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 Ξ term in Einsteinβs field equation
π ππ β1
2ππππ = Ξπππ + πππ
called the cosmological constant.
β’ Why is the observed value so many orders of
magnitude smaller than that expected in QFT?
β’ 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 βπ, Guendelman and Kaganovich
put another measure which is also a density and a total derivative. For
example constructing this measure 4 scalar fields π(π), where a = 1, 2, 3, 4:
Ξ¦ =1
4!νππππνπππππππ
(π)πππ(π)πππ
(π)πππ(π) = det π,π
π
with the total action:
π = ΰΆ± βπβ1 +Ξ¦β2
The variation from the scalar fields π(π) we get β2 = M = const.
Unified scalar DE-DM
For a scalar field theory with a new measure:
π = ΰΆ± βππ + βπ +Ξ¦ Ξ d4x
where Ξ = ππΌπ½π,πΌπ,π½. The Equations of Motion:
Ξ = π = ππππ π‘
ππ = 1 +Ξ¦
βππππ
πππ = πππΞ + 1 +Ξ¦
βπππππππ = πππΞ + πππππ
β’ 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
β’ 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 αΆπ = πΆ1 and.
A conserved current: π»πππ =
1
βπππ βπππ =
1
π3π
ππ‘π3π0 = 0
or π0 =πΆ3
π3. The complete set of the densities:
πΞ = αΆπ2 = πΆ1 πΞ = βπΞ
πd =πΆ3
π3αΆπ =
πΆ1πΆ3
π3πd = 0
The precise solution for Friedman equation π βΌαΆπ
π
2in this
case is:
πΞβπ =πΆ3
πΆ1
Ξ€1 3
sinh Ξ€2 33
2πΆ1π‘
Which helps us to reconstruct the original physical values:
ΩΠ=πΆ1π»0
ΩΠ=πΆ3 πΆ1π»0
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):
ππ‘π + π£ ππ₯π = π ππ€2π βΉ πππππ β Ξππ
π ππππππππ = π·ππ
π β distribution function, v β velocity, π β diffusion coefficient.
The current density and the energy momentum tensor πππ are definedas:
ππ = β βπΰΆ±πππ
π0ππ1 β§ ππ2 β§ ππ3
πππ = β βπΰΆ±πππππ
π0ππ1 β§ ππ2 β§ ππ3
ππ is a time-like vector field and πππ verifies the dominant and strongenergy conditions.
π»ππππ = 3πππ π»ππ
π = 0
The number of particles is conserved, but not the energy momentumtensor.
Connection to Cosmology Calogeroβs idea: πCDM. The cosmological constant is replaced by a
scalar filed, which would the source of the Cold Dark Matter stress
energy tensor:
π ππ β1
2ππππ = πππ + ππππ
π»ππππ = 3πππ
π»ππ = β3πππ
The value 3π measures the energy transferred from the scalar field to the
matter
per unit of time due to diffusion.
This modification applied βby handβ, 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:
ππ - dynamical space-time vector field , ππ;π = ππ£ππ β Ξπππ ππ in second
order formalism Ξπππ is Christoffel Symbol.
π(π)ππ
- stress energy tensor. The variation according to π gives a conserved
diffusive energy: π»ππ(π)ππ
= 0, in addition to π(πΊ)ππ
=πΏπ(π)
πΏπππ.
Dynamical time is as T.M.T for π(π)ππ
= πππΞ.
Ξ¦ =1
4!νππππνπππππππ
(π)πππ(π)πππ
(π)πππ(π)
π = ΰΆ±Ξ¦β1 π π = ΰΆ± βπππ;ππ(π)πππ4π₯
Diffusive energy We replace the dynamical space time vector ππ by a gradient of a
scalar filed π,π:
π π = ΰΆ± βππ,π;ππ(π)πππ4π₯
π - scalar filed , π,π;π = ππ£πππ β Ξπππ πππ, π(π)
ππ- stress energy tensor.
The variation according to π gives a non-conserved diffusive energy momentum tensor:
π»ππ(π)ππ
= ππ£ ; π»π£ππ£ = 0
The variation according to the metric gives a conserved stress energytensor, (which is familiar from Einstein eq. π πΊ
ππ£= π ππ β
1
2πππ π ) :
π(πΊ)ππ
=β2
βπ
πΏ βπβππΏπππ
Dynamical time with source
An action with no high derivatives, is to add a source coupled to ΟΞΌ:
π π = ΰΆ± βπππ;ππ(π)πππ4π₯ +
π
2ΰΆ± βπ ππ + πππ΄
2π4π₯
π ππ: π»ΞΌT(Ο)ΞΌΞ½
= Ο ΟΞ½ + πΞ½A
π π: Ο π»Ξ½ ΟΞ½ + πΞ½A = 0
One difference between those theories:
Here - π appears as a parameter
- in the higher derivative theory π appears as an integration of
constant.
