Diffusive DE & DM. David Benisty Eduardo Guendelman

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1 Diffusive DE & DM David Benisty Eduardo Guendelman arxiv: Int.J.Mod.Phys. D26 (2017) no.12, DOI: /S Eur.Phys.J. C77 (2017) no.6, 396 DOI: /epjc/s x arxiv:

2 Main problems in cosmology The vacuum energy behaves as the Λ term in Einstein s field equation called the cosmological constant. R μν 1 2 g μνr = Λg μν + T μν 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 g, Guendelman and Kaganovich put another measure which is also a density and a total derivative. For example constructing this measure 4 scalar fields φ (a), where a = 1, 2, 3, 4: Φ = 1 4! εμνρσ ε abcd μ φ (a) ν φ (b) ρ φ (c) σ φ (d) = det φ,j i with the total action: S = න gl 1 + ΦL 2 The variation from the scalar fields φ (a) we get L 2 = M = const.

4 Unified scalar DE-DM For a scalar field theory with a new measure: S = න gr + g + Φ Λ d 4 x where Λ = g αβ φ,α φ,β. The Equations of Motion: Λ = M = const j μ = 1 + Φ g μ φ T μν = g μν Λ Φ g μ φ ν φ = g μν Λ + j μ ν φ 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 φ ሶ = C 1 and. A conserved current: μ j μ = 1 g μ gj μ = 1 a 3 t a3 j 0 = 0 or j 0 = C 3 a3. The complete set of the densities: ρ Λ = φሶ 2 = C 1 p Λ = ρ Λ ρ d = C 3 φ ሶ = a 3 C 1 C 3 a 3 p d = 0 The precise solution for Friedman equation ρ case is: a Λ d = C 3 C 1 Τ 1 3 sinh 2Τ3 3 2 C 1t ሶ a a 2 in this Which helps us to reconstruct the original physical values: Ω Λ = C 1 H 0 Ω Λ = C 3 C 1 H 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): t f + v x f = σ 2 w f p μ μ f Γ i μν p μ p ν p if = D p f f distribution function, v velocity, σ diffusion coefficient. The current density and the energy momentum tensor T μν are defined as: j μ = g න f pμ p 0 dp 1 dp 2 dp 3 T μν = g න f pμ p ν p 0 dp 1 dp 2 dp 3 j μ is a time-like vector field and T μν verifies the dominant and strong energy conditions. μ T μν = 3σj ν μ j ν = 0 The number of particles is conserved, but not the energy momentum tensor.

8 Connection to Cosmology Calogero s idea: φcdm. The cosmological constant is replaced by a scalar filed, which would the source of the Cold Dark Matter stress energy tensor: R μν 1 2 g μνr = T μν + φg μν μ T μν = 3σj ν ν φ = 3σj ν The value 3σ 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! εμνρσ ε abcd μ φ (a) ν φ (b) ρ φ (c) σ φ (d) S = න ΦL 1 S χ = න g χ μ;ν T μν (χ) d 4 x χ λ - dynamical space-time vector field, χ μ;ν = v χ μ Γ λ μν χ λ in second order formalism Γ λ μν is Christoffel Symbol. T μν (χ) - stress energy tensor. The variation according to χ gives a conserved diffusive energy: μ T μν (χ) = 0, in addition to T μν (G) = δs(χ) δg μν. Dynamical time is as T.M.T for T μν (χ) = g μν Λ.

10 Diffusive energy We replace the dynamical space time vector χ μ by a gradient of a scalar filed χ,μ : S χ = න g χ,μ;ν T μν (χ) d 4 x χ - scalar filed, χ,μ;ν = v μ χ Γ λ μν λ χ, T μν (χ) - stress energy tensor. The variation according to χ gives a non-conserved diffusive energy momentum tensor: μ T μν (χ) = f v ; v f v = 0 The variation according to the metric gives a conserved stress energy tensor, (which is familiar from Einstein eq. T G μv = R μν 1 2 gμν R) : T μν (G) = 2 δ g gl m δg μν

11 Dynamical time with source An action with no high derivatives, is to add a source coupled to χ μ : S χ = න g χ μ;ν T μν (χ) d 4 x + σ 2 න g χ μ + μ A 2 d 4 x δχ λ : μν μ T (χ) = σ χ ν + ν A δa: σ ν χ ν + ν A = 0 One difference between those theories: Here - σ appears as a parameter - in the higher derivative theory σ appears as an integration of constant.

12 Symmetries If the matter is coupled through its energy momentum tensor as: T χ μν T χ μν + λg μν the process will not affect the equations of motion. In Quantum Field Theory this is normal ordering. χ χ + λ

13 A toy model We start with a simple action of one dimensional particle in a potential V(x). S = න Bሷ 1 2 mx2 ሶ + V x dt δb gives the total energy of a particle with constant power P: 1 2 mx2 ሶ + V x = E t = E 0 + Pt δx gives the condition for B: mxሷ B ሷ + mxሶ ሸB = V x ሷ B or ሸB B ሷ = 2V x 2m E t V x P 2 E t V x

14 A conserved Hamiltonian Momentums for this toy model: π x = L x ሶ = mxሶ Bሷ π B = L Bሶ d dt Π B = L ሷ B L B ሷ = d E t dt = E t The Hamiltonian (with second order derivative): H = 2 xπ ሶ x + Bπ ሶ B + BΠ ሷ B L = mxሶ Bሷ Bሶ E ሶ = π x m Π B V x + ሶ Bπ B 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) T μν (χ) = Λg μν with the kinetic k-essence term Λ = g αβ φ,α φ,β, where φ a scalar filed. The full theory: S = න 1 2 gr + g χ + 1 Λ d4 x when 8πG = c = 1.

