An Approach to Testing Dark Energy by Observations Collaborators : Chien-Wen Chen 陳建文 @ Phys,...

An Approach to Testing

Dark Energy by Observations

Collaborators : Chien-Wen Chen 陳建文 @ Phys, NTU Pisin Chen 陳丕燊 @ LeCosPA, NTU

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

2009/11/20 @ CosPA 2009, Melbourne

An Approach to Testing

Dark Energy by Observations

Collaborators : Chien-Wen Chen 陳建文 @ Phys, NTU Pisin Chen 陳丕燊 @ LeCosPA, NTU

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

2009/11/20 @ CosPA 2009, Melbourne

References

Je-An Gu, Chien-Wen Chen, and Pisin Chen, “A new approach to testing dark energy models by observations,”

New Journal of Physics 11 (2009) 073029 [arXiv:0803.4504].

Chien-Wen Chen, Je-An Gu, and Pisin Chen, “Consistency test of dark energy models,”

Modern Physics Letters A 24 (2009) 1649 [arXiv:0903.2423].

Concordance: = 0.73 , M = 0.27

Accelerating Expansion

(homogeneous & isotropic)

Based on FLRW Cosmology

Dark Energy

Observations (which are driving Modern Cosmology)

(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)

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

:

:

:

Reality : Many models survive

An Approach to Testing

Dark Energy Models

via Characteristic Q(z)

Gu, C.-W. Chen and P. Chen, New J. Phys. [arXiv:0803.4504] C.-W. Chen, Gu and P. Chen, Mod. Phys. Lett. A [arXiv:0903.2423]

Characteristic Q(z)

1. Q(z) is time-varying (i.e. dependent on z) in general.

2. Q(z) is constant within the model M (under consideration).

3. Q(z) plays the role of a key parameter within Model M.

4. Q(z) is a functional of the parametrized physical quantity P(z).

5. Q(z) can be reconstructed from data via the constraint on P(z).

6. dQ(z)/dz can also be reconstructed from data.

7. The (in)compatibility of the observational constraint of M dQ(z)/dz and the theoretical prediction of dQ(z)/dz : “0”tells the (in)consistency between data and Model M.

For each model, introduce a characteristic Q(z) with the following features:

Gu, CW Chen & P ChenarXiv:0803.4504

E.g., CDM

DE(z): energy density

wDE(z) = w0 + waz/(1+z)

Along a similar line of thought, focusing on CDM:

Sahni, Shafieloo and Starobinsky, PRD [0807.3548]:

Zunckel and Clarkson 2008, PRL101 [0807.4304]:

111)( 320

2 zHzHzOm

cDE zzQ 1)(

Q1(z)

Q2(z)

Q3(z)

:

Qi(z)

:

:

:

M1

M2

M3

:

Mi

:

:

:

Model

(parametrization)

DataP(z)

Constraints on

Parameters

Test the Consistency between Models and DataGu, CW Chen and P Chen, 2008

Characteristic Q

Qi [P(z),z]

in

Measure of Consistency M

M1

M2

M3

:

Mi

:

:

:

Model

(parametrization)

DataP(z)

Constraints on

Parameters

Test the Consistency between Models and DataGu, CW Chen and P Chen, 2008

Mi dQi (z)/dz

:

:

:

:

reconstr

uct

observationalconstraint

:

:

:

:theoretical

prediction: 0

consistentinconsistent

Q1(z)

Q2(z)

Q3(z)

:

Qi(z)

:

:

:

M1(z)

M2(z)

M3(z)

:

Mi(z)

:

:

:

in

Q1(z)

Q2(z)

Q3(z)

:

Qi(z)

:

:

:

M1(z)

M2(z)

M3(z)

:

Mi(z)

:

:

:

Measure of Consistency M

M1

M2

M3

:

Mi

:

:

:

Model

(parametrization)

DataP(z)

Constraints on

Parameters

Test the Consistency between Models and DataGu, CW Chen and P Chen, 2008

Mi dQi (z)/dz

:

:

:

:

reconstr

uct

observationalconstraint

:

:

:

