Diffusive DE amp DMDiffusive DE and DMEduardo Guendelman With my student David Benisty

Ben Gurion University

PRESENTED AT

MIAMI 2017 December 15

Problems in late cosmology

The vacuum energy behaves as the Λ term in Einsteinrsquos field equation

119877120583120584 minus1

2119892120583120584119877 = Λ119892120583120584 + 119879120583120584

called the cosmological constant

bull Why is the observed value so many orders of

magnitude smaller than that expected in QFT

bull Why is it of the same order of

magnitude as the matter density of

the universe at the present time

bull flatness taken care by inflation

Two Measure Theory

In addition to the regular measure minus119892 another measure which is also adensity and a total derivative For example constructing this measure outof a 4 index field strength We can also proceed by using 4 scalar fields120601(119886) where a = 1 2 3 4 and the jacobian of the mapping between thesescalar and coordinate space

Φ =1

4휀120583120584120588120590휀119886119887119888119889120597120583120601

(119886)120597120584120601(119887)120597120588120601

(119888)120597120590120601(119889) = det 120601119895

119894

And in the Two Measures Theory we consider the total action

119878 = නd4x minus119892ℒ1 + d4xΦℒ2

The variation from the scalar fields 120601(119886) we get ℒ2 = M = const

Unified scalar DE-DM leading to Lambda CDM

For a scalar field theory with a new measure

119878 = න minus119892119877 + minus119892 + Φ Λ d4x

where Λ = 119892120572120573120593120572120593120573 The Equations of Motion

Λ = 119872 = 119888119900119899119904119905

119895120583 = 1 +Φ

minus119892120597120583120593

119879120583120584 = 119892120583120584Λ + 1 +Φ

minus119892120597120583120593120597120584120593 = 119892120583120584Λ + 119895120583120597120584120593

bull Dark Energy and Dark Matter From Hidden Symmetry of Gravity Model with a Non-Riemannian Volume Form

European Physical Journal C75 (2015) 472-479 arXiv150802008

bull A two measure model of dark energy and dark matter Eduardo Guendelman Douglas Singleton Nattapong

Yongram arXiv12051056 [gr-qc]

Problems in late cosmology

The vacuum energy behaves as the Λ term in Einsteinrsquos field equation

119877120583120584 minus1

2119892120583120584119877 = Λ119892120583120584 + 119879120583120584

called the cosmological constant

bull Why is the observed value so many orders of

magnitude smaller than that expected in QFT

bull Why is it of the same order of

magnitude as the matter density of

the universe at the present time

bull flatness taken care by inflation

Two Measure Theory

In addition to the regular measure minus119892 another measure which is also adensity and a total derivative For example constructing this measure outof a 4 index field strength We can also proceed by using 4 scalar fields120601(119886) where a = 1 2 3 4 and the jacobian of the mapping between thesescalar and coordinate space

Φ =1

4휀120583120584120588120590휀119886119887119888119889120597120583120601

(119886)120597120584120601(119887)120597120588120601

(119888)120597120590120601(119889) = det 120601119895

119894

And in the Two Measures Theory we consider the total action

119878 = නd4x minus119892ℒ1 + d4xΦℒ2

The variation from the scalar fields 120601(119886) we get ℒ2 = M = const

Unified scalar DE-DM leading to Lambda CDM

For a scalar field theory with a new measure

119878 = න minus119892119877 + minus119892 + Φ Λ d4x

where Λ = 119892120572120573120593120572120593120573 The Equations of Motion

Λ = 119872 = 119888119900119899119904119905

119895120583 = 1 +Φ

minus119892120597120583120593

119879120583120584 = 119892120583120584Λ + 1 +Φ

minus119892120597120583120593120597120584120593 = 119892120583120584Λ + 119895120583120597120584120593

bull Dark Energy and Dark Matter From Hidden Symmetry of Gravity Model with a Non-Riemannian Volume Form

European Physical Journal C75 (2015) 472-479 arXiv150802008

bull A two measure model of dark energy and dark matter Eduardo Guendelman Douglas Singleton Nattapong

Yongram arXiv12051056 [gr-qc]

Two Measure Theory

In addition to the regular measure minus119892 another measure which is also adensity and a total derivative For example constructing this measure outof a 4 index field strength We can also proceed by using 4 scalar fields120601(119886) where a = 1 2 3 4 and the jacobian of the mapping between thesescalar and coordinate space

Φ =1

4휀120583120584120588120590휀119886119887119888119889120597120583120601

(119886)120597120584120601(119887)120597120588120601

(119888)120597120590120601(119889) = det 120601119895

119894

And in the Two Measures Theory we consider the total action

119878 = නd4x minus119892ℒ1 + d4xΦℒ2

The variation from the scalar fields 120601(119886) we get ℒ2 = M = const

Unified scalar DE-DM leading to Lambda CDM

For a scalar field theory with a new measure

119878 = න minus119892119877 + minus119892 + Φ Λ d4x

where Λ = 119892120572120573120593120572120593120573 The Equations of Motion

Λ = 119872 = 119888119900119899119904119905

119895120583 = 1 +Φ

minus119892120597120583120593

119879120583120584 = 119892120583120584Λ + 1 +Φ

minus119892120597120583120593120597120584120593 = 119892120583120584Λ + 119895120583120597120584120593

bull Dark Energy and Dark Matter From Hidden Symmetry of Gravity Model with a Non-Riemannian Volume Form

European Physical Journal C75 (2015) 472-479 arXiv150802008

bull A two measure model of dark energy and dark matter Eduardo Guendelman Douglas Singleton Nattapong

Yongram arXiv12051056 [gr-qc]

Unified scalar DE-DM leading to Lambda CDM

For a scalar field theory with a new measure

119878 = න minus119892119877 + minus119892 + Φ Λ d4x

where Λ = 119892120572120573120593120572120593120573 The Equations of Motion

Λ = 119872 = 119888119900119899119904119905

119895120583 = 1 +Φ

minus119892120597120583120593

119879120583120584 = 119892120583120584Λ + 1 +Φ

minus119892120597120583120593120597120584120593 = 119892120583120584Λ + 119895120583120597120584120593

bull Dark Energy and Dark Matter From Hidden Symmetry of Gravity Model with a Non-Riemannian Volume Form

European Physical Journal C75 (2015) 472-479 arXiv150802008

bull A two measure model of dark energy and dark matter Eduardo Guendelman Douglas Singleton Nattapong

Yongram arXiv12051056 [gr-qc]

