2. Dark Energy Phenomenology: Quintessence Potential Reconstruction Je-An Gu 顧哲安 National Center for Theoretical Sciences 2007/10/02 @ CYCU Collaborators : Chien-Wen Chen 陳建文 @ NTU Pisin Chen 陳丕燊 @ SLAC New : Yu-Hsun Lo 羅鈺勳 @ NTHU Qi-Shu Yan 晏啟樹 @ NTHU

3.  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)

4. Introduction (Basic Knowledge, Motivation, SN, SNAP)

5. Accelerating Expansion (homogeneous & isotropic) Based on FLRW Cosmology Concordance:   = 0.73,  M = 0.27

6. Supernova Cosmology Project: http://www-supernova.lbl.gov/

7. 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 d L (distance precision:  mag = 0.15 mag   d L /d L ~ 7%)  Spectral information  redshift z SN Ia Data: d L (z) [ i.e, d L,i (z i ) ] [ ~ x(t) ~ position (time) ] F: flux (energy/area  time) L: luminosity (energy/time) Distance Modulus  (z) (z) history

8. (can hardly distinguish different models) SCP (Perlmutter et. al.) Distance Modulus 1998

9. 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

10. Riess et al. astro-ph/0611572 2006

11. Riess et al. astro-ph/0611572

12. Supernova / Acceleration Probe (SNAP) z0~0.20.2~1.21.2~1.41.4~1.7 # of SN5018005015 observe ~2000 SNe in 2 years statistical uncertainty  mag = 0.15 mag  7% uncertainty in d L  sys = 0.02 mag at z =1.5  z = 0.002 mag (negligible)

14. Supernova Data Analysis ( Parametrization / Fitting Formula )

15. Observations / Data mapping 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  N  M analyses  models and M data set   analyses !! Reality: many models survive. Not so meaningful….

16. ( Reality : Many models survive )  Instead of comparing models and data (thereby ruling out models), Extract physical information about dark energy from data. Two Basic Questions about Dark Energy which should be answered first ? Is Dark Energy played by  ? i.e. w DE =  1 ? ? Is Dark Energy metamorphic ? i.e. w DE = 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.

17. Observations / Data mapping out the evolution history (e.g. SNe Ia, Baryon Acoustic Oscillation) Parametrization Fitting Formula (model independent ?) Dark Energy Info  w DE =  1 ?  w DE = const. ? analyzed by invoking

18.  Error Evaluation:Gaussian error propagation: [from d L (z) to w(z)] d L (z)  w(z) Parametrization / Fitting Formula : one example (polynomial fit of d L ) { d L (0) = 0  c 0 = 0 }  Best Fit: Minimizing the  2 function (function of c i ’s)  constraints on c i ’s

19. Two Parametrizations / Fits Fit 1 ( “  ” means dark energy ) Fit 2 (Linder)

20.  Best Fit: minimizing the  2 function (function of w i ’s)  constraints on w i ’s  Error Bar: Gaussian error propagation: Two Parametrizations / Fits (1) ; (2)

21. In the parametrization part ….. Two Basic Questions about Dark Energy which should be answered first ? Is Dark Energy played by  ? i.e. w DE =  1 ? ? Is Dark Energy metamorphic ? i.e. w DE = 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 w DE model w DE =  1 ? w DE = const. ?

22. Quintessence Model ( Potential Reconstruction: general formulae)

23. Friedmann-Lemaitre-Robertson-Walker (FLRW) Cosmology Homogeneous & Isotropic Universe : (Dark Energy)

24.  (from vacuum energy) Quintessence Candidates: 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 (Non-FLRW) for an inhomogeneous universe (based on FLRW)

25. FLRW + Quintessence Quintessence: dynamical scalar field  Action : Field equation: energy density and pressure : Slow evolution and weak spatial dependence  V(  ) dominates  w  ~  1  Acceleration How to achieve it (naturally) ? ?

