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Cosmological particle creation, conformal invariance and A. D. Sakharov’s induced gravity Victor Berezin, Vyacheslav Dokuchaev and Yury Eroshenko Institute for Nuclear Research of the Russian Academy of Sciences, Moscow Ginzburg Centennial Conference on Physics, May 29 – June 3, 2017 Moscow Outline I Introduction II Particle creation — phenomenology III Conformal symmetry IV Scalar field V The “good” hydrodynamical variables and equations of motion VI Vacuum equations “Empty” vacuum and “Dynamical” vacuum = Dirac sea I. Introduction Particle creation — quantum phenomenon Renormalization. Counter-terms. A. D. Sakharov — induced gravity Ya. B. Zel’dovich, A. A. Starobinsky, L. Parker, S. Fulling, V. N. Lukash, A. A. Grib, S. G. Mamaev, V. A. Mostepanenko Back reaction — problems Process of creation Global geometry: horizons, energy dominance violation Phenomenological description of particle creation: Eulerian coordinates, “creation law”, Weyl tensor — renormalizability Conformal gravity. History Nowadays — Higgs Mechanism Creatioon of universe from “nothing” — A. Vilenkin Initial states is conformally invariant — R. Penrose, G. ‘t Hooft Sign convention signature (+, −, −, −) ∂Γµ νσ ∂x λ R µνµσ R µνλσ = Rνσ = − ∂Γµ νλ ∂x σ + Γµκλ Γκνσ − Γµκσ Γκνλ R = g νσ Rνσ Γλµν — metric connections Units ~=c =1 [G ] = [`]2 = 1 [m]2 = [t]2 II. Particle creation — phenomenology Hydrodynamics of perfect fluid with varying number of particles Lagrange description is not adequate =⇒ worldline is varied without taking into account the process of its creation =⇒ Euler description Standard action: (J. R. Ray 1972) Z √ ε(X , n) −g dx + Z µ√ Shydro = − + λ2 X,µ u Z Z −g dx + √ λ0 (u µ uµ − 1) −g dx √ λ1 (nu µ );µ −g dx ε(X , n) — invariant energy density n — invariant particle number density u µ — four-velocity of particle flux X — auxiliary variable (numeration of worldlines) λ0 , λ1 , λ2 — Lagrange multipliers (nu µ );µ = 0 — particle number conservation law Dynamical variables: n, u µ , X Our proposal: to include into the formalism the “creation law” (nu µ );µ = Φ(inv ) 6= 0 (V. A. Berezin 1987) Fundamental result for (cosmological) particle creation Φ = βC 2 C µνλσ Shydro (Ya. B. Zel’dovich, A. A. Starobinsky 1977) — Weyl tensor Z Z √ √ = − ε(X , n) −g dx + λ0 (u µ uµ − 1) −g dx Z Z √ µ√ + λ2 X,µ u −g dx + λ1 ((nu µ );µ − βC 2 ) −g dx λ1 is defined up to an additive constant λ1 → λ1 + γ0 , γ0 = const Z Z √ √ µ 2 √ γ0 ((nu );µ − βC ) −g dx = γ0 ((n −g u µ ),µ − βC 2 −g ) dx Z → −γ0 β √ C 2 −g dx → conformal Weyl gravity III. Conformal invariance — the postulate “+” Additional symmetry 1 2 Creation of the universe from “nothing” A.Vilenkin, Ya. B. Zel’dovich & L.P.Grishchuk, S.W.Hawking... R. Penrose, G ’t Hooft “−” 1 2 Theory was rejected — absence of massive particles H. Weyl, A. Einstein 1921 However, the Higgs mechanism was unknown in 1921! G ’t Hooft 2014 δStot =0 ⇒ δΩ Conformal multiplier Ω may be considered as an independent variable gµν = Ω2 ĝµν , Stot = Sgrav + Smatter By definition: 1 2 Z δSmatter = 1 2 Z δSmatter = Let δg µν = − √ Tµν −g δg µν dx T̂µν p −ĝ δĝ µν dx 2 µν 2 ĝ δΩ = − g µν δΩ 3 Ω Ω Then, if 0 = δSmatter δSgrav =0 ⇒ δΩ Z δΩ √ = − Tµν g µν −g dx Ω ⇒ Tr (Tµν ) = 0 Let δg µν = Ω2 δĝ µν ⇒ T̂µν = Ω2 Tµν (T̂νµ = Ω4 Tνµ , T̂ µν = Ω6 T µν ) IV. What are those fields responsible for particle crreation? The simplest case — scalar field Z 1 2 2 √ 1 µ Sscalar = χ χµ − m χ −g dx 2 2 Conformal transformation (standard): δg µν = Ω2 δĝ µν , Sscalar 1 χ̂ Ω ⇒ Z 1 µ 1 1 χ̂2 1 2 2 2 √ µ µ = χ̂ χ̂µ − χ̂µ Ω + Ωµ Ω − m Ω χ −g dx 2 Ω 2 Ω2 2 χµ = χ,µ χµ = g µν χν χ= χ̂µ = ĝ µν χ̂ν Ωµ = g µν Ων How to make Sscalar to be conformally covariant? R 2 The known prescription: to add to Lagrangian 12 χ R — scalar curvature, constructed on metric gµν ⇒ √ R 1 1 µ χ χµ + χ2 − m2 χ2 −g dx 2 12 2 ! Z R̂ 2 1 2 2 2 p 1 µ χ̂ χ̂µ + χ − m Ω χ̂ = −ĝ dx 2 12 2 Z 1 Ωλ p −ĝ dx − χ̂2 2 Ω |λ Z Sscalar = The last term is the surface integral m2 = 0 — all is OK? But! The correct sign ⇐⇒ the incorrect sign Our choice: 1 2 R̂ 2 The “correct” sign for R̂: − 12 χ̂ The “incorrect” sign for kinetic term: − 12 χ̂µ χ̂µ “Explanatory note” Our scalar field χ is not “real” (fundamental). The already created particles are quanta of this field (phenomenology!) What remains — is a “vacuum” part of the scalar field, containing conformal anomaly, responsible for particle creation The wrong sign sign of kinetic term =⇒ absence of the law energy bound (C -field by Hoyle and Narlikar) Scalar filed χ is not a dynamical variable =⇒ there is no variation How could the field χ know about the conformal transformation? It is always possible to choose χ̂ = 1` ϕ, ϕ = Ω 1 (χ = Ω1 χ̂, [`] = [m] , ~ = c = 1) =⇒ ! Z 1 1 µ R̂ 2 1 2 4 p Sscalar = − 2 ϕ ϕµ + ϕ − m ϕ −ĝ dx ` 2 12 2 It appears ϕ4 ! (d = dim M = 4)! Conformal covariance p p ϕ(new ) = Ω̃ϕ(old), −g (old) = Ω̃4 −g (new ) Cosmological term or quintessence: 3m2 = Λ Independent variations of ϕ and ĝ µν Energy-momentum tensor 1 1 1 scalar = − 2 ϕµ ϕµ + 2 ϕσ ϕσ ĝµν − 2 m2 ϕ4 ĝµν T̂µν ` 2` 2` 1 1 2 σ − 2 ϕ (R̂µν − ĝµν R̂) − 2 (ϕϕν )|µ − (ϕϕ )|σ ĝµν 6` 2 1 1 2 scalar Tr Tµν = − 2 ϕϕσ|σ + 2 R̂ϕ2 − 2 m2 ϕ4 ` 6` ` V. The “good” hydrodynamical variables for hydrodynamical action (nu µ );µ = βC 2 ds 2 = ϕ2 dŝ 2 ⇒ ⇒ p √ C 2 −g = Ĉ 2 −ĝ , uµ = 1 µ û , ϕ √ û µ ûµ = 1, −g = ϕ4 uµ = ϕûµ p √ √ (nu µ );µ −g = (nu µ −g ),µ = (ϕ3 nu µ −ĝ ),µ Z p n̂ = nϕ3 −ĝ ⇒ N = n̂ d 3 x − inv ⇒ √ λ0 (u µ uµ − 1) −g dx Z Z √ µ√ + λ2 X,µ u −g dx + λ1 (nu µ );µ −g dx =⇒ Z Z p p n̂ = − ε X , 3√ ϕ4 −ĝ dx + λ0 (û µ ûµ − 1)ϕ4 −ĝ dx ϕ −ĝ Z Z p p µ 3 + λ2 X,µ û ϕ −ĝ dx + λ1 (n̂û µ ),µ − β Ĉ 2 −ĝ dx Z Shydro = − Shydro √ ε(X , n) −g dx + p −ĝ Z ε+p =n hydro T̂µν = ∂ε ∂n =⇒ |λ|σ 1 λσ (ε+p)ϕ ûµ ûν −pϕ ĝµν −4βϕ (λ1 Ĉµσνλ ) + λ1 Ĉµλνσ R̂ 2 4 4 2 hydro Tr Tµν = (ε − 3p)ϕ4 total Tr Tµν =− 1 2 1 ϕϕσ|σ + 2 R̂ϕ2 − 2 m2 ϕ4 + (ε − 3p)ϕ4 2 ` 6` ` Variation of ϕ: δStotal δϕ =0 =⇒ 1 σ 1 (ϕ |σ − R̂ϕ + 2m2 ϕ3 ) + (ε − 3p)ϕ3 = 0 2 ` 6 As it should be! VI. Vacuum equations Without “hats” (!): 1 hydro Rµν − gµν R − Λgµν = 8πGTµν 2 hydro Tµν = ( + p)uµ uν − pgµν − 4βBµν [λ1 ] 1 Bµν [λ1 ] = (λ1 Cµσνλ );λ;σ + λ1 Cµλνσ R λσ 2 “Empty” vacuum: =p=0 1 Λ Gµν − gµν = 0 16πG 16πG 1 Gµν = Rµν − gµν R 2 4α0 Bµν [λ1 ] + In terms of ϕ and ĝµν (gµν = ϕ2 ĝµν ): 1 |λ|σ λσ 0 = −4β (λ1 Ĉµσνλ ) + λ1 Ĉµλνσ R̂ 2 1 1 − 2 ϕ2 (R̂µν − ĝµν R̂) + 4ϕµ ϕν − ϕσ ϕσ ĝµν 6` 2 |σ −2ϕϕν|µ + 2ϕϕ|σ ĝµν − Λϕ4 ĝµν Trace: |σ ϕ|σ − 1 2 R̂ϕ − Λϕ3 = 0 6 3 Dirac sea Weyl tensor Cµσνλ 6= 0 Ingredients: particles of both positive and negative energies µ (±) , p(±) , ~v(±) , u(±) Hydrodynamical equations (±) (±) (±) (±) + p(±) uµ(±) + λ2 X,µ − n(±) λ1,µ = 0, ∂(±) (±) (±)σ − λ u = 0 2 ;σ ∂X (±) Complete compensations + = −− , u (+)µ (+) λ2 =u = ⇒ n+ = n− (−)µ (−) −λ2 (+) λ1 ⇒ ⇒ X (+) λ1,µ (−) + λ1 (+) p+ = −p− = X (−) ⇒ (−) −λ1,µ ⇒ = = const “Dynamical vacuum” equations 4α0 Bµν + 16πG Gµν − 1 Cµλνσ R λσ 2 1 = Rµν − gµν R 2 Λ Bµν = Cµσνλ;λ;σ + Gµν 1 Λ 16πG gµν = 0 ⇒ Bach tensor ⇒ Einstein tensor ⇒ Cosmological constant Thanks to everybody! The end Abstract We constructed a phenomenological model for cosmological particle creation. It allows to take into account the back reaction on the space-time metric not only of the energy-momentum tensor of the already created particles, but also the very process of the creation. Adopting the well-known fact that the particle production rate is proportional to the square of the Weyl tensor, we found that there is no more need to include, ad hoc, the conformal gravity term into the action integral, it is already there. Such a feature is in accordance with the idea of the induced gravity, first proposed by A. D. Sakharov. Assuming, then, the simplest possible (without self-interaction) action for the scalar field responsible for the particle creation, we found that the requirement for the local conformal transformation to be the fundamental physical law, leads not only to the induced scalar term in the action (Einstein-dilaton gravity), but also to the appearance of the self-interaction for the scalar field absent before. Moreover, the exponent of this self-interaction term (= 4) is due to the dimensionality of our space-time.