Which gives constant scalar filed ሶ120593 = 1198621 and

A conserved current 120571120583119895120583 =

1

minus119892120597120583 minus119892119895120583 =

1

1198863120597

12059711990511988631198950 = 0

or 1198950 =1198623

1198863 The complete set of the densities

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this

case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

ΩΛ =1198623 11986211198670

There have been some other

Unified Models of DEDM worth

mentioning for example the

Chaplygin gas see eg

Unification of dark matter and dark energy The Inhomogeneous Chaplygin gas

Neven Bilic Gary B Tupper Raoul D Viollier (Cape Town

U) Nov 2001 10 pp

Published in PhysLett B535 (2002) 17-21

There have been some other

Unified Models of DEDM worth

mentioning for example the

Chaplygin gas see eg

Unification of dark matter and dark energy The Inhomogeneous Chaplygin gas

Neven Bilic Gary B Tupper Raoul D Viollier (Cape Town

U) Nov 2001 10 pp

Published in PhysLett B535 (2002) 17-21

In that case there is also some

communication between DE

and DMWE ARE GOING TO CONSIDER A GENERALIZALION OF

OUR unified DEDM THAT ALSO INVOLVES DEDM

EXCHANGE THAT IS THOSE TWO COMPONENTS ARE

NOT GOING TO BE SEPARATELLY CONSERVED AND THE

WAY THEY WILL EXCHANGE ENERGY WILL BE IN A

DIFFUSIVE WAY SO WE NOW REVIEW A FEW NOTIONS

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 st Kinetic diffusion equation(Fokker Planck)

120597119905119891 + 119907 120597119909119891 = 120590 1205971199082119891 ⟹ 119901120583120597120583119891 minus Γ120583120584

119894 119901120583119901120584120597119901119894119891 = 119863119901119891

119891 minusdistribution function v ndash velocity 120590 ndash diffusion coefficient

The current density and the energy momentum tensor 119879120583120584 are definedas

119895120583 = minus minus119892න119891119901120583

11990101198891199011 and 1198891199012 and 1198891199013

119879120583120584 = minus minus119892න119891119901120583119901120584

11990101198891199011 and 1198891199012 and 1198891199013

119895120583 is a time-like vector field and 119879120583120584 verifies the dominant and strongenergy conditions

120571120583119879120583120584 = 3120590119895120584 120571120583119895

120584 = 0

The number of particles is conserved but not the energy momentumtensor

Connection to Cosmology Calogerorsquos Habarsquos idea 120601 CDM The cosmological constant is

replaced by a scalar filed which would the source of the Cold Dark

Matter stress energy tensor

119877120583120584 minus1

2119892120583120584119877 = 119879120583120584 + 120593119892120583120584

120571120583119879120583120584 = 3120590119895120584

120571120584120593 = minus3120590119895120584

The value 3120590 measures the energy transferred from the scalar field to the

matter

per unit of time due to diffusion

This modification applied ldquoby handrdquo and not from action principle

Alternative approach through a - Diffusive Energy Action A generalization of the non Riemannian volume form is required

1st step from metric independent

Volume form to Dynamicalspacetime

The basic result of can be expressed as a covariant conservation of a

stress energy tensor

120594120582 - dynamical space-time vector field 120594120583120584 = 120597119907120594120583 minus Γ120583120584120582 120594120582 in second

order formalism Γ120583120584120582 is Christoffel Symbol

119879(120594)120583120584

- stress energy tensor The variation according to 120594 gives a conserved

energy momentum tensor 120571120583119879(120594)120583120584

= 0 in addition to 119879(119866)120583120584

=120575119878(120594)

120575119892120583120584

Dynamical time is as TMT for 119879(120594)120583120584

= 119892120583120584Λ

Φ =1

4휀120583120584120588120590휀119886119887119888119889120597120583120601

(119886)120597120584120601(119887)120597120588120601

(119888)120597120590120601(119889)

119878 = නΦℒ1 119878 120594 = න minus119892 120594120583120584119879(120594)1205831205841198894119909

2nd step the Diffusive energy action principle

We replace the dynamical space time vector 120594120583 by a gradient of a scalar filed 120594120583

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909

120594 - scalar field 120594120583120584 = 120597119907120597120583120594 minus Γ120583120584120582 120597120582120594 119879(120594)

120583120584- stress energy tensor

The variation according to 120594 gives a non-conserved diffusive energy momentum tensor

120571120583119879(120594)120583120584

= 119891119907 120571119907119891119907 = 0

The variation according to the metric gives a conserved stress energytensor (which is familiar from Einstein eq 119879 119866

120583119907= 119877120583120584 minus

1

2119892120583120584 119877)

119879(119866)120583120584

=minus2

minus119892

120575 minus119892ℒ119898120575119892120583120584

Alternative formulation withouthigher derivative with mass like term

An action with no high derivatives is obtained by adding another term involving χμ

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909 +

120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120517120536120524 120571μT(χ)μν

= σ χν + 120597νA

120517119808 σ 120571ν χν + 120597νA = 0

One difference between those theories

Here - 120590 appears as a parameter

- in the higher derivative theory 120590 appears as an integration

constant

Symmetries

If the matter is coupled through its energy momentum tensor

as

119879 120594120583120584

rarr 119879 120594120583120584

+ 120582119892120583120584

the process will not affect the equations of motion In Quantum

Field Theory this is ldquonormal orderingrdquo

120594 rarr 120594 + 120582

A toy model

We start with a simple action of one dimensional particle in a

potential 119881(119909)

119878 = න ሷ1198611

2119898 ሶ1199092 + 119881 119909 119889119905

120575119861 gives the total energy of a particle with constant power P

1

2119898 ሶ1199092 + 119881 119909 = 119864 119905 = 1198640 + 119875119905

120575119909 gives the condition for B

119898 ሷ119909 ሷ119861 +119898 ሶ119909ሸ119861 = 119881prime 119909 ሷ119861 or ሸ119861

ሷ119861=

2119881prime 119909

2119898 119864 119905 minus119881 119909

minus119875

2 119864 119905 minus119881 119909

A conserved Hamiltonian

Momentums for this toy model

120587119909 =120597ℒ

120597 ሶ119909= 119898 ሶ119909 ሷ119861

120587119861 =120597ℒ

120597 ሶ119861minus119889

119889119905

120597ℒ

120597 ሷ119861= minus

119889

119889119905119864 119905

Π119861 =120597ℒ

120597 ሷ119861= 119864 119905

The Hamiltonian (with second order derivative)

ℋ = ሶ119909120587119909 + ሶ119861120587119861 + ሷ119861Π119861 minus ℒ = 119898 ሶ119909 ሷ119861 minus ሶ119861 ሶ119864 = 1205871199092

119898Π119861 minus 119881 119909 + ሶ119861120587119861

The action isnrsquot dependent on time explicitly so the Hamiltonian is

conserved

Interacting Diffusive DE ndash DM

The diffusive stress energy tensor in this theory

119879(120594)120583120584

= Λ119892120583120584

with the kinetic ldquok-essencerdquo term Λ = 119892120572120573120593120572120593120573

where 120593 ndash a scalar filed

The full theory

119878 = න1

2minus119892119877 + minus119892 ⊔ 120594 + 1 Λ 1198894119909

when 8120587119866 = 119888 = 1

(with high derivatives)

