Coupled Dark Energy and Dark Matter from dilatation symmetry.

Coupled Dark Energy and Dark Matter fromdilatation symmetry

Cosmological ConstantCosmological Constant- Einstein -- Einstein -

Constant Constant λλ compatible with all symmetries compatible with all symmetries Constant Constant λλ compatible with all compatible with all

observationsobservations No time variation in contribution to energy No time variation in contribution to energy

densitydensity

Why so small ? Why so small ? λλ/M/M44 = 10 = 10-120-120

Why important just today ?Why important just today ?

Cosmological mass scalesCosmological mass scales Energy densityEnergy density

ρρ ~ ( 2.4×10 ~ ( 2.4×10 -3-3 eV eV ))- 4- 4

Reduced Planck Reduced Planck massmass

M=2.44M=2.44×10×101818GeVGeV Newton’s constantNewton’s constant

GGNN=(8=(8ππM²)M²)

Only ratios of mass scales are observable !Only ratios of mass scales are observable !

homogeneous dark energy: homogeneous dark energy: ρρhh/M/M44 = 7 · = 7 · 10ˉ¹²¹10ˉ¹²¹

matter: matter: ρρmm/M/M4= 3 · 10ˉ¹²¹= 3 · 10ˉ¹²¹

Cosm. Const | Quintessence static | dynamical

QuintessenceQuintessenceDynamical dark Dynamical dark

energy ,energy ,

generated by scalargenerated by scalar

field field (cosmon)(cosmon)

C.Wetterich,Nucl.Phys.B302(1988)668, C.Wetterich,Nucl.Phys.B302(1988)668, 24.9.87 24.9.87P.J.E.Peebles,B.Ratra,ApJ.Lett.325(1988)LP.J.E.Peebles,B.Ratra,ApJ.Lett.325(1988)L17, 20.10.8717, 20.10.87

CosmonCosmon Scalar field changes its value even Scalar field changes its value even

in the in the present present cosmological epochcosmological epoch Potential und kinetic energy of Potential und kinetic energy of

cosmon contribute to the energy cosmon contribute to the energy density of the Universedensity of the Universe

Time - variable dark energy : Time - variable dark energy : ρρhh(t) decreases with time !(t) decreases with time !

V(V(φφ) =M) =M44 exp( - exp( - αφαφ/M )/M )

Two key features Two key features for realistic cosmologyfor realistic cosmology

1 ) Exponential cosmon potential and 1 ) Exponential cosmon potential and

scaling solutionscaling solution

V(V(φφ) =M) =M44 exp( - exp( - αφαφ/M ) /M )

V(V(φφ →→ ∞ ) → 0 ! ∞ ) → 0 !

2 ) Stop of cosmon evolution by 2 ) Stop of cosmon evolution by

cosmological triggercosmological trigger

e.g. growing neutrino quintessencee.g. growing neutrino quintessence

Evolution of cosmon fieldEvolution of cosmon field

Field equationsField equations

Potential V(Potential V(φφ) determines details of the ) determines details of the modelmodel

V(V(φφ) =M) =M4 4 exp( - exp( - αφαφ/M )/M )

for increasing for increasing φφ the potential the potential decreases towards zero !decreases towards zero !

exponential potentialexponential potentialconstant fraction in dark constant fraction in dark

energyenergy

can explain order of can explain order of magnitude magnitude

of dark energy !of dark energy !

ΩΩh h = 3(4)/= 3(4)/αα22

Asymptotic solutionAsymptotic solution

explain V( explain V( φφ → ∞ ) = 0 ! → ∞ ) = 0 !

effective field equations should effective field equations should have generic solution of this typehave generic solution of this type

setting : quantum effective action , setting : quantum effective action , all quantum fluctuations included:all quantum fluctuations included:investigate generic forminvestigate generic form

realized by fixed point realized by fixed point of runaway solutionof runaway solution

in higher dimensions :in higher dimensions :

dilatation symmetrydilatation symmetry

Cosmon and Cosmon and bolonbolon

Two scalar fields : common Two scalar fields : common origin from origin from dilatation symmetric fixed dilatation symmetric fixed point of point of higher dimensional theoryhigher dimensional theory