16 Equation of motions δχ - non trivial evolving dark energy: δφ - a conserved current: Λ = 0 j β = 2 χ + 1 φ,β δg μν - a conserved stress energy tensor: T μν (G) = g μν Λ + χ,σ Λ,σ Dark Energy + j μ φ,ν χ,μ Λ,ν χ,ν Λ,μ Dark Matter

17 F.L.R.W solution Λ = 0: 2φሶ φ ሷ = C 2 a 3 φሶ 2 = C 1 + C 2 න dt a 3 j β = 2 χ + 1 φ,β : χ ሶ = C 4 a a 3 න a3 dt C 3 2a 3 න dt φሶ T μν (G) - a conserved stress energy tensor: ρ Λ = ρ d = C 3 a 3 φሶ 2 + C 2 χሶ a 3 φሶ 2 C 2 a 3 p Λ = ρ Λ χሶ p d = 0

18 Asymptotic solution The field χሶ asymptotically goes to the value as De Sitter space a ~ e H0t : lim t χ ሶ = 1 a 3 න a3 dt = 1 3H 0 The asymptotic values of the densities are: ρ Λ = C 1 + C 2 න dt a 3 + C 2 a 3 χ ሶ = C 1 + O 1 a 6 ρ CDM = C 3 C 1 2C 2 3H 0 1 a 3 + O 1 a 6 The observable values: C 1 = Ω H Λ 0 C 3 C 1 2C 2 = H 3H 0 Ω d 0

19 Stability of the solutions More close asymptotically with ΛCDM : the dark energy become constant, and the amount of dark matter slightly change ρ CDM ~ 1 a 3. C 3 C 1 > 2C 2 for positive dust density. For C 3H 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 C 2 = 0. Which gives constant scalar filed φ ሶ = C 1 and φ ሷ = 0. ρ Λ = φሶ 2 = C 1 p Λ = ρ Λ ρ d = C 3 φ ሶ = a 3 C 1 C 3 a 3 p d = 0 The precise solution for Friedman equation ρ a Λ d = C 3 C 1 Τ 1 3 ሶ a a sinh 2Τ3 3 2 C 1t 2 in this case is: Which helps us to reconstruct the original physical values: Ω Λ = C 1 H 0 Ω d = C 3 C 1 H 0

21 Perturbative solution The scalar field has perturbative properties λ 1,2 1: λ 1 t, t 0 = C 2 න dt C 1 a 3 λ 2 t, t 0 = C 2 χሶ C 1 C 3 For a first order solution in perturbation theory: ρ Λ = C λ 1 + C 3 C 1 λ 2 + O 2 λ 1, λ 2 ρ CDM = C 1C 3 a λ 1 + λ 2 + O 2 λ 1, λ 2 For rising dark energy, dark matter amount goes lower ( C 2 < 0, C 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 gr + gχ μ;νt μν (χ) + σ 2 g χ μ + μ A 2 + gλ Where Λ = g αβ φ,α φ,β, and T μν (χ) = Λg μν. All the E.o.M are the same, except: T μν (G) = g μν Λ + χ,λ Λ,λ + 1 2σ Λ,λ Λ,λ + j μ φ,ν χ,μ Λ,ν χ,ν Λ,μ + 1 σ Λ,μ Λ,ν For the late universe both theories are equivalent, Λ,μ Λ,ν 1 a 6. For σ, the term σ g χ 2 μ + μ A 2 forces χ μ = μ A, and D.T becomes Diffusive energy with high energy.

23 ሶ ሶ Calogero s model φcmd Calogero put two stress energy tensor of DE-DM. Each stress energy tensor in non-conserved: μ T μν μν Λ = μ T Dust = 3σj ν, j ;ν ν = 0 For FRWM, this calculation leads to the solution: ρ Λ = C 1 + C 2 න dt a 3 ρ Dust = C 3 a 3 C 2t a 3 The two model became approximate for C 2 χ 1. Our asymptotic solution becomes with constant densities, because C 2 χ C 2, which makes the DE 3H 0 decay lower from φcmd, and DM evolution as ΛCMD.

24 Preliminary ideas on Quantization Taking Dynamical space time theory (with source), and by integration by parts: μν S = න gr + න g Λ න g χ μ T (χ);ν d 4 x + σ 2 න g χ μ + μ A 2 d 4 x δχ μ : ν T χ μν = f ν = σ χ ν + ν A into the action: S = න gr + න g g αβ φ α φ β 1 2σ න g f νf ν d 4 x + න g μ Af ν d 4 x The partition function considering Euclidean metrics (exclude the gravity terms): Ζ = න Dφ δ f ;μ μ exp 1 2σ න g f νf ν d 4 x න g g αβ φ α φ β We see that for σ < 0, there will a convergent functional integration, so this is a good sign for the quantum behavior of the theory.

25 ሶ Numerical solution By redefine the red shift as the changing variable: a z = a 0 z + 1 d dt = z + 1 H z d dz The physical behavior can be obtained by the deceleration parameter: H q = 1 H 2 = 1 K 1 + 3ω ah 2 type Dark energy Hyper - inflation dust Stiff w q -1 < -1 1/2 2

26 C 2 > 0 Solutions

27 C 2 < 0 Solutions

28 Bouncing solutions

29 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 T μν (x) has a source current, unlike the T μν (G) which is conserved. The non conservation of T μν (x) is of the diffusive form. Asymptotically stable solution ΛCDM is a fixed point. A bounce hyper inflation solution for the early universe. 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 σ < 0, and therefor the theory is a good property before quantizing the theory.

30 Future research A Stellar model Inflationary model? Quantum solutions Wheeler de Witt equation A solution for coincidence problem?

31 And this is only the beginning