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

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 st Kinetic diffusion equation(Fokker Planck)

120597119905119891 + 119907 120597119909119891 = 120590 1205971199082119891 ⟹ 119901120583120597120583119891 minus Γ120583120584

119894 119901120583119901120584120597119901119894119891 = 119863119901119891

119891 minusdistribution function v ndash velocity 120590 ndash diffusion coefficient

The current density and the energy momentum tensor 119879120583120584 are definedas

119895120583 = minus minus119892න119891119901120583

11990101198891199011 and 1198891199012 and 1198891199013

119879120583120584 = minus minus119892න119891119901120583119901120584

11990101198891199011 and 1198891199012 and 1198891199013

119895120583 is a time-like vector field and 119879120583120584 verifies the dominant and strongenergy conditions

120571120583119879120583120584 = 3120590119895120584 120571120583119895

120584 = 0

The number of particles is conserved but not the energy momentumtensor

Connection to Cosmology Calogerorsquos Habarsquos idea 120601 CDM The cosmological constant is

replaced by a scalar filed which would the source of the Cold Dark

Matter stress energy tensor

119877120583120584 minus1

2119892120583120584119877 = 119879120583120584 + 120593119892120583120584

120571120583119879120583120584 = 3120590119895120584

120571120584120593 = minus3120590119895120584

The value 3120590 measures the energy transferred from the scalar field to the

matter

per unit of time due to diffusion

This modification applied ldquoby handrdquo and not from action principle

Alternative approach through a - Diffusive Energy Action A generalization of the non Riemannian volume form is required

1st step from metric independent

Volume form to Dynamicalspacetime

The basic result of can be expressed as a covariant conservation of a

stress energy tensor

120594120582 - dynamical space-time vector field 120594120583120584 = 120597119907120594120583 minus Γ120583120584120582 120594120582 in second

order formalism Γ120583120584120582 is Christoffel Symbol

119879(120594)120583120584

- stress energy tensor The variation according to 120594 gives a conserved

energy momentum tensor 120571120583119879(120594)120583120584

= 0 in addition to 119879(119866)120583120584

=120575119878(120594)

120575119892120583120584

Dynamical time is as TMT for 119879(120594)120583120584

= 119892120583120584Λ

Φ =1

4휀120583120584120588120590휀119886119887119888119889120597120583120601

(119886)120597120584120601(119887)120597120588120601

(119888)120597120590120601(119889)

119878 = නΦℒ1 119878 120594 = න minus119892 120594120583120584119879(120594)1205831205841198894119909

2nd step the Diffusive energy action principle

We replace the dynamical space time vector 120594120583 by a gradient of a scalar filed 120594120583

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909

120594 - scalar field 120594120583120584 = 120597119907120597120583120594 minus Γ120583120584120582 120597120582120594 119879(120594)

120583120584- stress energy tensor

The variation according to 120594 gives a non-conserved diffusive energy momentum tensor

120571120583119879(120594)120583120584

= 119891119907 120571119907119891119907 = 0

The variation according to the metric gives a conserved stress energytensor (which is familiar from Einstein eq 119879 119866

120583119907= 119877120583120584 minus

1

2119892120583120584 119877)

119879(119866)120583120584

=minus2

minus119892

120575 minus119892ℒ119898120575119892120583120584

Alternative formulation withouthigher derivative with mass like term

An action with no high derivatives is obtained by adding another term involving χμ

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909 +

120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120517120536120524 120571μT(χ)μν

= σ χν + 120597νA

120517119808 σ 120571ν χν + 120597νA = 0

One difference between those theories

Here - 120590 appears as a parameter

- in the higher derivative theory 120590 appears as an integration

constant

Symmetries

If the matter is coupled through its energy momentum tensor

as

119879 120594120583120584

rarr 119879 120594120583120584

+ 120582119892120583120584

the process will not affect the equations of motion In Quantum

Field Theory this is ldquonormal orderingrdquo

120594 rarr 120594 + 120582

A toy model

We start with a simple action of one dimensional particle in a

potential 119881(119909)

119878 = න ሷ1198611

2119898 ሶ1199092 + 119881 119909 119889119905

120575119861 gives the total energy of a particle with constant power P

1

2119898 ሶ1199092 + 119881 119909 = 119864 119905 = 1198640 + 119875119905

120575119909 gives the condition for B

119898 ሷ119909 ሷ119861 +119898 ሶ119909ሸ119861 = 119881prime 119909 ሷ119861 or ሸ119861

ሷ119861=

2119881prime 119909

2119898 119864 119905 minus119881 119909

minus119875

2 119864 119905 minus119881 119909

A conserved Hamiltonian

Momentums for this toy model

120587119909 =120597ℒ

120597 ሶ119909= 119898 ሶ119909 ሷ119861

120587119861 =120597ℒ

120597 ሶ119861minus119889

119889119905

120597ℒ

120597 ሷ119861= minus

119889

119889119905119864 119905

Π119861 =120597ℒ

120597 ሷ119861= 119864 119905

The Hamiltonian (with second order derivative)

ℋ = ሶ119909120587119909 + ሶ119861120587119861 + ሷ119861Π119861 minus ℒ = 119898 ሶ119909 ሷ119861 minus ሶ119861 ሶ119864 = 1205871199092

119898Π119861 minus 119881 119909 + ሶ119861120587119861

The action isnrsquot dependent on time explicitly so the Hamiltonian is

conserved

Interacting Diffusive DE ndash DM

The diffusive stress energy tensor in this theory

119879(120594)120583120584

= Λ119892120583120584

with the kinetic ldquok-essencerdquo term Λ = 119892120572120573120593120572120593120573

where 120593 ndash a scalar filed

The full theory

119878 = න1

2minus119892119877 + minus119892 ⊔ 120594 + 1 Λ 1198894119909

when 8120587119866 = 119888 = 1

(with high derivatives)

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Kinetic diffusion on curved st Kinetic diffusion equation(Fokker Planck)

120597119905119891 + 119907 120597119909119891 = 120590 1205971199082119891 ⟹ 119901120583120597120583119891 minus Γ120583120584

119894 119901120583119901120584120597119901119894119891 = 119863119901119891

119891 minusdistribution function v ndash velocity 120590 ndash diffusion coefficient

The current density and the energy momentum tensor 119879120583120584 are definedas

119895120583 = minus minus119892න119891119901120583

11990101198891199011 and 1198891199012 and 1198891199013

119879120583120584 = minus minus119892න119891119901120583119901120584

11990101198891199011 and 1198891199012 and 1198891199013

119895120583 is a time-like vector field and 119879120583120584 verifies the dominant and strongenergy conditions