Cosmon – bolon - Cosmon – bolon - potentialpotential

Two characteristic Two characteristic behaviorsbehaviors

Bolon oscillates - Bolon oscillates -

if mass larger than H if mass larger than H

Bolon is frozen -Bolon is frozen -

if mass smaller that Hif mass smaller that H

Cosmon and Cosmon and bolonbolon

Dark Dark EnergyEnergy

Dark Dark MatterMatter

Early scaling solutionEarly scaling solution

dominated by cosmondominated by cosmonbolon frozen and negligiblebolon frozen and negligiblebolon mass increases during bolon mass increases during scaling solutionscaling solution

Bolon oscillationsBolon oscillations

ratio bolon mass / H increasesratio bolon mass / H increases bolon starts oscillating once mass bolon starts oscillating once mass

larger than Hlarger than H subsequently bolon behaves as Dark subsequently bolon behaves as Dark

MatterMatter matter radiation equality around matter radiation equality around

beginning of oscillationsbeginning of oscillations

Bolon oscillationsBolon oscillations

ww

Transition to matter Transition to matter dominationdomination

precise timing depends at this stage on initial precise timing depends at this stage on initial value of bolonvalue of bolon

Effective coupling betweenEffective coupling betweenDark Energy and Dark Dark Energy and Dark

mattermatter

Scaling solution for Scaling solution for coupled coupled

Dark EnergyDark Energy

Realistic quintessence needs late Realistic quintessence needs late modificationmodification

modification of cosmon – bolon potential modification of cosmon – bolon potential or growing neutrinos or …or growing neutrinos or …

Modification of potential for Modification of potential for large large χχ : :

independence of initial independence of initial conditionsconditions

matter - radiation equality depends now on matter - radiation equality depends now on parameters of potentialparameters of potential

Present bolon mass Present bolon mass corresponds to range on corresponds to range on

subgalactic scalessubgalactic scales

suppression of small scale Dark Matter suppression of small scale Dark Matter structures ?structures ?

conclusions (1)conclusions (1)

Bolon : new Dark Matter candidateBolon : new Dark Matter candidate

not detectable by local observations –not detectable by local observations –

direct or indirect dark matter direct or indirect dark matter searchessearches

perhaps observation by influence on perhaps observation by influence on subgalactic dark matter structuressubgalactic dark matter structures

Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy

Higher –dimensional Higher –dimensional dilatation symmetrydilatation symmetry

solvessolvescosmological constant cosmological constant

problemproblem

graviton and dilatongraviton and dilaton

dilatation symmetric effective actiondilatation symmetric effective action

simple examplesimple example

in general : in general : many dimensionless many dimensionless parameters characterize effective parameters characterize effective actionaction

dilatation dilatation transformationstransformations

is invariantis invariant

flat phaseflat phase

generic existence of solutions of generic existence of solutions of

higher dimensional field equations higher dimensional field equations withwith

effective effective four –dimensional gravityfour –dimensional gravity andand

vanishing cosmological constantvanishing cosmological constant

torus solutiontorus solutionexample : example : Minkowski space Minkowski space xx D-dimensional torus D-dimensional torus ξξ = const = const solves higher dimensional field equationssolves higher dimensional field equations extremum of effective actionextremum of effective action

finite four- dimensional gauge couplingsfinite four- dimensional gauge couplings dilatation symmetry spontaneously brokendilatation symmetry spontaneously broken

generically many more solutions in flat phase !generically many more solutions in flat phase !

massless scalars

dilaton

geometrical scalars ( moduli ) variation of circumference of tori change of volume of internal space bolon is associated to one such

scalar

Higher dimensional Higher dimensional dilatation symmetrydilatation symmetry

for arbitrary values of effective couplings within a certain for arbitrary values of effective couplings within a certain range :range :

higher dimensional dilatation symmetry implies existence higher dimensional dilatation symmetry implies existence of of

a large class of solutions with vanishing four –dimensionala large class of solutions with vanishing four –dimensional cosmological constant cosmological constant

all stable quasi-static solutions of higher dimensional field all stable quasi-static solutions of higher dimensional field equations , which admit a finite four-dimensional equations , which admit a finite four-dimensional

