Diffusive DE & DM Diffusve DE and DM. Eduardo Guendelman With my student David Benisty, Joseph Katz Memorial Conference, May 23, 2017

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1 Diffusive DE & DM Diffusve DE and DM Eduardo Guendelman With my student David Benisty, Joseph Katz Memorial Conference, May 23, 2017

2 Main problems in cosmology The vacuum energy behaves as the Λ term in Einstein s field equation called the cosmological constant. RR μμμμ 1 2 gg μμμμrr = Λgg μμμμ + TT μμμμ Why is the observed value so many orders of magnitude smaller than that expected in QFT? Why is it of the same order of magnitude as the matter density of the universe at the present time?

3 Two Measure Theory In addition to the regular measure gg, Guendelman and Kaganovich put another measure which is also a density and a total derivative. For example constructing this measure 4 scalar fields φφ (aa), where a = 1, 2, 3, 4: Φ = 1 4! εεμμμμμμμμ εε aaaaaaaa μμ φφ (aa) νν φφ (bb) ρρ φφ (cc) σσ φφ (dd) = det φφ,jj ii with the total action: SS = ggl 1 + ΦL 2 The variation from the scalar fields φφ (aa) we get L 2 = M = const.

4 Unified scalar DE-DM For a scalar field theory with a new measure: SS = ggrr + gg + Φ Λ d 4 x where Λ = gg αααα φφ,αα φφ,ββ. The Equations of Motion: TT μμμμ = gg μμμμ Λ + Λ = MM = cccccccccc jj μμ = 1 + Φ BB gg μμ φφ 1 + Φ BB gg μμ φφ νν φφ = gg μμμμ Λ + jj μμ νν φφ Dark Energy and Dark Matter From Hidden Symmetry of Gravity Model with a Non-Riemannian Volume Form European Physical Journal C75 (2015) arxiv: A two measure model of dark energy and dark matter Eduardo Guendelman, Douglas Singleton, Nattapong Yongram, arxiv: [gr-qc]

5 Which gives constant scalar filed φφ = CC 1 and. A conserved current: μμ jj μμ = 1 gg μμ ggjj μμ = 1 aa 3 tt aa3 jj 0 = 0 or jj 0 = CC 3 aa3. The complete set of the densities: ρρ Λ = φφ 2 = CC 1 pp Λ = ρρ Λ ρρ d = CC 3 φφ = aa 3 CC 1 CC 3 aa 3 pp d = 0 The precise solution for Friedman equation ρρ case is: aa Λ dd = CC 3 CC sinh CC 1tt aa aa 2 in this Which helps us to reconstruct the original physical values: Ω Λ = CC 1 HH 0 Ω Λ = CC 3 CC 1 HH 0

6 Velocity diffusion notion In General Relativity Diffusion may also play a fundamental role in the large scale dynamics of the matter in the universe. J. Franchi, Y. Le Jan. Relativistic Diffusions and Schwarzschild Geometry. Comm. Pure Appl. Math., 60 : , 2007; Z. Haba. Relativistic diffusion with friction on a pseudoriemannian manifold. Class. Quant. Grav., 27 : , ; 2010 J. Hermann. Diffusion in the general theory of relativity. Phys. Rev. D, 82: , 2010; S. Calogero. A kinetic theory of diffusion in general relativity with cosmological scalar field. J. Cosmo. Astro. Particle Phys , 2011

7 Kinetic diffusion on curved s.t Kinetic diffusion equation(fokker Planck): tt ff + vv xx ff = σσ 2 ww ff pp μμ μμ ff Γ ii μμμμ pp μμ pp νν pp iiff = DD pp ff ff distribution function, v velocity, σσ diffusion coefficient. The current density and the energy momentum tensor TT μμμμ are defined as: jj μμ = gg ff ppμμ pp 0 ddpp 1 ddpp 2 ddpp 3 TT μμμμ = gg ff ppμμ pp νν pp 0 ddpp 1 ddpp 2 ddpp 3 jj μμ is a time-like vector field and TT μμμμ verifies the dominant and strong energy conditions. μμ TT μμμμ = 3σσjj νν μμ jj νν = 0 The number of particles is conserved, but not the energy momentum tensor.