120571120583119879120583120584 = 3120590119895120584 120571120583119895

120584 = 0

The number of particles is conserved but not the energy momentumtensor

Connection to Cosmology Calogerorsquos Habarsquos idea 120601 CDM The cosmological constant is

replaced by a scalar filed which would the source of the Cold Dark

Matter stress energy tensor

119877120583120584 minus1

2119892120583120584119877 = 119879120583120584 + 120593119892120583120584

120571120583119879120583120584 = 3120590119895120584

120571120584120593 = minus3120590119895120584

The value 3120590 measures the energy transferred from the scalar field to the

matter

per unit of time due to diffusion

This modification applied ldquoby handrdquo and not from action principle

Alternative approach through a - Diffusive Energy Action A generalization of the non Riemannian volume form is required

1st step from metric independent

Volume form to Dynamicalspacetime

The basic result of can be expressed as a covariant conservation of a

stress energy tensor

120594120582 - dynamical space-time vector field 120594120583120584 = 120597119907120594120583 minus Γ120583120584120582 120594120582 in second

order formalism Γ120583120584120582 is Christoffel Symbol

119879(120594)120583120584

- stress energy tensor The variation according to 120594 gives a conserved

energy momentum tensor 120571120583119879(120594)120583120584

= 0 in addition to 119879(119866)120583120584

=120575119878(120594)

120575119892120583120584

Dynamical time is as TMT for 119879(120594)120583120584

= 119892120583120584Λ

Φ =1

4휀120583120584120588120590휀119886119887119888119889120597120583120601

(119886)120597120584120601(119887)120597120588120601

(119888)120597120590120601(119889)

119878 = නΦℒ1 119878 120594 = න minus119892 120594120583120584119879(120594)1205831205841198894119909

2nd step the Diffusive energy action principle

We replace the dynamical space time vector 120594120583 by a gradient of a scalar filed 120594120583

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909

120594 - scalar field 120594120583120584 = 120597119907120597120583120594 minus Γ120583120584120582 120597120582120594 119879(120594)

120583120584- stress energy tensor

The variation according to 120594 gives a non-conserved diffusive energy momentum tensor

120571120583119879(120594)120583120584

= 119891119907 120571119907119891119907 = 0

The variation according to the metric gives a conserved stress energytensor (which is familiar from Einstein eq 119879 119866

120583119907= 119877120583120584 minus

1

2119892120583120584 119877)

119879(119866)120583120584

=minus2

minus119892

120575 minus119892ℒ119898120575119892120583120584

Alternative formulation withouthigher derivative with mass like term

An action with no high derivatives is obtained by adding another term involving χμ

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909 +

120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120517120536120524 120571μT(χ)μν

= σ χν + 120597νA

120517119808 σ 120571ν χν + 120597νA = 0

One difference between those theories

Here - 120590 appears as a parameter

- in the higher derivative theory 120590 appears as an integration

constant

Symmetries

If the matter is coupled through its energy momentum tensor

as

119879 120594120583120584

rarr 119879 120594120583120584

+ 120582119892120583120584

the process will not affect the equations of motion In Quantum

Field Theory this is ldquonormal orderingrdquo

120594 rarr 120594 + 120582

A toy model

We start with a simple action of one dimensional particle in a

potential 119881(119909)

119878 = න ሷ1198611

2119898 ሶ1199092 + 119881 119909 119889119905

120575119861 gives the total energy of a particle with constant power P

1

2119898 ሶ1199092 + 119881 119909 = 119864 119905 = 1198640 + 119875119905

120575119909 gives the condition for B

119898 ሷ119909 ሷ119861 +119898 ሶ119909ሸ119861 = 119881prime 119909 ሷ119861 or ሸ119861

ሷ119861=

2119881prime 119909

2119898 119864 119905 minus119881 119909

minus119875

2 119864 119905 minus119881 119909

A conserved Hamiltonian

Momentums for this toy model

120587119909 =120597ℒ

120597 ሶ119909= 119898 ሶ119909 ሷ119861

120587119861 =120597ℒ

120597 ሶ119861minus119889

119889119905

120597ℒ

120597 ሷ119861= minus

119889

119889119905119864 119905

Π119861 =120597ℒ

120597 ሷ119861= 119864 119905

The Hamiltonian (with second order derivative)

ℋ = ሶ119909120587119909 + ሶ119861120587119861 + ሷ119861Π119861 minus ℒ = 119898 ሶ119909 ሷ119861 minus ሶ119861 ሶ119864 = 1205871199092

119898Π119861 minus 119881 119909 + ሶ119861120587119861

The action isnrsquot dependent on time explicitly so the Hamiltonian is

conserved

Interacting Diffusive DE ndash DM

The diffusive stress energy tensor in this theory

119879(120594)120583120584

= Λ119892120583120584

with the kinetic ldquok-essencerdquo term Λ = 119892120572120573120593120572120593120573

where 120593 ndash a scalar filed

The full theory

119878 = න1

2minus119892119877 + minus119892 ⊔ 120594 + 1 Λ 1198894119909

when 8120587119866 = 119888 = 1

(with high derivatives)

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Connection to Cosmology Calogerorsquos Habarsquos idea 120601 CDM The cosmological constant is

replaced by a scalar filed which would the source of the Cold Dark

Matter stress energy tensor

119877120583120584 minus1

2119892120583120584119877 = 119879120583120584 + 120593119892120583120584

120571120583119879120583120584 = 3120590119895120584

120571120584120593 = minus3120590119895120584

The value 3120590 measures the energy transferred from the scalar field to the

matter

per unit of time due to diffusion

This modification applied ldquoby handrdquo and not from action principle

Alternative approach through a - Diffusive Energy Action A generalization of the non Riemannian volume form is required

1st step from metric independent

Volume form to Dynamicalspacetime

The basic result of can be expressed as a covariant conservation of a

stress energy tensor

120594120582 - dynamical space-time vector field 120594120583120584 = 120597119907120594120583 minus Γ120583120584120582 120594120582 in second

order formalism Γ120583120584120582 is Christoffel Symbol

119879(120594)120583120584

- stress energy tensor The variation according to 120594 gives a conserved

energy momentum tensor 120571120583119879(120594)120583120584

= 0 in addition to 119879(119866)120583120584

=120575119878(120594)

120575119892120583120584

Dynamical time is as TMT for 119879(120594)120583120584

= 119892120583120584Λ

Φ =1

4휀120583120584120588120590휀119886119887119888119889120597120583120601

(119886)120597120584120601(119887)120597120588120601

(119888)120597120590120601(119889)

119878 = නΦℒ1 119878 120594 = න minus119892 120594120583120584119879(120594)1205831205841198894119909

2nd step the Diffusive energy action principle

We replace the dynamical space time vector 120594120583 by a gradient of a scalar filed 120594120583

