Dark Energy Phenomenology: Quintessence Potential Reconstruction

Dark Energy Phenomenology:Quintessence Potential Reconstruction

Je-An Gu 顧哲安National Center for Theoretical Sciences

2007/10/16 @ NTHU

Collaborators : Chien-Wen Chen 陳建文 @ NTU Pisin Chen 陳丕燊 @ SLAC

New blood : 羅鈺勳 @ NTHUQi-Shu Yan 晏啟樹 @ NTHU

(in alphabetical order)

Introduction (basic knowledge, motivation; SN; SNAP)

Quintessence -- potential reconstruction: general formulae

Summary

Content

Supernova Data Analysis (parametrization / fitting formula)

Quintessence Potential Reconstruction (from data via two parametrizations)

Introduction(Basic Knowledge, Motivation, SN, SNAP)

Accelerating Expansion

(homogeneous & isotropic)Based on FLRW Cosmology

Concordance: = 0.73 , M = 0.27

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 10100 5/

mMF m

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

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

Riess et al. astro-ph/0611572200625)pc/( 5 10 Mdlog L

2006 Riess et al. 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)

Supernova Data Analysis( Parametrization / Fitting Formula )

Observations / Datamapping out the evolution history

(e.g. SNe Ia , Baryon Acoustic Oscillation)

Phenomenology Data Analysis

Models / Theories(of Dark Energy)

• N models and 1 data set N analyses• N models and M data set NM analyses• models and M data set analyses !!

Reality: many models survive.

Not so meaningful….

( 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 / Datamapping out the evolution history

(e.g. SNe Ia , Baryon Acoustic Oscillation)

Parametrization Fitting Formula(model independent ?)

Dark Energy Info wDE = 1 ? wDE = const. ?

analyzedby invoking

Error Evaluation: Gaussian error propagation: [from dL(z) to w(z)]

5/)10)(ln()(

of sample simulated the ofmatrix covariant:

2

kLmagkL

iij

jiij

ji

zdzdc

cw

cw

w

dL(z) w(z)

Parametrization / Fitting Formula : one example

N

i

iiL zczd

1

)( (polynomial fit of dL) { dL(0) = 0 c0 = 0 }

2320

3220

/1)1(/2)1(3

311

rzHrrzHzw

m

mDE

cii

)1/()()( zzdzr L

zN

k kL

kfitLk

tL

i zdzdzdc

0

2exp'2

)()()(})({

Best Fit: Minimizing the 2 function (function of ci’s) constraints on ci’s

Two Parametrizations / Fits

zwwzw 10)( Fit 1

( “ ” means dark energy )

)1(1

)( 00 awwz

zwwzw aa

Fit 2

(Linder)

1 set , 83 i.e., , (flat) 0 Assume 0

20

0 aG

Hk c

aaaz 11 0

Best Fit: minimizing the 2 function (function of wi’s) constraints on wi’s

Error Bar: Gaussian error propagation:

zN

k kL

kfitLk

tL

i zdzdzdw

0

2exp'2

)()()(})({

5/)10)(()(

z)(1

ji

2

lnzdzd

ww

ww

w

kLmagkL

jiij

jiij

ji

zwww 10

m 1

z

z

mo

fitL

zzdzwz

zdH

zzd0

0

3

1)(13exp)1(

1)(

Two Parametrizations / Fits

)1(1

)( 00 awwz

zwwzw aa

(1) ; (2)

cii

In the parametrization part …..

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. ?

(We can never know which model is correct.)(What we can do is ruling out models.)

Which parametrization is capable of ruling out :

trivial incapable example:CDM model

trivial incapable example:const wDE model

wDE = 1? wDE = const. ?

Quintessence Model( Potential Reconstruction: general formulae)

Friedmann-Lemaitre-Robertson-Walker (FLRW) Cosmology

22

2

2222

1 )( :metric (RW) Walker-Robertson dr

rkdrtadtds

Homogeneous & Isotropic Universe :

)( pGaa

Gak

aa

33

43

82

2

)(

, costant with for

:onconservati momentum-energy

iwi

iiii

a

wwp

apdad

13

33

0310 ap :onaccelerati ,

(Dark Energy)

(from vacuum energy)

• Quintessence

Candidates: 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

(Non-FLRW)

for an inhomogeneous universe(based on FLRW)

FLRW + Quintessence

Quintessence: dynamical scalar field

VggdxS214Action :

013 222

2

V

atH

tField equation:

Vat

p

Vat

22

2

22

2

61

21

21

21

energy densityand pressure :

Slow evolution and weak spatial dependence V() dominates w ~ 1 Acceleration

How to achieve it (naturally) ??

