diffusive de & dm - miami
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Diffusive DE & DMDavid Benisty
Eduardo Guendelman
arXiv:1710.10588
Int.J.Mod.Phys. D26 (2017) no.12,
1743021
DOI: 10.1142/S0218271817430210
Eur.Phys.J. C77 (2017) no.6, 396 DOI:
10.1140/epjc/s10052-017-4939-xarXiv:1701.08667
Main problems in cosmology
The vacuum energy behaves as the ฮ term in Einsteinโs field equation
๐ ๐๐ โ1
2๐๐๐๐ = ฮ๐๐๐ + ๐๐๐
called the cosmological constant.
โข Why is the observed value so many orders of
magnitude smaller than that expected in QFT?
โข Why is it of the same order of
magnitude as the matter density of
the universe at the present time?
Two Measure Theory
In addition to the regular measure โ๐, Guendelman and Kaganovich
put another measure which is also a density and a total derivative. For
example constructing this measure 4 scalar fields ๐(๐), where a = 1, 2, 3, 4:
ฮฆ =1
4!ํ๐๐๐๐ํ๐๐๐๐๐๐๐
(๐)๐๐๐(๐)๐๐๐
(๐)๐๐๐(๐) = det ๐,๐
๐
with the total action:
๐ = เถฑ โ๐โ1 +ฮฆโ2
The variation from the scalar fields ๐(๐) we get โ2 = M = const.
Unified scalar DE-DM
For a scalar field theory with a new measure:
๐ = เถฑ โ๐๐ + โ๐ +ฮฆ ฮ d4x
where ฮ = ๐๐ผ๐ฝ๐,๐ผ๐,๐ฝ. The Equations of Motion:
ฮ = ๐ = ๐๐๐๐ ๐ก
๐๐ = 1 +ฮฆ
โ๐๐๐๐
๐๐๐ = ๐๐๐ฮ + 1 +ฮฆ
โ๐๐๐๐๐๐๐ = ๐๐๐ฮ + ๐๐๐๐๐
โข Dark Energy and Dark Matter From Hidden Symmetry of Gravity Model with a Non-Riemannian Volume Form
European Physical Journal C75 (2015) 472-479 arXiv:1508.02008
โข A two measure model of dark energy and dark matter Eduardo Guendelman, Douglas Singleton, Nattapong
Yongram, arXiv:1205.1056 [gr-qc]
Which gives constant scalar filed แถ๐ = ๐ถ1 and.
A conserved current: ๐ป๐๐๐ =
1
โ๐๐๐ โ๐๐๐ =
1
๐3๐
๐๐ก๐3๐0 = 0
or ๐0 =๐ถ3
๐3. The complete set of the densities:
๐ฮ = แถ๐2 = ๐ถ1 ๐ฮ = โ๐ฮ
๐d =๐ถ3
๐3แถ๐ =
๐ถ1๐ถ3
๐3๐d = 0
The precise solution for Friedman equation ๐ โผแถ๐
๐
2in this
case is:
๐ฮโ๐ =๐ถ3
๐ถ1
ฮค1 3
sinh ฮค2 33
2๐ถ1๐ก
Which helps us to reconstruct the original physical values:
ฮฉฮ =๐ถ1๐ป0
ฮฉฮ =๐ถ3 ๐ถ1๐ป0
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 : 187251, 2007;
Z. Haba. Relativistic diffusion with friction on a pseudoriemannian manifold. Class. Quant. Grav., 27 : 095021, 2010;
J. Hermann. Diffusion in the general theory of relativity. Phys. Rev. D, 82:
024026, 2010;
S. Calogero. A kinetic theory of diffusion in general relativity with cosmological scalar field. J. Cosmo. Astro. Particle Phys. 11 2011 ,016.
Kinetic diffusion on curved s.t Kinetic diffusion equation(Fokker Planck):
๐๐ก๐ + ๐ฃ ๐๐ฅ๐ = ๐ ๐๐ค2๐ โน ๐๐๐๐๐ โ ฮ๐๐
๐ ๐๐๐๐๐๐๐๐ = ๐ท๐๐
๐ โ distribution function, v โ velocity, ๐ โ diffusion coefficient.
The current density and the energy momentum tensor ๐๐๐ are definedas:
๐๐ = โ โ๐เถฑ๐๐๐
๐0๐๐1 โง ๐๐2 โง ๐๐3
๐๐๐ = โ โ๐เถฑ๐๐๐๐๐
๐0๐๐1 โง ๐๐2 โง ๐๐3
๐๐ is a time-like vector field and ๐๐๐ verifies the dominant and strongenergy conditions.
๐ป๐๐๐๐ = 3๐๐๐ ๐ป๐๐
๐ = 0
The number of particles is conserved, but not the energy momentumtensor.
