PROBLEMS OF VACUUM ENERGY AND DARK ENERGY A.D. Dolgov

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1 PROBLEMS OF VACUUM ENERGY AND DARK ENERGY A.D. Dolgov ITEP, , Moscow, Russia INFN, Ferrara 40100, Italy Luminy, September, 19-23,

2 alias: Lambda-term, Vacuum energy Traditional GR point of view: Lambda term (or cosmological constant) is GEOMETRICAL, while vacuum energy is PHYSICAL quantity. They do not have anything in common However, there is no way to distinguish between them. 2

3 PROBLEMS: 1. Theoretically: Λ. Mismatch between theory and data: ORDERS OF MAGNITUDE. 2. Majority point of view during long time and maybe even now: = 0 3. New independent pieces of data: EMPTY SPACE (ANTI)GRAVITATES 4. Close proximity of ρ vac = const to ρ c 1/t 2 exactly today. 5. If antigravitating substance is not vacuum energy then WHAT? 3

4 CONTENT 1. Definition and history. 2. Data in favor of ρ vac 0 (or ρ DE 0). 3. Cosmology with Λ Almost infinite contributions into ρ vac. 5. Possible ways out. 4

5 Biographical notes Name(s): Cosmological constant, Λ-term, vacuum energy or, maybe, dark energy. Date of birth: 1918 Father A. Einstein: The biggest blunder of my life (after Hubble s discovery of cosmological expansion). Many times assumed dead, probably erroneously. Well alive today. Still not safe - many want to kill it. 5

6 SOME MORE QUOTATIONS: LeMaittre: greatest discovery, worth to make Eisntein s name famous. Gamow: λ raises its nasty head again (after indications that quasars are accumulated near z = 2 in the 60s) 6

7 One parameter freedom in GR equations satisfying general covariance: R µν 1 2 g µνr g µν Λ = 8πG N T (m) µν Introduced to make the universe stationary, Λ counterweights gravitational attraction. However, the solution is unstable. Due to covariant conservation laws: D µ (R ν µ 1 ) 2 gµ ν R = 0 (1) D µ T µ (m) ν = 0. and the condition D µ g µ ν 0: Λ = const 7

8 Models with Λ = Λ(t) are not innocent, new fields to respect energy conservation condition are necessary. First attempts to make time-dependent Lambda, 1935 by Bronshtein (Leningrad); strongly criticized by Landau. 8

9 Modern point of view - instead of l.h.s. put Λ into r.h.s. and call it the vacuum energy: R µν 1 ( 2 g µνr = 8πG N T (m) ) µν + ρ vac g µν Normalized to the critical energy density ρ c = 3H 2 m 2 P l /8π: Ω v = ρ vac ρ c 9

10 RISE AND FALL OF LAMBDA-TERM 1. Birth: Ω v Hubble discovery of expansion, earlier Friedman solution: Ω v LeMaitre, De Sitter, later Eddington: one of the most important discoveries in GR. 4. Still non-zero Lambda is not accepted by majority. 5. QSO accumulation near z=2 explained by Ω v 1. Later rejected. 6. From 60s to the end of the Millennium Lambda was identically zero. Only a few treated it seriously, starting from Zeldovich. 10

11 7. End of 90s: a) Universe age crisis. b) Ω m = 0.3, while inflation predicts Ω tot = 1. c) Dimming of high redshift supenovae. d) LSS and CMBR well fit theory if Ω v 0.7. CONCLUSION: Ω v = 0.7 ρ vac GeV Ω m =

12 EVOLUTION OF VACUUM(-LIKE) ENERGY DURING COSMIC HISTORY 1. At inflation ρ vac ρ now v and was DOMINANT. But it was not real vacuum energy but vacuum-like energy of almost constant scalar field inflaton. 2. At GUT p.t. (if such era existed) δρ vac GeV 4 3. At electro-weak p.t. δρ vac 10 8 GeV 4 12

13 4. At QCD p.t. δρ vac 10 2 GeV 4 The magnitude of vacuum energies of gluon and chiral condensates are known from experiment! 13