gravitationalgravitational constant and non-zero value for the dilaton , have V=0constant and non-zero value for the dilaton , have V=0

self-tuning mechanismself-tuning mechanism

look for extrema of effective actionlook for extrema of effective actionfor general field configurationsfor general field configurations

warpingwarping

most general metric with maximal most general metric with maximal four – dimensional symmetryfour – dimensional symmetry

general form of quasi – static solutionsgeneral form of quasi – static solutions( non-zero or zero cosmological constant )( non-zero or zero cosmological constant )

effective four – dimensional effective four – dimensional actionaction

flat phase : flat phase : extrema of Wextrema of Win higher dimensions , those exist generically !in higher dimensions , those exist generically !

extrema of Wextrema of W

provide large class of solutions with provide large class of solutions with vanishing four – dimensional constantvanishing four – dimensional constant

dilatation transformationdilatation transformation

extremum of W must occur for W=0 !extremum of W must occur for W=0 ! effective cosmological constant is effective cosmological constant is

given by Wgiven by W

extremum of W must occur extremum of W must occur for W = 0for W = 0

for any given solution : rescaled metric and for any given solution : rescaled metric and dilaton is again a solutiondilaton is again a solution

for rescaled solution :for rescaled solution :

useuse

extremum condition :extremum condition :

extremum of W is extremum of W is extremum of effective extremum of effective

actionaction

effective four – dimensional effective four – dimensional cosmological constant cosmological constant

vanishes for extrema of Wvanishes for extrema of W

expand effective 4 – d - actionexpand effective 4 – d - action

in derivatives :in derivatives :

4 - d - field 4 - d - field equationequation

Quasi-static solutionsQuasi-static solutions

for arbitrary parameters of dilatation for arbitrary parameters of dilatation symmetric effective action :symmetric effective action :

large classes of solutions with extremum of large classes of solutions with extremum of W and WW and Wext ext = 0 are explicitly known ( = 0 are explicitly known ( flat flat phasephase ) )

example : example : Minkowski space Minkowski space xx D-dimensional D-dimensional torustorus

only for certain parameter regions : further only for certain parameter regions : further solutions without extremum of W exist : solutions without extremum of W exist :

( ( non-flat phasenon-flat phase ) )

sufficient condition for sufficient condition for vanishing cosmological vanishing cosmological

constantconstant

extremum of W extremum of W existsexists

self tuning in higher self tuning in higher dimensionsdimensions

involves infinitely many degrees of freedom !involves infinitely many degrees of freedom !

for arbitrary parameters in effective action : for arbitrary parameters in effective action : flat phase solutions are flat phase solutions are presentpresent

extrema of W existextrema of W exist

for flat 4-d-space : W is functional for flat 4-d-space : W is functional of internal geometry, independent of x of internal geometry, independent of x

solve field equations for internal metric and solve field equations for internal metric and σσ and and ξξ

Dark energyDark energy

if cosmic runaway solution has not yet if cosmic runaway solution has not yet reached fixed point :reached fixed point :

dilatation symmetry of field equations dilatation symmetry of field equations

not yet exactnot yet exact

“ “ dilatation anomaly “dilatation anomaly “

non-vanishing effective potential V in non-vanishing effective potential V in reduced four –dimensional theoryreduced four –dimensional theory

conclusions (2)conclusions (2)

cosmic runaway towards fixed point cosmic runaway towards fixed point maymay

solve the cosmological constant solve the cosmological constant problemproblem

andand

account for dynamical Dark Energyaccount for dynamical Dark Energy

effective dilatation effective dilatation symmetry insymmetry in

full quantum theoryfull quantum theory

realized for fixed pointsrealized for fixed points

Cosmic runawayCosmic runaway large class of cosmological solutions which never large class of cosmological solutions which never

reach a static state : runaway solutionsreach a static state : runaway solutions

some characteristic scale some characteristic scale χχ changes with time changes with time

effective dimensionless couplings flow with effective dimensionless couplings flow with χχ ( similar to renormalization group )( similar to renormalization group )

couplings either diverge or reach fixed pointcouplings either diverge or reach fixed point

for fixed point : for fixed point : exact dilatation symmetry exact dilatation symmetry of full of full quantum field equations and corresponding quantum field equations and corresponding quantum effective actionquantum effective action

approach to fixed pointapproach to fixed point

dilatation symmetry not yet realizeddilatation symmetry not yet realized dilatation anomalydilatation anomaly effective potential effective potential V(φ)V(φ) exponential potential reflects exponential potential reflects

anomalous dimension for vicinity of anomalous dimension for vicinity of fixed pointfixed point