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909

120594 - scalar field 120594120583120584 = 120597119907120597120583120594 minus Γ120583120584120582 120597120582120594 119879(120594)

120583120584- stress energy tensor

The variation according to 120594 gives a non-conserved diffusive energy momentum tensor

120571120583119879(120594)120583120584

= 119891119907 120571119907119891119907 = 0

The variation according to the metric gives a conserved stress energytensor (which is familiar from Einstein eq 119879 119866

120583119907= 119877120583120584 minus

1

2119892120583120584 119877)

119879(119866)120583120584

=minus2

minus119892

120575 minus119892ℒ119898120575119892120583120584

Alternative formulation withouthigher derivative with mass like term

An action with no high derivatives is obtained by adding another term involving χμ

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909 +

120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120517120536120524 120571μT(χ)μν

= σ χν + 120597νA

120517119808 σ 120571ν χν + 120597νA = 0

One difference between those theories

Here - 120590 appears as a parameter

- in the higher derivative theory 120590 appears as an integration

constant

Symmetries

If the matter is coupled through its energy momentum tensor

as

119879 120594120583120584

rarr 119879 120594120583120584

+ 120582119892120583120584

the process will not affect the equations of motion In Quantum

Field Theory this is ldquonormal orderingrdquo

120594 rarr 120594 + 120582

A toy model

We start with a simple action of one dimensional particle in a

potential 119881(119909)

119878 = න ሷ1198611

2119898 ሶ1199092 + 119881 119909 119889119905

120575119861 gives the total energy of a particle with constant power P

1

2119898 ሶ1199092 + 119881 119909 = 119864 119905 = 1198640 + 119875119905

120575119909 gives the condition for B

119898 ሷ119909 ሷ119861 +119898 ሶ119909ሸ119861 = 119881prime 119909 ሷ119861 or ሸ119861

ሷ119861=

2119881prime 119909

2119898 119864 119905 minus119881 119909

minus119875

2 119864 119905 minus119881 119909

A conserved Hamiltonian

Momentums for this toy model

120587119909 =120597ℒ

120597 ሶ119909= 119898 ሶ119909 ሷ119861

120587119861 =120597ℒ

120597 ሶ119861minus119889

119889119905

120597ℒ

120597 ሷ119861= minus

119889

119889119905119864 119905

Π119861 =120597ℒ

120597 ሷ119861= 119864 119905

The Hamiltonian (with second order derivative)

ℋ = ሶ119909120587119909 + ሶ119861120587119861 + ሷ119861Π119861 minus ℒ = 119898 ሶ119909 ሷ119861 minus ሶ119861 ሶ119864 = 1205871199092

119898Π119861 minus 119881 119909 + ሶ119861120587119861

The action isnrsquot dependent on time explicitly so the Hamiltonian is

conserved

Interacting Diffusive DE ndash DM

The diffusive stress energy tensor in this theory

119879(120594)120583120584

= Λ119892120583120584

with the kinetic ldquok-essencerdquo term Λ = 119892120572120573120593120572120593120573

where 120593 ndash a scalar filed

The full theory

119878 = න1

2minus119892119877 + minus119892 ⊔ 120594 + 1 Λ 1198894119909

when 8120587119866 = 119888 = 1

(with high derivatives)

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

1st step from metric independent

Volume form to Dynamicalspacetime

The basic result of can be expressed as a covariant conservation of a

stress energy tensor

120594120582 - dynamical space-time vector field 120594120583120584 = 120597119907120594120583 minus Γ120583120584120582 120594120582 in second

order formalism Γ120583120584120582 is Christoffel Symbol

119879(120594)120583120584

- stress energy tensor The variation according to 120594 gives a conserved

energy momentum tensor 120571120583119879(120594)120583120584

= 0 in addition to 119879(119866)120583120584

=120575119878(120594)

120575119892120583120584

Dynamical time is as TMT for 119879(120594)120583120584

= 119892120583120584Λ

Φ =1

4휀120583120584120588120590휀119886119887119888119889120597120583120601

(119886)120597120584120601(119887)120597120588120601

(119888)120597120590120601(119889)

119878 = නΦℒ1 119878 120594 = න minus119892 120594120583120584119879(120594)1205831205841198894119909

2nd step the Diffusive energy action principle

We replace the dynamical space time vector 120594120583 by a gradient of a scalar filed 120594120583

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909

120594 - scalar field 120594120583120584 = 120597119907120597120583120594 minus Γ120583120584120582 120597120582120594 119879(120594)

120583120584- stress energy tensor

The variation according to 120594 gives a non-conserved diffusive energy momentum tensor

120571120583119879(120594)120583120584

= 119891119907 120571119907119891119907 = 0

The variation according to the metric gives a conserved stress energytensor (which is familiar from Einstein eq 119879 119866

120583119907= 119877120583120584 minus

1

2119892120583120584 119877)

119879(119866)120583120584

=minus2

minus119892

120575 minus119892ℒ119898120575119892120583120584

Alternative formulation withouthigher derivative with mass like term

An action with no high derivatives is obtained by adding another term involving χμ

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909 +

120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120517120536120524 120571μT(χ)μν

= σ χν + 120597νA

120517119808 σ 120571ν χν + 120597νA = 0

One difference between those theories

Here - 120590 appears as a parameter

- in the higher derivative theory 120590 appears as an integration

constant

Symmetries

If the matter is coupled through its energy momentum tensor

as

119879 120594120583120584

rarr 119879 120594120583120584

+ 120582119892120583120584

the process will not affect the equations of motion In Quantum

Field Theory this is ldquonormal orderingrdquo

120594 rarr 120594 + 120582

A toy model

We start with a simple action of one dimensional particle in a

potential 119881(119909)

119878 = න ሷ1198611

2119898 ሶ1199092 + 119881 119909 119889119905

120575119861 gives the total energy of a particle with constant power P

1

2119898 ሶ1199092 + 119881 119909 = 119864 119905 = 1198640 + 119875119905

120575119909 gives the condition for B

119898 ሷ119909 ሷ119861 +119898 ሶ119909ሸ119861 = 119881prime 119909 ሷ119861 or ሸ119861

ሷ119861=

2119881prime 119909

2119898 119864 119905 minus119881 119909

minus119875

2 119864 119905 minus119881 119909

A conserved Hamiltonian

Momentums for this toy model

120587119909 =120597ℒ

120597 ሶ119909= 119898 ሶ119909 ሷ119861

120587119861 =120597ℒ

120597 ሶ119861minus119889

119889119905

120597ℒ

120597 ሷ119861= minus

119889

119889119905119864 119905

Π119861 =120597ℒ

120597 ሷ119861= 119864 119905

The Hamiltonian (with second order derivative)

ℋ = ሶ119909120587119909 + ሶ119861120587119861 + ሷ119861Π119861 minus ℒ = 119898 ሶ119909 ሷ119861 minus ሶ119861 ሶ119864 = 1205871199092