Connection to Cosmology Calogeroโs idea: ๐CDM. The cosmological constant is replaced by a
scalar filed, which would the source of the Cold Dark Matter stress
energy tensor:
๐ ๐๐ โ1
2๐๐๐๐ = ๐๐๐ + ๐๐๐๐
๐ป๐๐๐๐ = 3๐๐๐
๐ป๐๐ = โ3๐๐๐
The value 3๐ measures the energy transferred from the scalar field to the
matter
per unit of time due to diffusion.
This modification applied โby handโ, and not from action principle.
Another purpose - Diffusive Energy Action.
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:
๐๐ - dynamical space-time vector field , ๐๐;๐ = ๐๐ฃ๐๐ โ ฮ๐๐๐ ๐๐ in second
order formalism ฮ๐๐๐ is Christoffel Symbol.
๐(๐)๐๐
- stress energy tensor. The variation according to ๐ gives a conserved
diffusive energy: ๐ป๐๐(๐)๐๐
= 0, in addition to ๐(๐บ)๐๐
=๐ฟ๐(๐)
๐ฟ๐๐๐.
Dynamical time is as T.M.T for ๐(๐)๐๐
= ๐๐๐ฮ.
ฮฆ =1
4!ํ๐๐๐๐ํ๐๐๐๐๐๐๐
(๐)๐๐๐(๐)๐๐๐
(๐)๐๐๐(๐)
๐ = เถฑฮฆโ1 ๐ ๐ = เถฑ โ๐๐๐;๐๐(๐)๐๐๐4๐ฅ
Diffusive energy We replace the dynamical space time vector ๐๐ by a gradient of a
scalar filed ๐,๐:
๐ ๐ = เถฑ โ๐๐,๐;๐๐(๐)๐๐๐4๐ฅ
๐ - scalar filed , ๐,๐;๐ = ๐๐ฃ๐๐๐ โ ฮ๐๐๐ ๐๐๐, ๐(๐)
๐๐- stress energy tensor.
The variation according to ๐ gives a non-conserved diffusive energy momentum tensor:
๐ป๐๐(๐)๐๐
= ๐๐ฃ ; ๐ป๐ฃ๐๐ฃ = 0
The variation according to the metric gives a conserved stress energytensor, (which is familiar from Einstein eq. ๐ ๐บ
๐๐ฃ= ๐ ๐๐ โ
1
2๐๐๐ ๐ ) :
๐(๐บ)๐๐
=โ2
โ๐
๐ฟ โ๐โ๐๐ฟ๐๐๐
Dynamical time with source
An action with no high derivatives, is to add a source coupled to ฯฮผ:
๐ ๐ = เถฑ โ๐๐๐;๐๐(๐)๐๐๐4๐ฅ +
๐
2เถฑ โ๐ ๐๐ + ๐๐๐ด
2๐4๐ฅ
๐ ๐๐: ๐ปฮผT(ฯ)ฮผฮฝ
= ฯ ฯฮฝ + ๐ฮฝA
๐ ๐: ฯ ๐ปฮฝ ฯฮฝ + ๐ฮฝA = 0
One difference between those theories:
Here - ๐ appears as a parameter
- in the higher derivative theory ๐ appears as an integration of
constant.
Symmetries
If the matter is coupled through its energy momentum tensor
as:
๐ ๐๐๐
โ ๐ ๐๐๐
+ ๐๐๐๐
the process will not affect the equations of motion. In Quantum
Field Theory this is โnormal orderingโ.
๐ โ ๐ + ๐
A toy model
We start with a simple action of one dimensional particle in a
potential ๐(๐ฅ).
๐ = เถฑ แท๐ต1
2๐ แถ๐ฅ2 + ๐ ๐ฅ ๐๐ก
๐ฟ๐ต gives the total energy of a particle with constant power P:
1
2๐ แถ๐ฅ2 + ๐ ๐ฅ = ๐ธ ๐ก = ๐ธ0 + ๐๐ก
๐ฟ๐ฅ gives the condition for B:
๐ แท๐ฅ แท๐ต + ๐ แถ๐ฅแธ๐ต = ๐โฒ ๐ฅ แท๐ต or แธ๐ต
แท๐ต=
2๐โฒ ๐ฅ
2๐ ๐ธ ๐ก โ๐ ๐ฅ
โ๐
2 ๐ธ ๐ก โ๐ ๐ฅ
A conserved Hamiltonian
Momentums for this toy model:
๐๐ฅ =๐โ
๐ แถ๐ฅ= ๐ แถ๐ฅ แท๐ต
๐๐ต =๐โ
๐ แถ๐ตโ๐
๐๐ก
๐โ
๐ แท๐ต= โ
๐
๐๐ก๐ธ ๐ก
ฮ ๐ต =๐โ
๐ แท๐ต= ๐ธ ๐ก
The Hamiltonian (with second order derivative):
โ = แถ๐ฅ๐๐ฅ + แถ๐ต๐๐ต + แท๐ตฮ ๐ต โ โ = ๐ แถ๐ฅ แท๐ต โ แถ๐ต แถ๐ธ = ๐๐ฅ2
๐ฮ ๐ต โ ๐ ๐ฅ + แถ๐ต๐๐ต
The action isnโt dependent on time explicitly, so the Hamiltonian is
conserved.