Symmetries
If the matter is coupled through its energy momentum tensor
as:
π πππ
β π πππ
+ ππππ
the process will not affect the equations of motion. In Quantum
Field Theory this is βnormal orderingβ.
π β π + π
A toy model
We start with a simple action of one dimensional particle in a
potential π(π₯).
π = ΰΆ± α·π΅1
2π αΆπ₯2 + π π₯ ππ‘
πΏπ΅ gives the total energy of a particle with constant power P:
1
2π αΆπ₯2 + π π₯ = πΈ π‘ = πΈ0 + ππ‘
πΏπ₯ gives the condition for B:
π α·π₯ α·π΅ + π αΆπ₯αΈπ΅ = πβ² π₯ α·π΅ or αΈπ΅
α·π΅=
2πβ² π₯
2π πΈ π‘ βπ π₯
βπ
2 πΈ π‘ βπ π₯
A conserved Hamiltonian
Momentums for this toy model:
ππ₯ =πβ
π αΆπ₯= π αΆπ₯ α·π΅
ππ΅ =πβ
π αΆπ΅βπ
ππ‘
πβ
π α·π΅= β
π
ππ‘πΈ π‘
Ξ π΅ =πβ
π α·π΅= πΈ π‘
The Hamiltonian (with second order derivative):
β = αΆπ₯ππ₯ + αΆπ΅ππ΅ + α·π΅Ξ π΅ β β = π αΆπ₯ α·π΅ β αΆπ΅ αΆπΈ = ππ₯2
πΞ π΅ β π π₯ + αΆπ΅ππ΅
The action isnβt dependent on time explicitly, so the Hamiltonian is
conserved.
Interacting Diffusive DE β DM
The diffusive stress energy tensor in this theory:
π(π)ππ
= Ξπππ
with the kinetic βk-essenceβ term Ξ = ππΌπ½π,πΌπ,π½,
where π β a scalar filed.
The full theory:
π = ΰΆ±1
2βππ + βπ β π + 1 Ξ π4π₯
when 8ππΊ = π = 1.
(with high derivatives)
Equation of motions
πΏπ - non trivial evolving dark energy:
β Ξ = 0
πΏπ - a conserved current:
jΞ² = 2 β Ο + 1 Ο,Ξ²
πΏπππ - a conserved stress energy tensor:
π(πΊ)ππ
= πππ βΞ + π ,πΞ,π + πππ,π β π ,πΞ,π β π ,πΞ,π
Dark Energy Dark Matter
F.L.R.W solution
β Ξ = 0:
2 αΆπ α·π =πΆ2π3
βΊ αΆπ2 = πΆ1 + πΆ2ΰΆ±ππ‘
π3
ππ½ = 2 β π + 1 π,π½:
αΆπ =πΆ4π3
+1
π3ΰΆ±π3 ππ‘ β
πΆ32π3
ΰΆ±ππ‘
αΆπ
T(πΊ)ππ
- a conserved stress energy tensor:
πΞ = αΆπ2 +πΆ2
π3αΆπ πΞ = βπΞ
πd =πΆ3
π3αΆπ β 2
πΆ2
π3αΆπ πd = 0
Asymptotic solution The field αΆπ asymptotically goes to the value as De Sitter space π ~ ππ»0π‘:
limπ‘ββ
αΆπ =1
π3ΰΆ±π3 ππ‘ =
1
3π»0
The asymptotic values of the densities are:
πΞ = πΆ1 + πΆ2ΰΆ±ππ‘
π3+πΆ2π3
αΆπ = πΆ1 + π1
π6
πCDM = πΆ3 πΆ1 β2πΆ23π»0
1
π3+ π
1
π6
The observable values:πΆ1π»0
= ΩΠπΆ3 πΆ1 β2πΆ23π»0
= π»0Ξ©d
Stability of the solutions More close asymptotically with ΞπΆπ·π : the dark energy become
constant, and the amount of dark matter slightly change πCDM~1
π3.
πΆ3 πΆ1 >2πΆ2
3π»0for positive dust density. For πΆ2 < 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, βTaking 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 πΆ2 = 0.
Which gives constant scalar filed αΆπ = πΆ1 and α·π = 0.
