An Approach to Testing Dark Energy by Observations

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


Mi dQi (z)/dz

:::

:

reconstruct

observationalconstraint

:

:::

theoreticalprediction: 0

consistent

inconsistent

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):::


M1

M2

M3

:Mi

:::

Model

(parametrization)

DataP(z)

Constraints on

Parameters


Mi dQi (z)/dz

:::

:

reconstruct


:

:::


consistent

inconsistent

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

in

parameters:{m,w0,wa}

zzwwzw 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):::


Qexp

Qpower

Qinv-exp

Chaplygin:::

Model

(parametrization)

DataP(z)

Constraints on

Parameters

Test the Consistency between Models and Data

Mi dQi (z)/dz

:::

reconstruct


:::


consistent

inconsistent



in

zzwwzw a

1)( 0DE

Linder, 2003


CW Chen, Gu and P Chen, 2009

exp-inv:Quint. in 142

22

2

23

exp-inv MddV

VdVd

ddV

VzQ

CDM in DE zzQ

exp:Quint. in 1

1

exp MzddVzVzQ

law-power:Quint. in 1

1

2

22

power nzd

VdzddVzVzQ

Chaplygin in 1

1DEDE

Chaplygin

zdz

dzdzdw

zwzzQ

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):::


Qexp

Qpower

Qinv-exp

Chaplygin:::

Model

(parametrization)

DataP(z)

Constraints on

Parameters


Mi dQi (z)/dz

:::

reconstruct


:::


consistent

inconsistent

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

zwwzw a

1)( 0DE

Linder, PRL, 2003


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):::


Qexp

Qpower

Qinv-exp

Chaplygin:::

Model

(parametrization)

DataP(z)

Constraints on

Parameters


Mi dQi (z)/dz

:::

reconstruct


:::


consistent

inconsistent



in


zzwwzw a

1)( 0DE

Linder, 2003


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

CDM : = constant CDM in DE zzQ

95.4% C.L.

68.3% C.L.

consistent


dz

zdQz ii M

Quintessence: Exponential potential

exp:Quint. in 1

1

exp MzddVzVzQ

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

95.4% C.L.

68.3% C.L.

inconsistent


dz

zdQz ii M

Quintessence: Power-law potential

law-power:Quint. in 1

1

2

22

power nzd

VdzddVzVzQ


95.4% C.L.

68.3% C.L.

consistent


dz

zdQz ii M

Quintessence: Inverse-exponential potential

exp-inv:Quint. in 142

22

2

23

exp-inv MddV

VdVd

ddV

VzQ

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

95.4% C.L.

68.3% C.L.

consistent


dz

zdQz ii M

Generalized Chaplygin Gas

Chaplygin in 1

1DEDE

Chaplygin

zdz

dzdzdw

zwzzQ

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

95.4% C.L.

68.3% C.L.

consistent


dz

zdQz ii M

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

dz

zdQz ii M

Discriminative Power

between Dark Energy Models

exp:Quint. in

1

1

exp

M

zddVzVzQ

law-power:Quint. in

1

1

2

22

power

n

zd

VdzddVzVzQ

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


Gu, CW Chen and P Chen, 2009

from

zzwDE

1

2.08.0

M5

zzwDE

1

2.005.1

M6

zzwDE

1

5.06.0

M7

zzwDE

1

0.105.1

M8

zzwDE

1

5.01

M31 wwDE

M18.0DEw

M2

zzwDE

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

FiducialModelsM1,…,M8

simulation

in



indistinguishable

distinguishable

ModelMeasure of Consistency M

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

zzwwzw a

1)( 0DE

Linder, 2003


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.

0dz

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.

An Approach to Testing Dark Energy by Observations

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