FLRW + Quintessence

Quintessence: dynamical scalar field

VK

VKpw

KVKpVKt

21 where, , 2Assume

VggdxS214Action :

013 222

2

V

atH

tField equation:

Vat

p

Vat

22

2

22

2

61

21

21

21

energy densityand pressure :

Tracker Quintessence

V

n

nMV

4

Power-law :

MVV exp0 Exponential :

Quintessence Potential Reconstruction(general formulae)

00

0 20

0

30

2

00

,

121

11

1)(13exp)1(

38

38

1)(13exp

,

zVzVz

wzV

zzd

zHzzw

z

zzdzwzGGzH

zzdzwz

zw

z

z

mM

z

GHk c

83 i.e., , (flat) 0 Assume

20

0 aa

az 11 0

adaw

d)1(3

wK 121

21 2

wpVK

VK

cii

analyzed by invoking

Observations / Data

Parametrization

Dark Energy Info (z) (z) and w (z)

[for dL(z) , (z) , w (z) , …etc.]

00 , zVzVzQuint. Reconstruction

zwwzw 10 (1)

)1/()( 0 zzwwzw a (2)

Quintessence Reconstruction( from data via 2 parametrizations of w )

Yun Wang and Pia Mukherjee, Astrophys.J. 650 (2006) 1[astro-poh/0604051].

Data and Parametrizations

)1/()( 0 zzwwzw a (2)

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]

68% 95%

zwwzw 10 (1) 1ww

Data and Parametrizations

)1/()( 0 zzwwzw a (2)

zwwzw 10 (1)

Astier05 + WMAP3 + SDSS (68% confidence level)

394.0~751.0 , 829.0~151.1 10 ww

006.1~091.1 , 818.0~217.10 aww

: require 11 : ceQuintessen For w

)1/()( 0 zzwwzw a (2)

zwwzw 10 (1) 121 , 11 100 www

11 , 11 00 awww

)1/()( 0 zzwwzw a (2)

zwwzw 10 (1) 12.0~12.0 , 95.0~05.1 10 ww

08.0~08.0 , 93.0~07.10 aww

SNAP expectation (68% confidence level) (centered on CDM)

00

0 20

0

30

2

00

,

121

11

1)(13exp)1(

38

38

1)(13exp

,

zVzVz

wzV

zzd

zHzzw

z

zzdzwzGGzH

zzdzwz

zw

z

z

mM

z

Quintessence Potential Reconstruction(general formulae)

GHk c

83 i.e., , (flat) 0 Assume

20

0 aa

az 11 0

wK 121

21 2

wpVK

VK

cii

adaw

d)1(3

Quintessence Potential Reconstruction

zwwzw 10 (1) 394.0~751.0 , 829.0~151.1 10 ww

121 , 11 100 www12.0~12.0 , 95.0~05.1 10 ww(SNAP)

(Astier05)

(Quint.)

V (in unit of 0)

0 [ in unit of (8G/3)1/2 ]

73.0 , 27.0 M

1.2

1.4

0.8

0.10.5


(SNAP)

(Astier05)

(Quint.)

006.1~091.1 , 818.0~217.10 aww)1/()( 0 zzwwzw a (2)

11 , 11 00 awww08.0~08.0 , 93.0~07.10 aww

V (in unit of 0)

0 [ in unit of (8G/3)1/2 ]

73.0 , 27.0 M

1.2

1.6

0.8

0.10.5

Quintessence Potential Reconstruction zwwzw 10 (1) 394.0~751.0 , 829.0~151.1 10 ww

12.0~12.0 , 95.0~05.1 10 ww(SNAP)

(Astier05)

V (in unit of 0)

0 [ in unit of (8G/3)1/2 ]

(SNAP)

(Astier05) 006.1~091.1 , 818.0~217.10 aww)1/()( 0 zzwwzw a (2)08.0~08.0 , 93.0~07.10 aww

73.0 , 27.0 M

1.2

1.6

0.8

0.10.5


M

AVV A exp Exponential :

nn AMV 4 Power-law : ( n < 0 for Tracker )

.1 constddV

V

11

/1/)4(

1

1

ddV

dzd

dzdV

ddVVn

VnMddV

Vnnn

characteristic :

characteristic :


z

zwwzw 10 (1) 394.0~751.0 , 829.0~151.1 10 ww

121 , 11 100 www12.0~12.0 , 95.0~05.1 10 ww(SNAP)

(Astier05)

(Quint.)

