1. 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

2. 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

4. 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].

5. Concordance:   = 0.73,  M = 0.27 Accelerating Expansion (homogeneous & isotropic) Based on FLRW Cosmology Dark Energy Observations (which are driving Modern Cosmology)

6. (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)

7. 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

8. 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]

9. 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:0803.4504 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 [0807.3548]:  Zunckel and Clarkson 2008, PRL101 [0807.4304]:

10. 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

11. 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

12. 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

13. 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

14. 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

15. Testing DE Models: Results

16. 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

18. 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

19.  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

20. 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

21. 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

22. 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

23. Generalized Chaplygin Gas  generalized Chaplygin gas: p DE (z) =  A/  DE (z) , A>0,   1 95.4% C.L. 68.3% C.L. consistent CW Chen, Gu and P Chen, 2009

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

25. Discriminative Power between Dark Energy Models

26. 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)

27. 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

28. 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

29. Summary

30.  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

31.  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.)

32.  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.)

33. Thank you.