119898Π119861 minus 119881 119909 + ሶ119861120587119861

The action isnrsquot dependent on time explicitly so the Hamiltonian is

conserved

Interacting Diffusive DE ndash DM

The diffusive stress energy tensor in this theory

119879(120594)120583120584

= Λ119892120583120584

with the kinetic ldquok-essencerdquo term Λ = 119892120572120573120593120572120593120573

where 120593 ndash a scalar filed

The full theory

119878 = න1

2minus119892119877 + minus119892 ⊔ 120594 + 1 Λ 1198894119909

when 8120587119866 = 119888 = 1

(with high derivatives)

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

2nd step the Diffusive energy action principle

We replace the dynamical space time vector 120594120583 by a gradient of a scalar filed 120594120583

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909

120594 - scalar field 120594120583120584 = 120597119907120597120583120594 minus Γ120583120584120582 120597120582120594 119879(120594)

120583120584- stress energy tensor

The variation according to 120594 gives a non-conserved diffusive energy momentum tensor

120571120583119879(120594)120583120584

= 119891119907 120571119907119891119907 = 0

The variation according to the metric gives a conserved stress energytensor (which is familiar from Einstein eq 119879 119866

120583119907= 119877120583120584 minus

1

2119892120583120584 119877)

119879(119866)120583120584

=minus2

minus119892

120575 minus119892ℒ119898120575119892120583120584

Alternative formulation withouthigher derivative with mass like term

An action with no high derivatives is obtained by adding another term involving χμ

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909 +

120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120517120536120524 120571μT(χ)μν

= σ χν + 120597νA

120517119808 σ 120571ν χν + 120597νA = 0

One difference between those theories

Here - 120590 appears as a parameter

- in the higher derivative theory 120590 appears as an integration

constant

Symmetries

If the matter is coupled through its energy momentum tensor

as

119879 120594120583120584

rarr 119879 120594120583120584

+ 120582119892120583120584

the process will not affect the equations of motion In Quantum

Field Theory this is ldquonormal orderingrdquo

120594 rarr 120594 + 120582

A toy model

We start with a simple action of one dimensional particle in a

potential 119881(119909)

119878 = න ሷ1198611

2119898 ሶ1199092 + 119881 119909 119889119905

120575119861 gives the total energy of a particle with constant power P

1

2119898 ሶ1199092 + 119881 119909 = 119864 119905 = 1198640 + 119875119905

120575119909 gives the condition for B

119898 ሷ119909 ሷ119861 +119898 ሶ119909ሸ119861 = 119881prime 119909 ሷ119861 or ሸ119861

ሷ119861=

2119881prime 119909

2119898 119864 119905 minus119881 119909

minus119875

2 119864 119905 minus119881 119909

A conserved Hamiltonian

Momentums for this toy model

120587119909 =120597ℒ

120597 ሶ119909= 119898 ሶ119909 ሷ119861

120587119861 =120597ℒ

120597 ሶ119861minus119889

119889119905

120597ℒ

120597 ሷ119861= minus

119889

119889119905119864 119905

Π119861 =120597ℒ

120597 ሷ119861= 119864 119905

The Hamiltonian (with second order derivative)

ℋ = ሶ119909120587119909 + ሶ119861120587119861 + ሷ119861Π119861 minus ℒ = 119898 ሶ119909 ሷ119861 minus ሶ119861 ሶ119864 = 1205871199092

119898Π119861 minus 119881 119909 + ሶ119861120587119861

The action isnrsquot dependent on time explicitly so the Hamiltonian is

conserved

Interacting Diffusive DE ndash DM

The diffusive stress energy tensor in this theory

119879(120594)120583120584

= Λ119892120583120584

with the kinetic ldquok-essencerdquo term Λ = 119892120572120573120593120572120593120573

where 120593 ndash a scalar filed

The full theory

119878 = න1

2minus119892119877 + minus119892 ⊔ 120594 + 1 Λ 1198894119909

when 8120587119866 = 119888 = 1

(with high derivatives)

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Alternative formulation withouthigher derivative with mass like term

An action with no high derivatives is obtained by adding another term involving χμ

119878 120594 = න minus119892120594120583120584119879(120594)1205831205841198894119909 +

120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120517120536120524 120571μT(χ)μν

= σ χν + 120597νA

120517119808 σ 120571ν χν + 120597νA = 0

One difference between those theories

Here - 120590 appears as a parameter

- in the higher derivative theory 120590 appears as an integration

constant

Symmetries

If the matter is coupled through its energy momentum tensor

as

119879 120594120583120584

rarr 119879 120594120583120584

+ 120582119892120583120584

the process will not affect the equations of motion In Quantum

Field Theory this is ldquonormal orderingrdquo

120594 rarr 120594 + 120582

A toy model

We start with a simple action of one dimensional particle in a

potential 119881(119909)

119878 = න ሷ1198611

2119898 ሶ1199092 + 119881 119909 119889119905

120575119861 gives the total energy of a particle with constant power P

1

2119898 ሶ1199092 + 119881 119909 = 119864 119905 = 1198640 + 119875119905

120575119909 gives the condition for B

119898 ሷ119909 ሷ119861 +119898 ሶ119909ሸ119861 = 119881prime 119909 ሷ119861 or ሸ119861

ሷ119861=

2119881prime 119909

2119898 119864 119905 minus119881 119909

minus119875

2 119864 119905 minus119881 119909

A conserved Hamiltonian

Momentums for this toy model

120587119909 =120597ℒ

120597 ሶ119909= 119898 ሶ119909 ሷ119861

120587119861 =120597ℒ

120597 ሶ119861minus119889

119889119905

120597ℒ

120597 ሷ119861= minus

119889

119889119905119864 119905

Π119861 =120597ℒ

120597 ሷ119861= 119864 119905

The Hamiltonian (with second order derivative)

ℋ = ሶ119909120587119909 + ሶ119861120587119861 + ሷ119861Π119861 minus ℒ = 119898 ሶ119909 ሷ119861 minus ሶ119861 ሶ119864 = 1205871199092

119898Π119861 minus 119881 119909 + ሶ119861120587119861

The action isnrsquot dependent on time explicitly so the Hamiltonian is

conserved

Interacting Diffusive DE ndash DM

The diffusive stress energy tensor in this theory

119879(120594)120583120584

= Λ119892120583120584

with the kinetic ldquok-essencerdquo term Λ = 119892120572120573120593120572120593120573

where 120593 ndash a scalar filed

The full theory

119878 = න1

2minus119892119877 + minus119892 ⊔ 120594 + 1 Λ 1198894119909

when 8120587119866 = 119888 = 1

(with high derivatives)

