Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy
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Transcript of Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Page 1: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy

Page 2: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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 ?

Page 3: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 4: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Cosm. Const | Quintessence static | dynamical

Page 5: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 6: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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 )

Page 7: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 8: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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 !

Page 9: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 10: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 11: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Higher dimensional Higher dimensional dilatation symmetrydilatation symmetry

all stable quasi-static solutions of higher all stable quasi-static solutions of higher dimensional field equations , which admit dimensional field equations , which admit a finite four-dimensional gravitational a finite four-dimensional gravitational constant and non-zero value for the constant and non-zero value for the dilaton , have V=0dilaton , have V=0

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

higher dimensional dilatation symmetry higher dimensional dilatation symmetry implies vanishing cosmological constant implies vanishing cosmological constant

self-tuning mechanismself-tuning mechanism

Page 12: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 13: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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 )

Page 14: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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 !

Page 15: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 16: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

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dilatation dilatation transformationstransformations

is invariantis invariant

Page 18: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 19: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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 !

Page 20: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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 )

Page 21: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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 !

Page 22: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 23: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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 :

Page 24: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

actionaction

Page 25: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 26: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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 Wlarge classes of solutions with Wext ext = 0 = 0 are explicitly known ( are explicitly known ( flat phaseflat phase ) )

example : example : Minkowski space Minkowski space xx D- D-dimensional torusdimensional torus

only for certain parameter regions : only for certain parameter regions : further solutions with Wfurther solutions with Wext ext ≠ 0 exist : ( ≠ 0 exist : ( non-flat phasenon-flat phase ) )

Page 27: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

sufficient condition for sufficient condition for vanishing cosmological vanishing cosmological

constantconstant

extremum of W extremum of W existsexists

Page 28: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

effective four – dimensional effective four – dimensional theorytheory

Page 29: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

characteristic length characteristic length scalesscales

ll :scale of internal space :scale of internal space

ξξ : dilaton scale: dilaton scale

Page 30: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 31: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

effective potentialeffective potential

Page 32: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 33: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

phase diagramphase diagram

stable solutionsstable solutions

Page 34: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 35: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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 !

Page 36: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 37: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 38: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 39: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

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

Page 40: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Time dependent Dark Time dependent Dark Energy :Energy :

QuintessenceQuintessence What changes in time ?What changes in time ?

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

V : potential energy of scalar field or cosmological V : potential energy of scalar field or cosmological constantconstant

V/MV/M4 4 is observableis observable

Imagine the Planck mass M increases …Imagine the Planck mass M increases …

Page 41: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Cosmon and Cosmon and fundamental mass scalefundamental mass scale

Assume all mass parameters are Assume all mass parameters are proportional to scalar field proportional to scalar field χχ (GUTs, (GUTs, superstrings,…)superstrings,…)

MMpp~ ~ χχ , m , mprotonproton~ ~ χχ , , ΛΛQCDQCD~ ~ χχ , M , MWW~ ~ χχ ,… ,…

χχ may evolve with time : may evolve with time : cosmoncosmon mmnn/M : ( almost ) constant - /M : ( almost ) constant - observationobservation !!

Only ratios of mass scales are Only ratios of mass scales are observableobservable

Page 42: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

theory without explicit theory without explicit mass scalemass scale

Lagrange density:Lagrange density:

recall :recall :

Page 43: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

realistic theoryrealistic theory

χχ has no gauge interactions has no gauge interactions χχ is effective scalar field after is effective scalar field after

“integrating out” all other scalar “integrating out” all other scalar fieldsfields

Page 44: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

four dimensional four dimensional dilatation symmetrydilatation symmetry

Lagrange density:Lagrange density:

Dilatation symmetry forDilatation symmetry for

Conformal symmetry for Conformal symmetry for δδ=0=0 Asymptotic flat phase solution : Asymptotic flat phase solution : λλ = 0 = 0

Page 45: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Asymptotically vanishing Asymptotically vanishing effective “cosmological effective “cosmological

constant”constant” Effective cosmological constant ~ V/MEffective cosmological constant ~ V/M4 4

dilatation anomaly : dilatation anomaly : λλ ~ ( ~ (χχ//μμ) ) –A–A

V ~ (V ~ (χχ//μμ) ) –A –A χχ4 4 V/MV/M4 4 ~(~(χχ//μμ) ) –A –A

M = M = χχ

It is sufficient that V increases less It is sufficient that V increases less fast than fast than χχ4 4 !!

Page 46: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

CosmologyCosmology

Cosmology : Cosmology : χχ increases with time ! increases with time !

( due to coupling of ( due to coupling of χχ to curvature scalar to curvature scalar ))

for large for large χχ the ratio V/M the ratio V/M4 4 decreases to decreases to zerozero

Effective cosmological constant vanishes Effective cosmological constant vanishes asymptotically for large t !asymptotically for large t !

Page 47: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Weyl scalingWeyl scaling

Weyl scaling : gWeyl scaling : gμνμν→→ (M/ (M/χχ))2 2 ggμνμν , ,

φφ/M = ln (/M = ln (χχ 44/V(/V(χχ))))

Exponential potential : V = MExponential potential : V = M44 exp(- exp(-φφ/M)/M)

No additional constant !No additional constant !