73.0 , 27.0 M

1 2

2

1

d

dVVdz

d 1

2/1

0

0

3/8 of unit in

: of unit in :

G

V


(SNAP)

(Astier05)

(Quint.)

006.1~091.1 , 818.0~217.10 aww)1/()( 0 zzwwzw a (2)

11 , 11 00 awww08.0~08.0 , 93.0~07.10 aww

z

73.0 , 27.0 M

1 2

6

2

4

d

dVVdz

d 1

2/1

0

0

3/8 of unit in

: of unit in :

G

V


12.0~12.0 , 95.0~05.1 10 ww(SNAP)

(Astier05)

(SNAP)


73.0 , 27.0 M

z1 2

4

1

3

2

d

dVVdz

d 1Exponential V() disfavored.

2/1

0

0

3/8 of unit in

: of unit in :

G

V


12.0~12.0 , 95.0~05.1 10 ww(SNAP)

(Astier05)

(SNAP)


73.0 , 27.0 M

z1 2

0.4

0.2

0.2

0.4

d

dVVdz

d 1

2/1

0

0

3/8 of unit in

: of unit in :

G

V

Exponential V() disfavored.


M

AVV A exp Exponential :

nn AMV 4 Power-law : ( n < 0 for Tracker )

.1 constddV

V

11

/1/)4(

1

1

ddV

dzd

dzdV

ddVVn

VnMddV

Vnnn

characteristic :

characteristic :


z

zwwzw 10 (1) 394.0~751.0 , 829.0~151.1 10 ww

121 , 11 100 www12.0~12.0 , 95.0~05.1 10 ww(SNAP)

(Astier05)

(Quint.) dz

zdn

1

1

dzdV

ddV

ddV

dzdV

zn

73.0 , 27.0 M

1 2

1

0.2

0.2

dz

zdn


(SNAP)

(Astier05)

(Quint.)

006.1~091.1 , 818.0~217.10 aww)1/()( 0 zzwwzw a (2)

11 , 11 00 awww08.0~08.0 , 93.0~07.10 aww

z

1

1

dzdV

ddV

ddV

dzdV

zn

73.0 , 27.0 M

1 2

1

1

3

2

2


12.0~12.0 , 95.0~05.1 10 ww(SNAP)

(Astier05)

(SNAP)


z

1

1

dzdV

ddV

ddV

dzdV

zn

73.0 , 27.0 M

dz

zdn

1 2

1

1

2

Power-law V() consistent with data.


12.0~12.0 , 95.0~05.1 10 ww(SNAP)

(Astier05)

(SNAP)


z

1

1

dzdV

ddV

ddV

dzdV

zn

73.0 , 27.0 M

dz

zdn

1 2

1

0.5

0.5

Power-law V() consistent with data.


z

For power-law V(), 0.75 < n < 0.

zwwzw 10 (1) 394.0~751.0 , 829.0~151.1 10 ww

121 , 11 100 www12.0~12.0 , 95.0~05.1 10 ww(SNAP)

(Astier05)

(Quint.) zn

1

1

dzdV

ddV

ddV

dzdV

zn

73.0 , 27.0 M

1 2

1

0.2


zn

For power-law V(), 1 < n < 0.

(SNAP)

(Astier05)

(Quint.)

006.1~091.1 , 818.0~217.10 aww)1/()( 0 zzwwzw a (2)

11 , 11 00 awww08.0~08.0 , 93.0~07.10 aww

z

1

1

dzdV

ddV

ddV

dzdV

zn

73.0 , 27.0 M

1 2

1

1


zn

zwwzw 10 (1) 394.0~751.0 , 829.0~151.1 10 ww12.0~12.0 , 95.0~05.1 10 ww(SNAP)

(Astier05)

(SNAP)


For power-law V(), 0.75 < n < 0.For power-law V(), 1 < n < 0.

z

1

1

dzdV

ddV

ddV

dzdV

zn

73.0 , 27.0 M

1 2

1

1

Summary

Formulae for quintessence V() reconstruction presented.

Summary

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.

Exponential V() disfavored.

For power-law V() , (1) –0.75 < n < 0 ; (2) –1 < n < 0 .

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.

Documents

Dark Energy Phenomenology: Quintessence Potential Reconstruction