14 After inflation till almost the present epoch ρ vac was always sub-dominant ρ vac started to dominate energy density only recently at z

15 SOME SIMPLE FEATURES OF LAMBDA DOMINANT COSMOLOGY In homogeneous and isotropic case the energy-momentum tensor has the diagonal form: T ν µ = diag (ρ, p, p, p), For vacuum (because of Lorentz invariance of vacuum) T ν µ g ν µ = δ ν µ and T ν (vac) µ = diag (ρ, ρ, ρ, ρ), (2) i.e. p vac = ρ vac. (Maybe breaking of Lorentz can solve the problem?) 15

16 Covariant energy conservation: ρ = 3H (ρ + p) Hence ρ vac = const and H = const, i.e. a(t) exp(ht) Expansion is ACCELERATED: ä a = 4πG N (ρ + 3p) >

17 For normal vacuum: ρ v = const. However, there can be states with p ρ and ρ const. Slowly varying scalar field φ. Its energymomentum tensor is T µν = 2φ µ φ ν g µν [φ α φ α U(φ)] For negligible φ,µ : T µν g µν U(φ) If U (φ) 0 the field slowly evolves down to equilibrium point. If U (φ) = 0 but this is not the lowest minimum (or even maximum) the field can make quantum jump to the equilibrium. 17

18 New phenomenological parameter: w = p ρ For homogeneous scalar field: w = 2U(φ) φ 2 2U(φ) + φ 2 Hence for normal fields 1 < w < +1. Phantom : w < 1 is possible only in pathological theory, e.g. higher spin tensor fields with tt physical component (AD) or scalars with wrong sign of kinetic term (Caldwell). Ultimately will turn apart everything. 18

19 CONTRIBUTIONS TO VACUUM ENERGY 1. Bosonic vacuum fluctuations: d 3 k ω H b vac = k (2π) 3 2 a k a k + b k b k vac d 3 k = (2π) 3 ω k = 4 2. Fermionic vacuum fluctuations: d 3 k ω H f vac = k (2π) 3 2 a k a k b k b k vac d 3 k = (2π) 3 ω k = 4 Bosonic/fermionic cancellation - Ya.B.Z. prior to SUSY. 19

20 Supersymmetry: N b = N f and m b = m f, then ρ vac = 0 if the symmetry is UNBROKEN. Soft SUSY breaking necessarily leads to ρ vac 10 8 GeV 4 0 Broken SUGRA allows for ρ vac = 0 but the natural value is ρ vac m 4 Pl GeV 4 Phase transitions in the course of cosmological cooling δρ vac GeV 4 20

21 QCD is well established and experimentally verified science leads to conclusion that vacuum is not empty but filled with quark and gluon condensates: qq 0 G µν G µν 0 both having NEGATIVE vacuum energy ρ QCD vac ρ c 21

22 Vacuum condensate is destroyed by quarks and the proton mass is: m p = 2m u + m d ρ vac l 3 p m u m d 5 ev. Who adds the necessary donation to make the OBSERVED ρ vac > 0 and what kind of matter is it? 22

23 INTERMEDIATE SUMMARY 1. Known and huge contributions to ρ vac but unknown mechanism of their compensation down to (almost) zero. 2. Observed today ρ vac ρ c. WHY? 3. What is the nature of antigravitating matter? Consistent with w = 1, vacuum? Mostly only problems 2 and 3 are addressed: a) modification of gravity; b) new field (quintessence) leading to accelerated expansion. 23

24 Most probably all three problems are strongly coupled and can be solved only after adjustment of ρ vac down to ρ c is understood. POSSIBLE SOLUTIONS 1. Subtraction constant. 2. Anthropic principle. 3. Infared instability of massless fields (gravitons) in DS space-time. 4. Dynamical adjustment. 24

25 Dynamical adjustment, as axionic solution of strong CP problem: New field Φ (scalar of higher spin) coupled to gravity is necessary. 1) Vacuum energy condensate of Φ 2) ρ(φ) compensates original ρ vac. Byproducts of dynamical adjustment have many features of less ambitious models of modified gravity, e.g. explicit breaking of Lorentz invariance, and time dependent unstable background and stable fluctuations over it. 25