V(V(φφ) =M) =M4 4 exp( - exp( - αφαφ/M )/M )

cosmic runaway and the cosmic runaway and the problem of time varying problem of time varying

constantsconstants It is not difficult to obtain quintessence It is not difficult to obtain quintessence

potentials from higher dimensional ( or potentials from higher dimensional ( or string ? ) theoriesstring ? ) theories

Exponential form rather generic Exponential form rather generic

( after Weyl scaling)( after Weyl scaling) Potential goes to zero for Potential goes to zero for φφ →→ ∞∞ But most models show too strong time But most models show too strong time

dependence of constants !dependence of constants !

higher dimensional higher dimensional dilatation symmetrydilatation symmetry

generic class of solutions with generic class of solutions with

vanishing effective four-dimensional vanishing effective four-dimensional cosmological constantcosmological constant

andand

constant effective dimensionless couplingsconstant effective dimensionless couplings

effective four – dimensional effective four – dimensional theorytheory

characteristic length characteristic length scalesscales

ll : scale of internal space : scale of internal space

ξξ : dilaton scale: dilaton scale

effective Planck masseffective Planck mass

dimensionless , dimensionless , depends on internal geometry ,depends on internal geometry ,from expansion of F in Rfrom expansion of F in R

effective potentialeffective potential

canonical scalar fieldscanonical scalar fields

consider field configurations with rescaledconsider field configurations with rescaledinternal length scale and dilaton valueinternal length scale and dilaton value

potential and effective Planck mass depend on scalar fieldspotential and effective Planck mass depend on scalar fields

phase diagramphase diagram

stable solutionsstable solutions

phase structure of phase structure of solutionssolutions

solutions in flat phase exist for arbitrary solutions in flat phase exist for arbitrary values of effective parameters of higher values of effective parameters of higher dimensional effective actiondimensional effective action

question : how “big” is flat phase question : how “big” is flat phase ( which internal geometries and warpings ( which internal geometries and warpings

are possible beyond torus solutions )are possible beyond torus solutions )

solutions in non-flat phase only exist for solutions in non-flat phase only exist for restricted parameter rangesrestricted parameter ranges

self tuningself tuning

for all solutions in flat phase : for all solutions in flat phase :

self tuning of cosmological constant self tuning of cosmological constant to zero !to zero !

self tuningself tuning

for simplicity : no contribution of F to for simplicity : no contribution of F to VV

assume Q depends on parameter assume Q depends on parameter αα , , which characterizes internal which characterizes internal geometry:geometry:

tuning required : tuning required : andand

self tuning in higher self tuning in higher dimensionsdimensions

Q depends on higher dimensionalQ depends on higher dimensional

extremum condition extremum condition amounts to field equationsamounts to field equations

typical solutions depend on integration constants typical solutions depend on integration constants γγ

solutions obeying boundary condition exist :solutions obeying boundary condition exist :

fieldsfields

self tuning in higher self tuning in higher dimensionsdimensions

involves infinitely many degrees of freedom !involves infinitely many degrees of freedom !

for arbitrary parameters in effective action : for arbitrary parameters in effective action : flat phase solutions are flat phase solutions are presentpresent

extrema of W existextrema of W exist

for flat 4-d-space : W is functional for flat 4-d-space : W is functional of internal geometry, independent of x of internal geometry, independent of x

solve field equations for internal metric and solve field equations for internal metric and σσ and and ξξ

Dark energyDark energy

if cosmic runaway solution has not yet if cosmic runaway solution has not yet reached fixed point :reached fixed point :

dilatation symmetry of field equations dilatation symmetry of field equations

not yet exactnot yet exact

“ “ dilatation anomaly “dilatation anomaly “

non-vanishing effective potential V in non-vanishing effective potential V in reduced four –dimensional theoryreduced four –dimensional theory

EnEndd