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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
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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)
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Introduction (Basic Knowledge, Motivation, SN, SNAP)
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Accelerating Expansion (homogeneous & isotropic) Based on FLRW Cosmology Concordance: = 0.73, M = 0.27
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Supernova Cosmology Project: http://www-supernova.lbl.gov/
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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
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(can hardly distinguish different models) SCP (Perlmutter et. al.) Distance Modulus 1998
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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
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Riess et al. astro-ph/0611572 2006
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Riess et al. astro-ph/0611572
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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)
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Supernova Data Analysis ( Parametrization / Fitting Formula )
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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….
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( 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.
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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
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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
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Two Parametrizations / Fits Fit 1 ( “ ” means dark energy ) Fit 2 (Linder)
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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)
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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. ?
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Quintessence Model ( Potential Reconstruction: general formulae)
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Friedmann-Lemaitre-Robertson-Walker (FLRW) Cosmology Homogeneous & Isotropic Universe : (Dark Energy)
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(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)
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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) ? ?
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FLRW + Quintessence Quintessence: dynamical scalar field Action : Field equation: energy density and pressure :
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Tracker Quintessence Power-law : Exponential :
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Quintessence Potential Reconstruction (general formulae)
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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)
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Quintessence Reconstruction ( from data via 2 parametrizations of w )
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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)
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Data and Parametrizations (2) (1) Astier05 + WMAP3 + SDSS (68% confidence level) (2) (1) (2) (1) SNAP expectation (68% confidence level) (centered on CDM)
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Quintessence Potential Reconstruction (general formulae)
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Quintessence Potential Reconstruction Power-law : Exponential : 4 parameters 3 parameters ?
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Quintessence Reconstruction: allowed region (1) (in unit of 0 ) [ in unit of (8 G/3) 1/2 ] excluded by parametrization (1) 2 10 11 0.2
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Quintessence Reconstruction: allowed region (in unit of 0 ) [ in unit of (8 G/3) 1/2 ] (2) excluded by parametrization (2) 2.5 10 11 0.2
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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 11 0.2
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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
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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
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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
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Tracker Quintessence Exponential : Power-law : ( n < 0 for Tracker ) characteristic :
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Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (Quint.) 1 2 22 11
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Quintessence Potential Reconstruction (SNAP) (Astier05) (Quint.) (2) 1 2 66 22 44
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Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (SNAP) (Astier05) (2) 1 2 44 11 33 22 Exponential V( ) disfavored.
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Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (SNAP) (Astier05) (2) 1 2 0.4 0.2 0.2 0.4 Exponential V( ) disfavored.
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Tracker Quintessence Exponential : Power-law : ( n < 0 for Tracker ) characteristic :
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Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (Quint.) 1 2 11 0.2 0.2
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Quintessence Potential Reconstruction (SNAP) (Astier05) (Quint.) (2) 1 2 1 11 33 22 2
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Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (SNAP) (Astier05) (2) 1 2 1 11 22 Power-law V( ) consistent with data.
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Quintessence Potential Reconstruction (1) (SNAP) (Astier05) (SNAP) (Astier05) (2) 1 2 11 0.5 0.5 Power-law V( ) consistent with data.
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Quintessence Potential Reconstruction For power-law V( ), 0.75 < n < 0. (1) (SNAP) (Astier05) (Quint.) 1 2 11 0.2
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Quintessence Potential Reconstruction For power-law V( ), 1 < n < 0. (SNAP) (Astier05) (Quint.) (2) 1 2 11 1
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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 11 1
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Summary
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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.
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