26. FLRW + Quintessence Quintessence: dynamical scalar field  Action : Field equation: energy density and pressure :

27. Tracker Quintessence  Power-law :  Exponential :

28. Quintessence Potential Reconstruction (general formulae)

29. analyzed by invoking Observations / Data Parametrization Dark Energy Info    (z)    (z) and w  (z) [for d L (z),   (z), w  (z), …etc.] Quint. Reconstruction (1) (2)

30. Quintessence Reconstruction ( from data via 2 parametrizations of w  )

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

32. Data and Parametrizations (2) (1) Astier05 + WMAP3 + SDSS (68% confidence level) (2) (1) (2) (1) SNAP expectation (68% confidence level) (centered on  CDM)

33. Quintessence Potential Reconstruction (general formulae)

34. Quintessence Potential Reconstruction  Power-law :  Exponential : 4 parameters 3 parameters ?

35. Quintessence Reconstruction: allowed region (1) (in unit of  0 ) [ in unit of (8  G/3)  1/2 ] excluded by parametrization (1) 2 10 11  0.2

36. Quintessence Reconstruction: allowed region (in unit of  0 ) [ in unit of (8  G/3)  1/2 ] (2) excluded by parametrization (2) 2.5 10 11  0.2

37. Quintessence Reconstruction: allowed region (1) (2) (in unit of  0 ) [ in unit of (8  G/3)  1/2 ] excluded by 2 parametrizations excluded by 2 parametrizations 2 10 11  0.2

38. Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (Quint.) (in unit of  0 ) [ in unit of (8  G/3)  1/2 ] 1.2 1.4 0.8  0.1  0.5

39. Quintessence Potential Reconstruction (SNAP) (Astier05) (Quint.) (2) (in unit of  0 ) [ in unit of (8  G/3)  1/2 ] 1.2 1.6 0.8  0.1  0.5

40. Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (in unit of  0 ) [ in unit of (8  G/3)  1/2 ] (SNAP) (Astier05) (2) 1.2 1.6 0.8  0.1  0.5

41. Tracker Quintessence  Exponential :  Power-law : ( n < 0 for Tracker ) characteristic :

42. Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (Quint.) 1 2 22 11

43. Quintessence Potential Reconstruction (SNAP) (Astier05) (Quint.) (2) 1 2 66 22 44

44. Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (SNAP) (Astier05) (2) 1 2 44 11 33 22 Exponential V(  ) disfavored.

45. Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (SNAP) (Astier05) (2) 1 2 0.4  0.2 0.2  0.4 Exponential V(  ) disfavored.

46. Tracker Quintessence  Exponential :  Power-law : ( n < 0 for Tracker ) characteristic :

47. Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (Quint.) 1 2 11  0.2 0.2

48. Quintessence Potential Reconstruction (SNAP) (Astier05) (Quint.) (2) 1 2 1 11 33 22 2

49. Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (SNAP) (Astier05) (2) 1 2 1 11 22 Power-law V(  ) consistent with data.

50. Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (SNAP) (Astier05) (2) 1 2 11  0.5 0.5 Power-law V(  ) consistent with data.

51. Quintessence Potential Reconstruction For power-law V(  ),  0.75 < n < 0. (1) (SNAP) (Astier05) (Quint.) 1 2 11  0.2

52. Quintessence Potential Reconstruction For power-law V(  ),  1 < n < 0. (SNAP) (Astier05) (Quint.) (2) 1 2 11 1

53. Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (SNAP) (Astier05) (2) For power-law V(  ),  0.75 < n < 0. For power-law V(  ),  1 < n < 0. 1 2 11 1

54. Summary

55.  Formulae for quintessence V(  ) reconstruction presented. Summary  Quintessence V(  ) reconstructed by recent data (SNLS SN, WMAP CMB, SDSS LSS-BAO)  Some region of V(  ) excluded by the chosen parametrization reconstructed by recent data (bias?)  no turning back for quintessence field   Exponential V(  ) disfavored (?)  For power-law V(  ), (1) –0.75 < n < 0 ; (2) –1 < n < 0.  A model-indep approach to comparing (ruling out) Quintessence models is proposed, which involves characteristics of potentials. 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.