wa

w0

1

Linder

real

4.1x w1

6.0


6.0,8.0,0.1,2.1,4.1x w2.0x w

from 192 SNe Ia


wa

w0

Linder

real

4.1x w

1

6.0

2


6.0,8.0,0.1,2.1,4.1x w2.0x w

from 192 SNe Ia


wa

w0

Linder

real

4.1x w

1

6.0

3


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


A

w0

wa

w0

Gu, Lo & Yan Linder

real

real

4.1x w 1 6.0

2 2


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


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


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




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




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




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



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




wa

w0

4.1x w1

6.0

60.0,65.0,,00.1,,30.1,35.1,40.1x w 205.0x w




wa

w0

4.1x w

16.0

60.0,65.0,,00.1,,30.1,35.1,40.1x w 305.0x w




wa

w0

4.1x w

16.0

60.0,65.0,,00.1,,30.1,35.1,40.1x w 405.0x w




wa

w0

4.1x w

1

6.0

60.0,65.0,,00.1,,30.1,35.1,40.1x w 505.0x w



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


(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.

Documents

Dark Energy Phenomenology a new parametrization