Page 48: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

quantum fluctuations quantum fluctuations and and

dilatation anomalydilatation anomaly

Page 49: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Dilatation anomalyDilatation anomaly

Quantum fluctuations responsible both Quantum fluctuations responsible both forfor

fixed point and dilatation anomaly close fixed point and dilatation anomaly close to fxed pointto fxed point

Running couplings:Running couplings: hypothesishypothesis

Renormalization scale Renormalization scale μμ : ( momentum : ( momentum scale )scale )

λλ~(~(χχ//μμ) ) –A–A

Page 50: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Asymptotic behavior of Asymptotic behavior of effective potentialeffective potential

λλ ~ ( ~ (χχ//μμ) ) –A–A

V ~ (V ~ (χχ//μμ) ) –A –A χχ44

VV ~ ~ χχ 4–A 4–A

crucial : behavior for large χ !crucial : behavior for large χ !

Page 51: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Without dilatation – anomaly :V= const. Massless Goldstone boson = dilaton

Dilatation – anomaly :V (φ )Scalar with tiny time dependent mass : cosmon

Page 52: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

Dilatation anomaly and Dilatation anomaly and quantum fluctuationsquantum fluctuations

Computation of running couplings Computation of running couplings ( beta functions ) needs unified theory !( beta functions ) needs unified theory !

Dominant contribution from modes Dominant contribution from modes with momenta ~with momenta ~χχ ! !

No prejudice on “natural value “ of No prejudice on “natural value “ of anomalous dimension should be anomalous dimension should be inferred from tiny contributions at inferred from tiny contributions at QCD- momentum scale !QCD- momentum scale !

Page 53: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

quantum fluctuations and quantum fluctuations and naturalnessnaturalness

Jordan- and Einstein frame completely Jordan- and Einstein frame completely equivalent on level of effective action equivalent on level of effective action and field equations ( and field equations ( afterafter computation computation of quantum fluctuations ! )of quantum fluctuations ! )

Treatment of quantum fluctuations Treatment of quantum fluctuations depends on frame : Jacobian for depends on frame : Jacobian for variable transformation in functional variable transformation in functional integralintegral

What is natural in one frame may look What is natural in one frame may look unnatural in another frameunnatural in another frame

Page 54: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

quantum fluctuations quantum fluctuations and framesand frames

Einstein frame : quantum fluctuations Einstein frame : quantum fluctuations make zero cosmological constant look make zero cosmological constant look unnaturalunnatural

Jordan frame : quantum fluctuations Jordan frame : quantum fluctuations are at the origin of dilatation anomaly;are at the origin of dilatation anomaly;

may be key ingredient for may be key ingredient for solutionsolution of of cosmological constant problem !cosmological constant problem !

Page 55: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

fixed points and fluctuation fixed points and fluctuation contributions of individual contributions of individual

componentscomponents

If running couplings influenced by fixed points:If running couplings influenced by fixed points:individual fluctuation contribution can be huge individual fluctuation contribution can be huge

overestimate !overestimate !

here : fixed point at vanishing quartic coupling and here : fixed point at vanishing quartic coupling and anomalous dimension anomalous dimension VV ~ ~ χχ 4–A 4–A

it makes no sense to use naïve scaling argument to it makes no sense to use naïve scaling argument to infer individual contribution infer individual contribution VV ~ h ~ h χχ 4 4

Page 56: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

conclusionsconclusions

naturalness of cosmological constant naturalness of cosmological constant and cosmon potential should be and cosmon potential should be discussed in the light of dilatation discussed in the light of dilatation symmetry and its anomaliessymmetry and its anomalies

Jordan frameJordan frame higher dimensional settinghigher dimensional setting four dimensional Einstein frame and four dimensional Einstein frame and

naïve estimate of individual naïve estimate of individual contributions can be very misleading !contributions can be very misleading !

Page 57: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

conclusionsconclusions

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

Page 58: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

EnEndd

Page 59: Asymptotically vanishing cosmological constant, Self-tuning and Dark Energy.

A few referencesA few references

C.Wetterich , Nucl.Phys.B302,668(1988) , received 24.9.1987C.Wetterich , Nucl.Phys.B302,668(1988) , received 24.9.1987

P.J.E.Peebles,B.Ratra , Astrophys.J.Lett.325,L17(1988) , received 20.10.1987P.J.E.Peebles,B.Ratra , Astrophys.J.Lett.325,L17(1988) , received 20.10.1987

B.Ratra,P.J.E.Peebles , Phys.Rev.D37,3406(1988) , received 16.2.1988B.Ratra,P.J.E.Peebles , Phys.Rev.D37,3406(1988) , received 16.2.1988

J.Frieman,C.T.Hill,A.Stebbins,I.Waga , Phys.Rev.Lett.75,2077(1995)J.Frieman,C.T.Hill,A.Stebbins,I.Waga , Phys.Rev.Lett.75,2077(1995)

P.Ferreira, M.Joyce , Phys.Rev.Lett.79,4740(1997)P.Ferreira, M.Joyce , Phys.Rev.Lett.79,4740(1997)

C.Wetterich , Astron.Astrophys.301,321(1995)C.Wetterich , Astron.Astrophys.301,321(1995)

P.Viana, A.Liddle , Phys.Rev.D57,674(1998)P.Viana, A.Liddle , Phys.Rev.D57,674(1998)

E.Copeland,A.Liddle,D.Wands , Phys.Rev.D57,4686(1998)E.Copeland,A.Liddle,D.Wands , Phys.Rev.D57,4686(1998)

R.Caldwell,R.Dave,P.Steinhardt , Phys.Rev.Lett.80,1582(1998)R.Caldwell,R.Dave,P.Steinhardt , Phys.Rev.Lett.80,1582(1998)

P.Steinhardt,L.Wang,I.Zlatev , Phys.Rev.Lett.82,896(1999)P.Steinhardt,L.Wang,I.Zlatev , Phys.Rev.Lett.82,896(1999)