8 Connection to Cosmology Calogero s idea: φφcdm. The cosmological constant is replaced by a scalar filed, which would the source of the Cold Dark Matter stress energy tensor: RR μμμμ 1 2 gg μμμμrr = TT μμμμ + φφgg μμμμ μμ TT μμμμ = 3σσjj νν νν φφ = 3σσjj νν The value 3σσ measures the energy transferred from the scalar field to the matter per unit of time due to diffusion. This modification applied by hand, and not from action principle. Another purpose - Diffusive Energy Action.

9 From T.M.T to Dynamical time The basic result of T.M.T can be expressed as a covariant conservation of a stress energy tensor: Φ = 1 4! εεμμμμμμμμ εε aaaaaaaa μμ φφ (aa) νν φφ (bb) ρρ φφ (cc) σσ φφ (dd) SS = ΦL 1 SS χχ = gg χχ μμ;νν TT μμμμ (χχ) dd 4 xx χχ λλ - dynamical space-time vector field, χχ μμ;νν = vv χχ μμ Γ λλ μμμμ χχ λλ in second order formalism Γ λλ μμμμ is Christoffel Symbol. TT μμμμ (χχ) - stress energy tensor. The variation according to χχ gives a conserved energy momentum tensor: μμ TT μμμμ (χχ) = 0, in addition to TT μμμμ (GG) = δδss(χχ) δδgg μμμμ. Dynamical time is as T.M.T for TT μμμμ (χχ) = gg μμμμ Λ.

10 Diffusive energy We replace the dynamical space time vector χχ μμ by a gradient of a scalar filed χχ,μμ : SS χχ = gg χχ,μμ;νν TT μμμμ (χχ) dd 4 xx χχ - scalar filed, χχ,μμ;νν = vv μμ χχ Γ λλ μμμμ λλ χχ, TT μμμμ (χχ) - stress energy tensor. The variation according to χχ gives a non-conserved diffusive energy momentum tensor: μμ TT μμμμ (χχ) = ff vv ; vv ff vv = 0 The variation according to the metric gives a conserved stress energy tensor, (which is familiar from Einstein eq. TT GG μμμμ = RR μμμμ 1 2 ggμμμμ RR) : TT μμμμ (GG) = 2 δδ gg ggl mm δδgg μμμμ

11 Dynamical time with source An action with no high derivatives, is obtained by adding another term involving χ µ : SS χχ = gg χχ μμ;νν TT μμμμ (χχ) dd 4 xx + σσ 2 gg χχ μμ + μμ AA 2 dd 4 xx δδχχ λλ : µν µ T (χ) = σ χ ν + ν A δδaa: σ ν χ ν + ν A = 0 One difference between those theories: Here - σσ appears as a parameter - in the higher derivative theory σσ appears as an integration constant.

12 Symmetries If the matter is coupled through its energy momentum tensor as: TT χχ μμμμ TT χχ μμμμ + λλgg μμμμ the process will not affect the equations of motion. In Quantum Field Theory this is normal ordering. χχ χχ + λλ

13 A toy model We start with a simple action of one dimensional particle in a potential VV(xx). SS = BB 1 2 mmxx2 + VV xx dddd δδbb gives the total energy of a particle with constant power P: 1 2 mmxx2 + VV xx = EE tt = EE 0 + PPPP δδδδ gives the condition for B: mmxx BB + mmxx BB = VV xx BB or BB BB = 2VV xx 2mm EE tt VV xx PP 2 EE tt VV xx

14 A conserved Hamiltonian Momentums for this toy model: ππ xx = L xx = mmxx BB ππ BB = L BB dd dddd Π BB = L BB L BB = dd EE tt dddd = EE tt The Hamiltonian (with second order derivative): H = xxππ xx + BBππ BB + 2 BBΠ BB L = mmxx BB BB EE = ππ xx mm Π BB VV xx + BBππ BB The action isn t dependent on time explicitly, so the Hamiltonian is conserved.

