La constante cosmologique Marc Lachièze-Rey (APC, Paris) Λ Marc Lachièze-Rey – Grenoble 2008
Outline I Historical elements II Elements of cosmology III Observational evidences for accelerating universe IV Possible solutions Dark energy ? Modify gravity ? The genuine cosmological constant Not a « problem » but a possible solution The physics with lambda - Marc Lachièze-Rey – Grenoble 2008
Short history of Λ General relativity : • Einstein 1916 : no cosmological constant • Einstein 1917: first relativistic model --> gr with Λ • 1930 (Slipher, Hubble, Lemaître) : cosmic expansion --> Einstein renounces to Λ • age problem : Lemaître « saves » the “big bang” with Λ later : Cosmic distance Recalibration : Λ useless ? --> CDM paradigm (1970’s) : Λ = = 0 0 • age , galaxy formation --> CDM- Λ • SN’s observations confirm need for Λ • at the same moment, rejection of Λ --> dark energy ? Marc Lachièze-Rey – Grenoble 2008
Original GR (Einstein, 1916) Gravitation = geometry of space-time: metric g --> Riemann and Ricci tensors, Einstein tensor G The material content of the universe determines the geometry : Einstein Equation : G(g) = χ T T = energy-momentum tensor of material content = the source of gravitation Marc Lachièze-Rey, Grenoble 4 2008
Relativistic Cosmology (Einstein 1917) GR is an ideal tool for cosmology : cosmic model = a space-time describing the whole universe, solution of Einstein equations function of the average material content (matter, radiation, etc.) Einstein wants a cosmic model: • without spatial infinite : closed spatial sections • without spatial limit • static (expansion unknown in 1917) No such solution to Einstein equation as written --> Einstein modifies its equation Marc Lachièze-Rey, Grenoble 5 2008
New Einstein equation (1917) G(g) = χ T G(g) = χ T + Λ g New term Λ = cosmological constant - (no other term possible from mathematical consistency) - absolute constant - non material - repulsive effect - [almost] no « local » effet (in Solar System, galaxies, bh, …): Only at cosmological scales. --> Einstein cosmological model Static : attraction by matter balanced by repulsion by Λ space = a three-sphere : closed, no boundary ! Marc Lachièze-Rey, Grenoble 6 2008
Cosmic expansion (1930) • Observations by Vesto Slipher, Edwin Hubble --> Hubble law (1929) • Theoretical work by Georges Lemaître (“Hubble law” 1927) --> cosmic expansion --> no static model --> no need for Λ ? Marc Lachièze-Rey, Grenoble 7 2008
Cosmic expansion (1930) Marc Lachièze-Rey, Grenoble 8 2008
Lemaître (1931) : primordial atom (--> later : big bang) Marc Lachièze-Rey, Grenoble 9 2008
Lemaître (1931) : primordial atom (--> later : big bang) Age problem : wrong calibrations --> Age of the Universe < age of the Earth ! possible solution : Λ also, galaxy formation difficult without Λ : Lemaître « saves » the big bang with Λ Marc Lachièze-Rey, Grenoble 10 2008
• Controversy : Einstein equation with or without Λ ? (Einstein will get out of the cosmological debate) Marc Lachièze-Rey, Grenoble 11 2008
…History… • 1960’s : Cosmic distance Recalibration --> age problem resolved without Λ : Λ useless ? --> CDM big bang paradigm ( 1970’s) : Λ = Ω Λ = 0 Ω matter = 1 But Age of the universe: One needs Λ Galaxy formation --> idem --> CDM- Λ • 1990: Supernova observations (a classical cosmological test) --> the expansion accelerates, exactly as predicted by Λ confirms the need of Λ : CDM- Λ • concordance : also confirmed by other observations • More recently : « GR with Λ unsatisfactory » --> Search another explanation for cosmic data (must mimick Λ ) Marc Lachièze-Rey – Grenoble 2008
II Cosmic parameters • Cosmic Dynamics • Spatial curvature : • Cosmological constant Λ • Content = source of gravitation Marc Lachièze-Rey, Grenoble 13 2008
Cosmic dynamics • Expansion law: scale factor R(t) • expansion • rate Present Value = Hubble constant = H 0 ≈ 70 km /sec /Mpc < 0 acceleration • deceleration parameter > 0 deceleration -.55 - Third derivative --> parameter w - Beyond w = w(z)… Marc Lachièze-Rey, Grenoble 14 2008
Spatial curvature • • • k = sign • R C = spatial curvature radius • Fundamental • relation : • Marc Lachièze-Rey, Grenoble 15 2008