26 DYNAMICAL ADJUSTMENT Generic predictions: 1. Change exponential expansion to power law one. 2. Compensation of vacuum energy is not complete but only down to terms of the order of ρ c (t). 3. Non-compensated energy may have an unusual equation of state. Unfortunately, no realistic model found starting from

27 EXAMPLES OF ADJUSTMENT 1. Non-minimally coupled scalar field (AD, 1982): φ + 3H φ + U (φ, R) = 0 with e.g. U = ξrφ 2 /2. Solutions are unstable if ξr < 0. Asymptotically: φ t and DS turns into Friedman, but T µν (φ) Fg µν and the change of the regime is achieved due to weakening of gravitational coupling: G N 1/t 2 27

28 2. Vector field V µ (AD, 1985): L = η [F µν F µν /4 + (V;µ) µ 2] ( ) +ξrm 2 ln 1 + V2 m 2 Unstable solution: and V t t + c/t T µν (V t ) g µν + vanishing terms Logarithmic variation of gravitational coupling with time. 28

29 3. Second rank tensor field S µν (AD, 1994): L 2 = η 1 S αβ;γ S αγ;β + η 2 S α β;α Sγβ ;γ + η 3 S α α;β Sγ;β γ Components S tt and isotropic part of S ij δ ij are unstable: ( 2 t + 3H t 6H 2 )S tt 2H 2 s jj = 0 ( 2 t + 3H t 6H 2 )s tj = 0 ( 2 t + 3H t 2H 2 )s ij 2H 2 δ ij S tt = 0 where s tj = S tj /a(t) and s ij = S ij /a 2 (t). 29

30 Ill-defined theory with non-physical components, T tt and/or T ii becoming physical? Ogievetsky and Polubarinov: Photon and Notoph - gauge theory of scalar field described by t-component of vector V µ. 30

31 In all the cases after some period of exponential expansion DS is changed into Friedman and the dominant term in T µν g µν but G N is time-dependent. More important: in all the models above expansion rate is not related to the usual matter. 31

32 4. Scalar with crazy coupling to gravity (Mukohayama, Randall, 2003; AD, Kawasaki, 2003:) A = d 4 x [ g 1 2 (R + 2Λ) + F 1(R) Solution tends to + D µφd µ φ 2 R 2 U(φ, R) R ρ vac + U(φ) = 0 It has some nice features ( almost realistic ), H = 1/2t, etc but unstable. ] 32

33 Equation of motion for Φ: ( ) ] D µ [D µ 1 2 φ + U (φ) = 0. R GR equations for the trace, with F 1 = C 1 R 2 : ( ) 1 2 R + 3 (D α φ) 2 4 [U(φ) + ρ vac ] R ( ) 6D [2C (Dα φ) 2 ] 1 R = T µ µ R R 33

34 10 U' = H R < 0 R R < 0 H > 0 H < 0 φ t 34

35 10 1 U' = 2.5 H 0.1 R < 0 R < 0 φ H > 0 H < 0 φ > 0 φ < 0 R t 35

36 A desparate attempt to improve the model: (Dφ) 2 (Dφ)2 R R. R 2 36

37 1 U' = 2.5 (b) 0.1 H t U - ρ vac Figure 1: (a) Evolution of the scalar φ, Hubble h and the curvature R for the modified kinetic term. (b) Evolution of h = Ht and U ρ vac. t 37

38 More general action with scalar field (AD, Kawasaki, 2003) not yet explored: A = d 4 x g[ m 2 Pl (R + 2Λ)/16π +F 1 (R) + F 2 (φ, R)D µ φd µ φ +F 3 (φ, R)D µ φd µ R U(φ, R)] Moreover R µν and R µναβ can be also included. 38

39 CONCLUSION 1. Some compensating agent must exist! QCD demands that. 2. Quite natural to expect that ρ vac is not completely compensated and ρ ρ c 3. Realistic model is needed, it can indicate what is w: is it (-1) or different. 4. Theoretical prior: A STRANGE FORM OF ENERGY LIVES IN THE UNIVERSE it must be included into data analysis. 39