:theoretical

prediction: 0

consistentinconsistent

SN Ia (Constitution)CMB (WMAP 5)BAO (SDSS,2dFGRS)

in

parameters:{m,w0,wa}

z

zwwzw a

1)( 0DE

Linder, 2003

Chevallier&Polarski, 2001

Q1(z)

Q2(z)

Q3(z)

:

Qi(z)

:

:

:

M1(z)

M2(z)

M3(z)

M4(z)

M5(z)

:

:

:

Measure of Consistency M

Qexp

Qpower

Qinv-exp

Chaplygin

:

:

:

Model

(parametrization)

DataP(z)

Constraints on

Parameters

Test the Consistency between Models and Data

Mi dQi (z)/dz

:

:

:

reconstr

uct

observationalconstraint

:

:

:theoretical

prediction: 0

consistentinconsistent

SN Ia (Constitution)CMB (WMAP 5)BAO (SDSS,2dFGRS)

parameters:{m,w0,wa}

in

z

zwwzw a

1)( 0DE

Linder, 2003

Chevallier&Polarski, 2001

CW Chen, Gu and P Chen, 2009

exp-inv:Quint. in 14

2

22

2

23

exp-inv Md

dV

Vd

Vd

d

dV

VzQ

CDM in DE zzQ

exp:Quint. in 1

1

exp Mzd

dVzVzQ

law-power:Quint. in 1

1

2

22

power nzd

Vdz

d

dVzVzQ

Chaplygin in 1

1

DEDEChaplygin

zdz

dz

dz

dw

zw

zzQ

Characteristics Q(z) of 5 Models

CDM : = constant

Quintessence, exponential: V() = V1exp[/M1]

Quintessence, power-law: V() = m4nn

Quintessence, inverse-exponential: V() = V2exp[M2/]

generalized Chaplygin gas: pDE(z) = A/DE(z) , A>0, 1

CW Chen, Gu and P Chen, 2009Gu, CW Chen and P Chen, 2008

Testing DE Models: Results

Q1(z)

Q2(z)

Q3(z)

:

Qi(z)

:

:

:

M1(z)

M2(z)

M3(z)

M4(z)

M5(z)

:

:

:

Measure of Consistency M

Qexp

Qpower

Qinv-exp

Chaplygin

:

:

:

Model

(parametrization)

DataP(z)

Constraints on

Parameters

Test the Consistency between Models and Data

Mi dQi (z)/dz

:

:

:

reconstr

uct

observationalconstraint

:

:

:theoretical

prediction: 0

consistentinconsistent

SN Ia (Constitution)CMB (WMAP 5)BAO (SDSS,2dFGRS) z

zwwzw a

1)( 0DE

Linder, PRL, 2003

parameters:{m,w0,wa}

in

Gu, CW Chen and P Chen, 2008 CW Chen, Gu and P Chen, 2009

Q1(z)

Q2(z)

Q3(z)

:

Qi(z)

:

:

:

M1(z)

M2(z)

M3(z)

M4(z)

M5(z)

:

:

:

Measure of Consistency M

Qexp

Qpower

Qinv-exp

Chaplygin

:

:

:

Model

(parametrization)

DataP(z)

Constraints on

Parameters

Test the Consistency between Models and Data

Mi dQi (z)/dz

:

:

:

reconstr

uct

observationalconstraint

:

:

:theoretical

prediction: 0

consistentinconsistent

SN Ia (Constitution)CMB (WMAP 5)BAO (SDSS,2dFGRS)

CW Chen, Gu and P Chen, 2009

in

parameters:{m,w0,wa}

z

zwwzw a

1)( 0DE

Linder, 2003

Chevallier&Polarski, 2001

CDM: measure of consistency M dQ(z)/dz

CDM : = constant CDM in DE zzQ

95.4% C.L.

68.3% C.L.

consistent

CW Chen, Gu and P Chen, 2009

dz

zdQz i

i M

Quintessence: Exponential potential

exp:Quint. in 1

1

exp Mzd

dVzVzQ

Quintessence, exponential: V() = V1exp[/M1]

95.4% C.L.

