Dark Energy Phenomenology: Quintessence Potential Reconstruction Je-An Gu 顧哲安 National Center for Theoretical Sciences NTHU Collaborators : Chien-Wen Chen NTU Pisin Chen SLAC New blood : NTHU Qi-Shu Yan NTHU (in alphabetical order)

 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)

Introduction (Basic Knowledge, Motivation, SN, SNAP)

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

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

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

Fig.4 in astro-ph/ [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   =

Riess et al. astro-ph/

Riess et al. astro-ph/

Supernova / Acceleration Probe (SNAP) z0~0.20.2~1.21.2~1.41.4~1.7 # of SN 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 = mag (negligible)

Supernova Data Analysis ( Parametrization / Fitting Formula )

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….

( Reality : Many models survive )  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 ? 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.

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

 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

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

 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)

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. ?

Quintessence Model ( Potential Reconstruction: general formulae)

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

 (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)

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) ? ?

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

Tracker Quintessence  Power-law :  Exponential :

Quintessence Potential Reconstruction (general formulae)

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)

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

Yun Wang and Pia Mukherjee, Astrophys.J. 650 (2006) 1 [astro-poh/ ]. Data and Parametrizations (2) Astier05: 1yr SNLS: Astron.Astrophys.447 (2006) [astro-ph/ ] WMAP3: Spergel et al., Astrophys.J.Suppl.170 (2007) 377 [astro-ph/ ] SDSS(BAO): Eisenstein et al., Astrophys.J. 633 (2005) 560 [astro-ph/ ] 68%95% (1)

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

Quintessence Potential Reconstruction (general formulae)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Summary

 Formulae for quintessence V(  ) reconstruction presented. Summary  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.  Exponential V(  ) disfavored.  For power-law V(  ), (1) –0.75 < n < 0 ; (2) –1 < n < 0. 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.