An Approach to Testing Dark Energy by Observations Collaborators : Chien-Wen Chen Phys, NTU Pisin Chen 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 CosPA 2009, Melbourne

An Approach to Testing Dark Energy by Observations Collaborators : Chien-Wen Chen Phys, NTU Pisin Chen 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 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) [arXiv: ].  Chien-Wen Chen, Je-An Gu, and Pisin Chen, “Consistency test of dark energy models,” Modern Physics Letters A 24 (2009) 1649 [arXiv: ].

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 ↑ G μν ＝ 8πG N T μν Modification of Gravity Averaging Einstein Equations Extra Dimensions for an inhomogeneous universe  (from vacuum energy) Quintessence/Phantom (based on FLRW)

M 1 (O) M 2 (O) M 3 (X) M 4 (X) M 5 (O) M 6 (O) : Observations Data Data Analysis Models Theories 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: ] C.-W. Chen, Gu and P. Chen, Mod. Phys. Lett. A [arXiv: ]

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 Chen arXiv: E.g.,  CDM  DE (z): energy density w DE (z) = w 0 + w a z/(1+z) Along a similar line of thought, focusing on  CDM :  Sahni, Shafieloo and Starobinsky, PRD [ ]:  Zunckel and Clarkson 2008, PRL101 [ ]:

Q1(z)Q2(z)Q3(z):Qi(z):::Q1(z)Q2(z)Q3(z):Qi(z)::: M1M2M3:Mi:::M1M2M3:Mi::: Model (parametrization) Data P(z)P(z) Constraints on Parameters Test the Consistency between Models and Data Gu, CW Chen and P Chen, 2008 Characteristic Q Q i [ P(z),z ] in

Measure of Consistency M M1M2M3:Mi:::M1M2M3:Mi::: Model (parametrization) Data P(z)P(z) Constraints on Parameters Test the Consistency between Models and Data Gu, CW Chen and P Chen, 2008 M i  dQ i (z)/dz :::::: : reconstruct observational constraint : :::::: theoretical prediction: 0  consistent inconsistent Q1(z)Q2(z)Q3(z):Qi(z):::Q1(z)Q2(z)Q3(z):Qi(z)::: M1(z)M2(z)M3(z):Mi(z):::M1(z)M2(z)M3(z):Mi(z)::: in

Q1(z)Q2(z)Q3(z):Qi(z):::Q1(z)Q2(z)Q3(z):Qi(z)::: M1(z)M2(z)M3(z):Mi(z):::M1(z)M2(z)M3(z):Mi(z)::: Measure of Consistency M M1M2M3:Mi:::M1M2M3:Mi::: Model (parametrization) Data P(z)P(z) Constraints on Parameters Test the Consistency between Models and Data Gu, CW Chen and P Chen, 2008 M i  dQ i (z)/dz :::::: : reconstruct observational constraint : :::::: theoretical prediction: 0  consistent inconsistent SN Ia (Constitution) CMB (WMAP 5) BAO (SDSS,2dFGRS) in parameters: {  m,w 0,w a } Linder, 2003 Chevallier&Polarski, 2001

Q1(z)Q2(z)Q3(z):Qi(z):::Q1(z)Q2(z)Q3(z):Qi(z)::: M1(z)M2(z)M3(z)M4(z)M5(z):::M1(z)M2(z)M3(z)M4(z)M5(z)::: Measure of Consistency M  Q exp Q power Q inv-exp Chaplygin : Model (parametrization) Data P(z)P(z) Constraints on Parameters Test the Consistency between Models and Data M i  dQ i (z)/dz :::::: reconstruct observational constraint :::::: theoretical prediction: 0  consistent inconsistent SN Ia (Constitution) CMB (WMAP 5) BAO (SDSS,2dFGRS) parameters: {  m,w 0,w a } in Linder, 2003 Chevallier&Polarski, 2001 CW Chen, Gu and P Chen, 2009