68.3% C.L.

inconsistent

CW Chen, Gu and P Chen, 2009

dz

zdQz i

i M

Quintessence: Power-law potential

law-power:Quint. in 1

1

2

22

power nzd

Vdz

d

dVzVzQ

Quintessence, power-law: V() = m4nn

95.4% C.L.

68.3% C.L.

consistent

CW Chen, Gu and P Chen, 2009

dz

zdQz i

i M

Quintessence: Inverse-exponential potential

exp-inv:Quint. in 14

2

22

2

23

exp-inv Md

dV

Vd

Vd

d

dV

VzQ

Quintessence, inverse-exponential: V() = V2exp[M2/]

95.4% C.L.

68.3% C.L.

consistent

CW Chen, Gu and P Chen, 2009

dz

zdQz i

i M

Generalized Chaplygin Gas

Chaplygin in 1

1

DEDEChaplygin

zdz

dz

dz

dw

zw

zzQ

generalized Chaplygin gas: pDE(z) = A/DE(z) , A>0, 1

95.4% C.L.

68.3% C.L.

consistent

CW Chen, Gu and P Chen, 2009

dz

zdQz i

i M

Measure of Consistency for 5 DE ModelsCW Chen, Gu and P Chen, 2009

dz

zdQz i

i M

Discriminative Power

between Dark Energy Models

exp:Quint. in

1

1

exp

M

zd

dVzVzQ

law-power:Quint. in

1

1

2

22

power

n

zd

Vdz

d

dVzVzQ

Distinguish …

Quintessence, exponential:

V() = V1exp[/M1] Quintessence, power-law: V() = m4nn

Gu, CW Chen and P Chen, 2009

from

z

zwDE

1

2.08.0

M5

z

zwDE

1

2.005.1

M6

z

zwDE

1

5.06.0

M7

z

zwDE

1

0.105.1

M8

z

zwDE

1

5.01

M31 wwDE

M18.0DEw

M2

z

zwDE

1

5.11

M4

(8 models)

M1(z)

M2(z)

M3(z)

Qexp

Qpower

(parametrization)

DataP(z)

Constraints on

Parameters

Procedures

reconstruct2023 SNe (SNAP quality)CMB (WMAP5 quality)BAO (current quality)

Gu, CW Chen and P Chen, 2009

FiducialModels

M1,…,M8

simulation

in

observationalconstraint

theoreticalprediction: 0

indistinguishable

distinguishable

ModelMeasure of Consistency M

Mi dQi (z)/dzparameters:{m,w0,wa}

z

zwwzw a

1)( 0DE

Linder, 2003

Chevallier&Polarski, 2001

Distinguish from 8 models (M1–M8) Gu, CW Chen and P Chen, 2009

Exp. potential Power-law …

exp.

power-law

exp.

power-law

more slowly evolving wDE(z) faster evolving wDE(z)

O O O O

O O

O O

O O

O O

X X

X X

Summary

We proposed an approach to the testing of dark energy models

by observational results via a characteristic Q(z) for each model.

We performed the consistency test of 5 dark energy models: CDM, generalized Chaplygin gas, and 3 quintessence with exponential, power-law, and inverse-exponential potentials. The exponential potential is ruled out at 95.4% C.L. while the other 4 models are consistent with current data.

With the future observations and via our approach:

– Exponential potential: distinguishable from the 8 models (under consideration).

– Power-law potential: distinguishable from the models with faster evolving w(z) [M3,M4,M7,M8]; but NOT from those with more slowly evolving w(z) [M1,M2,M5,M6].

Summary and Discussions

The consistency test is to examine whether the condition necessary for a model is excluded by observations.

Our approach to the consistency test is simple and efficient because: For all models, Q(z) and dQ/dz are reconstructed from data via the observational constraints on a single parameter space that by choice can be easily accessed.

By our design of Q(z), the consistency test can be performed without the knowledge of the other parameters of the models.

Generally speaking, an approach invoking parametrization may be accompanied by a bias against certain models. This issue requires further investigation.

0

dz

zdQ

Summary and Discussions (cont.)

This approach can be applied to other DE models and other explanations of the cosmic acceleration.

The general principle of this approach may be applied to other cosmological models and even those in other fields beyond the scope of cosmology.

Summary and Discussions (cont.)

Thank you.