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Symmetries

If the matter is coupled through its energy momentum tensor

as

119879 120594120583120584

rarr 119879 120594120583120584

+ 120582119892120583120584

the process will not affect the equations of motion In Quantum

Field Theory this is ldquonormal orderingrdquo

120594 rarr 120594 + 120582

A toy model

We start with a simple action of one dimensional particle in a

potential 119881(119909)

119878 = න ሷ1198611

2119898 ሶ1199092 + 119881 119909 119889119905

120575119861 gives the total energy of a particle with constant power P

1

2119898 ሶ1199092 + 119881 119909 = 119864 119905 = 1198640 + 119875119905

120575119909 gives the condition for B

119898 ሷ119909 ሷ119861 +119898 ሶ119909ሸ119861 = 119881prime 119909 ሷ119861 or ሸ119861

ሷ119861=

2119881prime 119909

2119898 119864 119905 minus119881 119909

minus119875

2 119864 119905 minus119881 119909

A conserved Hamiltonian

Momentums for this toy model

120587119909 =120597ℒ

120597 ሶ119909= 119898 ሶ119909 ሷ119861

120587119861 =120597ℒ

120597 ሶ119861minus119889

119889119905

120597ℒ

120597 ሷ119861= minus

119889

119889119905119864 119905

Π119861 =120597ℒ

120597 ሷ119861= 119864 119905

The Hamiltonian (with second order derivative)

ℋ = ሶ119909120587119909 + ሶ119861120587119861 + ሷ119861Π119861 minus ℒ = 119898 ሶ119909 ሷ119861 minus ሶ119861 ሶ119864 = 1205871199092

119898Π119861 minus 119881 119909 + ሶ119861120587119861

The action isnrsquot dependent on time explicitly so the Hamiltonian is

conserved

Interacting Diffusive DE ndash DM

The diffusive stress energy tensor in this theory

119879(120594)120583120584

= Λ119892120583120584

with the kinetic ldquok-essencerdquo term Λ = 119892120572120573120593120572120593120573

where 120593 ndash a scalar filed

The full theory

119878 = න1

2minus119892119877 + minus119892 ⊔ 120594 + 1 Λ 1198894119909

when 8120587119866 = 119888 = 1

(with high derivatives)

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

A toy model

We start with a simple action of one dimensional particle in a

potential 119881(119909)

119878 = න ሷ1198611

2119898 ሶ1199092 + 119881 119909 119889119905

120575119861 gives the total energy of a particle with constant power P

1

2119898 ሶ1199092 + 119881 119909 = 119864 119905 = 1198640 + 119875119905

120575119909 gives the condition for B

119898 ሷ119909 ሷ119861 +119898 ሶ119909ሸ119861 = 119881prime 119909 ሷ119861 or ሸ119861

ሷ119861=

2119881prime 119909

2119898 119864 119905 minus119881 119909

minus119875

2 119864 119905 minus119881 119909

A conserved Hamiltonian

Momentums for this toy model

120587119909 =120597ℒ

120597 ሶ119909= 119898 ሶ119909 ሷ119861

120587119861 =120597ℒ

120597 ሶ119861minus119889

119889119905

120597ℒ

120597 ሷ119861= minus

119889

119889119905119864 119905

Π119861 =120597ℒ

120597 ሷ119861= 119864 119905

The Hamiltonian (with second order derivative)

ℋ = ሶ119909120587119909 + ሶ119861120587119861 + ሷ119861Π119861 minus ℒ = 119898 ሶ119909 ሷ119861 minus ሶ119861 ሶ119864 = 1205871199092

119898Π119861 minus 119881 119909 + ሶ119861120587119861

The action isnrsquot dependent on time explicitly so the Hamiltonian is

conserved

Interacting Diffusive DE ndash DM

The diffusive stress energy tensor in this theory

119879(120594)120583120584

= Λ119892120583120584

with the kinetic ldquok-essencerdquo term Λ = 119892120572120573120593120572120593120573

where 120593 ndash a scalar filed

The full theory

119878 = න1

2minus119892119877 + minus119892 ⊔ 120594 + 1 Λ 1198894119909

when 8120587119866 = 119888 = 1

(with high derivatives)

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

A conserved Hamiltonian

Momentums for this toy model

120587119909 =120597ℒ

120597 ሶ119909= 119898 ሶ119909 ሷ119861

120587119861 =120597ℒ

120597 ሶ119861minus119889

119889119905

120597ℒ

120597 ሷ119861= minus

119889

119889119905119864 119905

Π119861 =120597ℒ

120597 ሷ119861= 119864 119905

The Hamiltonian (with second order derivative)

ℋ = ሶ119909120587119909 + ሶ119861120587119861 + ሷ119861Π119861 minus ℒ = 119898 ሶ119909 ሷ119861 minus ሶ119861 ሶ119864 = 1205871199092

119898Π119861 minus 119881 119909 + ሶ119861120587119861

The action isnrsquot dependent on time explicitly so the Hamiltonian is

conserved

Interacting Diffusive DE ndash DM

The diffusive stress energy tensor in this theory

119879(120594)120583120584

= Λ119892120583120584

with the kinetic ldquok-essencerdquo term Λ = 119892120572120573120593120572120593120573

where 120593 ndash a scalar filed

The full theory

119878 = න1

2minus119892119877 + minus119892 ⊔ 120594 + 1 Λ 1198894119909

when 8120587119866 = 119888 = 1

(with high derivatives)

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Interacting Diffusive DE ndash DM

The diffusive stress energy tensor in this theory

119879(120594)120583120584

= Λ119892120583120584

with the kinetic ldquok-essencerdquo term Λ = 119892120572120573120593120572120593120573

where 120593 ndash a scalar filed

The full theory

119878 = න1

2minus119892119877 + minus119892 ⊔ 120594 + 1 Λ 1198894119909

when 8120587119866 = 119888 = 1

(with high derivatives)

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Equation of motions

120575120594 - non trivial evolving dark energy

⊔ Λ = 0

120575120593 - a conserved current

jβ = 2 ⊔ χ + 1 φβ

120575119892120583120584 - a conserved stress energy tensor

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594120590Λ120590 + 119895120583120593120584 minus 120594120583Λ120584 minus 120594120584Λ120583