Characteristics Q(z) of 5 Models   CDM :   = constant  Quintessence, exponential: V(  ) = V 1 exp [  /M 1 ]  Quintessence, power-law: V(  ) = m 4  n  n  Quintessence, inverse-exponential: V(  ) = V 2 exp [ M 2 /  ]  generalized Chaplygin gas: p DE (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):::Q1(z)Q2(z)Q3(z):Qi(z)::: M1(z)M2(z)M3(z)M4(z)M5(z):::M1(z)M2(z)M3(z)M4(z)M5(z)::: Measure of Consistency M  Q exp Q power Q inv-exp Chaplygin : Model (parametrization) Data P(z)P(z) Constraints on Parameters Test the Consistency between Models and Data M i  dQ i (z)/dz :::::: reconstruct observational constraint :::::: theoretical prediction: 0  consistent inconsistent SN Ia (Constitution) CMB (WMAP 5) BAO (SDSS,2dFGRS) Linder, PRL, 2003 parameters: {  m,w 0,w a } in Gu, CW Chen and P Chen, 2008CW Chen, Gu and P Chen, 2009

Q1(z)Q2(z)Q3(z):Qi(z):::Q1(z)Q2(z)Q3(z):Qi(z)::: M1(z)M2(z)M3(z)M4(z)M5(z):::M1(z)M2(z)M3(z)M4(z)M5(z)::: Measure of Consistency M  Q exp Q power Q inv-exp Chaplygin : Model (parametrization) Data P(z)P(z) Constraints on Parameters Test the Consistency between Models and Data M i  dQ i (z)/dz :::::: reconstruct observational constraint :::::: theoretical prediction: 0  consistent inconsistent SN Ia (Constitution) CMB (WMAP 5) BAO (SDSS,2dFGRS) CW Chen, Gu and P Chen, 2009 in parameters: {  m,w 0,w a } Linder, 2003 Chevallier&Polarski, 2001

 CDM: measure of consistency M   dQ  (z)/dz   CDM :   = constant 95.4% C.L. 68.3% C.L. consistent CW Chen, Gu and P Chen, 2009

Quintessence: Exponential potential  Quintessence, exponential: V(  ) = V 1 exp [  /M 1 ] 95.4% C.L. 68.3% C.L. inconsistent CW Chen, Gu and P Chen, 2009

Quintessence: Power-law potential  Quintessence, power-law: V(  ) = m 4  n  n 95.4% C.L. 68.3% C.L. consistent CW Chen, Gu and P Chen, 2009

Quintessence: Inverse-exponential potential  Quintessence, inverse-exponential: V(  ) = V 2 exp [ M 2 /  ] 95.4% C.L. 68.3% C.L. consistent CW Chen, Gu and P Chen, 2009

Generalized Chaplygin Gas  generalized Chaplygin gas: p DE (z) =  A/  DE (z) , A>0,   % C.L. 68.3% C.L. consistent CW Chen, Gu and P Chen, 2009

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

Discriminative Power between Dark Energy Models

Distinguish …  Quintessence, exponential: V(  ) = V 1 exp [  /M 1 ]  Quintessence, power-law: V(  ) = m 4  n  n Gu, CW Chen and P Chen, 2009 from M5M6M7M8 M3M1M2M4 (8 models)

M1(z)M2(z)M3(z)M1(z)M2(z)M3(z)  Q exp Q power (parametrization) Data P(z)P(z) Constraints on Parameters Procedures reconstruct 2023 SNe (SNAP quality) CMB (WMAP5 quality) BAO (current quality) Gu, CW Chen and P Chen, 2009 Fiducial Models M1,…,M8 simulation in observational constraint theoretical prediction: 0  indistinguishable distinguishable Model Measure of Consistency M M i  dQ i (z)/dz parameters: {  m,w 0,w a } 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 w DE (z)faster evolving w DE (z) OOOO OO OO OO OO XX XX

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. 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.