[genuine] cosmological Constant • Λ constant by definition (required by mathematical consistency) • In cosmic units : ( ≈ 0.7) • Fundamental length scale (constant) R Λ =( Λ ) -1/2 ∼ 3 Gpc Marc Lachièze-Rey, Grenoble 16 2008
Material content = source of gravitation : For any substance : • energy density ρ in cosmological units (density parameter) • pressure p cosmological influence depends on ρ +3p • equ. of state p = f( ρ ) , parametrized as Non relativistic matter ( = dust) p ≈ 0 : w = 0 Radiation p = ρ / 3 : w= 1 / 3 Nothing else known in physics Marc Lachièze-Rey – Grenoble 2008
( Equation of State) For a flat Universe: – Matter-dominated Universe � � R � 3 , R � t 2 /3 – Radiation-dominated Universe � � R � 4 , R � t 1/2 – Vacuum-dominated Universe � � R 0 , R � e Ht Marc Lachièze-Rey – Grenoble 2008
Exotic substance as a source of gravitation ? Accelerating ρ +3p < 0 ⇔ w < -1/3 ⇔ w = -1 : same cosmological effect than Λ w ∼ -1 : similar effects: exotic (dark) energy some physical basis ? see later Marc Lachièze-Rey – Grenoble 2008
Summary .98 < Ω total < 1.08 Ω rad ~ 5 10 -5 Ω baryons ~ 0.04 Ω matter ~ .3 ==> Ω Λ ~ .7 Even without SN’s observations Marc Lachièze-Rey – Grenoble 2008
III Observational evidences for accelerating Universe - age of the Universe - Galaxy formation - Cosmography : SNIa - HST, CMB, LSS : concordance -X-ray clusters - BAO (<-- SDSS) - cosmic shear -Sachs-Wolfe integrated effect Marc Lachièze-Rey – Grenoble 2008
Age of the universe The strongest the more direct historically the first evidence for lambda Marc Lachièze-Rey, Grenoble 22 2008
Age of the universe t U = « time » duration since the univers was « very small ». Finite by definition in big bang models. Function of the cosmic parameters Marc Lachièze-Rey, Grenoble 23 2008
t_ U > ages of oldest stars ( ∼ 12 Gyrs) ==> Λ ≠ 0 Galaxy formation • If Λ =0, no sufficient time for the galaxies to form (Lemaître, 1930’s) 1980’s --> CDM Λ paradigm Marc Lachièze-Rey, Grenoble 24 2008
Supernovas (SNIa) = luminosity distance measurements CLASSICAL COSMOLOGICAL TEST : Hyp : SNIa are standardizable candles L D lum = 4 � f Riess et al. 1998, AJ 116,, 1009 (High-Z SN Search) Perlmutter et al 1999, ApJ 517, 565 (SCP) Perlmutter 2003,Physics Today Marc Lachièze-Rey – Grenoble 2008
Supernova cosmology project (Knop, Perlmuter) Do we have good data, good interpretation of them ? • Are SNs good standard candles ? • May some substance interact with the photons and modify our perception of SN data? ••• But concordance Marc Lachièze-Rey – Grenoble 2008
(SN observations) • SuperNova Legacy Survey (SNLS) P. Astier,et al , A&A, 447, 31, (2006) • ”gold” data set of supernovae Riess et al. http://fr.arxiv.org/abs/astro-ph/0611572 • The ESSENCE Supernova Survey: In its first four years, 102 type Ia SNe, at z from 0.10 to 0.78. Marc Lachièze-Rey – Grenoble 2008
(Essence ) Marc Lachièze-Rey – Grenoble 2008
concordance Marc Lachièze-Rey – Grenoble 2008
WMAP Marc Lachièze-Rey – Grenoble 2008
CMB alone measurements are degenerate in Ω M & Ω Λ (requires h) --> Use CMB ( WMAP-3 (Spergel et al. 06) ) + something else something else • Different combinations of HST, SN, LSS, BAO, Shear, LTSW are consistent : Different combinations of HST, SN, LSS, BAO, Shear, LTSW are consistent : Overconstrained model : model : Overconstrained � � ~ 1 � � M ~ 0.7 • CMB + HST : Ω Λ = 0.758 +- 0.06 HST Key Project measurement of the Hubble constant ( Freedman et al. 2001, ApJ 553, 47) : h 100 = 0.72 +- 0.08 • CMB + SNLS : Ω Λ = 0.719 +- 0.03 Marc Lachièze-Rey – Grenoble 2008
Galaxy clusters Marc Lachièze-Rey – Grenoble 2008
Galaxy clusters S.W. Allen et al. 2002 (arXiv:astro-ph/0205007v1) X-ray gas mass fraction (in a sample of luminous X clusters <-- Chandra Observatory) as a function of z --> cosmological constraints Ω m = 0.30+0.04, Ω Λ = 0.95 assuming -a profile (FNW) - a cosmological scenario of cluster formation (gravitational instability) - the value of H 0 (<-- Hubble Key Project) (independent mass confirmation from gravitational lensing studies.) ρ =2500 ρ critical --> radius r 2500 : approximately constant value Marc Lachièze-Rey – Grenoble 2008
Marc Lachièze-Rey – Grenoble 2008
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