Interacting Diffusive DE โ DM
The diffusive stress energy tensor in this theory:
๐(๐)๐๐
= ฮ๐๐๐
with the kinetic โk-essenceโ term ฮ = ๐๐ผ๐ฝ๐,๐ผ๐,๐ฝ,
where ๐ โ a scalar filed.
The full theory:
๐ = เถฑ1
2โ๐๐ + โ๐ โ ๐ + 1 ฮ ๐4๐ฅ
when 8๐๐บ = ๐ = 1.
(with high derivatives)
Equation of motions
๐ฟ๐ - non trivial evolving dark energy:
โ ฮ = 0
๐ฟ๐ - a conserved current:
jฮฒ = 2 โ ฯ + 1 ฯ,ฮฒ
๐ฟ๐๐๐ - a conserved stress energy tensor:
๐(๐บ)๐๐
= ๐๐๐ โฮ + ๐ ,๐ฮ,๐ + ๐๐๐,๐ โ ๐ ,๐ฮ,๐ โ ๐ ,๐ฮ,๐
Dark Energy Dark Matter
F.L.R.W solution
โ ฮ = 0:
2 แถ๐ แท๐ =๐ถ2๐3
โบ แถ๐2 = ๐ถ1 + ๐ถ2เถฑ๐๐ก
๐3
๐๐ฝ = 2 โ ๐ + 1 ๐,๐ฝ:
แถ๐ =๐ถ4๐3
+1
๐3เถฑ๐3 ๐๐ก โ
๐ถ32๐3
เถฑ๐๐ก
แถ๐
T(๐บ)๐๐
- a conserved stress energy tensor:
๐ฮ = แถ๐2 +๐ถ2
๐3แถ๐ ๐ฮ = โ๐ฮ
๐d =๐ถ3
๐3แถ๐ โ 2
๐ถ2
๐3แถ๐ ๐d = 0
Asymptotic solution The field แถ๐ asymptotically goes to the value as De Sitter space ๐ ~ ๐๐ป0๐ก:
lim๐กโโ
แถ๐ =1
๐3เถฑ๐3 ๐๐ก =
1
3๐ป0
The asymptotic values of the densities are:
๐ฮ = ๐ถ1 + ๐ถ2เถฑ๐๐ก
๐3+๐ถ2๐3
แถ๐ = ๐ถ1 + ๐1
๐6
๐CDM = ๐ถ3 ๐ถ1 โ2๐ถ23๐ป0
1
๐3+ ๐
1
๐6
The observable values:๐ถ1๐ป0
= ฮฉฮ ๐ถ3 ๐ถ1 โ2๐ถ23๐ป0
= ๐ป0ฮฉd
Stability of the solutions More close asymptotically with ฮ๐ถ๐ท๐ : the dark energy become
constant, and the amount of dark matter slightly change ๐CDM~1
๐3.
๐ถ3 ๐ถ1 >2๐ถ2
3๐ป0for positive dust density. For ๐ถ2 < 0 cause higher dust
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, โTaking 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.
Late universe solution The familiar solution of non-interacting DE-DM solution is for ๐ถ2 = 0.
Which gives constant scalar filed แถ๐ = ๐ถ1 and แท๐ = 0.
๐ฮ = แถ๐2 = ๐ถ1 ๐ฮ = โ๐ฮ
๐d =๐ถ3
๐3แถ๐ =
๐ถ1๐ถ3
๐3๐d = 0
The precise solution for Friedman equation ๐ โผแถ๐
๐
2in this case is:
๐ฮโ๐ =๐ถ3
๐ถ1
ฮค1 3
sinh ฮค2 33
2๐ถ1๐ก
Which helps us to reconstruct the original physical values:
ฮฉฮ =๐ถ1๐ป0
ฮฉd =๐ถ3 ๐ถ1๐ป0
Perturbative solution
The scalar field has perturbative properties ๐1,2 โช 1:
๐1 ๐ก, ๐ก0 =๐ถ2๐ถ1เถฑ๐๐ก
๐3
๐2 ๐ก, ๐ก0 =๐ถ2
๐ถ1๐ถ3แถ๐
For a first order solution in perturbation theory:
๐ฮ = ๐ถ1 1 + ๐1 +๐ถ3
๐ถ1๐2 + ๐2 ๐1, ๐2
๐๐ถ๐ท๐ =๐ถ1๐ถ3๐3
1 +1
2๐1 + ๐2 + ๐2 ๐1, ๐2
For rising dark energy, dark matter amount goes lower ( ๐ถ2< 0, ๐ถ1,3,4 > 0). For decreasing dark energy, the amount of dark
matter goes up(all components are positive).
