Dark Energy Phenomenology: Quintessence Potential Reconstruction

1. Dark Energy Phenomenology: Quintessence Potential Reconstruction Je-An Gu顧哲安 National Center for Theoretical Sciences Collaborators : Chien-Wen Chen 陳建文@ NTU Pisin Chen 陳丕燊@ SLAC New blood : 羅鈺勳 @ NTHU Qi-Shu Yan 晏啟樹 @ NTHU (in alphabetical order) 2007/10/16 @NTHU

2. Content • Introduction(basic knowledge, motivation; SN; SNAP) • Supernova Data Analysis (parametrization / fitting formula) • Quintessence -- potential reconstruction: general formulae • Quintessence Potential Reconstruction (from data via two parametrizations) • Summary

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

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

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

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

8. 2004 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

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

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

11. Supernova / Acceleration Probe (SNAP) observe ~2000 SNe in 2 years statistical uncertainty mag = 0.15 mag  7% uncertainty in dL sys = 0.02 mag at z =1.5 z= 0.002 mag (negligible)

13. Supernova Data Analysis ( Parametrization / Fitting Formula )

14. 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….

15. ( Reality : Many models survive ) Is Dark Energy played by ? i.e. wDE = 1 ? Is Dark Energy metamorphic? i.e. wDE = const.? ? ? • Instead of comparing models and data (thereby ruling out models), • Extract physical information about dark energy from data • in model independent manner. Two Basic Questions about Dark Energy which should be answered first 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.

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

17. Parametrization / Fitting Formula : one example (polynomial fit of dL) { dL(0) = 0  c0 = 0 } • Best Fit: Minimizing the 2 function (function of ci’s)  constraints on ci’s • Error Evaluation: Gaussian error propagation: [from dL(z) to w(z)] dL(z)  w(z)

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

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

20. In the parametrization part ….. Which parametrization is capable of ruling out : wDE = 1 wDE = const. trivial incapable example: CDM model trivial incapable example: const wDE model ? ? ? ? Two Basic Questions about Dark Energy which should be answered first Is Dark Energy played by  ? i.e. wDE = 1 ? Is Dark Energy metamorphic? i.e. wDE = const.? (We can never know which model is correct.) (What we can do is ruling out models.)

21. Quintessence Model ( Potential Reconstruction: general formulae)

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

23. Candidates: Dark Geometryvs. Dark Energy Geometry Matter/Energy ↑ ↑ Dark Geometry Dark Matter / Energy Einstein Equations Gμν＝ 8πGNTμν • (from vacuum energy) • Quintessence • Modification of Gravity • Extra Dimensions • Averaging Einstein Equations (based on FLRW) for an inhomogeneous universe (Non-FLRW)

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

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

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

27. Quintessence Potential Reconstruction (general formulae)

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

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

30. Data and Parametrizations (1) (2) Yun Wang and Pia Mukherjee, Astrophys.J. 650 (2006) 1 [astro-poh/0604051]. 68% 95% 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]

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

32. Quintessence Potential Reconstruction (general formulae)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

49. Summary

50. Summary Formulae for quintessence V() reconstruction presented. Quintessence V() reconstructed by recent data. (SNLS SN, WMAP CMB, SDSS LSS-BAO) 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.  Exponential V() disfavored.  For power-law V() , (1) –0.75 < n < 0 ; (2) –1 < n < 0 .