15 Interacting Diffusive DE DM The diffusive stress energy tensor in this theory: (with high derivatives) TT μμμμ (χχ) = Λgg μμμμ with the kinetic k-essence term Λ = gg αααα φφ,αα φφ,ββ, where φφ a scalar filed. The full theory: SS = 1 2 ggrr + gg χχ + 1 Λ dd4 xx when 8ππππ = cc = 1.

16 Equation of motions δδδδ - non trivial evolving dark energy: δδφφ - a conserved current: Λ = 0 j β = 2 χ + 1 φ,β δδgg μμμμ - a conserved stress energy tensor: TT μμμμ (GG) = gg μμμμ Λ + χχ,σσ Λ,σσ Dark Energy + jj μμ φφ,νν χχ,μμ Λ,νν χχ,νν Λ,μμ Dark Matter

17 F.L.R.W solution Λ = 0: 2φφ φφ = CC 2 aa 3 φφ 2 = CC 1 + CC 2 dddd aa 3 jj ββ = 2 χχ + 1 φφ,ββ : χχ = CC 4 aa aa 3 aa3 dddd CC 3 2aa3 dddd φφ T μμνν (GG) - a conserved stress energy tensor: ρρ Λ = ρρ d = CC 3 aa 3 φφ 2 + CC 2 χχ aa 3 φφ 2 CC 2 aa 3 pp Λ = ρρ Λ χχ pp d = 0

18 Asymptotic solution The field χχ asymptotically goes to the value as De Sitter space aa ~ ee HH0tt : lim tt χχ = 1 aa 3 aa3 dddd = 1 3HH 0 The asymptotic values of the densities are: ρρ Λ = CC 1 + CC 2 dddd aa 3 + CC 2 aa 3 χχ = CC 1 + OO 1 aa 6 ρρ CDM = CC 3 CC 1 2CC 2 3HH 0 1 aa 3 + OO 1 aa 6 The observable values: CC 1 = Ω HH Λ 0 CC 3 CC 1 2CC 2 = HH 3HH 0 Ω d 0

19 Stability of the solutions More close asymptotically with ΛCCCCCC : the dark energy become constant, and the amount of dark matter slightly change ρρ CDM ~ 1 aa 3. CC 3 CC 1 > 2CC 2 for positive dust density. For CC 3HH 2 < 0 cause higher dust 0 density asymptotically, and there will be a positive flow of energy in the inertial frame to the dust component, but the result of this flow of energy in the local inertial frame will be just that the dust energy density will decrease a bit slower that the conventional dust (but still decreases). Explaining the particle production, Taking vacuum energy and converting it into particles" as expected from the inflation reheating epoch. May be this combined with a mechanism that creates standard model particles.

20 Late universe solution The familiar solution of non-interacting DE-DM solution is for CC 2 = 0. Which gives constant scalar filed φφ = CC 1 and φφ = 0. ρρ Λ = φφ 2 = CC 1 pp Λ = ρρ Λ ρρ d = CC 3 φφ = aa 3 CC 1 CC 3 aa 3 pp d = 0 The precise solution for Friedman equation ρρ aa Λ dd = CC 3 CC aa aa sinh CC 1tt 2 in this case is: Which helps us to reconstruct the original physical values: Ω Λ = CC 1 HH 0 Ω d = CC 3 CC 1 HH 0

21 Perturbative solution The scalar field has perturbative properties λλ 1,2 1: λλ 1 tt, tt 0 = CC 2 dddd CC 1 aa 3 λλ 2 tt, tt 0 = CC 2 χχ CC 1 CC 3 For a first order solution in perturbation theory: ρρ Λ = CC λλ 1 + CC 3 CC 1 λλ 2 + OO 2 λλ 1, λλ 2 ρρ CCCCCC = CC 1CC 3 aa λλ 1 + λλ 2 + OO 2 λλ 1, λλ 2 For rising dark energy, dark matter amount goes lower ( CC 2 < 0, CC 1,3,4 > 0). For decreasing dark energy, the amount of dark matter goes up(all components are positive).