Dark Energy Dark Matter

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

FLRW solution

⊔ Λ = 0

2 ሶ120593 ሷ120593 =11986221198863

⟺ ሶ1205932 = 1198621 + 1198622න119889119905

1198863

119895120573 = 2 ⊔ 120594 + 1 120593120573

ሶ120594 =11986241198863

+1

1198863න1198863 119889119905 minus

119862321198863

න119889119905

ሶ120593

T(119866)120583120584

- a conserved stress energy tensor

120588Λ = ሶ1205932 +1198622

1198863ሶ120594 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 minus 2

1198622

1198863ሶ120594 119901d = 0

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Asymptotic solution The field ሶ120594 asymptotically goes to the value as De Sitter space 119886 ~ 1198901198670119905

lim119905rarrinfin

ሶ120594 =1

1198863න1198863 119889119905 =

1

31198670

The asymptotic values of the densities are

120588Λ = 1198621 + 1198622න119889119905

1198863+11986221198863

ሶ120594 = 1198621 + 1198741

1198866

120588CDM = 1198623 1198621 minus2119862231198670

1

1198863+ 119874

1

1198866

The observable values11986211198670

= ΩΛ 1198623 1198621 minus2119862231198670

= 1198670Ωd

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Stability of the solutions More close asymptotically with Λ119862119863119872 the dark energy become

constant and the amount of dark matter slightly change 120588CDM~1

1198863

1198623 1198621 gt21198622

31198670for positive dust density For 1198622 lt 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 ldquoTaking 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 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Late universe solution The familiar solution of non-interacting DE-DM solution is for 1198622 = 0

Which gives constant scalar filed ሶ120593 = 1198621 and ሷ120593 = 0

120588Λ = ሶ1205932 = 1198621 119901Λ = minus120588Λ

120588d =1198623

1198863ሶ120593 =

11986211198623

1198863119901d = 0

The precise solution for Friedman equation 120588 simሶ119886

119886

2in this case is

119886Λminus119889 =1198623

1198621

Τ1 3

sinh Τ2 33

21198621119905

Which helps us to reconstruct the original physical values

ΩΛ =11986211198670

Ωd =1198623 11986211198670

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Perturbative solution

The scalar field has perturbative properties 12058212 ≪ 1

1205821 119905 1199050 =11986221198621න119889119905

1198863

1205822 119905 1199050 =1198622

11986211198623ሶ120594

For a first order solution in perturbation theory

120588Λ = 1198621 1 + 1205821 +1198623

11986211205822 + 1198742 1205821 1205822

120588119862119863119872 =119862111986231198863

1 +1

21205821 + 1205822 + 1198742 1205821 1205822

For rising dark energy dark matter amount goes lower ( 1198622lt 0 119862134 gt 0) For decreasing dark energy the amount of dark

matter goes up(all components are positive)

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Diffusive energy without higher derivatives

The full theory

ℒ =1

2minus119892119877 + minus119892120594120583120584119879(120594)

120583120584+120590

2minus119892 120594120583 + 120597120583119860

2+ minus119892Λ

Where Λ = 119892120572120573120593120572120593120573 and 119879(120594)120583120584

= Λ119892120583120584

All the EoM are the same except

119879(119866)120583120584

= 119892120583120584 minusΛ + 120594 120582Λ120582 +120783

120784120648120498120524120498120524 + 119895120583120593120584 minus 120594 120583Λ120584 minus 120594 120584Λ120583 +

120783

120648120498120525120498120642

For the late universe both theories are equivalent Λ120583Λ120584 sim1

1198866

For 120590 rarr infin the term 120590

2minus119892 120594120583 + 120597120583119860

2forces 120594120583 = minus120597120583119860 and DT

becomes Diffusive energy with high energy

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Comparison with Calogerorsquosand Habarsquos model φCMD

Calogero put two stress energy tensor of DE-DM Each stress energy

tensor in non-conserved

120571120583119879 Λ120583120584

= minus120571120583119879 Dust120583120584

= 3120590119895120584 119895120584120584 = 0

For FRWM this calculation leads to the solution

120588Λ = 1198621 + 1198622න119889119905

1198863

120588Dust =11986231198863

minus1198622119905

1198863

The two model became approximate for C2 ሶ120594 ≪ 1 Our asymptotic solution

becomes with constant densities because C2 ሶ120594 rarr1198622

31198670 which makes the DE

decay lower from 120593119862119872119863 and DM evolution as Λ119862119872119863

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Preliminary ideas on Quantization

Taking Dynamical space time theory (with source) and by integration by parts

119878 = න minus119892119877 + න minus119892Λ minus න minus119892120594120583119879(120594)120584120583120584

1198894119909 +120590

2න minus119892 120594120583 + 120597120583119860

21198894119909

120633120652120641 120571120584119879 120594120583120584

= 119891120584 = 120590 120594120584 + 120597120584119860 and put back into the action

119878 = න minus119892119877 +න minus119892 g120572120573120601120572120601120573 minus1

2120590න minus119892119891120584119891

1205841198894119909 + න minus119892 1205971205831198601198911205841198894119909

The partition function considering Euclidean metrics (exclude the

gravity terms)

Ζ = න119863120601 120575 119891120583120583exp

1

2120590න 119892 119891120584119891

1205841198894119909 minus න 119892 g120572120573120601120572120601120573

We see that for 120590 lt 0 there will a convergent functional integration so

this is a good sign for the quantum behavior of the theory By analytic

continuation you may define theory for the other sign of 120590

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

It is interesting to solve nume-

rically and show q the

DECCELERATION PARAMETER AS A FUNCTION OF

REDSHIFT

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

numerical results

DEDM AND BOUNCE

EXTENDING TO EARLY UNIVERSE

WE GET SUPERINFLATION

EXAMPLES

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Final Remarks

TMT - Unified Dark Matter Dark Energy The cosmological constant appearsas an integration constant

The Dynamical space time Theories ndash both energy momentum tensor areconserved

Diffusive Unified DE and DM ndash the vector field is taken to be the gradient of

a scalar the energy momentum tensor 119879(119909)120583120584

has a source current unlike the

119879(119866)120583120584

which is conserved The non conservation of 119879(119909)120583120584

is of the diffusive form

There is an integration constant 1198622 that controls how much model

deviates from the Lambda CDM ie how the Lambda CDM is deformedThis constant 1198622 measures how much we DEFORM our model from ΛCDM inthe sense Steinheimer talked about deforming theories

Asymptotically stable solution ΛCDM is a fixed point

For rising dark energy dark matter amount goes lower For decreasing darkenergy the amount of dark matter goes up

The partition function is convergent for 120590 lt 0 and therefor the theory is agood property before quantizing the theory

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

Ongoing research

Numerical solution for 1198622 ne 0 and using these to impose limits on 1198622 from data

A Stellar model spherically symmetric solutions

Unpublished results show appearance of linear potentials similar to Mannheim Conformal Gravity Theory the source is a non symmetric wormhole and the coefficient of linear term needed to explain rotation curves according to Mannheim and collaborators is proportional to the asymmetry parameter of the wormhole and the cosmological constant

T

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

references

Interacting Diffusive Unified

Dark Energy and Dark

Matter from Scalar Fields

David Benisty EI

Guendelman (Ben Gurion

U of Negev) Jan 30 2017

10 pp

Published in EurPhysJ C77

(2017) no6 396

DOI 101140epjcs10052-

017-4939-x

e-Print arXiv170108667

And essay to gravity

research foundation

awarded HonorableMention

And this is only the beginninghellip