πΞ = αΆπ2 = πΆ1 πΞ = βπΞ
πd =πΆ3
π3αΆπ =
πΆ1πΆ3
π3πd = 0
The precise solution for Friedman equation π βΌαΆπ
π
2in this case is:
πΞβπ =πΆ3
πΆ1
Ξ€1 3
sinh Ξ€2 33
2πΆ1π‘
Which helps us to reconstruct the original physical values:
ΩΠ=πΆ1π»0
Ξ©d =πΆ3 πΆ1π»0
Perturbative solution
The scalar field has perturbative properties π1,2 βͺ 1:
π1 π‘, π‘0 =πΆ2πΆ1ΰΆ±ππ‘
π3
π2 π‘, π‘0 =πΆ2
πΆ1πΆ3αΆπ
For a first order solution in perturbation theory:
πΞ = πΆ1 1 + π1 +πΆ3
πΆ1π2 + π2 π1, π2
ππΆπ·π =πΆ1πΆ3π3
1 +1
2π1 + π2 + π2 π1, π2
For rising dark energy, dark matter amount goes lower ( πΆ2< 0, πΆ1,3,4 > 0). For decreasing dark energy, the amount of dark
matter goes up(all components are positive).
Diffusive energy without higher derivatives
The full theory:
β =1
2βππ + βπππ;ππ(π)
ππ+π
2βπ ππ + πππ΄
2+ βπΞ
Where Ξ = ππΌπ½π,πΌπ,π½, and π(π)ππ
= Ξπππ .
All the E.o.M are the same, except:
π(πΊ)ππ
= πππ βΞ + π,πΞ,π +π
πππ²,ππ²,π + πππ,π β π,πΞ,π β π ,πΞ,π +
π
ππ²,ππ²,π
For the late universe both theories are equivalent, Ξ,πΞ,π βΌ1
π6.
For π β β, the term π
2βπ ππ + πππ΄
2forces ππ = βπππ΄, and D.T
becomes Diffusive energy with high energy.
Calogeroβs model ΟCMD Calogero put two stress energy tensor of DE-DM. Each stress energy
tensor in non-conserved:
π»ππ Ξππ
= βπ»ππ Dustππ
= 3πππ, π;ππ = 0
For FRWM, this calculation leads to the solution:
πΞ = πΆ1 + πΆ2ΰΆ±ππ‘
π3
πDust =πΆ3π3
βπΆ2π‘
π3
The two model became approximate for C2 αΆπ βͺ 1. Our asymptotic solution
becomes with constant densities, because C2 αΆπ βπΆ2
3π»0, which makes the DE
decay lower from ππΆππ·, and DM evolution as ΞπΆππ·.
Preliminary ideas on Quantization
Taking Dynamical space time theory (with source), and by integration by parts:
π = ΰΆ± βππ + ΰΆ± βπΞ β ΰΆ± βππππ(π);πππ
π4π₯ +π
2ΰΆ± βπ ππ + πππ΄
2π4π₯
πΉππ: π»ππ πππ
= ππ = π ππ + πππ΄ into the action:
π = ΰΆ± βππ + ΰΆ± βπ gπΌπ½ππΌππ½ β1
2πΰΆ± βπ πππ
ππ4π₯ + ΰΆ± βππππ΄πππ4π₯
The partition function considering Euclidean metrics (exclude the
gravity terms):
Ξ = ΰΆ±π·ππΏ π;ππexp
1
2πΰΆ± π πππ
ππ4π₯ β ΰΆ± π gπΌπ½ππΌππ½
We see that for π < 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 β shift as the changing variable:
π π§ =π 0
π§ + 1
π
ππ‘= β π§ + 1 π» π§
π
ππ§
The physical behavior can be obtained by the deceleration parameter:
π = β1 βαΆπ»
π»2 =1
21 + 3π 1 +
πΎ
ππ»2
qwtype
-1-1Dark energy
< -1Hyper - inflation
1/20dust
21Stiff
πΆ2 > 0 Solutions
πΆ2 < 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 β both energy momentum tensor
are conserved.
Diffusive Unified DE and DM β the vector field is taken to be the gradient
of a scalar, the energy momentum tensor π(π₯)ππ
has a source current,
unlike the π(πΊ)ππ
which is conserved. The non conservation of π(π₯)ππ
is of the
diffusive form.
Asymptotically stable solution ΞCDM 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 π < 0, and therefor the theory isa good property before quantizing the theory.
Future research
A Stellar model
Inflationary model ?
Quantum solutions β Wheeler de Witt equation
A solution for coincidence problem ?
And this is only the beginningβ¦