22 Diffusive energy without higher derivatives The full theory: L = 1 2 ggrr + ggχχ μμ;ννtt μμμμ (χχ) + σσ 2 gg χχ μμ + μμ AA 2 + ggλ Where Λ = gg αααα φφ,αα φφ,ββ, and TT μμμμ (χχ) = Λgg μμμμ. All the E.o.M are the same, except: TT μμμμ (GG) = gg μμμμ Λ + χχ,λλ Λ,λλ σσ ΛΛ,λλ ΛΛ,λλ + jj μμ φφ,νν χχ,μμ Λ,νν χχ,νν Λ,μμ + 11 σσ ΛΛ,μμ ΛΛ,νν For the late universe both theories are equivalent, Λ,μμ Λ,νν 1 aa 6. For σσ, the term σσ gg χχ 2 μμ + μμ AA 2 forces χχ μμ = μμ AA, and D.T becomes Diffusive energy with high energy.

23 Calogero s model φcmd Calogero put two stress energy tensor of DE-DM. Each stress energy tensor in non-conserved: μμ TT μμμμ μμμμ Λ = μμ TT Dust = 3σσjj νν, jj ;νν νν = 0 For FRWM, this calculation leads to the solution: ρρ Λ = CC 1 + CC 2 dddd aa 3 ρρ Dust = CC 3 aa 3 CC 2tt aa 3 The two model became approximate for C 2 χχ 1. Our asymptotic solution becomes with constant densities, because C 2 χχ CC 2, which makes the DE 3HH 0 decay lower from φφφφφφφφ, and DM evolution as ΛCCCCCC.

24 Preliminary ideas on Quantization Taking Dynamical space time theory (with source), and by integration by parts: μμμμ SS = ggrr + gg Λ gg χχ μμ TT (χχ);νν dd 4 xx + σσ 2 gg χχ μμ + μμ AA 2 dd 4 xx δδδδ μμ : νν TT χχ μμμμ = ff νν = σσ χχ νν + νν AA and put back into the action: SS = ggrr + gg g αααα φφ αα φφ ββ 1 2σσ gg ff ννff νν dd 4 xx + gg μμ AAff νν dd 4 xx The partition function considering Euclidean metrics (exclude the gravity terms): Ζ = DDDD δδ ff ;μμ μμ exp 1 2σσ gg ff ννff νν dd 4 xx gg g αααα φφ αα φφ ββ We see that for σσ < 0, there will a convergent functional integration, so this is a good sign for the quantum behavior of the theory.

25 Final Remarks T.M.T - Unified Dark Matter Dark Energy. The cosmological constant appears as an integration constant. The Dynamical space time Theories both energy momentum tensor are conserved. Diffusive Unified DE and DM the vector field is taken to be the gradient of a scalar, the energy momentum tensor TT μμμμ (xx) has a source current, unlike the TT μμμμ (GG) which is conserved. The non conservation of TT μμμμ (xx) is of the diffusive form. Asymptotically stable solution ΛCDM is a fixed point. For rising dark energy, dark matter amount goes lower. For decreasing dark energy, the amount of dark matter goes up. The partition function is convergent for σσ < 0, and therefor the theory is a good property before quantizing the theory.

26 Future research Numerical solution for CC 2 0 A Stellar model, Quantum solutions Wheeler de Witt equation A solution for coincidence problem

27 And this is only the beginning