Diffusive energy without higher derivatives
The full theory:
โ =1
2โ๐๐ + โ๐๐๐;๐๐(๐)
๐๐+๐
2โ๐ ๐๐ + ๐๐๐ด
2+ โ๐ฮ
Where ฮ = ๐๐ผ๐ฝ๐,๐ผ๐,๐ฝ, and ๐(๐)๐๐
= ฮ๐๐๐ .
All the E.o.M are the same, except:
๐(๐บ)๐๐
= ๐๐๐ โฮ + ๐,๐ฮ,๐ +๐
๐๐๐ฒ,๐๐ฒ,๐ + ๐๐๐,๐ โ ๐,๐ฮ,๐ โ ๐ ,๐ฮ,๐ +
๐
๐๐ฒ,๐๐ฒ,๐
For the late universe both theories are equivalent, ฮ,๐ฮ,๐ โผ1
๐6.
For ๐ โ โ, the term ๐
2โ๐ ๐๐ + ๐๐๐ด
2forces ๐๐ = โ๐๐๐ด, and D.T
becomes Diffusive energy with high energy.
Calogeroโs model ฯCMD Calogero put two stress energy tensor of DE-DM. Each stress energy
tensor in non-conserved:
๐ป๐๐ ฮ๐๐
= โ๐ป๐๐ Dust๐๐
= 3๐๐๐, ๐;๐๐ = 0
For FRWM, this calculation leads to the solution:
๐ฮ = ๐ถ1 + ๐ถ2เถฑ๐๐ก
๐3
๐Dust =๐ถ3๐3
โ๐ถ2๐ก
๐3
The two model became approximate for C2 แถ๐ โช 1. Our asymptotic solution
becomes with constant densities, because C2 แถ๐ โ๐ถ2
3๐ป0, which makes the DE
decay lower from ๐๐ถ๐๐ท, and DM evolution as ฮ๐ถ๐๐ท.
Preliminary ideas on Quantization
Taking Dynamical space time theory (with source), and by integration by parts:
๐ = เถฑ โ๐๐ + เถฑ โ๐ฮ โ เถฑ โ๐๐๐๐(๐);๐๐๐
๐4๐ฅ +๐
2เถฑ โ๐ ๐๐ + ๐๐๐ด
2๐4๐ฅ
๐น๐๐: ๐ป๐๐ ๐๐๐
= ๐๐ = ๐ ๐๐ + ๐๐๐ด into the action:
๐ = เถฑ โ๐๐ + เถฑ โ๐ g๐ผ๐ฝ๐๐ผ๐๐ฝ โ1
2๐เถฑ โ๐ ๐๐๐
๐๐4๐ฅ + เถฑ โ๐๐๐๐ด๐๐๐4๐ฅ
The partition function considering Euclidean metrics (exclude the
gravity terms):
ฮ = เถฑ๐ท๐๐ฟ ๐;๐๐exp
1
2๐เถฑ ๐ ๐๐๐
๐๐4๐ฅ โ เถฑ ๐ g๐ผ๐ฝ๐๐ผ๐๐ฝ
We see that for ๐ < 0, there will a convergent functional integration, so
this is a good sign for the quantum behavior of the theory.
Numerical solution By redefine the red โ shift as the changing variable:
๐ ๐ง =๐ 0
๐ง + 1
๐
๐๐ก= โ ๐ง + 1 ๐ป ๐ง
๐
๐๐ง
The physical behavior can be obtained by the deceleration parameter:
๐ = โ1 โแถ๐ป
๐ป2 =1
21 + 3๐ 1 +
๐พ
๐๐ป2
qwtype
-1-1Dark energy
< -1Hyper - inflation
1/20dust
21Stiff
๐ถ2 > 0 Solutions
๐ถ2 < 0 Solutions
Bouncing solutions
Final Remarks
T.M.T - Unified Dark Matter Dark Energy. The cosmological constant
appears as an integration constant.
The Dynamical space time Theories โ both energy momentum tensor
are conserved.
Diffusive Unified DE and DM โ the vector field is taken to be the gradient
of a scalar, the energy momentum tensor ๐(๐ฅ)๐๐
has a source current,
unlike the ๐(๐บ)๐๐
which is conserved. The non conservation of ๐(๐ฅ)๐๐
is of the
diffusive form.
Asymptotically stable solution ฮCDM 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 ๐ < 0, and therefor the theory isa good property before quantizing the theory.
Future research
A Stellar model
Inflationary model ?
Quantum solutions โ Wheeler de Witt equation
A solution for coincidence problem ?
And this is only the beginningโฆ